A Schema for Complex Time-Dependent Transformations Between Geodetic Reference Frames

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Surname/Family Name : Stanaway Given Name/s : Richard Frank Abbreviation for degree as give in the University calendar : PhD Faculty : Engineering School : Civil and Environmental Engineering A schema for complex time-dependent transformations between geodetic Thesis Title : reference frames

Abstract 350 words maximum: (PLEASE TYPE) GNSS point-positioning is becoming more precise and accessible to a wider spectrum of non-expert users. As a consequence, the issue of misalignment between GNSS positioning reference frames and spatial data reference frames is already apparent. Positions of ground-fixed features within GNSS frames are kinematic in nature due to global plate motions and other geodynamic effects such as seismic deformation and glacial isostatic adjustment. On the other hand, dense, high resolution spatial data such as imagery and point clouds are not intrinsically kinematic at the moment of acquisition. Misalignment of this data with global frames such as those used by GNSS is inevitable unless a rigorous time-dependent transformation accommodating the complexity of global deformation is applied. Spatial data defined by different epochs of a kinematic frame require time-dependent transformations within GIS in order to align the data correctly for the purpose of visualisation and precision analysis. The absence of a transformation schema to handle complex deformation is already impacting significantly on the integrity of high precision spatial data acquired at different epochs. This thesis develops and describes a schema for time-dependent transformations between geodetic reference frames in complex deformation environments. The schema can support geodetic applications including positioning, surveying and spatial data management. Precision requirements for reference frames in practice are assessed taking into consideration the impact of global and local deformations on precision at different spatial and temporal scales. Strategies for the realisation of stable local reference frames using plate motion models are also described. Case studies in Australia and New Zealand are presented showing the significant improvement in precision and reference frame longevity that can result by application of the schema for geodetic applications in complex deformation zones. The case studies are characterised by very different tectonic settings to highlight the flexibility of the schema.

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FOR OFFICE USE ONLY Date of completion of requirements for Award: 2 A schema for complex time-dependent transformations between geodetic reference frames

Richard Frank Stanaway

A thesis submitted for the degree of Doctor of Philosophy of the University of New South Wales

School of Civil and Environmental Engineering Faculty of Engineering

1st April 2019

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4 Abstract

GNSS point-positioning is becoming more precise and accessible to a wider spectrum of non-expert users. As a consequence, the issue of misalignment between GNSS positioning reference frames and spatial data reference frames is increasingly apparent. Positions of ground-fixed features within GNSS frames are kinematic in nature due to global plate motions and other geodynamic effects such as seismic deformation and glacial isostatic adjustment. On the other hand, dense, high resolution spatial data such as imagery and point clouds are not intrinsically kinematic at the moment of acquisition. Misalignment of this data with global reference frames such as those used by GNSS is inevitable unless a rigorous time-dependent transformation accommodating the complexity of global deformation is applied. Spatial data defined by different epochs of a kinematic frame require time-dependent transformations within GIS in order to align the data correctly for the purpose of visualisation and precision analysis. The absence of a transformation schema to handle complex deformation is already impacting significantly on the integrity of high precision spatial data acquired at different epochs.

This thesis develops and describes a schema for time-dependent transformations between geodetic reference frames in complex deformation environments. The schema can support geodetic applications including positioning, surveying and spatial data management. Precision requirements for reference frames in practice are assessed taking into consideration the impact of global and local deformations on precision at different spatial and temporal scales. Strategies for the realisation of stable local reference frames using plate motion models are also described.

Case studies in Australia and New Zealand are presented showing the significant improvement in precision and reference frame longevity that can result by application of the schema for geodetic applications in complex deformation zones. The case studies are characterised by very different tectonic settings to highlight the flexibility of the schema.

Acknowledgements

Firstly, I would like to thank my supervisor, Craig Roberts, for his very generous support and guidance from the very start. His encouragement and financial support for attendance at conferences and workshops has been very worthwhile and much appreciated, especially at some testing times. It has been quite a long journey in a sometimes fast evolving space. Chris Rizos and Bruce Harvey have also been very supportive with some great discussions over the years and also for their timely editing of draft chapters and papers.

I would also like to thank all of the geodetic staff at Land Information New Zealand for their support and collegiality over the last several years. Hopefully our regular collaborations, meetings and visits have been mutually beneficial. Especially, I would especially like to thank Graeme Blick, Chris Crook and Nic Donnelly for their great friendship, advice and support.

My research has led to some great opportunities for active involvement with Commission 5 of the FIG and Commission 1 of the IAG. It has been a great privilege to be chair of the IAG Working Group dealing with deformation and reference frames. These involvements have led to some great friendships with geodesists in Australia and around the world.

A very special thanks to Mum and Dad in Sydney and my mother-in-law, Noeline in Murray Bridge. They have provided wonderful hospitality and support over many extended thinking and writing sessions in both Sydney and Adelaide. The thesis writing wouldn’t have been possible without these monastic excursions. Also, an apology to many of my closer family and friends who I have not kept in contact with as much as we would have liked. I am looking forward to make up for the lost time with them.

Finally, I wish to deeply thank my wife Sandra and my two daughters Elsa and Meredith for their love, support and forbearance. They have put up with many of my absences (even at home), especially over the last few years and I can’t wait to give them back all of that now.

Table of Contents

List of Abbreviations and Acronyms ...... 11 Glossary of terms ...... 15 List of Figures ...... 21 List of Tables ...... 23 List of Publications and Presentations ...... 24 Chapter 1 Introduction ...... 27 1.1 GNSS Reference Frames ...... 27 1.2 Challenges of interfacing GNSS positioning and precise spatial data ...... 28 1.2.1 Misalignment of GNSS and spatial data reference frames ...... 28 1.2.2 Requirement for local frame stability for civil projects and urban areas ...... 29 1.2.3 Implications of geodynamic processes on precision GNSS users ...... 30 1.2.4 Drivers for reference frame modernisation ...... 30 1.2.5 The requirement for a rigorous time-dependent transformation strategy ...... 31 1.2.6 Current approaches to time-dependent transformations ...... 31 1.2.7 Characteristics of existing time-dependent transformation strategies ...... 31 1.3 Thesis aims ...... 32 1.4 Thesis outline ...... 33 Chapter 2 Earth deformation with relevance to geodetic reference frames ...... 35 2.1 Geodynamic processes on Earth ...... 35 2.1.1 Modes and causes of Earth deformation ...... 35 2.1.2 Stable tectonic plate motion ...... 36 2.1.3 Estimation of stable plate motion models (PMM) ...... 40 2.1.4 Glacial isostatic adjustment (GIA) ...... 42 2.1.5 Interseismic plate boundary deformation ...... 43 2.1.6 Coseismic deformation and volcanism ...... 45 2.1.7 Postseismic deformation ...... 48 2.1.8 Periodic and seasonal deformation ...... 49 2.1.9 A summary of geodynamics deformation effects with relevance to reference frames ...... 50 2.2 Site specific and localised deformation ...... 51 2.2.1 Regolith deformation and creep ...... 51 2.2.2 Stability of geodetic monuments ...... 52 2.2.3 Impact of localised deformation and monument stability on reference frame definition ...... 53 2.2.4 Localised deformation models ...... 53 2.3 Vertical specific deformation and subsidence ...... 56 2.4 Apparent deformation effects (artefacts) ...... 57 Chapter 3 Geodetic reference systems, transformations and precision requirements ...... 58 3.1 Geodetic reference systems and frames ...... 58 3.1.1 Coordinate reference systems (CRS) and ellipsoids ...... 58 3.1.2 Geodetic datums and reference frames (RF) ...... 59 3.1.3 Static reference frames ...... 60 3.1.4 Kinematic reference frames ...... 60 3.1.5 Limitations of static and kinematic reference frames in practice ...... 61 3.1.6 Plate-fixed reference frames (PFRF) ...... 62 3.1.7 Crust-fixed reference frames (CFRF) and semi-kinematic RF ...... 63 3.1.8 Local Reference Frames (LRF) ...... 63 3.1.9 The dual-frame approach ...... 64 3.2 Reference frame transformations ...... 64 3.2.1 Three parameter geocentric Cartesian transformation ...... 65 3.2.2 Seven parameter Bursa-Wolf geocentric Cartesian transformation ...... 65 3.2.3 14 parameter geodetic Cartesian transformation (time-dependent) ...... 66

8 3.2.4 Coplanar two parameter two dimensional transformation ...... 66 3.2.5 Block shift model (distortion grid) ...... 67 3.2.6 Coplanar six parameter two dimensional transformation ...... 67 3.2.7 Limitations of static RF transformation strategies ...... 67 3.2.8 Existing time-dependent transformations ...... 69 3.3 Positioning and dimensional tolerances in applied geodesy and surveying ...... 70 3.3.1 The distinction between dimensional and positional tolerance ...... 70 Chapter 4 The impact of deformation on reference frames ...... 72 4.1 The effect of plate rotation on terrestrial referemce frame in practice ...... 72 4.2 The effect of geodetic strain on reference frames ...... 75 4.3 The effect of coseismic displacement on reference frames ...... 78 4.4 The effect of postseismic deformation on reference frames ...... 80 4.5 Effects of localised deformation on reference frames ...... 82 4.6 Effect of vertical deformation on vertical tolerance limits ...... 83 Chapter 5 Kinematic models to support complex time-dependent transformation schema ...... 84 5.1 Structure of time-dependent transformation model formats ...... 84 5.1.1 Triangulated Irregular Networks (TIN) ...... 84 5.1.2 Discrete Global Grid systems and Polyhedral models ...... 84 5.1.3 Grid models ...... 85 5.2 Grid formats for geodetic applications ...... 85 5.2.1 Resolution and geometry considerations for geodetic data grids ...... 85 5.2.2 Model grid metadata, node and data units ...... 87 5.2.3 Site velocity and displacement grid formats ...... 88 5.2.4 Formatting and distribution of model grids ...... 88 5.3 Summary of coordinate and displacement conversions ...... 89 5.3.1 Ellipsoidal coordinate to geocentric Cartesian coordinate conversion ...... 89 5.3.2 Geocentric Cartesian coordinate to ellipsoidal coordinate conversion ...... 90 5.3.3 Conversion of geocentric Cartesian displacement or velocity to topocentric format ...... 90 5.3.4 Conversion of a topocentric displacement or velocity to a geocentric Cartesian format ...... 91 5.4 Bilinear interpolation ...... 91 5.5 Time-dependent transformation model formats within geodetic registries ...... 92 5.5.1 Standardisation of time-dependent transformations in geodetic registries ...... 93 Chapter 6 Formulation of a complex time dependent transformation schema ...... 95 6.1 Scenarios for complex time-dependent transformations ...... 95 6.2 Schema overview and structure ...... 95 6.2.1 Schema overview ...... 101 6.2.2 Intraframe propagation ...... 101 6.2.3 Interseismic displacement ...... 102 6.2.4 Coseismic displacement ...... 102 6.2.5 Postseismic displacement ...... 103 6.2.6 Supplementary displacement ...... 105 6.3 Generalised schema ...... 106 6.3.1 Schema case testing and logic flow ...... 106 6.3.2 Special cases ...... 110 6.3.3 Interframe transformation ...... 110 6.3.4 Estimation and correlation of uncertainty ...... 110 6.4 Schema examples for different RF definition transformations ...... 111 6.4.1 Transformation between kinematic RF ...... 111 6.4.2 Kinematic RF to semi-kinematic RF ...... 116 6.4.3 Kinematic RF to a static RF ...... 119 6.4.4 Semi-kinematic RF to Kinematic RF ...... 121 6.4.5 Semi-kinematic RF to Semi-kinematic RF across an earthquake event ...... 123 6.4.6 Semi-kinematic RF to static RF ...... 124 6.4.7 Static RF to kinematic RF ...... 125 6.4.8 Static RF to semi-kinematic RF ...... 128

6.4.9 Static RF to Static RF ...... 129 6.5 Management and evolution of time-dependent transformations ...... 129 6.6 Legal considerations with use of kinematic coordinates ...... 130 6.7 Height transformation schema ...... 131 Chapter 7 Case Studies ...... 132 7.1 Australian Case Study ...... 132 7.1.1 Development of a stable Australian PMM ...... 132 7.1.2 Application of SAPRF2014 in practice ...... 136 7.1.3 Development of an Australian site velocity grid ...... 136 7.2 New Zealand case study 1 – multiple coseismic events ...... 137 7.2.1 Tectonic setting of Cape Campbell ...... 137 7.2.2 GNSS point positioning test data ...... 138 7.2.3 Estimation of displacements from published models ...... 141 7.2.4 Estimation of the interseismic velocity ...... 141 7.2.5 Estimation of coseismic displacements for each significant earthquake ...... 142 7.2.6 Estimation of postseismic displacements ...... 145 7.2.7 Supplementary corrections to the deformation model ...... 147 7.2.8 Interframe transformation parameters ...... 148 7.2.9 Summary of test position data and model estimation for use in schema test case studies . 148 7.2.10 Kinematic RF (with no DM) to Kinematic RF (with a defined DM) ...... 151 7.2.11 Kinematic RF (with no DM) to a semi-kinematic RF ...... 153 7.2.12 Kinematic RF (with DM) to a static RF ...... 156 7.3 New Zealand case study 2 – multiple SSEs ...... 159 7.3.1 Tectonic setting of Gisborne ...... 159 7.3.2 GISB time-series analysis ...... 160 7.3.3 Estimation of GISB interseismic velocity from the NZGD2000 deformation model ...... 161 7.3.4 Deconvolution of the GISB timeseries into interseismic and SSE components ...... 162 7.3.5 Forward modelling using deconvolved SSE timeseries ...... 163 Chapter 8 Discussion and Recommendations ...... 165 References ...... 168

List of Abbreviations and Acronyms

ADM Absolute Deformation Model AGD Australian Geodetic Datum AGD66 Australian Geodetic Datum 1966 APREF Asia-Pacific Reference Frame ARGN Australian Regional GNSS Network ARP Antenna Reference Point ASCII American Standard Code for Information Interchange ATRF Australian Terrestrial Reference Frame AusPOS Geoscience Australia Online GPS Processing Service BEIDOU BeiDou Navigation Satellite System (China) BIM Building Information Modelling BP Before Present CBD Central Business District CF Coordinate Frame CFRF Crust-fixed Reference Frame CFRS Crust-fixed Reference System CM Centre of Mass CORS Continuously Operating Reference Station CRS Celestial Reference System CRS Coordinate Reference System (refer p. 58) CSRS Canadian Spatial Reference System CVG Canadian Velocity Grid DCDB Digital Cadastral Database DM Deformation Model DORIS Détermination d'Orbite et Radiopositionnement Intégré par Satellite DTM Digital Terrain Model DTRF Deutsches Geodätisches Forschungsinstitut Reference Frame ECEF Earth-centred Earth-fixed ED50 European Datum 1950 EPSG European Petroleum Survey Group ESA European Space Agency ETRF European Terrestrial Reference Frame ETRF89 European Terrestrial Reference Frame 1989 GA Geoscience Australia GALILEO Global Navigation Satellite System (European Union) GBAS Ground Based Augmentation System

GDA2020 Geocentric Datum of Australia 2020 GDA94 Geocentric Datum of Australia 1994 GDAL Geospatial Data Abstraction Library GEONET GNSS Earth Observation Network System (Japan) GeoNet Geological Hazard Monitoring Network (New Zealand) GGOS Global Geodetic Observing System GGRF Global Geodetic Reference Frame GIA Glacial Isostatic Adjustment GIS Geographical Information System GLONASS GLObal NAvigation Satellite System (Russia) GNS GNS Science (New Zealand) GNSS Global Navigation Satellite Systems GPS Global Positioning System (USA) GRS80 Geodetic Reference System 1980 GTRF Galileo Terrestrial Reference Frame HTDP Horizontal Time-Dependent Positioning HYDL Hydrological Loading ICSM Intergovernmental Committee on Surveying and Mapping (Australia and New Zealand) IERS International Earth Rotation and Reference Systems Service IGb08 IGS Reference Frame 2008 revised IGS International GNSS Service IGS08 IGS Reference Frame 2008 IGS14 IGS Reference Frame 2014 INSAR Interferometric Synthetic Aperture Radar IOGP International Association of Oil and Gas Producers IRM International Reference Meridian IRNSS Indian Regional Navigation Satellite System IRP International Reference Pole ISO International Standards Organisation ITRF International Terrestrial Reference Frame ITRF2008 International Terrestrial Reference Frame 2008 ITRF2014 International Terrestrial Reference Frame 2014 ITRF92 International Terrestrial Reference Frame 1992 ITRF96 International Terrestrial Reference Frame 1996 ITRS International Terrestrial Reference System JPL Jet Propulsion Laboratory JTRF Jet Propulsion Laboratory Terrestrial Reference Frame LGM Last Glacial Maximum

LiDAR Light Detection and Ranging LINZ Land Information New Zealand LRF Local Reference Frame LSC Least-squares collocation LU Local Uncertainty MDT Mean Dynamic Topography MSL Mean Sea Level NAD27 North American Datum 1927 NAD83 North American Datum 1983 NGS National Geodetic Survey (USA) NMEA National Marine Electronics Association NNR No-Net Rotation NRCan Natural Resources Canada NRTK Network Real-Time Kinematic NTAL Non-tidal Atmospheric Loading NTOL Non-tidal Ocean Loading NTv2 National Transformation version 2 NZGD49 New Zealand Geodetic Datum 1949 NZGD2000 New Zealand Geodetic Datum 2000 OPUS Online Positioning User Service (USA-NGS) OTL Ocean Tide Loading PFRF Plate-Fixed Reference Frame PFRS Plate-Fixed Reference System PGR Post-Glacial Rebound PMM Plate Motion Model PNT Positioning, Navigation and Timing ppb parts-per-billion ppm parts-per-million PPP Precise Point Positioning PU Positional Uncertainty PV Position Vector PZ90 Parametry Zemli 1990 QGIS open source GIS software QZSS Quasi-Zenith Satellite System (Japan) RF Reference Frame RINEX Receiver Independent Exchange Format RRF Regional Reference Frame RTCM Radio Technical Commission for Maritime Services

RTK Real-Time Kinematic RU Relative Uncertainty SAPRF2014 Stable Australian Plate Frame 2014 SBAS Satellite Based Augmentation System SDI Spatial Data Infrastructure SET Solid Earth Tide SI Système International SLR Satellite Laser Ranging SPP Single Point Positioning SSE Slow-slip Event TIN Triangulated Irregular network TRANSIT Satellite navigation system (discontinued) TRF Terrestrial Reference Frame TRS Terrestrial Reference System UN United Nations UPS Universal Polar Stereographic USA United States of America USGS United States Geological Survey UTM Universal Transverse Mercator VLBI Very-Long-Baseline Interferometry WGS 84 World Geodetic System 1984

Glossary of terms

Absolute deformation (geodesy) A change in shape of the Earth's crust with respect to a geocentric terrestrial reference system. Absolute deformation can be described by models of plate rotation, site velocities, coseismic displacement and postseismic terms.

Celestial Reference System (CRS) A reference system used in astronomy to describe the positions of celestial bodies.

Coordinate Reference System (CRS) A reference system used to describe geodetic, ellipsoidal, topocentric or projected coordinates.

Coseismic displacement Near instantaneous displacement arising from seismic activity (e.g., earthquakes). In the context of applied geodesy it refers to displacement of bedrock or solid regolith at the Earth's surface.

Crust (Earth) The uppermost layer of the solid Earth the thickness of which varies according to tectonic position. Crustal thickness varies from near zero to 10 km for oceanic crust and up to 70 km thick for continental crust underneath the Himalayas. The Earth's crust is divided into tectonic plates which move over the underlying mantle. The boundary zones between tectonic plates are deforming zones.

Crust-fixed reference frame (CFRF) A set of geodetic monuments fixed to the Earth's surface (crust) with Cartesian coordinates related to either a plate-fixed reference system (e.g., a stable tectonic plate) or a terrestrial reference system. The crust-fixed reference frame (CFRF) typically has coordinates defined at a reference epoch. Epoch specific displacements (coseismic deformation, antenna displacement and postseismic parameters) are handled by displacement models or patches which are applied to the frame retrospectively at the reference epoch.

Crust-fixed reference system (CFRS) A reference system defined by crustal bedrock. A crust-fixed system is typically used within plate boundary zones and deforming zones where no single tectonic plate is suitable for adoption as a plate- fixed reference system. The crust-fixed reference system is typically realised by a model of interseismic site velocities with respect to the predominant terrestrial or plate-fixed reference system within the plate boundary zone.

Deformation (geodesy) A change in shape, position or orientation of the Earth's crust with respect to a time invariant terrestrial reference system.

Deformation model (geodesy) A model of site velocities and displacements that define the change in shape of the Earth's crust. The model can be referenced to a plate-fixed reference system (relative displacement) or a terrestrial (Earth fixed) reference system.

Deformation patch A model of coseismic displacements applied to pre-earthquake coordinates in order to estimate post- earthquake coordinates within the same reference frame. The format of the patch is typically a regular grid of topocentric displacements at a fixed epoch representing a deformation event.

Deforming zones Portions of the Earth's crust which are undergoing deformation. Deforming zones are mostly located near plate boundaries where relative movement of adjacent tectonic plates induces crustal strain. Deforming zones can also be located where glacial isostatic adjustment, volcanic activity or water abstraction are occurring.

Displacement Movement of a fixed feature on the Earth's surface with respect to time invariant reference system.

Distortion (geodesy) The change in shape of a reference frame from its initial realisation where no deformation is modelled. In applied geodesy, distortion usually arises from measurement imprecision and positional uncertainty rather than deformation in stable plate settings.

Distortion grid A grid of displacements from known to truth. In most instances the displacement is due to improved measurement or positional uncertainty rather than deformation. If the displacement results from deformation that is not known or modelled then the distortion grid can conflate both distortion and deformation components.

Dual-frame geodetic system A situation where both a semi-kinematic reference frame and kinematic reference frame are used concurrently. The semi-kinematic frame is used for surveying and mapping where coordinate stability during periods of interseismic stability is important. The kinematic frame is used for satellite point positioning, semi-kinematic frame maintenance and integrity monitoring. Deformation models enable interframe transformations.

Dynamic datum A misnomer used informally to describe a kinematic reference frame.

Earth-fixed datum or reference frame An alternative name for a time-dependent terrestrial reference frame. Also referred to as a kinematic reference frame or informally by the misnomer "dynamic datum".

Geocentre The centre of mass (CM) of the Earth.

Geodetic Datum A means of describing the orientation and scale of an ellipsoid and geodetic coordinate system with respect to the Earth. A geocentric datum is a datum whose origin coincides with the centre of mass of the Earth, the geocentre.

Glacial isostatic adjustment (GIA) Movement of the Earth's crust as a response to loading or unloading of ice sheets and glaciers. GIA has also been referred to as post-glacial rebound. The displacement is predominantly vertical with a smaller horizontal component.

Interseismic velocity The linear rate of change (or displacement rate) of a position between episodes of non-linear seismic deformation (coseismic and postseismic deformation). It can also be referred to as a secular velocity.

Kinematic reference frame A reference frame where the motion of monuments within the frame are described with respect to a time invariant reference system.

Mantle The portion of the Earth between the crust and the core, composed largely of periodite and constituting most of the volume of the Earth. The mantle is largely solid but viscosity is apparent over geological time scales.

Plate-fixed reference frame (PFRF) A set of geodetic monuments fixed to the Earth's surface with coordinates related to a plate fixed reference system (e.g., a stable tectonic plate). The plate-fixed reference frame (PFRF) typically has terrestrial reference frame coordinates defined at a reference epoch. The velocities and displacements are related to the stable plate with the secular plate motion velocity removed.

Plate-fixed reference system (PFRS) A reference system defined by the pole of rotation of a rotating tectonic plate. The pole of rotation is classically described by the latitude and longitude of the pole of rotation and the angular velocity (Euler theorem). The rotation can also be described by the rate of rotation of the Cartesian axes of the rotating plate with respect to a terrestrial reference system.

Postseismic displacement Elastic creep, slow slip displacement and viscoelastic relaxation that occurs after an earthquake and associated coseismic deformation. Postseismic displacement can be represented by a combination of logarithmic, exponential parameters and transient velocity changes. Postseismic displacement can be evident for decades following large earthquakes.

Reference ellipsoid An ellipse rotated about its semi-minor axis to form a shape that approximates the figure of the Earth. The semi-minor axis is aligned with the Cartesian z-axis and the semi-major axis is coplanar with the equator formed by a plane including the x and y axes.

Reference frame (geodesy) A realisation of a geodetic reference system by a network of physical geodetic monuments including GNSS CORS, VLBI, SLR, Doris and gravity stations. (Refer also to Terrestrial Reference Frame)

Relative displacement A displacement of a monument with respect to the stable portion of an adjacent tectonic plate.

Semi-kinematic reference frame A reference frame in two parts. The static part of the frame is defined by the coordinates of geodetic monuments at a reference epoch. The kinematic part of the frame is a deformation model defined within a crust-fixed, plate-fixed or terrestrial reference system. The deformation model is comprised of interseismic velocity models and seismic displacement models that enables transformations between a kinematic reference frame and the static part of the frame.

Site velocity The rate of change of position of a location on the Earth's surface with respect to a geodetic reference system (refer also to interseismic velocity).

Slow-slip events Nearly aseismic deformation occurring over longer time periods (hours to months).

Static datum or frame A geodetic datum or frame whose coordinates are invariant with time, irrespective of any deformation that may occur or be occurring within the frame.

Strain (geodesy) Strain in terms of geodesy is a change in length between any two points. Strain is unitless and is described as a ratio.

Strain rate (geodesy) The rate of change of length between any two points.

Surface creep The geophysical process where the Earth's surface moves with respect to underlying bedrock under the force of gravity. Creep is evident on steeper slopes with soil, clay or loose regolith. Geodetic monuments fixed to these surface materials will move over time but not reflect underlying tectonic processes.

Tectonic plate An internally stable portion of the Earth's crust that moves over the underlying mantle.

Terrestrial reference frame (TRF) A set of geodetic monuments fixed to the Earth's surface with Cartesian coordinates related to a terrestrial reference system (TRS). The TRF typically has coordinates defined at a reference epoch and also time dependent parameters that define the motion of the monuments within the TRS. Time dependent parameters can include site velocities (interseismic deformation) and epoch specific displacements (coseismic deformation, antenna displacement and postseismic parameters).

Terrestrial Reference System (TRS) A system uniquely describing the coordinates of points (using the International System of Unit for length, the metre) with respect to the geocentre. The system is primarily defined by three Cartesian axes (x, y and z) whose origin is the Geocentre and where the z axis is aligned with the International Reference Pole (North Pole), x axis is aligned with the equator at the International Reference Meridian (0° E) and the y axis is orthogonal to both at 90°E in a right-handed system. The axes co-rotate with the Earth on contemporary time scales.

Topocentric coordinates Three dimensional coordinates in an East, North, Up (E,N,U) system related to a point on the Earth's surface, typically defined at the ellipsoid surface.

List of Figures

Figure 2.1 Map of principal tectonic plates ...... 36 Figure 2.2 Euler Pole defining a plate rotation ...... 37 Figure 2.3 Second invariant of strain rate from the Global Strain Rate Model (v. 2.1) ...... 38 Figure 2.4 Glacial Isostatic Adjustment ...... 42 Figure 2.5 Schematic representation of a plate boundary deformation zone ...... 44 Figure 2.6 Site velocity changes within plate boundary deformation zones...... 44 Figure 2.7 NZGD2000 Deformation Model - absolute interseismic velocity model ...... 45 Figure 2.8 Coseismic displacement grid - example, Christchurch, New Zealand ...... 47 Figure 2.9 Estimating coseismic displacement from sparse campaign measurements ...... 48 Figure 2.10 Temporal and spatial domain of terrestrial deformation ...... 50 Figure 2.11 Localised deformation of geodetic monuments ...... 52 Figure 2.12 Surface deformation separated from coseismic crustal deformation ...... 55 Figure 2.13 Vertical deformation in urban areas due to groundwater changes ...... 56 Figure 2.14 Effect of differential subsidence on urban structures ...... 57 Figure 3.1 Representation of a CRS using the ITRS as an example ...... 59 Figure 4.1 Effect of plate rotation on PU for different applications ...... 73 Figure 4.2 Effect of plate rotation on GNSS vectors and networks ...... 74 Figure 4.3 Years before dimensional tolerances are exceeded for differing strain rate scenarios ...... 76 Figure 4.4 Years until PU tolerances are exceeded for differing strain rates and specifications ...... 76 Figure 4.5 Effect of interseismic strain on dimensional tolerances within plate boundary zones ...... 77 Figure 4.6 Effect of coseismic deformation on the cadastral boundaries and geodetic referencing ...... 78 Figure 4.7 Modelled global coseismic deformation resulting from the Mw 9.0 Sumatra earthquake ...... 79 Figure 4.8 Slow-slip event deformation, example is GISB, New Zealand East component ...... 80 Figure 4.9 Example of postseismic deformation and associated positional tolerance ...... 81 Figure 4.10 Linear approximations of postseismic deformation ...... 81 Figure 4.11 Localised deformation example – regolith creep ...... 83 Figure 5.1 Nested structure of grid models to accommodate variable resolution ...... 86 Figure 5.2 Bilinear interpolation of gridded data ...... 91 Figure 6.1 Typical kinematic RF transformation scenarios with no deformation model...... 96 Figure 6.2 Typical kinematic RF transformation scenarios with a deformation model...... 97 Figure 6.3 Typical semi-kinematic RF transformation scenarios ...... 98 Figure 6.4 Typical static RF transformation scenarios ...... 99 Figure 6.5 Point motion trajectory and intraframe propagation in a complex deformation zone ...... 100 Figure 6.6 General schema for complex time-dependent transformation between kinematic RF ...... 108 Figure 6.7 Transformation between two kinematic RF with no DM ...... 112 Figure 6.8 Transformation from a kinematic RF with no DM to a kinematic RF with a DM ...... 113 Figure 6.9 Transformation from a kinematic RF with a DM to a kinematic RF with no DM ...... 114 Figure 6.10 Transformation between two kinematic RF with a DM ...... 115 Figure 6.11 Transformation from a kinematic RF with no DM to a semi-kinematic RF ...... 116

Figure 6.12 Transformation from a kinematic RF with a DM to a semi-kinematic RF ...... 117 Figure 6.13 Transformation from a kinematic RF with no DM to a static RF ...... 119 Figure 6.14 Transformation from a kinematic RF with a DM to a static RF ...... 120 Figure 6.15 Transformation from semi-kinematic RF to a kinematic RF with no DM ...... 121 Figure 6.16 Transformation from semi-kinematic RF to a kinematic RF with a DM ...... 122 Figure 6.17 Transformation from between two semi-kinematic RF ...... 123 Figure 6.18 Transformation from semi-kinematic RF to a static RF ...... 124 Figure 6.19 Transformation from a static RF to a kinematic RF with no DM ...... 126 Figure 6.20 Transformation from a static RF to a kinematic RF with a DM ...... 127 Figure 6.21 Transformation from a static RF to a semi-kinematic RF ...... 128 Figure 6.22 Transformation between two static RF ...... 129 Figure 7.1 ARGN and AuScope CORS selection used to estimate a stable Australia PMM ...... 133 Figure 7.2 ITRF site velocities for selected ARGN and AuScope stations ...... 134 Figure 7.3 Velocity residuals for stable Australian plate ...... 135 Figure 7.4 Kaikōura earthquake, November 2016, tectonic setting and observed displacements ...... 137 Figure 7.5 Location of test point CMBL and interseismic velocity model data ...... 142 Figure 7.6 Location of second New Zealand case study ...... 159 Figure 7.7 GeoNET – CORS station GISB ...... 160 Figure 7.8 GPS time-series for CORS showing slow-slip events and trending interseismic velocities .. 161 Figure 7.9 Comparison between NZGD2000 DM and observed SSE displacements ...... 162 Figure 7.10 GISB timeseries with interseismic signal minimised to highlight SSE displacements ...... 163

List of Tables

Table 2.1 Recent plate motion models ...... 39 Table 2.2 ITRF2014 Plate Motion Model ...... 40 Table 2.3 Modes of deformation, their magnitude, duration and extent ...... 50 Table 3.1 Indicative horizontal dimensional tolerances and uncertainties ...... 71 Table 3.2 Indicative Positional Uncertainty (PU) tolerances ...... 71 Table 4.1 Number of years until 3D PU tolerance is exceeded for GNSS vectors ...... 74 Table 4.2 Number of years before NRTK positioning tolerance is exceeded due to strain ...... 77 Table 5.1 Summary of appliction of grid formats ...... 93 Table 6.1 Determining the epoch of the schema interframe transformation epoch ...... 107 Table 7.1 Pre-earthquake sequence observed coordinates at epoch 2008.0 for CMBL ...... 139 Table 7.2 Post-earthquake sequence observed coordinates at epoch 2019.167 for CMBL ...... 140 Table 7.3 Tabulated NZGD2000 and derived NZGD49 data for test point CMBL ...... 140 Table 7.4 ITRF96 Interseismic velocity model node data used for test point CMBL ...... 141 Table 7.5 July 2013 Cook Strait earthquake coseismic displacement model for CMBL ...... 143 Table 7.6 August 2013 Lake Grassmere earthquake coseismic displacement model for CMBL ...... 143 Table 7.7 November 2016 Kaikōura earthquake coseismic displacement model for CMBL ...... 143 Table 7.8 First correction for Kaikōura earthquake coseismic displacement model ...... 144 Table 7.9 Second correction for Kaikōura earthquake coseismic displacement model ...... 144 Table 7.10 2016 Kaikōura earthquake first postseismic displacement model for CMBL ...... 146 Table 7.11 2016 Kaikōura earthquake second postseismic displacement model for CMBL ...... 146 Table 7.12 2016 Kaikōura earthquake supplementary displacement model for CMBL ...... 147 Table 7.13 ITRF2014 to ITRF96 (LINZ) PV notation transformation parameters ...... 148 Table 7.14 ITRF2014 to ITRF96 (LINZ) CF notation transformation parameters ...... 148 Table 7.15 Interseismic velocity model node data used for GISB ...... 161 Table 7.16 GISB tabulated NZGD2000 coordinates ...... 161

List of Publications and Presentations

Stanaway, R., and Roberts, C.A., A Simplified Parameter Transformation Model from ITRF2005 to any Static Geocentric Datum (e.g., GDA94), Proceedings of the International Global Navigation Satellite Systems Society Symposium IGNSS2009 1-3 December 2009, Gold Coast, Queensland, Australia, 2009

Stanaway, R., and Roberts, C.A., CORS Network and Datum Harmonisation in the Asia-Pacific Region, Proceedings of the FIG Congress, 11-16 April 2010, Sydney, Australia, 2010.

Stanaway, R., and Roberts, C.A., Rigid Plate Transformations to Support PPP and Absolute Positioning in Africa, Proceedings of the FIG Working Week, Marrakech, Morocco, 18-22 May 2011, 2011

Stanaway, R., and Roberts, C.A., ITRF Transformations in Deforming Zones to Support CORS-NRTK Applications, Proceedings of the International Global Navigation Satellite Systems Society (IGNSS) Symposium 15-17 November 2011, Sydney, Australia, 2011.

Stanaway, R., The Future of Geodetic Datums, SSSI Land Surveying Commission – National Conference, 18-21 April 2012.

Stanaway, R., Roberts, C., Blick, B., and Crook, C., Four Dimensional Deformation Modelling, the link between International, Regional and Local Reference Frames, Proceedings of the FIG Working Week, Rome, Italy, 6-10 May 2012, 2012.

Stanaway, R., Dynamic (kinematic) datums - International Trends and Future Challenges, CRCSI Project 1.02 – Next Generation Australian and New Zealand Datum meeting, Canberra, Australia, 16th July, 2012.

Stanaway, R., GDA2020 Deformation Model - a semi-kinematic approach to datum modernisation, NSW LPI datum modernisation workshop, Bathurst, NSW, Australia, 26th October 2012.

Stanaway, R., Roberts, C.A., Blick, G., and Crook, C., Four Dimensional Deformation Modelling to support Datum development in Australia and New Zealand, Presentation at LINZ, Wellington, New Zealand, 14th February, 2013.

Stanaway, R., A Deformation Model to support a Next Generation Australian Geodetic Datum, Surveying and Spatial Sciences Conference 2013, Canberra, Australia,15–19 April 2013.

Stanaway, R., A template for the development of a modernised geodetic infrastructure in Pacific Island states, FIG Pacific Small Island Developing States Symposium, Fiji, 18-20 September, 2013.

Stanaway, R., Roberts, C.A., and Blick, G.; Realisation of a Geodetic Datum Using a Gridded Absolute Deformation Model (ADM), in Rizos, C., and Willis, P. (eds.), Earth on the Edge: Science for a Sustainable Planet, International Association of Geodesy Symposia, Vol. 139, 259-256, Proceedings of the IAG General Assembly, Melbourne, Australia, June 28 - July 2, 2011, doi:10.1007/978-3-642- 37222-3_34, 2013.

Blick, G. (ed.), Crook, C., Donnelly, N., Fraser, R., Lilje, M., Martin, D., Rizos, C., Roman, D., Sarib, R., Soler, T., Stanaway, R., and Weston, N., Reference Frames in Practice Manual, FIG Publication 64, FIG Commission 5 Working Group 5.2, May 2014.

Stanaway, R., Deformation Modelling to support the Papua New Guinea Geodetic Datum 1994 (PNG94), Proceedings of the FIG Congress, 16-21 June, Kuala Lumpur, Malaysia, 2014.

Wallace, L. M., Ellis, S., Little, T., Tregoning, P., Palmer, N., Rosa, R., Stanaway, R., Oa, J., Nidkombu, E., and Kwazi, J., Continental breakup and UHP rock exhumation in action: GPS results from the Woodlark Rift, Papua New Guinea, Geochemistry Geophysics Geosystems, 15, 4267–4290, doi:10.1002/2014GC005458, 2014.

Stanaway, R., The Next-Generation Australian Geodetic Datum Benefits and Challenges, Institution of Surveyors Victoria Regional Conference, Wangaratta, Victoria, Australia, 18th April 2015.

Stanaway R., and Roberts C.; A High-Precision Deformation Model to Support Geodetic Datum Modernisation in Australia. in: Rizos C., Willis P. (eds.), IAG 150 Years. International Association of Geodesy Symposia, Vol. 143, Springer, Proceedings of the IAG Scientific Assembly, Potsdam, Germany, September 1-6, 2013, doi:10.1007/1345_2015_31, 2015.

Stanaway, R., Roberts, C.A., Rizos, C., Donnelly, N., Crook, C., and Haasdyk, J.; Defining a Local Reference Frame Using a Plate Motion Model and Deformation Model, in: van Dam, T. (ed.), REFAG 2014, International Association of Geodesy Symposia Vol. 146, Springer, Proceedings of the IAG Commission 1 Symposium Kirchberg, Luxembourg, 13–17 October, 2014, doi:10.1007/1345_2015_147, 2015.

Donnelly, N., Crook, C., Stanaway, R., Roberts, C., Rizos, C., and Haasdyk, J., A Two-Frame National Geospatial Reference System Accounting for Geodynamics, in: van Dam, T. (ed.), REFAG 2014, International Association of Geodesy Symposia Vol. 146, Springer, Proceedings of the IAG Commission 1 Symposium Kirchberg, Luxembourg, 13–17 October, 2014, doi:10.1007/1345_2015_188, 2015.

Stanaway, R., Roberts, C.A., Tregoning, P., McClusky, S., Koulali, A., Wallace, L., and Rosa, R., Development of a geodetic deformation model for Papua New Guinea (PNG), 26th General Assembly of the International Union of Geodesy and Geophysics (IUGG), Prague, Czech Republic, 22 June - 2 July, 2015.

Koulali, A., Tregoning, P., McClusky, S., Stanaway, R., Wallace, L., and Lister, G., New Insights into the present-day kinematics of the central and western Papua New Guinea from GPS, Geophysical Journal International, 202, 993–1004 doi: 10.1093/gji/ggv200, 2015.

Stanaway, R., A template for the development of a modern national reference frame, FIG/IAG/UN-GGIM- AP/UN-ICG/NZIS Technical Seminar - Reference Frame in Practice, Christchurch, New Zealand, 1-2 May 2016.

Stanaway, R., Plate tectonics, GDA2020 and the future of positioning, SSSI Qld – Northern Group Conference – Townsville, Queensland, Australia, 28-29 October 2016.

Stanaway, R., Practical implementation of time-dependent reference frames, FIG/IAG/UN-ICG/HKMO Technical Seminar - Reference Frame in Practice, Istanbul, Turkey, 4-5 May 2018.

Stanaway, R., Reference Frames, Transformations and GIS, FIG/IAG/UN-GGIM-AP/ICG Technical Seminar - Reference Frame in Practice, Hanoi, Vietnam, 20-21 April 2019.

Chapter 1 Introduction

Satellite navigation and positioning are becoming ubiquitous in contemporary society. Almost all aspects of everyday life and commerce are becoming increasingly reliant on satellite positioning, navigation and timing (PNT) systems. The accuracy and precision of these systems have improved dramatically since the development of the TRANSIT Doppler system in 1964 (Danchik, 1998). Global Navigation and Satellite Systems (GNSS) which now broadly encompass the USA’s Global Positioning System (GPS), Russia’s GLONASS, Europe’s GALILEO, China’s BEIDOU, Japan’s QZSS and India’s IRNSS now routinely provide 3 - 8 metre positioning accuracy in Single Point Positioning (SPP) mode using broadcast ephemerides (e.g., ESA, 2019; US Government, 2008). This accuracy can be improved to ~1 metre by means of GNSS augmentation including; Satellite-based Augmentation Systems (SBAS) (GMV, 2016) and Ground-based Augmentation Systems (GBAS). Precise Point Positioning (PPP) services including; OmniSTAR (OmniSTAR, 2019), StarFire (NAVCOM, 2019), CenterPoint RTX (Trimble, 2019), TerraStar (Novatel, 2019), Seastar (Fugro, 2019) and Atlas (Hemisphere, 2019) can provide 2-50 centimetre positioning accuracy in near real-time to subscribers of those services. Post-processed PPP services such as the Canadian Spatial Reference System (CSRS) PPP service (NRCan, 2019) can provide positioning accuracy of < 1 centimetre in static mode depending upon observation time and observing conditions.

Differencing of the carrier phase can improve relative positioning precision between two or more GNSS sensors in static mode to within 5 mm over thousands of km using geodetic grade multi-frequency GNSS receivers and sensors. Several agencies provide free online carrier-phase differencing processing services including, OPUS (NGS, 2019), AusPOS (GA, 2019a) and Scout (SCRIPPS, 2019).

It is anticipated that by 2025, a wide spectrum of non-expert users of GNSS positioning devices (e.g., GNSS enabled smartphones) enabled with SBAS, Real-Time Kinematic (RTK) or improved SPP such as the implementation of the new higher resolution L5/E5 interoperable signals (e.g., Roberts, 2011) will have access to real-time absolute positioning accuracy better than 20 cm. Improvements to GNSS orbit modelling, availability of near real-time precise orbits, interoperable GNSS, improved clocks and processing software will further support this outlook.

1.1 GNSS Reference Frames

GNSS orbits and positions are intrinsically defined within geocentric reference systems whose origins are the centre-of-mass of the Earth, the geocentre, which also defines the fundamental origin of satellite orbit models. Reference systems for the different GNSS constellations include the World Geodetic System 1984 (WGS 84) for GPS and PZ-90 for GLONASS. Both of these systems are now aligned with the International Terrestrial Reference System (ITRS) which is realised on the Earth’s surface by the International Terrestrial Reference Frame (ITRF). The latest realisation of ITRF is ITRF2014 (Altamimi et al., 2016). The

International GNSS Service (IGS) has also released a GNSS realised frame IGS14 that is closely aligned with ITRF2014 (IGS, 2017).

Importantly, coordinates estimated by the GNSS systems within any of these global reference frames are kinematic in nature due to large scale geodynamic effects including plate tectonic activity and associated horizontal and vertical displacements of the Earth's surface due to seismic deformation, volcanism, glacial isostatic adjustment (GIA) and localised ground instability. GNSS reference frames are necessarily kinematic in nature to prevent degradation of orbit products arising from relative displacement of stations within GNSS tracking networks as a result of these geodynamic processes.

1.2 Challenges of interfacing GNSS positioning and precise spatial data

The rapid improvement in mass-market positioning precision is presenting new challenges to users and managers of spatial data where GNSS positions are used in conjunction with non-GNSS aligned reference frame based spatial data. In general, spatial data infrastructure (SDI) is still largely referenced to ground- fixed reference frames (static geodetic datums as defined later in Chapter 2) where coordinates of stable features are not expected to change significantly as a function of time unless localised deformation or network readjustments have occurred. The widespread assumption is that these current generation ground-fixed reference frames are precisely aligned with GNSS reference frames over time. In reality, the crust on which the ground-fixed reference frame is realised is moving over the Earth's asthenospheric mantle (Turcotte and Schubert, 2002). Temporal and spatial stability of a coordinate system is a necessary precondition for stacking of spatial data acquired at different epochs to be correctly aligned (Stanaway and Roberts, 2009). In a kinematic reference frame such as those used intrinsically by GNSS, coordinates of spatial data acquired at different epochs are displaced as a function of time and the data site velocity due to plate tectonics and other geodynamic effects. This displacement is undesirable unless the site motion is estimated with significant precision and also modelled rigorously within spatial data software.

1.2.1 Misalignment of GNSS and spatial data reference frames

Difficulties therefore arise where GNSS derived precise positions in terms of a kinematic global reference frame are used erroneously in the context of existing spatial data defined within ground-fixed frames. Geographical Information System (GIS) software used to store, manage and visualise spatial data is still not generally capable of temporally transforming data represented at different epochs of a kinematic frame to a common epoch for the purpose of alignment, visualisation and comparative analysis. Repeat surveys using GNSS kinematic reference frames as little as a few months apart are adversely affected by coordinate changes due to tectonic motion, unless the position is transformed to a common epoch or local reference frame (e.g., Stanaway and Roberts, 2011).

To overcome the issue of misalignment of GNSS precise positioning with ground-fixed coordinates and spatial data, many of the positioning services described earlier provide coordinates in both kinematic

ITRF and local static geodetic datums where global plate motions are intrinsically modelled. For example, Geoscience Australia's AusPOS service provides coordinates in three different reference frames; ITRF2014, Geocentric Datum of Australia 2020 (GDA2020) and its predecessor the Geocentric Datum of Australia 1994 (GDA94) within the stable portion of the Australian Plate. Trimble's CenterPoint RTX post- processing service (Trimble, 2019) allows users to select a tectonic plate for representation of ITRF coordinates at a reference epoch (currently 2010.0 using ITRF2014) using embedded Plate Motion Models (PMM) but this is only useful within the stable portions of plate models defined within the processing engine. No standard model format or approach currently exists to transform ITRF2014 data within plate boundary or other rapidly deforming zones including, Japan, China, Western USA, Chile, Greece, New Zealand and Indonesia. New Zealand's PositioNZ-PP service (Pearson et al., 2015) uses a plate-boundary deformation model to transform ITRF positions to the New Zealand Geodetic Datum 2000 (NZGD2000). The NRCan-PPP and OPUS services currently provide coordinates in either ITRF2014 or the North American Datum 1983 (NAD83) realised separately in Canada by NAD83(CSRS) and NAD83(2011)/SPCS in the United States of America (USA) using recent velocity models.

1.2.2 Requirement for local frame stability for civil projects and urban areas

The implications of mass-market access to precise ITRF positioning capability is significant as the international geodetic community, as well as vendors of GNSS hardware and associated software, do not currently have a uniform or rigorous approach to handle the kinematic nature of ITRF coordinates. The kinematic nature of GNSS positions within ITRF and WGS 84 is poorly understood by land surveyors, engineers and other spatial science professionals who are considered the technical custodians of these datasets. For example, ad hoc realisations of ITRF derived from precise positioning services are often used as a basis for an operational datum for enduring civil projects in the absence of a widely adopted or mandated local reference frame. Current civil projects are typically complex with a lifetime measured in several decades, involving many diverse stakeholders, spatial data managers, spatial data software and surveyors over the project lifetime. Such projects require a strictly mandated, stable and precise coordinate system as a fundamental basis for integrity of design, construction and maintenance over the project lifespan.

Where the epoch of ITRF is not clearly specified in documentation or metadata, it inevitably leads to data misalignments, positioning errors and confusion as the project progresses. This approach is commonplace in countries or regions with a sparse or non-existent network of geodetic monuments and Continuously Operating Reference Stations (CORS). Repeated use of PPP is further complicated by interseismic tectonic deformation, which can be up to 80 mm yr-1 in magnitude, and seismic deformation which can result in up to several metres of movement almost instantaneously. Unless subsequent positions are corrected for tectonic deformation (whether secular interseismic, episodic, or both), continued use of PPP will degrade the precision of localised geodetic networks as a function of time and as a consequence lead to costly spatial integrity issues with cadastral surveys and civil works.

1.2.3 Implications of geodynamic processes on precision GNSS users

The implications for specialist users of GNSS post-processing software products and CORS networks are also important. For example, a limitation of a conventional static geodetic datum arises from the processing of long GNSS baselines. If the static coordinates of a reference station are held fixed for baseline processing and network adjustment, rigid plate rotation of a long baseline or network will degrade the precision of the point computation as a function of time (e.g., Dawson and Woods, 2010), even if baseline lengths do not change. In order for surveys undertaken at different epochs to be combined or integrated within a kinematic frame, a time-dependent transformation model has to be applied rigorously, or be embedded within the metadata. This time-dependent transformation is necessarily complex in tectonically active regions. Furthermore, for the model to be applied correctly, all data has to be correctly time-tagged with the epoch of acquisition.

The implications of kinematics are also quite significant considering that global PPP services are often used for precision navigation and steering applications which are typically defined by ground-fixed coordinates (e.g., precision agriculture, automated mining, machine control guidance systems and other high precision navigation applications such as autonomous driving).

The use of CORS and processing of long GNSS baselines are also adversely affected by global and local deformations. For example, Network Real Time Kinematic (NRTK) GNSS requires < 15 mm a priori precision of the CORS station coordinates (Ramm and Hale, 2004) in order to correctly model tropospheric and orbit biases within the network coverage area. This precision is difficult to attain in a deforming zone if the CORS coordinates are fixed indefinitely or if a deformation model is not applied to ITRF coordinates. In order for positions from precision GNSS techniques to maintain consistency within a static local reference system, a time-dependent transformation strategy is also required to relate instantaneous kinematic ITRF coordinates to the local system.

1.2.4 Drivers for reference frame modernisation

There are two competing drivers for adoption of a kinematic reference frame for operational use. This conflict is between the current user requirements for a conventional static datum where coordinates are invariant with time; and the complexities of geodynamics where coordinates are in reality changing constantly. This conflict inevitably leads to inconsistent coordinates of ground-fixed features. On the one hand, there is the need for a kinematic and high precision global reference frame to compute precise GNSS orbits and to monitor real-time changes in the Earth (e.g., Global Geodetic Observing System - GGOS) (Plag and Pearlman, 2009). On the other hand, there is the need for coordinate consistency within a localised reference frame to support integration of spatial data acquired over long periods of time (e.g., cadastral surveys, infrastructure surveys, Building Information Modelling (BIM), land management, spatial data

30 management, mapping and precision navigation) where coordinate stability (and by definition positioning repeatability) are essential (e.g., Stanaway and Roberts, 2009).

1.2.5 The requirement for a rigorous time-dependent transformation strategy

At present, the latest realisation of ITRF, ITRF2014, fulfils the role of a high precision global scientific reference frame, but in the absence of time-dependent transformation tools the kinematic nature of ITRF at present precludes practical adoption by the majority of users and real-world applications. Rather, a semi-kinematic or dual-frame approach is increasingly preferred as a working compromise, to overcome the limitations of both kinematic and static reference frames, while still retaining the benefits of both.

1.2.6 Current approaches to time-dependent transformations

Presently there are two approaches to resolving the misalignment between kinematic frames used for GNSS positioning and spatial data reference frames. One approach is to transform existing spatial data sets defined in a ground-fixed frame forward to the epoch of a GNSS precise position (a forward time- dependent transformation). Another approach is to transform GNSS precise positions back to a local ground-fixed frame (a reverse time-dependent transformation). Both approaches make use of a conformal time-dependent transformation such as a 14 parameter transformation e.g., Dawson and Woods (2010) or a time-dependent displacement grid derived from an ITRF site velocity model e.g., Stanaway and Roberts, (2015), Crook et al. (2016) and LINZ (2019). The conformal parametric approach is only suitable in stable plate settings such as the Australian continent whereas the velocity model approach can be used in any tectonic setting. The computational overhead of transforming large volumes of spatial data "on-the-fly" can be a limitation with transforming ground-fixed spatial data to the epoch of positioning, particularly for very large raster files such as high resolution imagery and dense point clouds acquired by laser scanning technologies. The reverse approach is presently in more widespread use (e.g., in geodetic analysis software) as it is well suited to current generation geodetic reference frames which are inherently fixed to the Earth's crust at a defined reference epoch. In any case, either approach must be used to combine and analyse spatial data acquired at different epochs of ITRF. If epoch information is unknown or absent in metadata, the difference between a GNSS precise position and a precise spatial data base can be significant.

1.2.7 Characteristics of existing time-dependent transformation strategies

Ground-fixed or Local Reference Frames (LRF) fixed to stable portions of the Earth's crust are ideally suited to support spatial data integration over longer periods of time. Tectonic site velocities are minimised with respect to the local frame. However, there remains the issue of how GNSS precise positions relate to spatial data defined in such a ground-fixed frame and the 14 parameter transformation and gridded deformation model approaches each have their limitations. Fourteen parameter transformations include scale and scale-rate parameters which, if non-zero, implicitly define uniformly distributed strain of the target local frame. Gridded site velocity models (or kinematic representations of

31 deformation models) can better accommodate localised and variable deformation, however they may be inefficient over large areas of stable tectonic plates unless the model architecture supports variable resolution grids such as the National Transformation version 2 (NTv2) format developed by Natural Resources Canada or GeoTIFF. NTv2 is now widely used globally for high precision grid based geodetic transformations but is limited to only two dimensions. For tectonically stable regions, a plate motion model (PMM) (e.g., Altamimi et al., 2017) can be used to transform GNSS point positions to a geocentric LRF. Where higher precision is required, a supplementary residual velocity model can also be applied if intraplate deformation is significant. In the USA, intraplate deformation models have been used in Horizontal Time-Dependent Positioning (HTDP) software (Snay, 1999; Pearson and Snay, 2012) since 2000.

The main advantage of a PMM is that it is inherently distortion and strain free as it is defined only by the rotation of a stable portion of the tectonic plate. The other advantage of a PMM is that localised or intraplate deformation is more clearly visualised and analysed where the rigid plate rotation signal component is removed.

Continental and Regional Reference Frames (RRF) fixed to tectonic plates such as the European Terrestrial Reference Frame 1989 (ETRF89) (Boucher and Altamimi, 1992), NAD83 (Schwarz, 1989), GDA94 (Steed, 1995) and GDA2020 (ICSM, 2018) have been defined from different realisations of ITRF, and 14 parameter transformations are required to transform positions within these RRF to ITRF at a specified epoch. Furthermore, deformation within these plate-fixed RRF is evident as non-zero station velocities for stations in deforming zones within the RRF.

1.3 Thesis aims

The aims of this thesis are to:

● Examine the effect of geodynamic processes on reference frames used in practice, taking into consideration precision and tolerance requirements for different geodetic applications

● Describe how a plate motion model can be used to realise a local reference frame and associated time dependent transformation to a terrestrial reference frame

● Develop a rigorous and traceable time-dependent transformation schema that can be applied for reference frame transformations in the most complex tectonic settings that includes estimation of uncertainty

● Develop a lexicon of terminology relevant for next generation reference frames and transformations

1.4 Thesis outline

This thesis develops and describes a robust and flexible schema for describing and defining complex time- dependent reference frame transformations. The schema enables combination of spatial data acquired at different epochs and also alignment of GNSS precise point positions with stale spatial data. The research is directly relevant for the implementation of kinematic reference frames for non-scientific applications such as surveying and mapping. There are many acronyms, abbreviations and new terminology introduced in this thesis. To assist readers, tables of abbreviations, acronyms and symbols are provided in the front content of the thesis including a glossary of terms providing short descriptions of terminology introduced or adopted in this thesis.

Chapter 2 is a review of the different modes of Earth deformation that can impact on reference frames, positioning, spatial data and transformations. The chapter also shows how time dependent transformation parameters can be estimated from plate motion and other geodynamical models. The primary mode of deformation globally is that of plate tectonics. Inversion of observed site velocities can be used to estimate plate motion models, residual velocities and associated uncertainties. Within plate boundary zones, interseismic coupling along the plate boundary interface induces geodetic strain that results in changes to stable plate velocities within these zones. Fault locking models can then be used to estimate these velocity changes. Strain accumulation is released during earthquakes, slow-slip events and subsequent postseismic deformation. Models of these deformation events are used to estimate associated displacements and velocity changes. Other geodynamic processes that impact on reference frames and surface motions such as glacial isostatic adjustment and water abstraction are also described.

Temporal and spatial modes of Earth deformation are then described in the context of geodetic reference frames taking into consideration their impact on positioning and dimensional tolerances for different users of a reference frame. Finally, the effects of localised deformation and monument instability are discussed.

Chapter 3 summarises general concepts of geodetic reference systems and reference frames. The hierarchy of reference frames is described. Limitations of current approaches to reference frames in practice are described with some recommendations proposed. Current transformation schema are also evaluated, again with a discussion of some of the limitations of current approaches. Positioning and dimensional tolerance requirements for a diverse range of user requirements are also examined with regard to the effects of deformation.

Chapter 4 assesses in detail the impact of global and local deformation on positional and dimensional tolerances with a view to estimating temporal validity of reference frames and transformation parameters between reference frames. A detailed analysis of the impact of coseismic deformation is provided. Localised deformation and vertical deformation are also discussed. This approach is essential to provide

33 guidance for geodetic agencies and developers of reference frames on the utility and longevity of reference frames in practice.

In Chapter 5, the formatting and presentation of time dependent transformations is described. Standardisation options for International Standards Organisation (ISO) and the European Petroleum Survey Group (EPSG) geodetic registries are also proposed.

Chapter 6 develops and describes a holistic time dependent transformation schema that can be applied in practice. Subschemata for different scenarios and geodetic applications are developed including precise positioning, geodetic analysis, GIS and mapping. Evolution of time-dependent transformation models are also discussed with a view to continuity and traceability of reference frames over time as are legal considerations for kinematic coordinates.

Chapter 7 shows how the schema described in this thesis can be applied in practice with three case studies. The first case study is from a stable plate setting and describes the development of a stable Australian Plate model that has been already been used as a template for the development of the recently released GDA2020 (ICSM, 2018) and the proposed ATRF. The second case study is from the Northern part of the South Island of New Zealand, a plate boundary region with complex non-linear deformation effects. The third case study in the Gisborne region of New Zealand is used to show application of the schema in a plate boundary region characterised by frequent slow slip events.

Chapter 2 Earth deformation with relevance to geodetic reference frames

The core aim of this thesis is to develop a robust schema for time-dependent transformations between reference frames that handle complex deformation across diverse spatial and temporal scales. The primary aim of the schema is to provide a framework for transformation of positioning and spatial data within a complex deformation environment which characterises the Earth’s surface for the purpose aligning datasets and positioning. In this chapter, Earth deformation modes are characterised with relevance to their effect on positioning precision and distortion of reference frames.

2.1 Geodynamic processes on Earth

A classical definition of deformation implies a change in shape, in other words dimensional changes within a solid body, shape or surface. In the context of applied geodesy the concept of deformation should be extended to any displacement of a feature on the Earth’s surface relative to the underlying mantle. Deformation of the Earth’s surface should be characterised on different temporal and spatial scales so that the time evolution of the Earth’s surface can be described and modelled with sufficient detail to support the requirements for the highest precision positioning and mapping and applications dependent upon them.

The motion of any feature on the Earth's surface can be described using temporal and spatial models. Temporal deformation can be linear, periodic, episodic and non-linear in character, on different time scales and with a wide range of spatial magnitudes. In order to develop a holistic schema to consider all modes of Earth deformation, these are categorised and characterised according to their temporal and spatial nature and expected deformation or displacement.

2.1.1 Modes and causes of Earth deformation

The most significant deformation on the modern Earth can be described by plate tectonic models, where stable segments of the lithosphere (tectonic plates) rotate over the underlying upper mantle driven by geodynamic processes such as slab pull in subduction zones. Plates collide and rift apart resulting in episodic seismic deformation when strain is released during earthquakes. In addition to plate tectonic processes, volcanic deformation is related to magma movement within the Earth’s crust which also results in episodic surface deformation. Loading and unloading of the Earth’s crust by glaciation also results in significant deformation on different time scales depending upon the size and areal extent of the ice load and glaciation cycle. Tidal forces generated by the interaction of other bodies in the solar system, predominantly the Moon and the Sun, cause periodic and predictable deformation of the Earth with solid earth tides (SET) (IERS, 2010). Other tides that cause smaller deformation are long-period tides and pole tides. Movement of water and the atmosphere also load the crust episodically resulting in small scale (but

35 not insignificant for geodesy) deformation such as ocean tide loading, non-tidal ocean loading (e.g., from storm surges), hydrological loading and atmospheric loading (IERS, 2010).

2.1.2 Stable tectonic plate motion

The Earth’s surface is divided up into a number of stable tectonic plates less than 125 km thick, moving slowly relative to one another over the underlying asthenospheric mantle, typically at rates of less than 80 mm yr-1 (IERS, 2010). Plates rift apart at mid-ocean ridges (e.g., the Mid-Atlantic Ridge) or continental rift zones (e.g., African Rift Valley) and collide to form either mountain ranges (e.g., the European Alps and Himalayas), or deep ocean trenches (e.g., the Mariana Trench) where a denser oceanic plate subducts beneath a more buoyant lithosphere. Plates also slide laterally past one another (e.g., the San-Andreas Fault, New Zealand Alpine Fault and Anatolian Faults). The whole Earth can be divided into several major tectonic plates (USGS, 2019a) (Figure 2.1), however within or near major plate boundary zones, smaller microplates and crustal blocks can define zones of localised rigidity within a broad plate boundary deforming zone.

Figure 2.1 Map of principal tectonic plates (USGS, 2019a)

Plate motion is conventionally defined by the axis and rate of rotation of a theoretically rigid plate (defined geometrically as an irregular edged portion of a spherical cap) about a pole of plate rotation or Euler pole and distinct from the Earth's pole of rotation in space (Figure 2.2). A spherical Earth is assumed in the model of plate tectonics since the flattening of the Earth is small. The assumption that the tectonic plate is a spherical rather than an ellipsoidal cap results in insignificant variations of site velocity and strain estimates (Drewes, 2009). An absolute Euler pole of rotation of a tectonic plate is defined by the

36 axis of rotation with respect to a No-net rotation (NNR) free Earth reference system (described later in this section). Plate rotation can be described in two ways: a. latitude (F) and longitude (L) of the Euler pole and rate of rotation (w) typically described in degrees per million years, or b. rotation of the rigid plate about the Earth’s three Cartesian axes typically in radians per million

years (Wx, Wy, and Wz).

Figure 2.2 Euler Pole defining a plate rotation

Rotation rates about the Cartesian axes can be computed from the Euler pole definition using Eq. 2.1 (where F, L, and w are converted to radians)

W=cos(FL )cos( )w x W=cos(FL )sin( )w y W=sin(F )w (2.1) z

Approximately 94% of the Earth’s land surface area can be considered to be on a stable tectonic plate (including microplates and rigid blocks) (Figure 2.3). This percentage is derived from Kreemer et al. (2014) who identify areas defined as being stable where the second invariant strain rates are < 1x10-9 yr-1.

Figure 2.3 Second invariant of strain rate from the Global Strain Rate Model (v. 2.1) (Kreemer et al., 2014). The white areas show stable plate regions where strain rates are insignificant.

Significant intraplate earthquakes do occur episodically (e.g., New Madrid, Missouri, USA, 1811-1812; Bhuj, Gujarat, India, 2001; Tennant Creek, Australia, 1987-1988) (USGS, 2019b), however such large earthquakes are rare on centennial timescales (Stein, 2007).

A NNR condition is implied where the rotation of all tectonic plates (plate area described in steradians multiplied by the plate rotation rate) and deforming zones sum to zero (Argus and Gordon, 1991). An absolute NNR condition is realised where there is no rotation integrated over the whole Earth.

The upper mantle, over which the Earth’s lithospheric or tectonic plates slide, represents the component of the Earth’s structure that has been most coupled with the Earth’s rotation throughout recent geological history (arbitrarily defined in a neotectonic sense as being the Quaternary era or the last ~2.5 million years). Mantle plumes originating from the lower mantle and/or core-mantle boundary may impinge on the lithosphere at volcanic hotspots (e.g., the Hawaiian-Emperor seamount chain) and provide a means of estimating motion of the overlying plate relative to the upper mantle which is inaccessible for direct observation. The effect of asthenospheric hotspot drag also needs to be considered. The Pacific Plate has a number of hotspot traces, and inversion of these has enabled the absolute motion of the Pacific Plate to be estimated and used as a reference plate for global plate rotation models (e.g., Gripp and Gordon, 2002).

Until the 1990s, models of absolute plate motion had been largely derived geophysically from analysis of these hotspot traces, paleomagnetism of oceanic crust on the sea floor, inversion of seismic slip vectors, and other geophysical studies (De Mets et al. 1990; 2009). These methods give an indication of plate motions averaged over geological time scales of millions to hundreds of millions of years.

Space geodetic techniques have been sufficiently precise over the last 30 years to measure contemporary site motion and therefore to estimate site velocities on the Earth’s surface directly. Contemporary plate motion (actual) models have been estimated by a combination of inversion of these site velocities and geophysical models where there are a lack of observed geodetic data (Sella et al., 2002; Bird, 2003; Argus et al., 2011; Kreemer et al., 2014; Drewes, 2017; Altamimi et al., 2017) (Table 2.1).

Author/ Plate No. of No. of Fixed Input Reference Model Rigid Deforming Plate Data Plates Zones De Mets et al.(1990) NUVEL-1 14 0 Pacific Geological De Mets et al.(1994) NNR-NUVEL-1A 14 0 Absolute Geological Sella et al.(2002) REVEL2000 19 0 Absolute mostly GPS Bird (2003) PB2002 52 13 Pacific Geol. + Geod. Argus et al., (2011) NNR-MORVEL56 56 0 Absolute Geol. + Geod. Kreemer et al.(2014) GSRM 2.1 50 contiguous Absolute Geodetic Drewes (2017) APKIM2014 11 contiguous Absolute Geodetic Altamimi et al.(2017) ITRF2014-PMM 11 0 Absolute Geodetic Table 2.1 Recent plate motion models

Current plate motions estimated from space geodetic techniques limited to the last 30 years of observations do not necessarily agree with plate motions derived over geological periods. For example,

Kreemer et al., (2006) and Tregoning et al., (2013) show that major (> Mw 9) earthquakes result in observable coseismic deformation thousands of kilometres from the earthquake epicentre. Therefore, it is reasonable to assume that inversion of plate models from geological observations implicitly includes cumulative far-field coseismic and postseismic deformation over geological timescales. The difference between interseismic site velocities and derived interseismic plate motion models can therefore be expected to be slightly different from geologically estimated site velocities and plate motion models where far-field seismic deformation is implicitly included in the estimation process. As a consequence a small difference between different interpretations of the NNR condition can be evident.

ITRF2014 absolute rotation poles (Altamimi et al., 2017) are listed in Table 2.2 for reference. A limitation of the plate model approach is how deforming zones between defined stable plates are treated (Kreemer et al., 2014 and Drewes et al., 2013).

Plate Euler pole of rotation Equivalent Cartesian angular velocity F L w Wx Wy Wz (°) (°) (° Ma-1) (rad Ma-1) (rad Ma-1) (rad Ma-1) Antarctica 58.8 -127.4 0.219 -0.001202 -0.001571 0.003272 Arabia 51.2 -6.7 0.515 0.005595 -0.000659 0.007001 Australia 32.4 38.1 0.631 0.007321 0.005730 0.005890 Eurasia 55.1 -99.1 0.261 -0.000412 -0.002574 0.003733 India 51.6 -0.2 0.516 0.005595 -0.000024 0.007049 Nazca 45.8 -102.2 0.629 -0.001614 -0.007486 0.007869 N. America -5.2 -88.0 0.194 0.000116 -0.003365 -0.000305 Nubia 49.7 -80.8 0.267 0.000480 -0.002977 0.003554 Pacific -62.6 111.3 0.679 -0.001983 0.005076 -0.010516 S. America -19.1 -131.9 0.119 -0.001309 -0.001459 -0.000679 Somalia 47.7 -98.7 0.332 -0.000587 -0.003849 0.004286 Table 2.2 ITRF2014 Plate Motion Model (from Altamimi et al., 2017)

2.1.3 Estimation of stable plate motion models (PMM)

Estimation of Euler poles of tectonic plates using space geodetic and geophysical observations is well documented (De Mets et al., 1990; Kreemer et al., 2014; Argus et al., 2011, Altamimi et al., 2016 and Drewes, 2017). Geodetic estimation of PMM is achieved by least-squares inversion of site velocities estimated from time-series analysis of absolute positions of monuments over time. Euler Poles for recent interseismic plate models can be defined using space geodetic techniques by least-squares inversion of n sites with TRF site velocities estimated from analysis of the TRF site time-series using Eq. 2.2 adapted from Goudarzi et al., (2014). This approach is valid where estimated rotations are small.

TT-1 ΩAWAAWLplate = ()() (2.2)

! éù0 ZY11- éùX1 êú êú-ZX0 Y! êú11 êú1 êú! éùwx êúYX11- 0 Z1 êú êú where, Ω = êúw A = and L = plateêú y êú!!! êú" êúw êú0 ZY- êúX! ëûz êúnn êún ! êú-ZXnn0 êúYn êúYX- 0 êúZ! ëûnn ITRF ëûn ITRF and,

Wplate is the Euler Pole (rotation rate of axes wx wy wz in rad yr-1) A is the design matrix of ITRF Cartesian site coordinates (m)

X1 Y1 Z1 to Xn Yn Zn W is the weight matrix (if applicable) L is the observation matrix of ITRF site velocities (m yr-1) ! ! ! ! ! ! X1 Y1 Z1 to X n Yn Z n

The Euler Pole can also be expressed using Eq. 2.3 where the pole rotation rates are described in rad yr-1 and positive is anti-clockwise about the pole.

222 wwwwplate=++ x y z w F=tan-1 z plate 22 wwxy+

wy Lplate = (2.3) wx Site velocities (m yr-1) for any specific point can be computed directly from the Euler Pole model using Eq. 2.4:

! éùXZYéùwwYZ- êú YXZ! =êúww- (2.4) êúêúZX êú! êú ëûZYXëûwwXY-

The Euler Pole can also be expressed as a time-dependent three parameter conformal transformation (Stanaway et al., 2015) using Eq. 2.5 where the rotation rates in rad yr-1.

r!xx=-w r!yy=-w r!zz=-w (2.5)

(Note: Eq. 2.5 uses the coordinate frame (CF) rotation convention. If the position vector (PV) notation convention is used, the signs of rotation rates and derived rotations are reversed.)

The rotation rates in Eq., (2.5) can be also expressed conventionally as arcseconds per year (arcsec yr-1) using Eq. 2.6. 648000w (rad yr-1 ) r 1 = (2.6) (arcsec yr- ) p Velocity residuals are computed using Eq. 2.7.

vA=ΩLplate - (2.7)

The reference standard deviation for the Euler Pole inversion is computed using Eq. 2.8.

vT Wv = (2.8) So r where r is the degree of freedom (r = 2n - 3), where n is the number of stations used in the inversion.

The standard deviation of each of the rotation parameters is derived by scaling the square-root of T -1 diagonal components of the variance-covariance matrix or inverted normal matrix (A WA) by So.

Site velocities are typically defined during the interseismic period, so any known coseismic and postseismic displacements should be isolated from the time-series inversion for each site. Seasonal and draconitic signals should also be removed, especially for shorter duration time series or where coloured noise is evident (e.g., Santamaría-Gómez et al., 2011). Strain accumulation arising from locked faults near a site should also be modelled using elastic half-space models in order to estimate interseismic back-slip (McCaffrey, 2002). This aspect is described in more detail later in this chapter. GIA deformation signals with respect to underlying stable plate motion can also be isolated in a similar way.

2.1.4 Glacial isostatic adjustment (GIA)

The other long period and near-secular mode of Earth deformation is that arising from the ongoing loading and unloading of ice sheets during major glaciation cycles resulting in glacial isostatic adjustment (GIA) also known in many earlier studies as post-glacial rebound (PGR). GIA continues for some time after changes in the surface load due to viscoelastic relaxation mechanisms. The response times are related to the rheological characteristics of the crust and the magnitude and distribution of the load (Figure 2.4).

Figure 2.4 Glacial Isostatic Adjustment

The last major period of glaciation was between 110,000 and 12,000 years before present (BP), peaking at 26,500 years BP an epoch known as the Last Glacial Maximum (LGM) (Lambeck and Rouby, 2014). During the LGM much of the higher latitudes were covered in ice sheets up to 4 km thick. The ice load resulted in over a hundred metres of downward deflection of the crust at the centre of the ice sheet. The volume of water bound up in the ice-sheets resulted in a decrease in relative sea levels of approximately 125 m, also decreasing the load on the oceanic lithosphere. The melting of the Laurentide and Fennoscandian ice sheets and ice retreat elsewhere at the end of the last glacial period has resulted in an unloading of the crust and an increase in global sea levels of 125 m (Lambeck and Rouby, 2014). The ice sheet loss and associated redistribution of mass around the Earth has resulted in ongoing vertical displacement of the

42 surface in response of up to 11 mm yr-1 vertical uplift in the Gulf of Bothnia (Lidberg et al., 2010), and 15 mm yr-1 vertical uplift near Hudson Bay close to the centre of former ice sheets. The uplift reduces to zero at a hinge line surrounding the area of former glaciation. Beyond this hinge line, subsidence of up to 2 mm yr-1 is evident where ductile asthenospheric material moves back into the uplifting zone beneath the former ice load. GIA also has a horizontal deformation component of up to 2 mm yr-1 due to crustal expansion associated with the uplift and is oriented away from the location of glaciation relative to the stable plate. On centennial time-scales displacements associated with the LGM are highly linear. Recent and rapid loss of ice sheets outside the polar regions has also resulted in significant GIA.

GNSS derived site velocities in GIA affected areas can be inverted to estimate the location and size of the load if the underlying stable plate velocity and tectonic uplift rates are isolated from the observed site velocity. Conversely, the stable PMM requires removal of GIA signals from observed site velocities in order to estimate the PMM, so some iteration and further geophysical modelling is required to deconvolve the deformation signal into plate motion/tectonic uplift and GIA components. Other geological and geophysical techniques can be used to refine the extent, size and duration of the ice load and the combination of geodetic and geological observations to model ice history is an active area of research (Lambeck and Rouby, 2014; Purcell et al., 2016; Peltier et al., 2015).

From an applied geodetic perspective, considerable research has also been undertaken to develop GIA models that can be used in practice to support geodetic applications and surveying. The Fennoscandian 3D GIA model NKG2016LU has been implemented by the Nordic nations (Vestøl et al., 2016) and a North American 3D model has also been developed (Snay et al., 2016; NRCan, 2019). These 3D models are in the form of a grid of residual site velocities (E, N and Up) with respect to the underlying stable North American Plate.

2.1.5 Interseismic plate boundary deformation

The boundary zone between any mobile tectonic plates or crustal blocks can be characterised by complex deformation where strain is accumulated along the interface (or block bounding fault) between the two plates or crustal blocks during the interseismic period (Figure 2.5). Strain increases within the deforming zone if the plate or block bounding fault is locked. Much of the strain is released during earthquakes or slow slip events. Plate boundary zone deformation has long been an intensive area of research in light of the hazards and damage caused by earthquakes in these regions.

Figure 2.5 Schematic representation of a plate boundary deformation zone

The degree of locking for any segment of a plate or block bounding fault can vary between a fully locked or coupled state (coupling ratio of 1) and a freely slipping state (coupling ratio of 0) (McCaffrey, 2002) (Figure 2.6). The theoretical slip rate (coupling ratio of 0) for each fault segment is estimated by differencing the TRF site velocities estimated using the PMM for each interfacing plate at the fault segment.

Figure 2.6 Site velocity changes within plate boundary deformation zones. Fully locked boundary (l) and freely slipping (r).

Within plate boundary zones, analysis of GNSS site velocity residuals with respect to predicted stable plate velocities for that site can suggest locking of nearby plate bounding faults or ongoing postseismic deformation at the site location. Velocity residuals can also be indicative of localised instability of the antenna mount or creep which can be verified by local site tie measurements.

Strain accumulation arising from locked plate or crustal block bounding faults near a site can be modelled using elastic half-space models in order to estimate interseismic back-slip and to form an a priori interseismic velocity model (McCaffrey, 2002). The estimated back-slip velocity would need to be added

44 to the observed velocity in order to estimate a stable plate motion for that site. Alternatively, an Euler pole for an elastically deforming rotating block can be estimated using observed velocities and defined plate bounding faults using the approach described in McCaffrey (2002) and applied using TDEFNODE software (McCaffrey, 2015).

Fault locking models can be refined by kriging or least-squares collocation (LSC) of differences between observed and modelled site velocity residuals. This approach has been used to develop the HTDP software used in the USA for time-dependent transformations (Snay, 1999; Pearson and Snay, 2012). The site velocity model can be represented as a grid of velocity residuals with respect to the stable plate. Alternatively, an Absolute Deformation Model (ADM) (Stanaway et al., 2013) can be formed by combination of a plate-fixed residual site velocity model and a PMM represented in grid format to represent the velocity field directly in terms of a TRF such as the ITRF. The ADM approach has been used in the development of the New Zealand Geodetic Datum 2000 Deformation Model (LINZ, 2019) (Figure 2.7).

Figure 2.7 NZGD2000 Deformation Model - absolute interseismic velocity model (coarse representation) (Government of New Zealand, 2019)

2.1.6 Coseismic deformation and volcanism

Unpredictable episodic deformation is typically associated with earthquakes. During an earthquake differential movement occurs along a fault plane referred to as coseismic displacement. The deformation can be very large (up to tens of metres for Mw9+ earthquakes) and happens over a very short period of time, typically minutes.

GNSS observations on ground fixed monuments provide the most direct measurement of coseismic deformation however other remote sensing techniques and slip dislocation models are also required to develop a better spatially distributed model of coseismic surface deformation in affected areas. For recent large earthquakes interferometric synthetic aperture radar (InSAR) has been widely used to observe coseismic surface deformation at high resolution (e.g., Hamling et al., 2017). InSAR initially provides a one-dimensional model of deformation (observed displacement along the radar line of sight) with orbit passes made before and after the deformation event typically in both ascending and descending mode. If different viewing geometries can be achieved (e.g., ascending and descending orbits) over the same location a 3D surface displacement model can be estimated. The geometry of InSAR view angles generally provides higher resolution of coseismic uplift or subsidence.

Vegetation coverage and landslides hinder coherence of interferograms so the application of InSAR is limited in densely vegetated areas unless permanent scatterers are placed. High resolution and geodetically controlled imagery (e.g., from aerial surveys and drones) and LiDar can also be used to build up a model of coseismic displacement by comparing spatial differences between identified objects in images and laser ground-strike (bare earth) models pre and post-earthquake. In addition to these direct measurements of deformation, slip dislocation models can be used to estimate surface deformation. The dislocation model is derived from seismological observations (moment tensor diagrams and magnitude) and crustal rheology models. The epicentre (location and depth of the earthquake) and slip geometry are estimated from the seismic network and the slip is then estimating using elastic half-space models (e.g., Okada, 1985 and 1992). In general, uncertainty of the epicentral location and depth together with assumptions of the slip geometry lead to some uncertainty of the estimation of the surface deformation. GNSS surface observations and InSAR can therefore be used to refine the epicentral location and slip geometry. The combination of all these techniques can be used to generate a coseismic displacement model (patch) that is generally in the format of a grid of displacements with respect to the underlying stable plate (Figure 2.8).

Figure 2.8 Coseismic displacement grid - example, Christchurch, New Zealand (Crook et al., 2016). Vectors show modelled coseismic displacements. Crosses indicate the grid nodes.

There are two other considerations when using direct observation methods for developing a coseismic displacement model. The main consideration is that both interseismic and postseismic deformation are isolated from the observed shifts, particularly if there is a large interval between the epochs of acquisition (Figure 2.9). This is especially important for campaign-style GNSS measurement of geodetic monuments. Site velocities estimated for these stations need to be used to estimate the station position at the time of the earthquake from which the coseismic deformation can be estimated.

Figure 2.9 Schematic illustration of the approach used to estimate coseismic displacement from temporally sparse campaign measurements

Removal of the postseismic deformation signal is more problematic, particularly if there is some delay in obtaining observations after an earthquake when some postseismic afterslip deformation would have occurred. The other consideration is site stability of stations used to model the coseismic deformation. Unless stations used are fixed to bedrock the displacement may actually include components of regolith deformation arising from rapid surface creep (lateral spreading) and liquefaction. Also, localised surface ruptures near geodetic stations can also introduce local uncertainty.

Deformation arising from movement of magma near the Earth’s crust can also result in significant localised surface deformation that is episodic in character. Geodetic sensors are placed in high risk zones to monitor this deformation and provide early warning for volcano monitoring teams (e.g., Dzurisin, 2007).

2.1.7 Postseismic deformation

Non-linear deformation is associated with postseismic deformation after large earthquakes and is typically represented by a combination of logarithmic, exponential and ramp functions with a decay time related to the geometry of the fault and rheology of the crust in the deformation environment. Postseismic deformation is further complicated by aftershock sequences. It can take several decades for earthquake affected locations to return to a secular interseismic velocity (e.g., Wang et al., 2007; Suito and Freymueller, 2009). Similarly, slow slip events (SSE) emulate the coseismic deformation of a larger single

48 earthquake but over a much longer period (from hours to weeks) with clusters of microseismic activity within a slipping fault zone over the period.

As described earlier, large earthquakes (Mw > 9.0) can also result in observable far-field deformation up to 5,000 km from the earthquake epicentre (Kreemer et al., 2006; and Tregoning et al., 2013). Whether this far-field deformation and postseismic relaxation should result in permanent coordinate changes depends upon the magnitude and tolerance of the local reference frame affected by it, and is discussed in more detail in Chapter 3.

For individual sites where the postseismic deformation can be observed directly and continuously, site specific functions can be fitted to the observed time series. This approach has been used in the development of postseismic decay terms in the most recent ITRF2014 realisation (Altamimi et al., 2016; Lercier, 2014). While this approach works well for CORS and geodetic stations that are regularly observed by campaign-style observations during the postseismic deformation phase, a model needs to be developed in order to estimate postseismic deformation in much the same way as coseismic deformation is estimated. The model can also be in a grid format with amplitude and decay time values for each grid node, e.g., as used in HTDP software (Pearson and Snay, 1999).

2.1.8 Periodic and seasonal deformation

Periodic deformation is mostly attributed to tidal effects such as solid Earth tides (SET), ocean tide loading (OTL) and seasonal effects such as atmospheric tides and hydrological loading. Other periodic effects include surface deformation arising from seasonal (annual) atmospheric and hydrological mass redistribution between the Southern and Northern hemispheres. This motion is evident as a periodic signal in the time series of station positions. These effects are generally quite predictable or can be modelled a posteriori using meteorological data such as monsoon or snow precipitation.

Semi-diurnal and diurnal SET and OTL deformations are modelled and can be estimated a priori, and must be applied for global and continental scale geodetic analyses, for example within GNSS baseline processing and PPP. Tidal deformations are predominantly vertical with a smaller horizontal component. Higher resolution OTL models are used in areas with complex coastline shapes adjacent to shallow continental shelves and with large tidal variations. Diurnal and semi-diurnal tidal deformation effects that can be modelled, and their vertical magnitudes are listed as follows (IERS, 2010):

Solid Earth Tide < 300 mm Ocean Tide Loading < 100 mm Atmospheric Tides < 2 mm

Many periodic deformations cannot be precisely modelled a priori due to high levels of temporal and spatial variability of the loading drivers. Lower precision predictions can be used in the absence of observations using short-term forecast models. Meteorological, gravimetric, tidal and satellite altimetry observations can be used to estimate the loading effects at a specific location, and should be modelled a posteriori using real-time gravimetric data and meteorological observations for the highest precision geodetic analysis. Vertical magnitude of these variable loading effects are listed below.

Non-tidal atmospheric loading (NTAL) < 10 mm Hydrological loading (HYDL) (e.g., monsoon, dense snow cover) < 30 mm Non-tidal ocean loading (NTOL) (e.g., storm surges) < 15 mm

2.1.9 A summary of geodynamics deformation effects with relevance to reference frames

Different modes of deformation, their magnitude, stability and extent are shown in Table 2.3 and illustrated in Figure 2.10.

3D Displacement Stability Cause Extent (km) or rates period (yr) Stable plate motion < 100 mm yr-1 10,000 300-8,000 Tectonic Uplift < 10 mm yr-1 1,000 50-1,000 Glacial Isostatic Adjustment < 10 mm yr-1 1,000 500-3,000 Interseismic strain < 50 mm yr-1 100 50-500 Magmatic diapirism < 20 mm yr-1 1,000 20-200 Water table changes < 250 mm yr-1 10 <200 Volcanism < 10,000 mm 0.01 <30 Coseismic Deformation < 10,000 mm 0.00001 <1000 Slow-slip Deformation < 300 mm 0.1 <100 Postseismic Deformation < 1,000 mm 10 <1000 Table 2.3 Modes of deformation, their magnitude, duration and extent

Figure 2.10 Temporal and spatial domain of terrestrial deformation (from Stanaway et al., 2012)

The high spatial and temporal variability of deformation can warrant updates to a RF if the deformation exceeds positioning and dimensional tolerances in affected areas. Coordinate updates also ensure that conformity is maintained in areas of significant deformation (e.g., fault scarps across property boundaries and service infrastructure easements).

Large shallow earthquakes can often result in significant surface rupturing and highly localised and variable displacements. Such displacements can be quantified using similar techniques used for measuring coseismic deformation described in Section 2.1.6.

It is reasonable to assume that any seismic deformation should result in a change of coordinates of geodetic infrastructure, especially where fault ruptures displace property boundaries. For example, consider the case where two geodetic monuments reference two adjoining cadastral parcel corners. An earthquake results in a lateral displacement of a boundary line. A change in coordinates (in any reference frame) for both the boundary corners and monuments can be expected. To distinguish seismically affected coordinates from those defined in the same pre-earthquake RF, the post-earthquake coordinates are described with the updated RF version to identify that a correction patch has been applied (Crook et al., 2016).

2.2 Site specific and localised deformation

2.2.1 Regolith deformation and creep

Plate motion is defined by the motion of stable bedrock forming the Earth’s crust. Overlying much of the crustal bedrock is regolith which includes clay, gravel, loose rock, sand, dust and soil. Regolith deformation is often independent of the underlying bedrock due to changes in water content and surface creep due to gravity. Examples include regolith overlying aquifers, permafrost and thick clay sub-soil layers which can uplift or subside substantially depending upon the soil or clay moisture content or permafrost decay. Subsidence of many urban areas and sedimentary basins can often be very substantial, for example, Jakarta and Mexico City are subsiding at up to 200 mm yr-1 (Abidin et al., 2015) as the aquifers beneath the cities become depleted over time. The depletion is typically caused by ongoing abstraction of groundwater by pumping that is not replenished due to the impermeability of much of the built-up area overlying the basin. In the San Joaquin Valley of California subsidence of 9 m has occurred over a period of 52 years (USGS, 2019c) as aquifers have become depleted for irrigation purposes. Extractive underground mining, coal-seam gas extraction and tunnelling are also notable causes of ground subsidence over localised areas on specific timescales related to the period of extraction. Smaller magnitudes of horizontal displacement are also often associated with any vertical subsidence.

Surface creep can also be significant, particularly with clay rich or poorly consolidated soils. Advanced and accelerating creep often becomes a landslide. Large structures can also load and compact the regolith causing localised subsidence. Geodetic monuments located in these settings can have significant localised deformation over time (Figure 2.11). For example, a monument fixed to sub-soil on a steep slope, or a tower-mounted GNSS antenna. Haasdyk and Roberts (2013), show examples of a trigonometric station located over underground mining operations subsiding by 1.6 m with 0.6 m of associated horizontal displacement over a period of seven years. The same study also describes a grain silo-mounted station showing 30 centimetre horizontal displacement over a period of 10 years.

Figure 2.11 Localised deformation of geodetic monuments (example showing regolith creep due to gravity)

During large earthquakes, even bedrock can be subject to extensive localised faulting and surface ruptures that are not necessarily representative of deeper crustal bedrock deformation. Liquefaction of regolith during seismic shaking can result in substantial surface displacement known as lateral spreading (USGS, 2019d).

2.2.2 Stability of geodetic monuments

Geodetic monuments themselves are often locally unstable (e.g., Hudnut et al., 1996) and can be displaced as a result of earthquake shaking, poor construction, mounting, windshear and thermal expansion of a supporting structure or antenna mast, or disturbance (e.g., a vehicle colliding with an antenna mast or a ship colliding with a tide gauge CORS for example). A review of a large number of geodetic monuments that form antenna mounts for CORS antennas indicate a potential for instability and residual displacement with respect to underlying crustal bedrock. Unless this displacement is verified and quantified, usage of the monument for estimation of site velocities used for inversion of stable plates and strain modelling will introduce uncertainty. Any localised deformation or monument stability can be verified by precision site tie surveys, redundant CORS arrays and continued integrity monitoring for antenna stability.

2.2.3 Impact of localised deformation and monument stability on reference frame definition

In terms of RF definition, localised deformation and monument instability are undesirable and should not be conflated with crustal deformation driven by secular tectonic processes and earthquakes. The displacement of a monument with respect to the local crustal structure can be quantified by repeated tie surveys between the monument and a local fiducial network of stations fixed to or anchored to bedrock. If the monument is a CORS, this can be achieved by repeat tie static GNSS measurements, or by reflectorless distance and angle measurements to the antenna mount from a stable monitoring network provided that the network is in close proximity to the CORS. The amount of displacement from the reference epoch position is then applied as a specific site correction for network analysis.

Site displacement can be episodic (e.g., a single disturbance event or offset), seasonal, “random walk” or secular (with a localised site velocity). The style and magnitude of localised deformation needs to be considered if the tolerances for positioning requirements are exceeded.

How local site displacement is described and used in modelling warrants some discussion. In terms of RF definition an assumption is made that localised deformation is zero. Where localised secular displacement is evident, the local site velocity should be removed from the RF site velocity component prior to PMM or regional DM estimation. Where the localised deformation is seasonal or random in nature, the temporal deformation signal should also be removed for RF analysis, or the velocity uncertainty scaled to accommodate the noise spectrum.

2.2.4 Localised deformation models

Models of localised surface deformation with respect to underlying crustal bedrock can be developed by repeat surveys of monitoring marks within the deforming area, or by using remote sensing techniques which include high resolution imagery. Displacements of surface features can also be derived from imagery analysis. These displacements can then be used to estimate a gridded local deformation model using kriging techniques.

If secular motion of CORS with respect to the underlying crust is observable by repeat site ties or redundant CORS arrays located on nearby bedrock sites, then a localised site velocity correction should be applied if the CORS is used for PMM estimation. If the localised CORS motion is episodic or “random walk” then an epoch-specific correction should be applied prior to estimation of the affected CORS site velocity and subsequent PMM estimation. This correction model can be in the form of 3D shifts over a given period of time at a specified epoch interval.

If a CORS is subject to localised deformation, this can have adverse impacts on positioning applications that utilise the CORS if the localised deformation is not modelled correctly as holding coordinates and elevations fixed would propagate modelling errors into Network RTK (NRTK). Integrity monitoring

53 software would be expected to be used for any active CORS to warn network operators of antenna disturbance or deformation outside specified tolerances. Optimally, CORS positions should be defined in terms of the latest realisation of a Global TRF at the current epoch and then transformed to any desired local reference frame (RF). Ideally, this would be the most recent weekly solution using IGS final orbits. Alternatively, the ITRF position could be predicted using the CORS site velocity (including any localised deformation velocity signal). Consequently any localised deformation relative to the local RF would be implicit with any derived transformation.

In many instances, however, CORS may be defined or fixed in terms of a local RF. As described previously, this approach can already have adverse impacts on positioning accuracy with increasing distance from the CORS, regardless of whether or not there is localised deformation of the CORS antenna monument. If localised deformation is evident, then the CORS position should be constantly updated in terms of a local RF in order to prevent propagation of errors from the affected CORS in network processing and single base RTK positioning.

Passive geodetic monuments are by definition not continuously monitored, so their use in local geodetic surveys and wider area deformation monitoring should be treated with some degree of caution. Many geodetic stations, especially lower hierarchy stations, are not necessarily constructed to high standards and may also be poorly anchored to the ground or not connected to bedrock. Such stations can exhibit significant localised movement from their initial surveyed positions within a RF over time. Station movement between subsequent geodetic connections to the parent geodetic network and local RF is typically not modelled. Realistic positional uncertainties (PU) can be assigned to unstable stations based upon repeat surveys over time, with the PU being scaled to accommodate maximum observed localised deformation. If the movement is secular in nature (e.g., the Earth surface above underground mines or coal seam gas extraction) coordinates can be updated at regular intervals whenever positioning tolerances are expected to be exceeded. Alternatively, a new station name can be assigned to the unstable station that includes the epoch of measurement in the station name.

While this thesis focuses on modelling of crustal deformation (deep seated bedrock movement) there is a need to distinguish this movement from actual surface deformation being defined here as movement of regolith or artificial structure with respect to the underlying crust (Figure 2.12). Monitoring of these surfaces is more in the realm of conventional deformation monitoring which is largely outside the scope of this thesis. For example, deformation monitoring of landslides, dam walls and structures.

Figure 2.12 Surface deformation (e.g., liquefaction and lateral spreading) separated from coseismic crustal deformation, 56 metre (0.0005° latitude interval) grid from Christchurch, New Zealand (Tonkin and Taylor, 2015). Rapid creep towards watercourses is evident.

Nevertheless, there is a practical requirement for surface deformation models after large earthquakes. The aim of these models is to differentiate surface movement caused by liquefaction, lateral spreading and accelerated creep from movement of the underlying crust. The distinction between these two forms of deformation is important for redefinition of cadastral boundaries after earthquakes. Lateral spreading is often significant and highly localised, but not deeply seated. As a consequence, movement of occupational cadastral boundaries due to highly localised lateral spreading (e.g., one side boundary of a land parcel) is usually recommended practice (Grant et al., 2015). Whether cadastral boundaries should be redefined in areas of lateral spreading is a matter of judgement for a cadastral surveyor. If lateral spreading is extensive and consistent (in other words whole cadastral parcels have moved coherently) then occupational boundaries can continue to be used to define the cadastre. If lateral spreading is localised to one smaller part of a boundary of a cadastral parcel, then the boundary would need to be reinstated in its original position relative to the unaffected boundaries of the affected parcel and abutting parcels.

In earthquake-affected areas, crustal deformation models need to have surface motion attributable to landslides and liquefaction isolated from the model. A separate model would be created for these surface deformations with respect to the underlying deformed crust. This separate surface deformation model could be used for boundary reinstatement.

2.3 Vertical specific deformation and subsidence

Vertical deformation should be handled differently from horizontal deformation, as changes in elevation can have a far more significant impact on civil engineering and hydrology studies such as flood modelling. Specifically, gradient tolerances can be very fine for large-scale hydrology projects, so the need for high precision vertical deformation models to support these projects is real. Flood modelling and the projected impact of sea level changes are also dependent on high-precision vertical models in terms of a global RF. For these reasons, vertical deformation model resolution may differ from horizontal deformation model resolution.

Secular vertical deformation is usually associated with tectonic uplift and subsidence or GIA and can be up to 15 mm yr-1 in magnitude. Vertical strain rates associated with these processes are usually small however, as the effects are distributed over regional spatial scales. Secular deformation on shorter temporal and spatial scales is a more pressing concern in many urban areas especially where ground water is abstracted either by anthropogenic or natural causes (Figure 2.13). Rapid differential subsidence induces very high bending strain rates, especially near the boundaries of aquifers that have a major impact on the integrity of built infrastructure (Figure 2.14).

Figure 2.13 Vertical deformation in urban areas due to groundwater changes (example: Jakarta vertical deformation rates 1991-2010) from Abidin et al., 2015

Figure 2.14 Effect of differential subsidence on urban structures (example: Mexico City) photo, Josh Haner, New York Times, 2017

Surfacing of large portions of an urban area prevents natural seepage of surface water into the underlying regolith and water table due to the higher impermeability of these surfaces. Rain water is channelled away and regolith moisture levels gradually decline resulting in variable subsidence over time. Areas of subsidence are also more prone to flooding. Various techniques can be used to model vertical deformation with high resolution including: (1) continuous GNSS (CORS), (2) campaign-style GNSS measurements on monitoring points, (3) repeat levelling surveys, (4) repeat Light Detection and Ranging (LiDAR) scanning and (5) repeat InSAR. All of these techniques require connection to an external fiducial reference frame outside the zone of deformation. A gridded vertical deformation model can then be derived for use in RF definition.

2.4 Apparent deformation effects (artefacts)

Apparent deformation effects are usually artefacts of geodetic analysis. These do not represent real deformation of the monument but can be a major source of error in precise geodetic analysis and site velocity estimation. Time series spectral analysis, precision site tie surveys and redundant CORS arrays can usually identify and mitigate these effects. Apparent deformation can be attributed to a wide variety of sources, including: - antenna modelling errors - antenna failure - incorrect antenna height measurement - antenna type metadata error - change of antenna cable - debris on antenna such as snow, animal deposits, moss - tropospheric modelling errors (including local temporal weather effects) - ionospheric modelling errors - orbit modelling errors (e.g., unmodelled solar radiation pressure effects) - seasonal multipath effects (e.g., deciduous trees near antenna) - variable multipath (e.g., tree growth, construction near antenna)

Chapter 3 Geodetic reference systems, transformations and precision requirements

In this chapter, the concepts of geodetic reference systems, geodetic datums and current transformation strategies between reference frames are reviewed. The positioning and tolerance requirements for reference frames are also analysed. Finally, an appraisal is made of the magnitude of deformation that can be accommodated within a reference frame before an update is warranted. The appraisal considers the highest tolerance positioning and surveying specifications required of a reference frame.

3.1 Geodetic reference systems and frames

A geodetic coordinate reference system (CRS) is the fundamental basis for describing positions (coordinates) and their evolution with time on Earth. Unfortunately there is a clash in the use of this acronym within geodesy as a Celestial Reference System (CRS) is also used in higher order geodesy. The usage of CRS to refer to coordinate systems is now in widespread use in surveying and GIS promoted by geodetic registries (e.g., EPSG, 2009) and this usage will be followed here. CRS are also referred to synonymously as terrestrial reference systems (TRS) in some literature and standards (e.g., IERS, 2010). A geodetic datum is strictly speaking a single point whose coordinates, zenith, orientation to another point and a scale define or realise a CRS. In some instances a network of stations collectively realise a geodetic datum such as the European Datum 1950 (ED50). A geodetic reference frame is a more semantically correct definition for a network of stations which realise a CRS rather than a single station.

3.1.1 Coordinate reference systems (CRS) and ellipsoids

Conventionally, a CRS or TRS is a three-dimensional coordinate system fixed to and co-rotating with the solid body of the Earth with an origin close to the centre of mass of the Earth, the geocentre (Figure 3.1). The x axis is aligned with the equator at 0 degrees longitude, the y axis at 90 degrees East longitude and the z axis at the North pole forming a right-handed Cartesian system. These axes also define the origin of an ellipsoid which best fits the Earth's surface at a gravitational equipotential approximating global mean sea level (MSL). Coordinates can be described either in geocentric Cartesian format [ X Y Z ] or ellipsoidal format ( f l h ) for an ellipsoid which is defined for the CRS. The ellipsoid is defined by its semi-major axis (a) and flattening (f) from which the semi-minor axis (b) is determined.

Figure 3.1 Representation of a CRS using the ITRS as an example

Until the era of space geodesy, astronomical methods were used to define a CRS and associated ellipsoid which best approximated the shape of the Earth at a local or regional level. As a consequence, many astronomically determined CRS are substantially offset from the true geocentre, for example the Australian Geodetic Datum (AGD) whose CRS origin is offset from the true geocentre by ~200 m (ICSM, 2018). The advent of space geodetic techniques has enabled the geocentre to be sensed with very high precision by inversion of satellite orbit trajectories. Combination of different space geodetic techniques including: (1) Satellite laser ranging (SLR), (2) Very-long-baseline Interferometry (VLBI), (3) GPS, and (4) Détermination d'Orbite et Radiopositionnement Intégré par Satellite (DORIS) have led to the development of the International Terrestrial Reference System (ITRS) (IERS, 2010). The key characteristics and definition of the ITRS are:

• The ITRS axes co-rotate with the body of the Earth with no net rotation of the Earth’s surface • The geocentre is the centre of mass of the whole Earth system (including oceans and atmosphere) • The reference meridian (0° Longitude) is defined as the International Earth Rotation and Reference Systems Service (IERS) Reference Meridian (IRM) • The pole is defined as the IERS Reference Pole (IRP)

3.1.2 Geodetic datums and reference frames (RF)

A CRS is realised by either a single point, orientation of the coordinate axes and scale or a network of geodetic stations or monuments whose coordinates define a RF and refer to a geodetic datum. Classically, the origin of a geodetic datum has been defined by fixing the coordinates of a geodetic origin station. In the era before space geodesy, datum origin coordinates would have been estimated by astronomical observations, known to have poor absolute precision. The datum would have been extended by more relatively precise terrestrial measurements (angles, distances and levelling) and an astronomically

59 determined reference azimuth at the datum origin. Consequently, before the era of space geodesy, CRS were inherently both plate and ground fixed.

3.1.3 Static reference frames

Until direct measurement of plate tectonic motions was possible, geodetic RF did not include models of site motion and were considered to be static datums or reference frames. As such, origin coordinates were invariant with time, regardless of any geodynamic processes impacting on movement of the origin in a global frame.

3.1.4 Kinematic reference frames

The development of space geodetic positioning techniques such as GNSS, SLR, VLBI and DORIS has resulted in very precise positioning capability, now approaching a few mm for any location on the Earth (IERS, 2019a). Station movement resulting from geodynamic processes is now very evident using space geodetic techniques, even over short differences between positioning epochs. A corollary of that sensitivity is that any station motion within the global tracking network of ground based sensors is required to be modelled with high precision to prevent degradation of orbit predictions, ephemerides and post-processed orbits for GNSS, SLR and DORIS. The requirement for a unified and necessarily kinematic RF to support space geodetic studies and applications is essential. The ITRF is a kinematic frame realised by the coordinates and motion of a global network of geodetic monuments using different space geodetic techniques and site ties between co-located ground sensors. The first release of ITRF was in 1988 with ITRF88 and the current (13th) realisation being ITRF2014 released in January 2016 (Altamimi et al., 2016). Starting with the fifth release of ITRF (ITRF92), site velocities have been included in the realisation of the frame to account for secular tectonic motion of each station that defines the ITRF. ITRF2014 is defined by the station coordinates and station velocities at a reference epoch of 2010.0 (1st January 2010). Velocities of stations are assumed to be constant between defined epochs. To accommodate coseismic offsets (which can be up to several metres in magnitude), site disturbance and instrument changes, the ITRF allows for coordinate steps at defined epochs corresponding with a seismic event or equipment change. Postseismic relaxation parameters have been included in ITRF2014 for the first time to better model this aspect of deformation (Lercier, 2014; Altamimi et al., 2016). These non-secular deformations are only described for specified sites affected by seismic deformation which realise the ITRF.

Other terrestrial RF have also been defined using different realisation criteria such as the Epoch Reference Frame (Drewes et al., 2013; Bloßfeld et al., 2015), the Jet Propulsion Laboratory (JPL) Terrestrial Reference Frame (JTRF) (Abbondanza et al., 2020), the Deutsches Geodätisches Forschungsinstitut Reference Frame (DTRF) (Bloßfeld et al., 2020) and more specialised RF used in global plate motion studies (e.g., Tregoning et al., 2013). The most widely used geodetic reference frame is WGS 84 used by GPS. The current realisation of WGS 84 (G1762) is aligned at sub-centimetre precision with ITRF2008. Other GNSS RF such as the IGS14 (GNSS realisation of ITRF2014), Galileo Terrestrial Reference Frame

(GTRF) and the Glonass reference frame, Parametry Zemli 1990 (PZ-90) whose current realisation is PZ- 90.11 are all now very closely aligned with recent or current realisations of ITRF. The United Nations (UN) global geodetic reference frame (GGRF) is currently realised by the ITRF.

Many modern geocentric regional and local reference frames are aligned to different realisations of ITRF where an epoch of ITRF is adopted as the basis for realising the coordinate of a regional or local frame.

3.1.5 Limitations of static and kinematic reference frame in practice

Clearly, a kinematic RF is a precise representation of the dynamic Earth surface at any given epoch and in theory should be used as the basis for all positioning, surveying and mapping purposes. Deformation of the Earth is the main limitation of a static RF in practice for two main reasons. First, fixing the RF where internal deformation is occurring will eventually result in dimensional tolerances being exceeded for surveys if the RF coordinates are fixed and not updated as a result of the deformation. Second, position data from GNSS positioning techniques will not align correctly with data derived from static RF data due to tectonic rotation of the RF and other geodynamic effects as a function of time.

Another limitation of a static RF arises from rotation of the frame with ITRS as a result of tectonic plate movement (e.g., Dawson and Woods, 2010). GNSS baselines are measured in terms of ITRS and so need to be transformed to the epoch of the static frame by a time-dependent transformation to prevent inconsistent positioning over time. This is usually only evident over longer baselines on rapidly rotating plates. For example, a typical tectonic plate rotation rate is 0.5° Ma-1 which means that a 50 kilometre plate-fixed baseline will rotate at ~0.002 arcsec yr-1. This equates to 1.6 mm yr-1 translation at the end of the baseline. A typical national RF may have a life-span of 20+ years which equates to a ~30 mm positioning error for typical longer range baselines.

In practice, however, there are a number of limitations of kinematic RFs which restrict their adoption by national geodetic agencies and managers of spatial data. These limitations are mostly related to inconsistency or lack of international standards for describing point motion as well as the lack of implementation of kinematic parameters to handle the deformation precisely in positioning and mapping software such as GIS. The nature of spatial data is itself also a limitation. Spatial data is largely a product of surveying and remote sensing techniques. Land surveys, point clouds and imagery are all essentially snapshots of a set of points or features on a dynamic surface at a given epoch in time (the measurement epoch or epoch of acquisition or survey) and therefore the data are intrinsically static in nature. If the RF for these surveys is kinematic (for instance ITRF using an untransformed GNSS PPP solution) then it becomes impossible to combine or align surveys from different epochs unless the motion between the different epochs is precisely known or modelled within the software used to visualise or analyse the data.

There are a number of different approaches to handle these limitations. The first and more commonly used approach at present is to transform the kinematic RF (ITRF or WGS 84) coordinates of GNSS positions to a fixed reference epoch or static datum using a time-dependent transformation applied within the positioning or GNSS processing software. The second but less widely used approach is for digital spatial data to be transformed to the epoch of GNSS positioning to ensure correct alignment. This approach requires the correct time-dependent transformation technique to be applied within GIS or survey software and this is not widely available at present (Stanaway et al., 2015). Also, transforming large raster imagery files and dense point cloud data can involve significant computational overhead. Regardless of which approach is used, spatial data within a kinematic RF will require a common epoch for combination using a forward or reverse time-dependent transformation to be applied to data from different acquisition or measurement epochs.

Enabling alignment of positioning and spatial data in a complex deformation environment (now the whole Earth) is the main premise of this thesis. A kinematic RF in practice is ultimately the optimum approach to ensure continuous alignment of GNSS positions and spatial data over the deforming surface of the Earth. For this goal to become achievable, the complexity of the deformation and motion must be handled robustly and consistently by all software that handles spatial data within a kinematic frame.

3.1.6 Plate-fixed reference frames (PFRF)

To support mapping and surveying activities in regions or countries within tectonically stable plates, a RF fixed to the stable portion of the tectonic plate can provide an alternative to a NNR based kinematic or dynamic RF overcoming the limitations of both static and kinematic RF. The magnitude of site velocities in a kinematic RF such as ITRF can be quite significant (up to 80 mm yr-1) and this motion impacts on the practical adoption of a kinematic RFs as described previously. A plate-fixed frame is different from an NNR frame such as ITRF in that the rotation rate of the tectonic plate is internally set to zero. For a non- deforming (rigid) tectonic plate, all site velocities in a plate-fixed frame would be zero, a condition which supports repeatability of coordinates at different epochs for stable features on the plate. In reality, all stable tectonic plates have some minor intraplate deformation and plate boundary zones have deformation arising from interseismic strain accumulation across the plate boundary zone. This deformation is evident with non-zero site velocities. An example of a plate-fixed reference frame is the ETRF89 (Boucher and Altamimi, 1992) fixed to the stable portion of the Eurasian plate. GDA94 and GDA2020 are also examples of continental or regional plate fixed RFs. Most locations in central, eastern and western Europe have < 1 mm yr-1 site velocities within ETRF, however plate fixed site velocities start to increase in southern Europe (Italy and Greece) towards the African and Anatolian Plate boundaries. Locations in Scandinavia and Finland are subject to GIA which is predominantly vertical deformation but with a smaller horizontal deformation component. In the Australian example plate-fixed site velocities are typically less than 0.5 mm yr-1.

3.1.7 Crust-fixed reference frames (CFRF) and semi-kinematic RF

To support surveying and mapping in plate boundary zones a kinematic representation of a deformation model (interseismic velocity model) can be used to estimate secular interseismic deformation between epochs. Within these deforming zones the strategy of fixing RF coordinates results in distortion of geodetic networks with fixed coordinates. This is due to rapid deformation within the network due to interseismic displacement of up to 50 mm yr-1 and coseismic displacement after larger earthquakes of several metres. Stations that realise a crust-fixed RF may not necessarily have site velocities assigned to them if they are estimated from a deformation model. This type of frame has also been called a semi- dynamic datum (or semi-kinematic datum) and has been adopted in the Western states of the USA (Snay, 1999; Pearson and Snay, 2012) and New Zealand (New Zealand Geodetic Datum 2000 (NZGD2000)), (Blick et al., 2006). A CFRF is originally defined at its reference epoch and the kinematic component is realised by an interseismic velocity or deformation model. The deformation model enables geodetic observations at the epoch of measurement to be transformed to the crust-fixed frame (forming quasi- observations) using the deformation model. This approach enables contemporary geodetic measurements which are strain free to be used within a strained static RF realisation. If interseismic strain rates are significant, the reference epoch and RF coordinates need to be updated when the interseismic strain exceeds certain dimensional tolerances. Updates or coseismic deformation patches applied to the reference epoch are also required if there is significant seismic activity, again impacting on dimensional tolerances post-earthquake.

3.1.8 Local Reference Frames (LRF)

Local reference frames (LRF) include many current national static geodetic datums that support mapping and cadastral surveys as well as project datums or grids used for mining or civil engineering projects. The connection between LRF and regional RF or ITRF may be unknown but in practice the precise transformation is important to support GNSS positioning within the LRF, for example GNSS-RTK. The transformation from the LRF to a fixed epoch of ITRF can be estimated by least-squares fit of LRF and ITRF coordinates or by using site transformation or calibration tools within GNSS software. Another approach is to design an LRF that is a fixed epoch realisation of ITRF so that the transformation is solely a time-dependent one without scaling or rotation. Site transformation or localisation algorithms used in RTK software realise a temporal LRF in order to fit the GNSS reference frame to a local coordinate system. This approach is somewhat ad hoc, and the geometry of the stations used and their uncertainty impact on the quality of the estimated transformation. Furthermore, these site transformations are not usually platform independent for use in other proprietary GNSS equipment or GIS and mapping systems. An important consideration for LRF is the stability of the LRF origin. This ensures that all spatial data (feature surveys and designs) are consistent with surveying and positioning techniques that interact with the data, for example, setting out civil works based on a design. In instances where local deformation monitoring is important (e.g., for structural monitoring), a stable non-deforming fiducial network within the LRF is

63 essential for measuring the deformation. In tectonically stable regions a LRF can theoretically encompass any area which is wholly within the stable portion of a tectonic plate. Such regions where LRF can be realised include: Great Britain, Ireland, France, Australia, Borneo, South Africa, Brazil, Eastern USA, Southern India, Germany, The Netherlands, Belgium and the many small island states.

3.1.9 The dual-frame approach

An emerging strategy for managing complex deformation working static RF is the dual-frame approach (Donnelly et al., 2015). Coordinates of the static RF (or fixed reference epoch of a kinematic RF) are adopted as a standard for spatial data combination at the reference epoch of the static RF. Positioning is accomplished within the same kinematic RF, for example in ITRF2014 with the ITRF2014 positions propagated to the static or semi-kinematic frame reference epoch to enable alignment between positioning and mapping. The dual-frame approach differs slightly from a semi-kinematic RF in that both ITRF and the local static or semi-kinematic RF are “gazetted” at a government jurisdictional level, whereas with a fully semi-kinematic approach, only a single RF is adopted associated with a defined deformation model. In either case a site velocity or deformation model is used in the same way.

3.2 Reference frame transformations

Geodetic transformations are required to enable coordinates in one datum or RF to be transformed into another datum or RF. The transformation is applied to spatial data or positions in the source RF (RFA) so that they can be overlain or aligned in the context of the target RF (RFB). Approaches to geodetic transformations are diverse and the optimal transformation strategy depends on factors such as the spatial extent of the transformation, time-dependence of the RF, the uncertainty of coordinates in the two RFs and the geometry of the two RFs.

Transformations of static RF can be broadly classified into geodetic or topocentric categories. A geodetic transformation involves transformation of Cartesian coordinates [ X Y Z ] that are related to the origin of the RF at or near the geocentre. A topocentric transformation relates to transformations of topocentric coordinates [ E N ] on a surface (or projection plane) tangential to the RF reference ellipsoid. The topocentric transformation approach is useful for modelling and describing complex horizontal surface deformations such as coseismic and postseismic displacements which are typically less than 20 m in magnitude on centennial time scales. This approach will be described in greater detail in Chapter 6.

Transformations that include a kinematic RF require time-dependent transformation parameters. These are currently in the form of static parameters defined at an inter-frame transformation reference epoch

(t0) and rates for each parameter. The rates are multiplied by the epoch difference between the epoch of data for each RF to compute parameters for a specific epoch.

Commonly used transformation approaches are summarised in the following sections.

3.2.1 Three-parameter geocentric Cartesian transformation

This simple transformation is used if there are no rotations and scale changes between the source and target RF. It can be used for high-precision transformations between homogeneous ITRS aligned RF within stable plate regions and is given by Eqs. 3.1 and 3.2:

XXTBA=+AB (3.1) or,

éùXXt éù éùx êúYYt=+ êú êú êú êú êúy êúZZt êú êú ëûBAAB ëû ëûz or in linear form:

XXtBA=+x

YYtBA=+y

ZZtBA=+z (3.2) where XA, YA and ZA are the Cartesian coordinates in the source RF (RFA), where XB, YB and ZB are the Cartesian coordinates estimated in the target RF (RFB), and tx, ty and tz are the Cartesian translation parameters between RFA and RFB.

3.2.2 Seven parameter Bursa-Wolf geocentric Cartesian transformation

This is the most widely used conformal transformation for geodetic transformations and models translations, rotation and scale changes between RF defined at the origin of the source RF using Eqs. 3.3 and 3.4. A conformal transformation between any two homogeneous RF in theory should have a singular scale factor. In practice, however, a scale parameter is often estimated where a bias or coloured noise is evident if one of both RF. In essence, the scale parameter models any systematic bias.

The Bursa-Wolf modification assumes that co-rotations between the axes can be generalised as a single rotation for small rotations (less than 10 arcsec). Where rotations are larger than 10 arcsec the full combination of the three rotation matrices should be defined around each axis (e.g., as described in Harvey, 1986). Eqs. 3.3 and 3.4 use the Coordinate Frame (CF) notation. For Position Vector (PV) notation the rotation parameters need to be reversed in sign (multiplied by -1).

XTBA=++AB (1s ) RXBA or (using CF notation):

éùXt éùxzéù1 rrX- yéù êú êú êúêú Yt=++(1 srrY )- 1 (3.3) êú êúyzxêúêú êúZt êú êú rr- 1 êú Z ëûBAB ëûzyëûxAB ëûA

or in linear form (using CF notation):

XtBA=++xz(1 sXrYrZ ) ( + A-yA)

YtBA=++yzx(1 srXYrZ ) (- +A + A)

ZtBA=++zyx(1 srXrYZ ) ( - A+A) (3.4)

where XA, YA and ZA are the Cartesian coordinates in the source RF (RFA), where XB, YB and ZB are the Cartesian coordinates estimated in the target RF (RFB), and tx, ty and tz are the Cartesian translation parameters between RFA and RFB rx, ry and rz are the Cartesian rotation parameters (in rad) between RFA and RFB (using CF notation. For PV notation the sign of the rotation parameters is reversed) s is scale difference between RFA and RFB.

3.2.3 14 parameter geodetic Cartesian transformation (time-dependent)

Time dependence can be applied to a seven parameter transformation by defining rate terms t!x , t!y , t!z , r!x , r!y , r!z and s! for each of the parameters defined at a fixed reference epoch t0. The transformation parameters at the epoch of transformation (t1) are then estimated by Eq. 3.5.

qt()1010=+ qt () ( t- t ) q! (3.5) where q is the parameter tx, ty, tz, rx, ry, rz or s at t0 and q! is the parameter rate of change t!x , t!y , t!z , r!x , r!y , r!z or s! . The 14 parameter model can be used for a wide range of scenarios where selected use of the parameters are made. For example, a stable plate rotation can be defined by only the rotation rates with zero for all other parameters. A block shift rate (in a localised area) can use translation rates with zero for all other parameters.

3.2.4 Coplanar two parameter two dimensional transformation

This is the most basic transformation used to transform coordinates in a planar or topocentric CRS sharing a common projection plane. It can be represented simply as a two-dimensional translation or block shift (tE tN ) using Eqs. 3.6 and 3.7:

éùEE éù éùtE êú=+ êú êú (3.6) ëûNNBA ëû ëûtN or in linear form: EEt=+ BAE

NNtBA=+N (3.7)

3.2.5 Block shift model (distortion grid)

A gridded topocentric translation model (1D or 2D only) (e.g., NTv2) is widely used where one or both datums are heterogeneous in nature. Equations 3.6 and 3.7 used for a topocentric translation are applied here, however the translation parameters are estimated by interpolation of a grid model of translations for each node. The block shift model is also widely used for representation of coseismic displacement models.

3.2.6 Coplanar six parameter two dimensional transformation

The plane six parameter transformation is a very commonly used method for transformation between two coordinate systems in a localised area and involves a plane translation, rotation and scale change. The transformation is fully reversible and suitable for very large rotations (e.g., between a mapping grid and a site, engineering or mine grid). The scale parameter can also unitise the scale factor if one of the coordinate systems is a mapping grid with a non unity scale factor for example, a Universal Transverse Mercator (UTM) projection and if elevation dependent scale factors are significant (e.g., in high elevation regions). Local transformation origin, rotation and scale can be parameterised to form a six parameter topocentric transformation in Eqs. 3.8 and 3.9:

éùE éù(EtA0 +E ) écosqq sin ù é(EEA- A0 ) ù =+êúk ê ú (3.8) êúN Nt+ê-sinqq cos úNN- ëûB ëû( A0 N ) ë û ë( A A0 ) û which in linear form is: EE=++ tkéùcosqq EE- +sin NN- BA0E ëû( AA0) ( AA0) NN=++ tkéù--sinqq EE+cos NN- (3.9) BA0N ëû( AA0) ( AA0)

where, EA and NA are the coordinates in the source CRS, EB and NB are the estimated coordinates in the target CRS. EA0 and NA0 are the coordinates of the transformation origin in the source CRS. tE and tN are the translations between the two CRS and ϴ is the rotation angle between the CRS.

3.2.7 Limitations of static RF transformation strategies

Most geodetic datums and frames have a defined set of transformation parameters to WGS 84 to assist with positioning and GIS where different RF are used in conjunction with each other. The hub transformation principle allows for any datum or RF with a transformation to WGS 84 to be transformed to any other RF that also has a defined transformation to WGS 84. In essence, a forward transformation to WGS 84 is applied to the source RF followed by a reverse transformation from WGS 84 to the target RF. Or put more simply, an interframe transformation is computed by subtracting the target datum WGS 84 transformation from the source datum WGS 84 transformation. The hub transformation principle is widely used within GIS software, particularly if specific interframe transformation parameters are not

67 estimated or made available. The principle uses a widely used international frame such as WGS 84 as a hub RF.

A very significant limitation with this approach is that the epoch of WGS 84 (which is itself a kinematic reference frame) is not defined in current transformation pipelines, a problem highlighted by Evers and Knudsen (2017). Any transformation between a static and kinematic RF requires time-dependent parameters, otherwise the precision of the transformation will degrade as a function of time. If the kinematic RF epoch is standardised for all static RF to kinematic RF transformations the problem is largely overcome but this approach is currently not defined in geodetic registries. In practice what happens is that ITRF derived RF and static RF usually have null transformations to WGS 84 since WGS 84 and ITRF are both truly geocentric at the centimetre level and the kinematic component has, until recently, not been considered to be significant for most GIS applications. Consequently, WGS 84 precise positions are transformed to a static datum with the epoch component ignored. This is highly undesirable and problematic for precise surveys where static RF positioning uncertainties less than 2 m are required.

Australia has recently gazetted GDA2020 (ICSM, 2018) which is a realisation of ITRF2014 projected to epoch 2020.0. GDA2020 supersedes GDA94 which is a realisation of ITRF92 at epoch 1994.0. There is a direct 7-parameter transformation between GDA94 and GDA2020 (EPSG: 8048) with a few centimetre uncertainty. But importantly, the transformations between GDA94 and WGS 84 (EPSG: 1120) and GDA2020 and WGS 84 (EPSG: 8450) were initially both null transformations with 3 m uncertainty. If the hub principle is applied for GDA94 and GDA2020 transformations then there will be no change in coordinates and elevations when in fact there should be approximately 1.6 m change in coordinates and 0.1 m change in ellipsoid height. While this difference is strictly beneath the threshold of the 3 metre uncertainty defined for the transformation, the possibility of the hub transformation being used inadvertently without considering more accurate transformations and uncertainties is very high. This transformation precision issue is driving the deprecation of many transformation parameters of later realisations of static RFs to WGS 84. In 2018, the EPSG published time-dependent transformation parameters between GDA2020 and WGS 84, with WGS 84 assumed to be fully coincident with ITRF2014 at the mm level (EPSG, 2019).

3.2.8 Existing time-dependent transformations

From the previous discussion it is apparent that time-dependent transformations are required between static and kinematic RF or between different kinematic RF if precision of transformation in the target RF is important. Several different approaches have been used by geodetic agencies since the 1990s.

The most widely used time-dependent conformal transformation is the 14-parameter transformation described in Section 3.2.3. The 14-parameter transformation is sometimes referred to as a 15-parameter transformation if the reference epoch (which is not estimable) is also considered a parameter. The 14- parameter transformation allows for time evolution of a 7-parameter transformation for example as a result of tectonic plate rotation.

14-parameter transformations are widely used in GNSS post-processing services such as AusPOS which currently transforms ITRF2014 (realised by IGS14) coordinates to both GDA94 and GDA2020, NRCan transforms IGS14 coordinates at the epoch measurement to NAD83(CSRS) using National Epoch Transformation (NET) v6.0 by applying the Canadian GNSS velocity grid (CVG) v6.0 (NRCan, 2019). OPUS uses HTDP to transform IGS14 coordinates to NAD83.

The 14-parameter model works well in stable plate settings such as Australia, Western Europe and Eastern USA, however it does have limitations in plate boundary regions where deformation is not uniformly distributed or is highly localised. The limitation is due to the fact that any localised deformation is distributed during the least-squares inversion of the 14-parameter transformation.

Triangulated or gridded deformation models overcome the limitations of conformal parametric approaches because localised and plate boundary deformation can be isolated within the model. In regions of localised deformation, or where positioning tolerances are smaller (e.g., urban areas), a denser deformation grid can be developed (Winefield et al., 2010).

In plate boundary regions such as the western states of the USA and New Zealand, different transformation strategies are used. In the USA, a software package HTDP is used for complex time- dependent transformations in the deforming areas. In New Zealand, a gridded deformation model (in effect a site velocity model) is used for time-dependent transformations between ITRF and NZGD2000. The application and formats of these models are quite different and not widely implemented within GIS and other spatial software as there is yet no international standard format for site velocity models.

3.3 Positioning and dimensional tolerances in applied geodesy and surveying

An important contribution of this thesis addresses the extent to which deformation of a RF can be accommodated before tolerance limits of defined geodetic applications are exceeded. This provides some guidance to custodians and managers of RFs as to when RF updates are required as well as the optimum form of transformation. For example, personal navigation and many GIS applications typically only require a precision of a few metres, routinely attainable with inexpensive GNSS enabled handheld devices in SPP mode. At this level of precision, no deformation modelling or transformation is usually necessary provided that the underlying map base is referenced to an ITRF aligned local RF, realised within the previous 50 years or so. On the other hand, high-precision deformation monitoring may have a tolerance of just a few mm (e.g., structural engineering).

3.3.1 The distinction between dimensional and positional tolerance

Dimensional tolerances are mostly governed by engineering requirements (e.g., bridge and dam construction, structural engineering, manufacturing, plant facilities) and also definition of cadastral boundaries defined by occupations, internal angles and distances. Dimensions and internal geometry for these applications can be essentially free of external RF constraint. Positional tolerances (e.g., for digital coordinated cadastres) on the other hand are RF dependent, with any dimensions derived by inversion of two positions or cadastral corners. Dimensional uncertainty in this instance is a function of the positional uncertainty of each location taking into considering any scale factor issues regarding the projection surface for positional coordinates. In many cadastral and engineering surveys, dimensional tolerances have precedence over positioning tolerances with regard to external connections to a RF.

With the advent of precision GNSS positioning application of the principle of uncertainty has become more complex. Local Uncertainty (LU), Relative Uncertainty (RU) and Positional Uncertainty (PU) are new terms used in specification documents (ICSM, 2014). These concepts are discussed more completely in Roberts et al., (2009). In practical terms, RU can be described as the dimensional uncertainty between a network of adjoining points (e.g., points on a bridge span, corners of a cadastral boundary, or a survey network in a regional town), whereas PU is the uncertainty of a point with respect to the datum or RF.

There are different conventions and standards for describing linear dimensional tolerances depending upon their application. A linear ratio (1:n) is commonly used (e.g., 1:10,000), however the parts-per- million (ppm) convention is also in widespread use as a definition of tolerance. Geodetic strain, on the other hand is typically defined as a change per unit length with geodetic strain rates described in terms of 1∙E-9 yr-1. The relationship between the different terms is as follows:

tolerance (1:n) = 1,000,000/tolerance (ppm) linear ratio (yr-1) = 1,000,000,000 / strain rate (E-9 yr-1) tolerance (ppm) = 1,000,000/tolerance (1:n) ppm yr-1 = strain rate / 1,000 strain rate = ppm yr-1∙1,000 or 1,000,000,000 / linear ratio yr-1 Tables 3.1 and 3.2 show typical dimensional and positional tolerances for a variety of spatial applications.

Relative Uncertainty (RU) Equivalent typical Linear Component Positional fixed maximum Uncertainty Application precision Equiv. project in as a at project component strain dimension ppm ratio extents (mm) E-9 (m) (m) High-precision 2 10 100,000 10,000 100 0.005 engineering Structural 3 20 50,000 20,000 500 0.015 engineering Civil Engineering 10 200 5,000 200,000 1,000 0.200 Civil Earthworks 100 400 2,500 400,000 1,000 0.500 CBD Cadastral / BIM 5 15 75,000 13,333 100 0.010 Urban/Cadastral 15 40 25,000 40,000 200 0.025 (Suburban) Rural Residential 25 50 20,000 50,000 1,000 0.075 Cadastral Rural Cadastral 50 50 20,000 50,000 2,000 0.150 Table 3.1 Indicative horizontal dimensional tolerances and uncertainties (fixed, ppm component and equivalent geodetic strain) at 1s

Positional Application Uncertainty (PU) (m) Precision Agriculture (inter-row steering) 0.025 Automated Mining 0.05 Feature survey and Site Plan 1:250 scale 0.05 Automated Driving – lane control 0.1 Underground utility maps 1:500 scale 0.1 Airborne Laser scanning (LiDar) 0.1 Digital Cadastral Database (DCDB) 0.2 Urban Services maps 1:1,000 autonomous car navigation 0.3 Outer-urban services GIS/maps 1:2,500 scale 0.5 Aircraft Instrument Landing 0.7 Suburban planning GIS/maps 1:5,000 scale/asset mapping 1.0 City GIS/Maps 1:10,000 scale 2.0 Personal Navigation 5.0 GIS/Topographic Maps 1:25,000 scale 5.0 GIS/Topographic Maps 1:50,000 scale 10.0 Table 3.2 Indicative Positional Uncertainty (PU) tolerances at 1s

The dimensional and positional uncertainties described here do not consider the effects of deformation (continuous or episodic) of the frame in which the uncertainties for the applications reside. These are analysed in Chapter 4.

Chapter 4 The impact of deformation on reference frames

Earth deformation on a wide spectrum of time scales, spatial extent and magnitude has a major impact on the application of RFs in practice, particularly for applications where positioning and dimensional tolerances are very high.

As described earlier in Section 3.1, a RF is a network of stations that realise a CRS. Each station will usually have a PU assigned to it, conventionally described as an uncertainty with respect to the RF at 95% confidence. If the station or RF are theoretically not deforming, the PU at the RF reference epoch will remain constant through time unless later surveys or network adjustments result in a change in coordinates or PU. For many contemporary RFs and geodetic datums, hierarchies of class and order have been assigned to geodetic control indicating the precision of the connection to the RF (order) or adjoining stations in the network (class), (Roberts et al., 2009; ICSM, 2014).

However, if the RF is deforming as a function of time, then PU for RF station coordinates become unrealistic once the deformation exceeds the PU. As a consequence, constraining coordinates in network adjustments and NRTK algorithms introduces distortion and uncertainty into other RF stations that may not be affected by the deformation. A stable and internally self-consistent RF is a fundamental requirement for precision land surveys, NRTK operations and derived spatial data. So, how much deformation can an RF accommodate to support the highest precision applications before RF coordinates have to be updated?

This chapter examines five cases where earth deformation impacts on RF precision:

● Stable plate motion within a TRF - where vectors between positions on different plates are changing in dimension due to plate rotation

● Interseismic motion associated with plate boundary, intraplate and GIA deformation between episodes of seismicity and subsequent postseismic deformation

● Coseismic displacement (episodic deformation) associated with seismic activity

● Postseismic deformation that follows on from coseismic displacement

● Localised deformation which can be both secular and episodic

4.1 The effect of plate rotation on terrestrial reference frame in practice

The most significant external deformation effect of local and plate-fixed RF is that of rigid plate rotation. In theory, there is no strain within a rotating, rigid tectonic plate as the interior geometry of a geodetic network fixed to a rigid plate is not deforming. In practice, however, the kinematics of a rotating plate

72 within a terrestrial RF can be considered to be a form of deformation of the Earth’s crust, or absolute deformation if one considers that vectors between points on adjoining stable plates are changing with time (Stanaway et al., 2013). The plate rotation effect is an especially important consideration when using GNSS techniques for positioning, as GNSS natively uses a terrestrial RF and GNSS vectors are defined within a TRS/TRF and not a plate-fixed RF if IGS orbits are used for vector processing.

There are two major effects of plate rotation on the usage of GNSS within a plate-fixed RF. The first and most obvious one is that GNSS point positions for any location on Earth defined in a TRF will change with time due to the effect of plate rotation. The most rapid secular velocities are in the western part of the Pacific Plate where secular site velocities can be up to 80 mm yr-1 (from IERS, 2019a). Unless the GNSS derived TRF coordinates are transformed into a plate-fixed RF, TRF coordinates will need to be referenced with the epoch of measurement in order to be useful for practical applications. If the TRF and plate-fixed RF coordinates are assumed to be the same then the validity of this assumption rapidly degrades as the difference between the measurement and reference epoch of the plate-fixed RF increases with time. For applications where positioning tolerances are high, the assumption may be only valid for a period of months (Figure 4.1). Rapidly rotating plates such as the Australian Plate (where site velocities are typically 60-70 mm yr-1) may require updates to a plate-fixed RF every three years for even lower precision applications if a suitable time-dependent transformation from a TRF is not applied.

Figure 4.1 Effect of plate rotation (represented by derived TRF site velocities) on PU for different applications in the absence of a time-dependent transformation (from Stanaway et al., 2012)

The second effect of plate rotation is that of misalignment of TRF GNSS vectors with an underlying plate- fixed RF over time (Figure 4.2). Rotation of a geodetic network on a stable plate does not impact on constituent baseline vector length or network scale but the rotation effect can be significant for longer baselines defined in terms of ITRS. The more rapid the plate rotation, the more noticeable the effect will be on GNSS network precision using local reference stations with plate-fixed RF coordinates. The plate rotation effect is 3D in nature with errors in ellipsoid heights also evident if TRF baseline vectors are not rotated to a plate-fixed RF. The equivalent topocentric error is dependent upon the vector geometry and location within the plate.

Commercial GNSS post-processing software packages and NRTK network operators need to apply a time- dependent rotation of GNSS vectors in ITRF to a plate-fixed RF within their processing engines. Not doing this will result in degradation of accuracy of GNSS post-processing with a plate-fixed RF over time. Table 4.1 shows typical scenarios for different tectonic plates where a 15 mm 3D precision limit is exceeded on a 30 km GNSS vector if the plate rotation effect is ignored.

Figure 4.2 Effect of plate rotation on GNSS vectors and networks

Number of years before Rotation Rate Application 15 mm 3D PU for rover GNSS °Myr-1 exceeded at 30 km range from CORS Pacific 0.68 42 Australian 0.63 45 Eurasian 0.26 110 North American 0.19 151 South American 0.12 239 South Bismarck 8.00 3 Table 4.1 Number of years until 3D PU tolerance is exceeded if GNSS vectors are not rotated into a plate-fixed RF

Many smaller microplates and rigid crustal blocks within deforming zones have very rapid rates of rotation. For example, the South Bismarck Plate in Papua New Guinea rotates at 8° Myr-1. On this microplate, a 30 km GNSS vector defined in ITRS will rotate over a microplate fixed RF introducing a positioning and heighting error of 4 mm yr-1 in a plate-fixed RF if the effect of plate rotation is not considered.

These plate-rotation effects can be overcome by using a TRF at the epoch of measurement for GNSS post- processing or NRTK processing and then applying a time-dependent transformation such as a PMM to the GNSS vectors.

4.2 The effect of geodetic strain on reference frames

Within stable tectonic plates away from plate boundaries, geodetic strain rates are typically less than 1∙E-9 yr-1 (from Kreemer et al., 2014) which implies that the life span of a fixed reference epoch of a local RF or plate-fixed RF can theoretically be hundreds of years in terms of the most stringent dimensional tolerance specifications described in Chapter 2. The continental part of the Australian Plate for example, is highly stable with strain rates of less than 0.1∙E-9 yr-1 which equates to a dimensional stability on crustal bedrock of better than 1:10,000,000,000 (1 mm per 10,000 km per year).

Within deforming plate boundary zones the assumption of stability is somewhat different. Secular geodetic strain rates within plate boundary zones during interseismic periods can exceed 500∙E-9 yr-1. Figure 4.3 indicates the number of years before different survey tolerances (described in Chapter 2) are exceeded within a plate boundary zone. Figure 4.4 shows a comparative relationship between PU and deformation rates for different specification of PU at a range of 30 km (a typical maximum baseline length between a rover GNSS and CORS/NRTK station).

Figure 4.3 Number of years before dimensional tolerances are exceeded for differing strain rate scenarios and specifications

Figure 4.4 Number of years until PU tolerances are exceeded for differing strain rates and specifications

For example, in New Zealand, geodetic strain rates within the plate boundary zone (e.g., along the Alpine fault and within Wellington city) are typically up to 300∙E-9 yr-1 in magnitude and can exceed 500∙E-9 yr-1 in central Wellington, which means that for local control surveys in those areas a redefinition of the local

76 crust-fixed CRS is required every 70 years during the interseismic period if a dimensional tolerance of 1:50,000 is to be maintained (Figure 4.5). Cadastral surveys may need to be redefined dimensionally every 170 years or so in order to maintain 1:20,000 precision even in the absence of any seismic deformation.

Figure 4.5 Effect of interseismic strain on dimensional tolerances within rapidly deforming plate boundary zones. A 10 km baseline would change by more than 50 mm in 10 years in the red zone. (Wellington, New Zealand example from Donnelly et al., 2015)

In order to maintain the desired 15 mm PU tolerance on a typical 30 km baseline (with an equivalent tolerance of 0.5 ppm) CORS/NRTK networks would need regular coordinate updates depending upon the geodetic strain rate in the network area (Table 4.2).

geodetic Typical deformation setting Number of years before 2nd (fast-moving strike-slip plate boundary) 15 mm PU for rover GNSS invariant (Note: as an illustrative guide only, as strain exceeded at 30 km range from strain rate rate is highly variable) CORS (E-9 yr-1) 0.5 interior of stable tectonic plate 1000.0 5 diffuse deformation zones 100.0 10 between 300-400 km from plate boundary 50.0 20 between 200-300 km from plate boundary 25.0 50 between 150-200 km from plate boundary 10.0 100 between 100-150 km from plate boundary 5.0 200 between 50-100 km from plate boundary 2.5 500 within 50 km of fast-moving plate boundary 1.0 Table 4.2 Number of years before NRTK positioning tolerance is exceeded as a result of geodetic strain

4.3 The effect of coseismic displacement on reference frames

Coseismic deformation arising from earthquakes has an instant and sometimes significant effect on a CRS. In many instances, use of pre-earthquake coordinates and elevations for post-earthquake surveys is not advisable, especially for larger area surveys where significant differential deformation has occurred. The motivation is obvious when one considers the case where a cadastral parcel and associated geodetic referencing is bisected by a displaced fault (Figure 4.6).

Figure 4.6 Effect of coseismic deformation on cadastral boundaries and associated geodetic referencing (l), field boundary offset resulting from Darfield earthquake, 2010, New Zealand (r) (Photo: GNS Science, NZ, 2010)

Post-earthquake coordinates and elevations should be referenced differently (identifying the CRS patch model used) to distinguish them uniquely, even if the underlying CRS and reference epoch are still maintained. The effect of coseismic deformation tends to diminish with increasing distance from the earthquake epicentre. Coseismic displacement models (CRS patches) can be developed to enable transformation of pre-earthquake positions to post-earthquake positions and vice-versa. Cumulative combinations of patches can be used to transform data across multiple earthquake events. The spatial extent of coseismic patch models warrants discussion. For local surveys using local NRTK/CORS networks the extent of the patch can be minimised where the derived coseismic deformation is beneath the dimensional tolerance thresholds illustrated in Table 4.2. Recent CRS patches developed for earthquakes in New Zealand have a coseismic displacement model threshold of 30 mm in order to minimise the spatial extent of the model for most practical applications.

The main limitation of setting a coseismic deformation threshold for patch models is that a threshold cut- off error is introduced into the CRS at the edge of the patch if the patch correction is applied to a survey that extends across the patch boundary. This limitation also applies to PPP (using a TRF) if the patch is not

78 applied near the boundary and a possible error is introduced. The limitation can be overcome by padding or buffering the model with null values or transitioning the modelled displacements near the patch boundary to zero using a ramping function. At present, the precision of real-time PPP can accommodate a 30 mm uncertainty, however PU for post-processed PPP using static GNSS is now typically at 10 mm for > 8 hour observations and this precision motivates the requirement to extend coseismic patches to accommodate a much higher threshold for precise transformations from a TRF to a local RF. Models suggest that recent great earthquakes result in coseismic deformation at the 4 mm level some thousands of kilometres from the epicentre (Kreemer et al. 2006) (Figure 4.7). The modelled coseismic displacements are supported by analysis of CORS time-series (Tregoning et al., 2013). Such a large areal extent of coseismic deformation can even warrant the development of a global coseismic patch model for the highest precision positioning tolerances (< 3 mm).

Figure 4.7 Modelled global coseismic deformation (shown in mm) resulting from the Mw 9.0 Sumatra earthquake (Kreemer et al., 2006)

Coseismic CRS patches, potentially at a global scale for TRFs, will ultimately need to have much greater coverage in order to support continuing improvement to real-time PPP precision in the context of a plate- fixed or crust-fixed RF.

Slow slip events (SSE) are another form of coseismic deformation but with a period of hours to weeks in duration. This mode of deformation is potentially difficult to manage for applied geodetic applications due to the frequency, magnitude and duration of the associated deformation (Figure 4.8). The optimal approach is to use a piecewise velocity model which will model slow-slip affected sites to a high degree of precision. The piecewise velocity model would be comprised of interseismic velocities between SSEs and slow slip velocities during SSEs. The next level of approximation would be to model a time series with interseismic velocities and regular coseismic patches whenever SSEs occur. In this case, the coseismic patch would be defined at the mean epoch of the SSE with SSE deformation either side of the mean epoch conflated into the patch. If the SSE events are small in magnitude, the overall time series can be averaged to include both interseismic and SSE deformation. This approach is only satisfactory if the desired

79 tolerance limits for the model are satisfied. Another workable approach is to maintain a cumulative patch model which models all non-secular displacement, including SSE between the reference epoch and defined epoch. The model would be updated after each significant SSE.

Figure 4.8 Slow-slip event displacement example for GISB, New Zealand (East component) (GNS, 2018)

4.4 The effect of postseismic deformation on reference frames

Postseismic deformation is a transitional mode of deformation between near instant coseismic deformation resulting from an earthquake and the eventual reversion to secular interseismic deformation some time after the earthquake. There are two main components of postseismic deformation. The first component is afterslip deformation which is generally associated with aseismic elastic creep near a displaced fault which can be modelled with a logarithmic decay function. Afterslip deformation is generally on shorter time scales (< 2 yr). The second major component of postseismic deformation is viscoelastic relaxation which is a much longer term response to larger earthquakes where deformation of the viscous upper mantle has occurred. This mode of deformation can be modelled by a stretched or composite exponential decay function. The Mw 9.5 earthquake in Valdivia, Chile on the 22nd May 1960 and the Mw 9.2 earthquake in Alaska on 27th March 1964 still show observable viscoelastic relaxation fifty years after the events (Freymueller et al., 2008; Pearson et al., 2013b). The Mw 9.2 Sumatran-Andaman earthquake on 26th December 2004 and the Mw 9.1 Tōhoku earthquake 11th March 2011 show similar ongoing postseismic deformation effects and recent analyses support this (Pollitz et al., 2006; Ozawa et al., 2011 and Tobita, 2016). Modelling of postseismic deformation is further complicated by larger aftershocks, slow-slip events and ongoing viscoelastic deformation from historical earthquakes.

In many cases postseismic deformation itself will result in geodetic strain rates exceeding many positioning and dimensional tolerances. In this situation, a postseismic model is required for coordinate and elevation updates (Figure 4.9).

Figure 4.9 Example of postseismic deformation and associated positional tolerance. Site is SEDD, New Zealand (GNS, 2018).

Postseismic deformation can also be linearly approximated by either supplementary displacement models (in the form of coseismic patches) when a dimensional or PU tolerance is exceeded during the decay phase. In this case, a quarantine period is also recommended to allow for aftershock and postseismic deformation immediately after the mainshock. This approach may be warranted if positioning or GIS software cannot be configured for implementation of logarithmic or exponential functions. For later stages of the postseismic decay, the viscoelastic deformation can be approximated as a linear velocity (Figure 4.10), or a ramp function. These options are discussed in more detail in Chapter 6 which describes implementation of gridded postseismic deformation models.

Figure 4.10 Linear approximations of postseismic deformation for same example as in Figure 4.9

4.5 Effects of localised deformation on reference frames

This is perhaps the most difficult mode of deformation to manage with regard to RF in practice. In principle, deformation of a RF should relate to movement of the crust defined geologically and structurally as bedrock underlying any regolith or rigid artificial surface not anchored directly to bedrock. Chapter 2 discussed the numerous forms and causes of localised deformation where existing surfaces can deform as a response to localised effects such as surface loading, downhill creep and soil moisture content. Regolith covers the majority of the land surface of the Earth and most urban areas and agricultural land are located on regolith which can be up to several hundred metres in depth in ancient sedimentary basins (Wilford et al., 2016). These are also the locations of the majority of geodetic surface infrastructure, much of which is unlikely to be directly anchored to bedrock. Deformation of the regolith surface with respect to underlying bedrock should be distinguished from stable plate motion using a separate model showing relative displacement with respect to underlying plate motion. In areas with low surface gradients, absence of aquifers and low soil moisture content, regolith deformation rates are likely to be insignificant.

In principle, localised deformation should be differentiated and modelled separately from underlying crustal deformation resulting from plate tectonics or GIA as the cause of deformation is often not related to geodynamics. It is interesting to speculate what effect unmodelled regolith deformation has on the definition of ITRF and inversion of PMM, since many ITRF geodetic sensors are located within sedimentary basins. The ITRF2014 PMM (Altamimi et al., 2017) has eliminated many sites in the plate mode inversion where residual velocities are beyond a 3 mm at 3s threshold. Investigation of the site location for these suggests a sedimentary basin setting which may explain a component of these variances. As with other modes of deformation, implementation of a localised deformation model is governed by dimensional and positional tolerances. Examples of where a localised deformation model might be implemented include ground surfaces above underground mining operations, coal seam gas fields, water aquifers, tunnelling projects or urban areas on sedimentary basins. Localised deformation also affects any geodetic control located within the deformation area, so some care is required when control affected this way are included in regional scale adjustments or geodynamic modelling. Other examples include stations located on steeper regolith slopes. The effects of downhill creep and ground instability can be very high resulting in the stations moving under the force gravity over time (Figure 4.11).

Figure 4.11 Localised deformation example – regolith creep (image – Khattak, 2019)

By extension the effect of localised deformation can be applied to the stability of geodetic monuments including CORS stations on potentially unstable structures.

4.6 Effect of vertical deformation on vertical tolerance limits

Vertical tolerances are treated differently from horizontal tolerances due to the effect of gravity on engineering and hydrology. Small elevation errors can have significant effects on the integrity of civil structures, high-speed rail, machinery operation (e.g., conveyors) and hydrological calculations. In absolute terms, vertical deformation of the Earth’s surface also has implications for flooding risk and the effects of sea level change in coastal zones. For localised surveys, relative vertical tolerances are important whereas for regional scale surveys absolute vertical tolerances are the main consideration.

For GNSS surveys, the choice of geoid model is also an important factor. There is considerable variation between different models in terms of absolute and relative precision with respect to local gravitational equipotential surfaces, especially in mountainous regions. Another related factor is the effect of mean dynamic topography (MDT) of nearby ocean surfaces. There is usually some offset of up to 1.8 m between a global geoid (defined as an equipotential surface approximating global MSL) and local MSL due to thermal expansion of the ocean column and permanent ocean currents (Andersen and Knudsen, 2009).

With regards to vertical deformation, relative deformation is the main consideration for civil works and machinery. If there is differential deformation across a site due to localised ground instability this will clearly impact on the integrity of structures on the site and engineering tolerances will need to be considered. In absolute terms, tectonic uplift and GIA can impact on regional scale engineering projects dependent upon precision hydrology (e.g., canals) and low-lying coastal areas prone to flooding.

Chapter 5 Kinematic models to support complex time-dependent transformation schema

For each of the different modes of deformation described in Chapter 2, models of displacement and velocity are required to form the basis of time-dependent transformations. This chapter describes how these models can be represented, formatted, interpolated and uncertainty estimated. Suggestions are also proposed for the standardisation of time-dependent transformations within geodetic registries such as those hosted by the ISO or the EPSG now part of the International Association of Oil and Gas Producers (IOGP).

5.1 Structure of time-dependent transformation model formats

A number of considerations influence the structure and representation of models to describe time- dependent transformations. The first consideration is the structure of the model which can be either parametric or non-parametric (e.g., raster grids and vector based models). In the context of this thesis, parametric models are suitable for describing stable PMMs using a minimum of three parameters being the rotation rates (Section 2.1.3), but can be augmented to a full 14 parameter model to account for other inter RF translations, rotations and scale differences (Section 3.2.3). For more complex models (such as for plate boundary or localised deformation), parametric models become inefficient due to the high spatial variability of site velocities and deformation within plate boundary zones.

5.1.1 Triangulated Irregular Networks (TIN)

Triangulated Irregular Networks (TINs) are an efficient way to describe complex surface deformation characterised by variable geometry and resolution, especially near fault boundaries. TINs require supplementary breakline or topology models in order force the tessellation format of the TIN to better represent fault geometry using standard triangulation techniques. This process has long been implemented in surface modelling, GIS and surveying software to represent Digital Terrain Models (DTM) with breakline models used to define changes of grade and other linear terrain features (e.g., Uren and Price, 2010). This forcing prevents interpolation of the DTM across linear features such as faults leading to inaccuracies in the interpolation. A limitation of using TIN models for describing deformation within software is the lack of a generic standard for presentation of a TIN, associated breaklines and other data.

5.1.2 Discrete Global Grid systems and Polyhedral models

Polyhedral tessellation models are also well suited to describe or sample curvilinear surfaces such as a reference ellipsoid and are gaining widespread adoption within GIS for minimal distortion spatial representation, referencing and mapping (e.g., Sahr, 2011).

5.1.3 Grid models

Grid and raster models are now widely used in geophysics and applied geodesy and are the most straightforward form of representation. Distortion models for heterogeneous geodetic datum transformations are a good example of the application of transformation grids including; North American Datum 1927 (NAD27) to NAD83 (Dewhurst, 1990), AGD66 to GDA94 (Collier, 2002), GDA94 to GDA2020 and New Zealand Geodetic Datum 1949 (NZGD49) to NZGD2000 (Crook and Pearse, 2001). Grid models are well suited for platform independent distribution of data and efficient computing and are increasingly a de facto standard for GIS and geodetic data presentation (e.g., geoid models and high accuracy transformations).

The source data used to form a grid model is usually irregular, for example observed site velocities and displacements. The simplest technique to generate a model from irregular source data is to generate a TIN which natively preserves the source geometry and data. Various techniques can be employed to estimate a grid model representation of irregular source data. The most widely used technique in geophysics is least squares collocation (LSC) or kriging, although other techniques can be used such as splines in tension (Wessel and Bercovici, 1998). In deforming zones, inversion of elastic crustal block rotations can be used to generate a grid model of secular site velocities (e.g. Pearson et al., 2013a).

The schema presented in Chapter Six is therefore based upon grid representations of displacement, velocities and other deformations. The structure and format of these grid formats are discussed here.

5.2 Grid formats for geodetic applications

5.2.1 Resolution and geometry considerations for geodetic data grids

Due to the geometric curvature of the Earth’s surface data grids, they cannot be equally spaced for entire global coverage. The simplest representation of the Earth is an ellipsoidal grid with equal spacing of latitude and longitude intervals which is the equivalent of a plate carrée or equirectangular projection used in cartography. The actual spacing between nodes is regular along the equator but as one approaches the poles meridian convergence leads to irregular spacing ratio between equal latitude and longitude intervals. To overcome this, longitude intervals can be changed in higher latitudes but this results in an irregular grid for large area coverage. For smaller area coverage, the grid spacing can be different for longitude and latitude to maintain more equal grid spacing.

Another form of representation is to use projected grids (e.g., transverse Mercator, conic or stereographic projections). Projected grids provide good spacing geometry in high latitude and polar regions, but are not suitable for coverage of larger areas of the Earth surface due to grid cell distortion near the edge of the model.

The effect of a planar assumption for grids on an ellipsoidal or spherical system is insignificant for grids of less than 1 degree spacing (Stanaway et al., 2012). For a one degree global grid, interpolated velocities differed by less than 0.01 mm yr-1 from velocities estimated directly from a PMM using Euler's theorem. Even for a 10 degree grid the degradation of interpolation precision is still < 0.12 mm compared with a PMM.

The resolution (spacing) of grids is another consideration. The grid resolution has to be sufficient to accurately represent the data being modelled using a suitable interpolation method such as bilinear interpolation. As a general rule, the grid resolution should be commensurate with, or finer than the source data used to generate a grid representation using LSC or kriging techniques. If the grid resolution is coarser than the source data, interpolation will lead to over generalisation and inaccuracies. The other related consideration of grid size is the precision requirement for the end user which is usually a trade off between sampling resolution, required precision, grid size and extent.

A grid format approach that is considered to be the most suitable for high accuracy geodetic transformations is a nested variable resolution grid structure that can accommodate the variable resolution and accuracy of the sampling data (Figure 5.1). When this grid structure is interrogated, the highest resolution grid is used. With variable resolution grids a buffer of one grid interval is recommended around high resolution grids to prevent inconsistencies with interpolation of the neighbouring lower resolution grid along the different resolution grid boundaries. The buffer edge values are generated by interpolation of the adjacent (coarser) grid. This approach has been widely used in nested grid models used in numerical weather modelling (e.g., Koch and McQueen, 1987) and prevents discontinuities with interpolation between the two models by ramping the resolution between the nested coarse parent and fine sub-grids.

Figure 5.1. Nested structure of grid models to accommodate variable resolution

Another consideration is discontinuity at the model edge. For example, coseismic deformation models may extend to a tolerance cut off limit related to user requirements. At the edge of the model the discontinuity can approach this cut off. An approach similar to multiple resolution grids could be adopted where a model buffer of zero values is added to the coseismic model in order to ramp the discontinuity from the tolerance limit to zero.

Three dimensional geodetic grids can be disarticulated into horizontal and vertical grids of different resolutions and extents to suit different user requirements.

5.2.2 Model grid metadata, node and data units

Some consideration is warranted of the format of deformation and site velocity grids used for time- dependent transformations, specifically CRS and coordinate formats for nodes and units which describe displacement and site velocity data.

The most common format for describing nodes in geodetic grids is ellipsoidal coordinates with decimal degree units. In the past, some simplifications have been made, for example, using positive longitudes in the Western hemisphere, or positive latitudes in the Southern hemisphere for models restricted to those hemispheres. Obviously this approach is not suitable for generic or global ellipsoidal grids and the practice should be discouraged in favour of a generic sign convention (negative longitudes in the Western hemisphere and negative latitudes in the Southern hemisphere). For high precision and high resolution grids the RF of the grid should be defined in the model header and metadata. Geocentric Cartesian grids are also unsuitable for surface model representation since such a grid surface is not aligned with the ellipsoid or projected surface.

For smaller areas, high latitude and polar regions, projected CRS are well suited for representation of geodetic grids due to meridian convergence of ellipsoidal grids in these areas. Again, the projected CRS would need to be clearly defined in the model header and metadata. For example in Antarctica, Universal Polar Stereographic (UPS) projection can be used for model grids over the Antarctic continent.

The next consideration is how the displacements and velocities together with their associated uncertainties, are described for each node. The primary output of geodetic processing and analysis are geocentric Cartesian coordinates and derived Cartesian displacements or velocities. This format is not intrinsically useful for visualisation and modelling of surface displacements on an ellipsoidal surface and are usually transformed into a local topocentric frame (North, East and Up) tangential to the ellipsoid. Topocentric displacements and velocities can also be transformed into projected CRS at the ellipsoid surface if required. Conversion to a locally-used projected CRS maybe useful for further computations in that CRS bearing in mind projection and elevation dependent scale factors of displacements and velocities.

Displacement and velocity units would normally be in the Système international (SI) system to conform to IERS and ISO conventions (mm, m and mm yr-1 and m yr-1), however alternative units can be used, for example, decimal latitude/longitude differences and rates at the ellipsoid surface or ellipsoidal arc seconds. These units would be specified in the model header or metadata.

5.2.3 Site velocity and displacement grid formats

Presently there is no standard presentation and format for site velocity and displacement grids and their associated uncertainties. Grids can be of varying resolutions (grid intervals) using different CRSs (e.g., decimal ellipsoidal coordinates, projected CRS, geocentric Cartesian) for node values and node velocities can be also be described in different units (e.g., topocentric E,N,U, mm yr-1, m yr-1, lat yr-1, long yr-1)

By convention, geocentric Cartesian coordinates are used to describe positions within a global RF, especially in the cases of GNSS processing and conformal 3D geodetic transformations. As displacement velocity and seismic displacement models are conventionally described within a local topocentric coordinate system [ E N U ] the schema presented here maintains consistency with this convention. The topocentric system used to describe velocities and displacements should also be consistent with the ITRS for its orientation and also be referenced to the GRS80 ellipsoid surface for correct scale (IERS, 2010).

The only limitation of the topocentric representation of displacement is that in very high latitude regions curvature of the parallel (line of constant latitude) away from the prime meridian becomes excessive closer to the poles requiring a polar stereographic representation of displacement. As these regions are largely either covered by sea ice or mobile ice sheets, the impact on the application of the schema is not considered further other than to state that the stereographic format can be used to describe ice motion for cryospheric studies in polar regions.

Geodetic conformal transformations result in non-zero covariance terms in the variance-covariance matrices which show correlation of the uncertainties between the estimated transformed coordinates. The schema allows for the combination of conformal transformations with non-conformal components (e.g., episodic displacements) that are effectively decorrelated using interpolation of a grid model. For the purposes of this schema, which describes a complex transformation incorporating a conformal transformation component, a topocentric translation is derived from a conformal transformation for combination with other topocentric displacement terms. With a time-dependent conformal transformation, the time-dependent topocentric displacement can also be derived.

5.2.4 Formatting and distribution of model grids

The most elementary grid formats are space, tab or comma separated American Standard Code for Information Interchange (ASCII) text, however to support faster computing, binary formats are more efficient. A large number of geodetic grid formats are in current use, many of which are in binary format

88 compiled from ASCII source data using open source code executables. Commonly used formats include: National Transformation version 2 (NTv2), NADCON 5.0, CTABLE, NZGD2000 Deformation Model, GTX (vertical) and HDF5 formats. NTv2 developed by Natural Resources Canada is at present widely used for representation of horizontal geodetic distortion grids. While limited to just 2D or 1D data, NTv2 does have some useful advantages: (1) it supports multiple resolution grids and (2) uncertainty information can be assigned to each of the two data fields for each node. NTv2 has been widely adopted outside Canada and is currently used in Australia, New Zealand, Germany, UK, Spain and Spain (ESRI, 2019) for heterogeneous datum transformations or datum distortion models in grid format. NTv2 is essentially a file of horizontal grid shift values which are interpolated using bilinear interpolation. NTv2 has also been used for geoid representation, e.g., Ausgeoid2020 (GA, 2019b).

Without any modification, NTv2 grids can also be used to represent coseismic displacement models. NTv2 grids could also be adapted to represent site velocities and potentially also postseismic deformation terms of amplitude and decay time along with their associated uncertainties. Interpolated values from the NTv2 site velocity or postseismic grid would be multiplied by an epoch difference to derive a displacement correction and uncertainty.

5.3 Summary of coordinate and displacement conversions

Coordinate, displacement and velocity conversions are summarised below. Considering that complex time-dependent transformations include displacement and velocity models typically defined in a topocentric system, displacements and velocities can then be converted to a topocentric system using the equations described in the following sections.

5.3.1 Ellipsoidal coordinate to geocentric Cartesian coordinate conversion

Ellipsoidal coordinates [ f l h ] can be converted to geocentric Cartesian coordinates [ X Y Z ] using Eq. 5.1. Xh=+()coscosnfl Yh=+(nfl )cos sin

2 Zveh=ëûéù(1- )+ sin f (5.1) where f, l and h are the ellipsoidal latitude, longitude and height respectively and the radius of curvature in the prime vertical n is expressed using Eq. 5.2.

a n = (5.2) 1- e22 sin f where a is the ellipsoid semi-major axis and e is the ellipsoid eccentricity. The eccentricity is derived from the ellipsoidal flattening using Eq. 5.3.

eff22=2 - (5.3)

5.3.2 Geocentric Cartesian coordinate to ellipsoidal coordinate conversion

The inverse conversion transforms Cartesian coordinates to the ellipsoid using Eq. 5.4. éùZfea(1- )+23 sin µ -1 f = tan êú23 ëû(1--fpea )( cosµ )

-1 æöY l = tan ç÷ èøX

hp=+cosff Z sin-- a 1 e22 sin f (5.4) where,

pXY=+22

rpZ=+22

2 -1 éùZeaéù µ =tanêúêú( 1-f ) + ëûprëû

5.3.3 Conversion of geocentric Cartesian displacement or velocity to topocentric format

A geocentric Cartesian velocity can be converted to a topocentric velocity using Eqs. 5.5 and 5.6. The same equation can be used to convert displacements, where the displacement is substituted for the velocity.

éùEX!!éù-sinll cos 0 éù êú êú !!êú (5.5) êúNY=êú--sinfl cos sin fl sin cos fêú êú!!êú ëûUZëûêúcosfl cos cos fl sin sin fëû or in linear form, EXY!!!=-sinll+ cos NXYZ!!!!=--sinfl cos sin fl sin+ cos f

UXYZ!!!!=++cosfl cos cos fl sin sin f (5.6)

Eqs. 5.5 and 5.6 can also be used to convert a Cartesian displacement to a topocentric displacement. The Cartesian velocity terms are simply substituted with Cartesian displacements to estimate the topocentric displacement.

5.3.4 Conversion of a topocentric displacement or velocity to a geocentric Cartesian format

A topocentric velocity can be converted to a geocentric Cartesian velocity using the Eqs. 5.7 and 5.8. The same equation can be used to convert displacements, where the displacement is substituted for the velocity.

éùXE!!éù--sinlflfl sin cos cos cos éù êú êú !!êú (5. 7) êúYN=êúcoslflfl- sin sin cos sin êú êú!!êú ëûZUëûêú0cossinffëû or in linear form: XE!!=--sinlflfl sin cos N !+ cos cos U !

YE!!=coslflfl- sin sin N !+ cos sin U !

ZNU!!!=+cosff sin (5.8)

Eqs. 5.7 and 5.8 can also be used to convert a topocentric displacement to a Cartesian displacement. The topocentric velocity terms are simply substituted with topocentric displacements to estimate the Cartesian displacement.

5.4 Bilinear interpolation

Bilinear interpolation is a standard operation for estimating values from gridded data sets as illustrated in Figure 5.2. In geodetic transformations it is currently used with transformation grids that include heterogeneous data (e.g., grid shifts between CRS). It is also used to estimate coseismic displacements and interseismic velocities in deformation models (e.g., LINZ, 2019).

Figure 5.2 Bilinear interpolation from gridded data

The equations used to estimate values (interseismic velocities in this case) for a point P are estimated using (Eqs. 5.9 to 5.11). Topocentric velocities for the four nodes (SW, SE, NW and NE) of the grid cell containing P are provided.

!! !! EEPS=+r NEE( NS-) !! !! NNPS=+r NNN( NS-) (5.9) where,

!! ! ! EENNWNENW=+r EEE( -)

!! ! ! NNNNWNENW=+r ENN( -)

!! ! ! EESSWSESW=+r EEE( -)

!! ! ! (5.10) NNSSWSESW=+r ENN( -) and,

EE- NN- rE = PSW and rN = P SW (5.11) EESE- SW NNNW- SW

5.5 Time-dependent transformation model formats within geodetic registries

There is currently a large diversity in formats and standards for geodetic operations (e.g., transformations, projections and terminology). Geodetic registries such as IOGP’s EPSG registry and the recently released ISO TC/211 registry and standard, ISO 19111:2019 Geographic information-Referencing by coordinates (ISO, 2019) aim to classify and define geodetic and cartographic data formats and operations to ensure replicability and traceability in a platform or software independent way. GIS and geodetic transformation software platforms such as PROJ and FME utilise these registries as authoritative sources.

At present, standardisation and implementation of time dependent transformation formats within geodetic registries, GIS, positioning software and surveying software is quite limited or generalised. There are numerous approaches and formats that have evolved to suit particular jurisdictional needs in a somewhat ad hoc way. As a consequence of this, application of generic time-dependent transformation models within GIS and positioning software is also limited.

There is also ongoing debate as to whether the use of a site velocity model is a form of inter-RF transformation or an operation solely within a single time-dependent RF. While a site velocity model propagates coordinates within a common RF between different epochs in instances where deformation is complex (non-conformal) it remains the only way to relate kinematic RFs such as ITRF to a static RF in these environments where conformal and parametric transformations are not suitable.

The two main approaches for describing time-dependent transformations are parametric (e.g., time- dependent conformal transformation) and interpolative (e.g., interpolation of gridded models). For the different modes of deformation described in this chapter the most appropriate choice of model is shown in Table 5.1.

Deformation Mode Parametric/Grid Velocity/displacement/other Stable Plate Motion Parametric and derived grid model Velocity Plate boundary zone Grid model Velocity interseismic motion Coseismic Deformation Grid Model Displacement Postseismic deformation Grid Model Amplitude/decay time/ramp Localised deformation Grid model Displacement Tectonic Uplift and GIA Grid model Velocity Localised subsidence Grid model Velocity Table 5.1. Summary of application of deformation grid formats

5.5.1 Standardisation of time-dependent transformations in geodetic registries

The principal geodetic registry currently in use is the EPSG (now part of IOGP) largely funded by the petroleum industry. The EPSG Registry is the principal authoritative repository of geodetic transformations, projections and datum definitions used in GIS, positioning and surveying software. The role of the ESPG is currently in the process of being replicated by the ISO through TC/211 (https://registry.isotc211.org/).

Time-dependent transformations are gradually being classified within the EPSG registry. EPSG accommodates time-dependent 14 parameter transformations (7 parameters at a reference epoch and their associated rates of change) with codes 1053 and 1056 for the two different rotation conventions. In EPSG documentation (EPSG, 2019) it is also referred to as a 15 parameter transformation if the epoch difference (δt) is considered a parameter, however δt is not estimable as is it used as an input variable into the transformation. As shown in Section 2.1.3 a PMM can be described by a 14 parameter transformation with most of the parameters set to zero, the plate motion being solely described by the 3 rotation rate parameters. This approach has been adopted for GDA2020 to ITRF2014 transformations in Australia (ICSM, 2018) following the approach described in Chapter 2 (Stanaway et al., 2015).

The EPSG Registry currently supports site velocity models (referred to as point motion models in EPSG) in grid format, however coseismic and postseismic models are not yet defined or supported by EPSG. The format of site velocity models is also not yet formalised within EPSG. GIS and software developers together with some jurisdictions have implemented time-dependent transformations outside the scope of EPSG. New Zealand uses the NZGD2000 Deformation Model which includes a grid of ITRF96 topocentric site velocities. PROJ have implemented some time dependent transformations in their open source software used within GDAL, however implementation of velocity grids has not yet been adopted within standard GIS software.

The most recent EPSG registry entries (and their EPSG codes) used to describe time-dependent transformations include:

• NTv2 OperationMethod 9615 • Point motion by grid (Canada NTv2_Vel) OperationMethod 1070 • Point Motion (ellipsoidal) 1067 • Point Motion (geocentric Cartesian) 1064 • NADCON 5.0 (2D) 1074 biquadratic • NADCON 5.0 (3D) 1075 biquadratic • OSNT (UK) 9633

Time-dependent transformations will also need to be implemented rigorously in real-time data streams such as Radio Technical Commission for Maritime Services (RTCM) and National Marine Electronics Association (NMEA).

Chapter 6 Formulation of a complex time dependent transformation schema

The primary aim of this thesis is to show how complex time dependent transformations between kinematic, semi-kinematic and static RF can be described, defined and robustly applied within geodetic applications such as GNSS positioning and GIS. The objective is to develop a holistic transformation schema that can be adapted by geodetic agencies and registries in order to provide guidance to developers of positioning software and GIS as well as spatial practitioners and users of precise spatial data such as civil engineers.

The schema describes a rigorous transformation logic flow or pipeline between any RF which handles complex deformation phenomena such as GIA, coseismic and postseismic deformation. Application of the schema enables spatial data and GNSS positions acquired at different epochs to be correctly aligned for visualisation and analysis, even within very complex deforming zones. The schema also describes how uncertainties of transformed data are estimated in order to provide guidance to precision users of that data.

6.1 Scenarios for complex time-dependent transformations

Typical scenarios and examples covering most geodetic applications that can utilise this schema are illustrated and described below in Figs. 6.1 to 6.4. Solid lines indicate modelled displacement with time. Dotted lines show implicit displacement (e.g., untracked displacement in a kinematic RF).

6.2 Schema overview and structure

A time-dependent transformation schema to support spatial and positioning applications such as those illustrated in Figs. 6.1 to 6.4 needs to accommodate and model all of the modes of deformation described in detail in earlier chapters, where the deformation is significant enough to exceed defined tolerances for a given spatial application. Any non-linear or episodic deformation event significant enough to impact on the highest tolerance requirements of spatial end users necessitates a non-linear transformation schema that models both secular (interseismic) and non-secular (episodic or patch) components.

Kinematic RFA (no DM) to Kinematic RFA (no DM) to Kinematic RFB (no DM) Kinematic RFB (with DM)

Without a deformation model associated with A GNSS PPP position (XA) in a kinematic RFA either RF, a transformation is only possible at (e.g., ITRF2014) of a ground control point the epoch of transformation. If either the (GCP) used for imagery acquired at a later date source or target position are located on a (XB) in a different kinematic RFB (e.g., stable plate and the stable PMM is used, the ETRF89) with a deformation model in position in the stable plate will be constant. Southern Italy. Kinematic RFA (no DM) to Kinematic RFA (no DM) to Semi-kinematic RFB Static RFB

A GNSS PPP position (XA) in a kinematic RFA A GNSS PPP position (XA) in a kinematic RFA (e.g., ITRF2014) of a rural cadastral reference (e.g., ITRF2014) on a 1970s era sub-surface mark for a cadastral survey defined in a semi- utility marker defined in a static RFB (e.g., kinematic RFB (XB) (e.g., NZGD2000) near AGD66) near Tennant Creek, NT, Australia Waiau, New Zealand. with coordinates (XB) unchanged after the 1988 earthquake. As the earthquake deformation is not modelled explicitly in ITRF2014, the coseismic deformation will be the error in the transformation. Figure 6.1 Typical kinematic RF transformation scenarios where no deformation model is defined in terms of the RF. Plots show point motion of a ground fixed point (X) in both source and target RF. The interframe transformation reference epoch is indicated by (t0). The epoch of the point in RFA is shown by (tA) and the epoch of the point in RFB is shown by (tB).

Kinematic RFA (with DM) to Kinematic RFA (with DM) to Kinematic RFB (no DM) Kinematic RFB (with DM)

A road design in a kinematic RFA (e.g., ETRF89 A survey control point (XA) in a kinematic RFA with a deformation model) in Southern Italy to with a deformation model (e.g., a hypothetical be set out by PPP at a later date (XB) in a ITRF2030) transformed to (XB) in a different different kinematic RFB (e.g., ITRF2014). kinematic RFB with a deformation model (e.g., a hypothetical ETRF2025). Kinematic RFA (with DM) to Kinematic RFA (with DM) to Semi-kinematic RFB Static RFB

The surveyed location (XA) in a kinematic RFA The surveyed location (XA) in a kinematic RFA (e.g., a hypothetical ITRF2030 with a global (e.g., a hypothetical ITRF2030 with a global deformation model) of a rural cadastral deformation model) converted to a static RFB reference mark for a cadastral survey defined (e.g., NZGD49) to align with imagery captured in a semi-kinematic RFB (XB) (e.g., NZGD2000) in that RF. corrected for the first earthquake near Waiau, New Zealand. Figure 6.2 Typical kinematic RF transformation scenarios where a deformation model is defined in terms of the RF. Plots show point motion of a ground fixed point (X) in both source and target RF. The interframe transformation reference epoch is indicated by (t0). The epoch of the point in RFA is shown by (tA) and the epoch of the point in RFB is shown by (tB).

Semi-kinematic RFA to Semi-kinematic RFA to Kinematic RFB (no DM) Kinematic RFB (with DM)

Infrastructure set-out point (XA) from a design A cadastral survey control point (XA) surveyed in a semi-kinematic RF (RFA) (e.g., NZGD2000) in a semi-kinematic RF (RFA) (e.g., NZGD2000) before the latest earthquake correction to be before the latest earthquake correction to be set out by GNSS PPP (XB) in a kinematic RF transformed to (XB) in a hypothetical (RFB) (e.g., ITRF2014) after an earthquake. kinematic NZGD2030 with an associated deformation model (RFB). Semi-kinematic RFA to Semi-kinematic RFA to Semi-kinematic RFB Static RFB

Infrastructure set-out point (XA) from a design Total station survey using a passive geodetic in a semi-kinematic RF (RFA) (e.g., NZGD2000) control point (XA) redefined in a semi- redefined after an earthquake to be aligned kinematic RF (RFA) (e.g., PNG94) after an with a pre-earthquake as-built survey in earthquake to relocate old survey marks (XB) another semi-kinematic RF (RFB) at (XB). defined in a static RF (RFB) (e.g., AGD66). Figure 6.3 Typical semi-kinematic RF transformation scenarios. Plots show point motion of a ground fixed point (X) in both source (zero RF motion during the interseismic period) and target RF. The interframe transformation reference epoch is indicated by (t0). The epoch of the point in RFA is shown by (tA) and the epoch of the point in RFB is shown by (tB).

Static RFA to Static RFA to Kinematic RFB (no DM) Kinematic RFB (with DM)

Line end point (XA) defined in a static RF (RFA) The same line end point (XA) defined in a static (e.g., GDA94) map base defined before an RF (RFA) (e.g., GDA94) map base defined earthquake sequence used in conjunction with before an earthquake sequence used in an autonomous vehicle or precision agriculture conjunction with an autonomous vehicle or positioning using GNSS PPP at (XB) in a precision agriculture positioning using GNSS kinematic RF (RFB) (e.g., ITRF2014). The NRTK at (XB) in a kinematic RF (RFB) (e.g., absence of a deformation model will result in a ATRF2014) that includes models of earthquake positioning error due to the unmodelled displacements. earthquake displacement. Static RFA to Static RFA to Semi-kinematic RFB Static RFB

Historic survey control coordinates (XA) Classical datum transformation between defined in a static RF (RFA) (e.g., NZGD49) to two static datums (e.g., between XA GDA94 be transformed to a semi-kinematic RF and XB in GDA2020). (RFB) (XB) for a survey control database (SCDB) update. Figure 6.4 Typical static RF transformation scenarios. Plots show point motion of a ground fixed point X in both source (no coordinate changes regardless of point displacement) and target RF. t0 is the interframe transformation reference epoch. tA is the epoch of the point in RFA and tB is the epoch of the point in RFB.

The current best practice approach is to define a trajectory model for a specific geodetic monument, typically a CORS forming part of the TRF realisation (Figure 6.5). Bevis and Brown (2014) provide an excellent description of a site trajectory model. ITRF2014 is currently defined by the coordinates of monuments at a reference epoch (2010.0 in the case of ITRF2014), site velocities, coseismic or other offsets at specific epochs and from ITRF2014 onwards, a postseismic model (Altamimi et al., 2016) for affected stations. The trajectory components are estimated a posteriori from time-series analysis. This approach however, is not suited to passive geodetic monuments or spatial data such as raster data where a time-series of position within the RF is sparse or non-existent and where secular and episodic displacements are required to be estimated from models. Non-linear trajectory models are adapted in this schema using grid models which can be interpolated to estimate motions for sites that do not have a direct observation history. The process is described in this thesis as an intraframe propagation. Displacements and associated uncertainties within the RF are propagated from a specific epoch in order to estimate coordinates and uncertainties at a different epoch.

Figure 6.5 Simplified point motion trajectory and intraframe propagation within a complex deformation zone. X is a fixed point (e.g., a geodetic monument or CORS). X(t0) is the point at the reference epoch t0. X(t1) is the same point at a later epoch t1. Slow slip events (SSE) and other transient velocity changes in excess of interseismic velocities can be accommodated in the postseismic model.

100

6.2.1 Schema overview

The general schema has three main components:

1. An intraframe propagation (if required) in the source RF (RFA) comprising secular interseismic displacement, coseismic displacement, postseismic displacement and

a supplementary displacement or correction defined in RFA

2. A transformation between RFA and RFB at the interframe transformation epoch

3. An intraframe propagation (if required) in the target RF (RFB) comprising secular interseismic displacement, coseismic displacement, postseismic displacement and

a supplementary displacement or correction defined in RFB

This general schema covers a potentially complex scenario for transformation between two RFs where both the source and target RF have different deformation models associated with them that may be uncorrelated. In this instance, the interframe transformation would be defined by either a non-time- dependent conformal or a grid transformation which is only valid at t0. This situation does not currently exist, but is likely to become a feature of RFs in the future as time-dependent RFs and their associated deformation models evolve or become superseded.

Currently, the more common scenario is where either the source or target RF (or both) are global kinematic RFs such as ITRF. ITRF is an Earth-fixed RF and therefore any surface deformation is not explicitly modelled (except for specific core RF stations that have interseismic velocities and seismic trajectories and derived PMM). Transformations that involve such RF are typically defined by a time- dependent conformal transformation such as the 14 parameter transformations published between different realisations of ITRF (IERS, 2019b). Unmodelled surface deformation is expected to be fully correlated between global kinematic frames that are Earth-fixed or plate-fixed. In this case, the interframe transformation is performed at the epoch of the RF (source or target) where there is no associated deformation model (e.g., ITRF).

A semi-kinematic RF in principle models out secular interseismic displacement but accommodates seismic displacements as permanent updates to the RF applied retrospectively at the original reference epoch. A static RF does not include any model of displacement.

6.2.2 Intraframe propagation

Intraframe propagation is a process whereby coordinates are estimated at different epochs within a single RF that may be subject to both secular and episodic displacements. The propagation process can be used for intraframe transformations, for example, to estimate coordinates in a deforming frame at a reference epoch for transformation to another RF. A non-linear propagation or transformation model is composed of secular interseismic motion, episodic and supplementary displacement components.

101

6.2.3 Interseismic displacement

The secular component of the propagation is the interseismic displacement (D) between two epochs (t1 and t2) described in decimal year units, estimated from the interseismic velocity model which is conventionally in a topocentric format using Eq. 6.1. By convention t2 > t1 with the velocities being described in a forward direction. The interseismic site velocity ( D! ) is estimated by bilinear interpolation (using Eqs. 5.9 to 5.11) from an interseismic velocity model.

DD=()tt-! (6.1) ()tt12® 21

The interpolation estimates the topocentric interseismic velocity components East, North and Up, ( E! , N! and U! ) of ( D! ) using Eq. 6.2 as follows: DED()= ( t- t ) E! ()tt12® 21 DND()= ( t- t ) N! ()tt12® 21 DUD()= ( t- tU )! (6.2) ()tt12® 21

In a stable interseismic tectonic setting the interseismic displacement will account for nearly all of the displacement in the absence of any seismic deformation. For example, D can be used to translate an instantaneous position in ITRF to a reference epoch (t0) used to define a LRF, or to another epoch of ITRF. New Zealand has used this approach since 2000 with the adoption of NZGD2000, which incorporates an interseismic velocity model (horizontal topocentric components only) defined in ITRF96, referred to as the NZGD2000 deformation model (Blick et al., 2006).

6.2.4 Coseismic displacement

The non-secular component of the intraframe propagation handles episodic deformation caused by earthquake activity or other deformation events. These events are by definition not predictable and so need to be modelled and implemented after each event if the deformation exceeds positioning and dimensional tolerances in an affected area. Coseismic deformation can be represented as a grid of displacements defined for each deformation event (earthquake). In New Zealand these displacement grids are known as reverse patches to the defined RF which is currently NZGD2000. The coseismic displacement model may quantify a permanent change to the RF definition excluding any interseismic deformation since the reference epoch of the RF. The patch can also be used as a soak to incorporate minor reference frame translations, unmodelled deformations and biases arising from imprecision in interseismic velocity models and network adjustments following the RF reference epoch. In the case of numerous deformation events occurring after the reference epoch, the displacements derived from each deformation patch for each event are aggregated to estimate the overall displacements caused by all

102 earthquakes since the reference epoch. For SSE, the summation can be used in the schema as a single component for all SSE between the reference epoch and the epoch of the latest summation. Coseismic displacement models can be formatted and operated in much the same way as grid distortion models.

The coseismic displacement component of an intraframe propagation is a summation of the coseismic displacements SC between any two epochs. The coseismic displacement for an event is estimated by bilinear interpolation of the highest resolution coseismic displacement grid file or patch for each particular event. The summation of coseismic displacements between t1 and t2 is computed using Eq. 6.3 where the displacements are estimated from the interpolation of each coseismic displacement model that applies to the point between the two epochs.

n CCCC=+++... (6.3) å ()tt12® 1 2 n tt12³C>

The coseismic displacement file (patch) is conventionally described as a grid of topocentric displacements [ DE DN DU ] for a given event epoch. By convention the displacements are time forward, with the displacement added for post-event estimation. The summation of coseismic displacements between any two epochs for each topocentric component of the displacement is estimated using Eq. 6.4.

n DEC() =DEC()=D EC ()+D EC ()++! D EC () ()tt12® å 1 2 n tCt12³>

n DNC() =DNC()=D NC ()+D NC ()++! D NC () ()tt12® å 1 2 n tCt12³>

n DUC() =DUC()=D UC ()+D UC ()++! D UC () (6.4) ()tt12® å 1 2 n tCt12³>

6.2.5 Postseismic displacement

Postseismic deformation is the other non-linear displacement that occurs subsequent to coseismic deformation (which is modelled as an instantaneous displacement). Postseismic deformation can also be significant and there is currently no standardised approach in applied geodesy to handle this form of deformation beyond individual site motion trajectory models. Application of gridded postseismic deformation models will be outlined here to enable this component of non-secular deformation to be incorporated into an intraframe propagation and transformation schema. A common approach in earthquake affected areas undergoing significant postseismic deformation is to simply update the RF whenever a tolerance threshold is reached by monitoring of network site motion time series or to conflate the postseismic deformation with the source event coseismic patch (LINZ, 2019). An exponential decay model has been implemented in Alaska after the Mw 7.9 earthquake of 3rd November 2002 (Pearson et al.,

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2013b). Slow-slip events (SSE) can also be handled as a form of postseismic deformation using a temporal velocity change parameter or slope function. After large earthquakes and subsequent aftershocks or slow- slip events, some latency in the formulation and release of coseismic and postseismic models can be beneficial in order to account for any aftershock deformation and other transient motion which may be non-linear in character.

Postseismic decay functions incorporate a combination of logarithmic and exponential functions to model the decay using input amplitude and decay time. The logarithmic function is dominant during the afterslip (elastic) deformation phase and the exponential function in the later viscoelastic relaxation phase (e.g., Lercier, 2014). Modelled postseismic displacement is displacement in excess of any interseismic and coseismic displacement. In other words, the trending interseismic velocity is removed from the observed postseismic displacement for modelling use. This approach enables continuous estimation of interseismic displacement within the postseismic decay period to enable transformation between frames (e.g., from ITRF to a plate-fixed RF). The amplitude and decay time terms are interpolated from the postseismic grid model of parameters which are defined for each topocentric component E N and U; epoch of estimation post-earthquake (t1), epoch of the earthquake (tq), amplitude of the logarithmic component (AL), amplitude of the exponential component (AE), decay time of the logarithmic component (tL), decay time of the exponential component (tE) and any transient velocity correction (e.g., for SSE), d E! , d N! and dU! . SSE can be modelled as a postseismic deformation simply by applying only the transient velocity correction and setting the amplitude terms to zero in the postseismic grid model. Postseismic deformations in a topocentric frame [P(E), P(N) and P(U)] for each event are then estimated using Eq. 6.5. A termination epoch (tmax) for each time component of the model can be applied to limit the temporal extent of each postseismic model component.

tt- æö- 1 q æött1 - q t E ()E ! PE()=++ ALE ()ln1 E ç÷A ()1 Eç÷- e+d Et()1 - t q t ()E ç÷ èøL èø

tt- æö- 1 q æött1 - q t E ()N ! PN()=++ ALE ()ln1 N ç÷A ()1 Nç÷- e+d Nt()1 - t q t ()N ç÷ èøL èø

tt- æö- 1 q æött1 - q t E ()U ! PU()=++ ALE ()ln1 U ç÷A ()1 Uç÷- e+d U() t1 - t q (6.5) t ()U ç÷ èøL èø

The postseismic displacement components of an intraframe propagation are a summation of the postseismic displacements SP between any two epochs. The postseismic displacement for each topocentric component is estimated using Eq. 6.5 with parameters estimated by bilinear interpolation of the postseismic parameter grid file for each event.

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n P is the summation of postseismic displacements between t1 and t2 using Eq. 6.6 where å ()tt12® tt12³P> the displacements are estimated from interpolation of each component of the postseismic displacement parameter model that applies to the point between the two epochs and applying the postseismic function described previously using Eq. 6.5.

n PPPP=+++... (6.6) å ()tt12® 12 n tt12³P>

As the postseismic displacements computed using Eq. 6.5 are conventionally described in a topocentric system for a given epoch Eq. 6.6 can be expressed using Eq. 6.7 for each term.

n DEP() =DEP()=D EP ()+D EP ()++ ...D EP () ()tt12® å 1 2 n tPt12³>

n DNP() =DNP()=D NP ()+D NP ()++ ...D NP () ()tt12® å 1 2 n tPt12³>

n DUP() =DUP()=D UP ()+D UP ()++ ...D UP () (6.7) ()tt12® å 1 2 n tPt12³>

6.2.6 Supplementary displacement

The deformation model components are assumed to be deeply-seated crustal displacements (e.g., underlying bedrock), but, as described in Chapter 2 and Chapter 4 there are numerous instances of highly localised surface deformation caused by regolith movement, surface creep, surface ruptures and subsidence. These can be modelled as localised displacements differential from displacement of the underlying crust. The supplementary displacement can also be used to define or include minor reference frame adjustments, interseismic model corrections and other corrections.

The supplementary displacement component of an intraframe propagation is a summation of the other

n displacements SS between any two epochs. S is the summation of supplementary corrections å ()tt12® tt12³C> between t1 and t2 using Eq. 6.8.

n SSSS=+++... (6.8) å ()tt12® 12 n tt12³S>

The supplementary displacement file is conventionally described as a grid of topocentric displacements

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[ DE DN DU ] for a given epoch in a time forward direction. The summation of supplementary displacements between any two epochs for each topocentric component of the displacement is estimated using Eq. 6.9.

n DES() =DES()=D ES ()+D ES ()++ ...D ES () ()tt12® å 1 2 n tSt12³>

n DNS() =DNS()=D NS ()+D NS ()++ ...D NS () ()tt12® å 1 2 n tSt12³>

n DUS() =DUS()=D US ()+D US ()++ ...D US () (6.9) ()tt12® å 1 2 n tSt12³>

6.3 Generalised schema

The most complex time-dependent transformation scenario will be used as a basis to formulate the schema described in this thesis. Such a scenario can be described by transformation of a point in a source

RF (RFA) that includes models of numerous seismic deformation events to a target RF (RFB) that may use different models applied as shown in Figure 6.6. Transformations for less complex scenarios can be achieved by eliminating zero value terms, or scaling the parameters by zero not relevant for a more simplified transformation. If the source or target RF are implicitly kinematic (e.g., ITRF) and do not embed any velocity or deformation model then the more simplified approach described previously can be used using a conformal transformation at the transformation epoch. This is explained in more detail later.

6.3.1 Schema case testing and logic flow

The general schema logic flow is summarised as follows:

Step 1. Identify the interframe transformation epoch tAB from one of the following cases which are also summarised in Table 6.1:

Case 1: RFA and RFB both have deformation models for intraframe propagation

tAB = t0

Case 2: RFA and RFB both do not have deformation models for intraframe propagation (e.g., ITRF)

tAB = tA

Case 3: RFA has a deformation model but RFB does not

tAB = tB

Case 4: RFA does not have a deformation model but RFB does

tAB = tA

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Target RF ®

kinematic

Source RF ¯ tAB = Kinematic (no deformation model) Kinematic (with deformation model) Semi static

Kinematic (no deformation model) tA tA tA t0 Kinematic (with deformation model) tB t0 t0 t0 Semi-kinematic tB t0 t0 t0 static t0 t0 t0 t0 Table 6.1. Determining the epoch of the schema interframe transformation epoch tAB

Step 2. If required, estimate the intraframe propagation within the source RF (RFA) between tA

and tAB comprising: secular displacement DA, coseismic deformation SCA, postseismic

deformation SPA and if required a supplementary displacement SSA.

Step 3. Transform the propagated position in RFA to RFB using the interframe transformation

at epoch tAB.

Step 4. If required, estimate the intraframe propagation within the target RF (RFB) between tAB

and tB comprising: secular displacement DB, coseismic deformation SCB, postseismic

deformation SPB and if required a supplementary displacement SSB.

Equations are shown in vector notation where each vector can represent either geodetic Cartesian or topocentric coordinates, velocities or displacements. Where velocities or displacements are modelled or described in topocentric terms East (E) and North (N) and Up (U) they may need to be transformed into a geodetic Cartesian form [X Y Z] using Eqs. 5.7 and 5.8 for use in the schema equation. The propagation equations (6.10 to 6.12) are used in the schema where t1 and t2 are represented as tA, tB or tAB depending upon the frame being propagated.

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Figure 6.6 General schema for a complex time-dependent transformation between two kinematic RF (RFA and RFB) that have associated deformation models. Fixed point X (e.g., a geodetic monument) is defined in terms of both RFA (XA) and RFB (XB) at different epochs tA and tB respectively. t0 is the reference epoch for the conformal component of the interframe transformation between RFA and RFB.

The general schema formulation uses Eqs. 6.10 to 6.12: XXXX=+fmm[()]()D+D (6.10) BABAt (AB) A A B B where,

XB is the point estimated in the target RF (RFB) at epoch tB,

f is the conformal transformation operation on the point in RFA after propagation ABt (AB)

to the interframe transformation epoch using tAB[XA+mA(DXA)].

XA is the point in the source RF (RFA) at epoch tA,

mA is the sign of the intraframe propagation within RFA.

mA =1 for a forward propagation (tA £ tAB)

mA = -1 for a reverse propagation (tA > tAB).

DXA is the intraframe propagation within RFA using Eq. 6.11

(always defined in a time forward sense from the model components),

mB is the sign of the intraframe propagation within RFB.

mB =1 for a forward propagation (tB ³ tAB)

mB = -1 for a reverse propagation (tB < tAB).

DXB is the intraframe propagation within RFB using Eq. 6.12 (always defined in a time forward sense from the model components).

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nnn DXD= + C + P + S (6.11) A(ttAAB® )ååå A A A ttAAB®®® tt AAB tt AAB

nnn DXD= + C + P + S (6.12) B(ttBAB® )ååå B B B ttBAB®®® tt BAB tt BAB

where,

DA is the secular displacement between tA and tAB estimated from the interseismic velocity ! DA interpolated from the interseismic velocity model using Eqs. 5.9 to 5.11.

n is the summation of forward coseismic displacements between t and t using Eq. 6.3 å CA A AB ttAAB® where the displacements are estimated from the interpolation of each coseismic displacement model that applies to the point between the two epochs.

n is the summation of forward postseismic displacements between t and t using Eq. 6.6 å PA A AB ttAAB® where the displacements are estimated from interpolation of each component of the postseismic displacement parameter model that applies to the point between the two epochs and applying the postseismic function using Eq. 6.5.

n is the summation of forward supplementary displacements between t and t using Eq. å SA A AB ttAAB® 6.8

Similarly, for the target frame RFB, the displacements terms are estimated in the same way where:

DB is the forward secular displacement between tB and tAB estimated from the interseismic ! velocity DB interpolated from the interseismic velocity model using Eq. 6.1

n is the summation of forward coseismic displacements between t and t using Eq. 6.3, å CB B AB ttBAB®

n is the summation of forward postseismic displacements between t and t using Eq. 6.6 å PB B AB ttBAB® where the displacements are estimated from interpolation of each component of the postseismic displacement parameter model that applies to the point between the two epochs and applying the postseismic function using Eq. 6.5.

n is the summation of forward supplementary displacements between t and t using Eq. å SB B AB ttBAB® 6.8.

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The coordinate and model format need to be consistent for all operations. If the deformation model components are described in a topocentric format then the propagation component will need to be converted to the input coordinate format required by the conformal or grid transformation operation.

6.3.2 Special cases

Where a displacement model coincides with an epoch used in the transformation (t0, tA or tB), the model is applied instantaneously at the epoch in order to avoid multiple coordinate values for a single epoch.

6.3.3 Interframe transformation

The interframe transformation operation fm[()]XX+D is defined between RFA and RFB at the ABt (AB) A A A interframe transformation epoch tAB. The transformation is applied to the position in the source RFA after it has been propagated to epoch tAB. The interframe transformation may be applied either by a conformal transformation (static or time-dependent) or by interpolation of a transformation grid (e.g., NTv2). The epoch of tAB is determined using the case testing criteria described in Section 6.3.1 also shown in Table 6.1.

6.3.4 Estimation and correlation of uncertainty

The approach used to estimate uncertainty in the schema is simplistic, using a standard root mean square error propagation. A limitation of this approach is that correlations between estimates of the displacement components are not estimated or described. Correlation from the conformal transformation can be estimated using known uncertainties of the transformation parameters and input coordinates. The combination of interpolated displacements from grids of temporally and spatially uncorrelated deformation events make realistic estimates of correlation difficult. This aspect warrants further research.

The uncertainties for each topocentric component of the transformation are expressed simplistically using Eq. 6.13:

2 22 2 2222 sssEEECEPEDETEDECEPBAA=+D( ) + sD A( ) + sD A( ) + sD ABBBB( ) + sD( ) + sD( ) + sD( )

2 22 2 2222 sssNNNCNPNDNTNDNCNPBAA=+D( ) + sD A( ) + sD A( ) + sD ABBBB( ) + sD( ) + sD( ) + sD( ) sssUU=+2 D UCUPUDUTUDUCUP22+ sD + sD 2+ sD 2222+ sD + sD + sD BA A( ) A( ) A( ) ABBBB( ) ( ) ( ) ( ) (6.13) where, (C) is the topocentric component of the summation of coseismic displacement, (P) is the topocentric component of the summation of postseismic displacement, (D) is the topocentric component of the uncertainty of secular displacement estimated from the velocity model, and,

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(T) is the topocentric component uncertainty from the interframe transformation.

Only the components used in the propagation and interframe transformation are used, with zeros for unused components. In some instances the supplementary corrections can be used to improve the precision of propagation by rectifying known biases in the model at the projected reference epoch. In such cases, the velocity model error component can be set to zero, or a negative correlation function can be applied. Also, redefinition or readjustment of a RF at a more current epoch can warrant a reset of the epoch from which uncertainty is propagated.

6.4 Schema examples for different RF definition transformations

For each of the transformation scenarios described and illustrated in Figures 6.1 to 6.4, a more detailed application of the generalised schema is expressed in more detail in this section. Supplementary displacements S are not shown (or are zero) in the illustrated examples for the purposes of clarity, but in practice may be included in real world applications.

6.4.1 Transformation between kinematic RF

Transformations between kinematic RFs are commonly used between different realisations of ITRF, other GNSS reference frames and kinematic plate-fixed frames such as the ETRF. The standard approach is a 14 parameter transformation with no explicit deformation model used in the frame definition. In the future, deformation models are likely to be used to model intraframe displacements within these frames, especially in deforming zones which are also subject to seismic activity. These models can be used to propagate coordinates within a kinematic frame to enable interframe transformations between different epochs. Interframe transformations between kinematic RFs without a deformation model to estimate displacements between epochs are only possible at the epoch of transformation (Figure 6.7). The schema approaches differ depending upon whether either or both kinematic RFs have defined deformation models using the case testing outlined previously. In the case where the source RF is a kinematic RF with no deformation model (e.g., ITRF2014) and the target RF is a kinematic one defined by a deformation model (Figure 6.8), the deformation in both frames would be correlated at any common epoch. In other words, the magnitudes of the modelled displacements and velocities in the target RF would implicitly be the same in the source RF for a common point on the ground. These different cases are described in more detail below.

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Figure 6.7 Transformation between two kinematic RF at different epochs where no deformation model is defined for either RF. In this case the transformation can only be performed at the epoch of the source RF tA = tAB (e.g., by a 14 parameter transformation).

In the case shown in Figure 6.7, there is no defined deformation model to estimate intraframe propagation, so the time-dependent interframe transformation can only be estimated at the epoch of the source RF position where tA = tAB.

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Figure 6.8 Transformation between a kinematic RF with no deformation model to a kinematic RF defined by a deformation model at a different epoch

In the case illustrated in Figure 6.8, a deformation model is defined for the target kinematic RF, so a transformation is possible between different epochs. The interframe transformation epoch tAB = tA since

RFA has no defined deformation model. In the example shown, tB > tAB so the propagation sign mB = 1 (forward propagation in the target RF).

Since tA = tAB DXA =0 (from 6.11)

nnn DXD= + C + P + S (from 6.12) B(ttBAB® )ååå B B B ttBAB®®® tt BAB tt BAB so the simplified schema is expressed using Eq. 6.14: XXX=+fm[] (D ) (6.14) BABAt (AB) B B

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Figure 6.9 Transformation between a kinematic RF defined by a deformation model to a kinematic RF not defined by a deformation model at a different epoch

In the case illustrated in Figure 6.9 the interframe transformation epoch tAB = tB since there is no deformation model defined in the target RF. In the example shown, tA < tAB so the propagation sign mA = 1 (forward propagation).

nnn DXD= + C + P + S (from 6.11) A(ttAAB® )ååå A A A ttAAB®®® tt AAB tt AAB

Since tB = tAB DXB =0 (from 6.12) so the simplified schema is expressed using Eq. 6.15 is: XXX=+fm[()]D (6.15) BABAt (AB) A A

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Figure 6.10 Transformation between two kinematic RF at different epochs where both RF are defined by deformation models

In the case illustrated in Figure 6.10 the interframe transformation epoch tAB = t0 since both source and target RF have defined deformation models which may not be correlated. In the example shown, tA > tAB so the propagation sign mA = -1 (reverse propagation) and tB > tAB so the propagation sign mB = 1 (forward propagation).

nnn DXD= + C + P + S (from 6.11) A(ttAAB® )ååå A A A ttAAB®®® tt AAB tt AAB

nnn DXD= + C + P + S (from 6.12) B(ttBAB® )ååå B B B ttBAB®®® tt BAB tt BAB so the full schema is expressed using Eq. 6.10 repeated below: XXXX=+fmm[()]()D+D (from 6.10) BABAt (AB) A A B B

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6.4.2 Kinematic RF to semi-kinematic RF

As it enables kinematic RF coordinates from real-time GNSS PPP applications to be aligned with a semi- kinematic RF, this is likely to be the most widely used transformation until full kinematic capability is possible in spatial data management.

Figure 6.11 Transformation from a kinematic RF with no defined deformation model to a semi-kinematic RF. The dashed green line shows the equivalent motion of the kinematic RF used to realise the semi-kinematic RF with the interseismic displacement set to zero in the RF.

In the case illustrated in Figure 6.11 the interframe transformation epoch tAB = tA since there is no defined deformation model for the source kinematic RF. The semi-kinematic frame is realised by a parent kinematic frame so the ground motion between the two kinematic frames is correlated (as shown) allowing time-dependent conformal transformations between the two RFs. A key difference with the schema applied for a semi-kinematic RF is that the interseismic displacement is modelled between t0 and tAB, not tB and tAB. Another important point to note is that if the semi-kinematic RF is updated at the reference epoch to account for later seismic displacement, then the seismic displacement terms are not applied for reverse propagations. Only the interseismic displacement will be propagated.

In the example shown, tB < tAB so the propagation sign mB = -1 (reverse propagation).

Since tA = tAB DXA =0 (from 6.11)

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If the point coordinates in the semi-kinematic RF have not been updated to account for the intervening seismic displacement (as illustrated) then Eq. 6.16 is used:

nnn DXD= + C + P + S (6.16) B()tt0AB® ååå B B B ttBAB®®® tt BAB tt BAB If the coordinates have been updated retrospectively to account for all seismic displacements then Eq. 6.17 is used:

n DXD= + S (6.17) B()tt0AB® å B ttBAB® so the simplified schema is expressed using Eq. 6.14: XXX=+fm[] (D ) (from 6.14) BABAt (AB) B B

Figure 6.12 Transformation from a kinematic RF defined by a deformation model to a semi-kinematic RF

In the case illustrated in Figure 6.12 the interframe transformation epoch tAB = t0 since there is potentially no correlation between the two frames at different epochs from the interframe transformation reference epoch. A key difference with the schema applied for a semi-kinematic RF is that the interseismic displacement is not propagated within a semi-kinematic RF between t0 and tB. Another important point to note is that if the semi-kinematic RF is updated at the reference epoch to account for all later seismic displacement up to the epoch in RFA, then all seismic displacement terms are applied for forward propagations.

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In the example shown, tA > tAB so the propagation sign mA = -1 (reverse propagation).

nnn DXD= + C + P + S (from 6.11) A(ttAAB® )ååå A A A ttAAB®®® tt AAB tt AAB

In the example shown, tB > tAB so the propagation sign mB = 1 (forward propagation). If the point coordinates in the semi-kinematic RF have not been updated to account for later seismic displacement (as illustrated) then Eq. 6.18 is used:

nnn (6.18) DXCPSBBBB=ååå + + ttBAB®®® tt BAB tt BAB

If the coordinates have been updated retrospectively to account for all seismic displacements then Eq. 6.19 is used:

nnn (6.19) DXCPSBBBB=ååå + + ttMAX®®® AB ttMAX AB ttMAX AB

where tMAX is the greater of tA or tB. so the full schema is expressed using Eq. 6.10 repeated here: XXXX=+fmm[()]()D+D BABAt (AB) A A B B

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6.4.3 Kinematic RF to a static RF

This approach is widely used in stable tectonic settings where a static RF is used in conjunction with GNSS positioning. The transformation enables GNSS point positions in ITRF to be transformed to a static RF fixed to the stable plate using a 14 parameter transformation or PMM. A limitation of this approach is that intraframe deformation represented as differential point motion (e.g., near plate boundaries and areas of subsidence) and seismic displacements are not applied and result in transformation errors (Figure 6.13).

Figure 6.13 Transformation from a kinematic RF with no defined deformation model to a static RF. No intraplate or seismic displacements are modelled which can result in an error in the transformed coordinate in the static RF.

In the case illustrated in Figure 6.13 the interframe transformation is only valid at the interframe transformation epoch tAB = t0 unless a time dependent transformation that tracks the plate movement is used and if there is no intraplate deformation or seismic displacement after t0.

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Figure 6.14 The same transformation from a kinematic RF to a static RF using a defined deformation model. Interseismic and seismic displacements are modelled, improving the accuracy of the transformation in affected areas.

The kinematic to static RF transformation can be improved by using a deformation model for the kinematic RF (Figure 6.14). The deformation model could be either Earth-fixed (absolute) or referenced to a plate-fixed RF thus minimising the magnitude of the interseismic velocities and displacements. In the case illustrated, the interframe transformation epoch tAB = t0 to enable intraframe propagation modelling of all displacements to the interframe transformation reference epoch between the kinematic RF and static RF. As the target RF is static, there is, by definition, no displacement modelled from the reference epoch. There may be adjustments made to the static RF after the reference epoch and these can be modelled using the supplementary displacement term.

In the example shown, tA > tAB so the propagation sign mA = -1 (reverse propagation).

nnn DXD= + C + P + S (from 6.11) A(ttAAB® )ååå A A A ttAAB®®® tt AAB tt AAB

Since tB = tAB DXB =0 (from 6.12) so the simplified schema is expressed using Eq. 6.15: XXX=+fm[()]D (from 6.15) BABAt (AB) A A

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6.4.4 Semi-kinematic RF to kinematic RF

This transformation is typically used where engineering design or cadastral data defined in a semi- kinematic RF are to be set out using GNSS PPP techniques.

Figure 6.15 Transformation from a semi-kinematic RF to a kinematic RF with no defined deformation model. The dashed blue line shows the equivalent motion of the kinematic RF used to realise the semi-kinematic RF with the interseismic displacement set to zero.

In the case illustrated in Figure 6.15 the interframe transformation epoch tAB = tB since there is no defined deformation model for the target kinematic RF. The semi-kinematic frame is realised by a parent kinematic frame so the ground motion between the two kinematic frames is correlated allowing time- dependent conformal transformations between the them. As with the earlier case, a key difference with the schema applied for a semi-kinematic RF is that the interseismic displacement is modelled between t0 and tAB, not tA and tAB. It is important to state again that if the semi-kinematic RF is updated at the reference epoch to account for later seismic displacement, then the seismic displacement terms are not applied for forward propagations. Only the interseismic displacement will be propagated.

In the example shown, tA < tAB so the propagation sign mA = 1 (forward propagation).

The source RF is semi-kinematic so the interseismic displacement must be modelled from the reference epoch. If the point coordinates in the semi-kinematic RF have not been updated to account for the intervening seismic displacement (as illustrated) then Eq. 6.20 is used:

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nnn DXD= + C + P + S (6.20) A()tt0AB® ååå A A A ttAAB®®® tt AAB tt AAB

If the coordinates have been updated retrospectively to account for all seismic displacements then Eq. 6.21 is used:

n DXD= + S (6.21) A()tt0AB® å A ttAAB®

Since tB = tAB DXB =0 (from 6.12) so the simplified schema is expressed using Eq. 6.15: XXX=+fm[()]D (from 6.15) BABAt (AB) A A

Figure 6.16 Transformation from a semi-kinematic RF to a kinematic RF with a defined deformation model

In the case illustrated in Figure 6.16 the interframe transformation epoch tAB = t0 since deformation models that may not be correlated are defined for both frames. If the semi-kinematic RF is updated at the reference epoch to account for the seismic displacement, then all seismic displacement terms are applied for the propagation in the source RF.

In the example shown, tA > tAB so the propagation sign mA = -1 (reverse propagation).

If the point coordinates in the semi-kinematic RF have not been updated to account for the intervening seismic displacement between tA and tB (as illustrated) then Eq. 6.22 is used:

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nnn (6.22) DXCPSAAAA=ååå + + ttAAB®®® tt AAB tt AAB

If the coordinates have been updated retrospectively to account for all seismic displacements then Eq. 6.23 is used:

nnn (6.23) DXCPSAAAA=ååå + + ttMAX®®® AB ttMAX AB ttMAX AB where tMAX is the greater of tA or tB.

nnn DXD= + C + P + S (from 6.12) B(ttBAB® )ååå B B B ttBAB®®® tt BAB tt BAB so the full schema is expressed using Eq. 6.10 repeated below: XXXX=+fmm[()]()D+D BABAt (AB) A A B B

6.4.5 Semi-kinematic RF to Semi-kinematic RF across an earthquake event

There may be instances in the future where two different semi-kinematic RF are used in conjunction with each other, for example if a semi-kinematic RF is redefined in terms of another parent kinematic RF.

Figure 6.17 Transformation between two semi-kinematic RF across a deformation event

The transformation between two semi-kinematic RF is nominally straightforward, however, can be complicated if stale semi-kinematic data not corrected for later coseismic displacements are transformed. Displacements in the two semi-kinematic RFs are not necessarily correlated which means that interframe

123 transformations are only possible at the interframe transformation reference epoch. In the example shown in Figure 6.17 a current position in the source RF is transformed to pre-earthquake position in the target RF. The interframe transformation epoch tAB = t0. In the example shown, the position is described in the latest update of the source RF. As tA > tAB the propagation sign of mA = -1 (reverse propagation in RFA) and tB > tAB so the propagation sign mB = 1 (forward propagation in RFB).

nnn (from 6.22) DXCPSAAAA=ååå + + ttAAB®®® tt AAB tt AAB

nnn (from 6.18) DXCPSBBBB=ååå + + ttBAB®®® tt BAB tt BAB If the position is required in the target semi-kinematic RF at the latest update accounting for all seismic deformation up to the later epoch tB, then:

nnn (from 6.19) DXCPSBBBB=ååå + + ttMAX®®® AB ttMAX AB ttMAX AB where tMAX is the greater of tA or tB, so the full schema is expressed using Eq. 6.10: XXXX=+fmm[()]()D+D (from 6.10) BABAt (AB) A A B B

6.4.6 Semi-kinematic RF to static RF

This transformation is typically used where a semi-kinematic RF supersedes a static RF and where newer surveys need to be related to earlier spatial data defined in the static RF.

Figure 6.18 Transformation from a semi-kinematic RF to a static RF

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In the case illustrated in Figure 6.18 the interframe transformation is only defined at epoch t0 = tAB. An important point to note is that the position in target static RF is the position not displaced by seismic events after the reference epoch. The use of the same interframe transformation at tA will mean that the position in the target frame will also be implicitly updated which may cause confusion as to how the static coordinates are interpreted, for example across a fault scarp.

In the example shown, tA > tAB so the propagation sign mA = -1 (reverse propagation).

nnn (from 6.11) DXCPSAAAA=ååå + + ttAAB®®® tt AAB tt AAB

Since tB = tAB DXB =0 (from 6.12) so the simplified schema is expressed using Eq. 6.15 is: XXX=+fm[()]D (from 6.15) BABAt (AB) A A

6.4.7 Static RF to kinematic RF

This is a commonly used transformation applied to align pre-earthquake data defined in a static RF with current GNSS positioning within a kinematic RF such as ITRF. As with the kinematic to static RF transformation, careful choice of the time-dependent transformation is required. For a static RF fixed to stable tectonic plate, a 14 parameter transformation that tracks the rotation of the plate and embeds a PMM is commonly used. As described in Chapter 3, the limitation of this approach is that intraplate deformation is not modelled and can result in an error in the transformed position (Figure 6.19). The limitation is largely overcome by using an absolute or plate-fixed deformation model (Figure 6.20).

Another issue to be considered is the effect of unmodelled deformation on stale (pre earthquake) coordinates defined in a static frame. Coordinates in the static frame may be updated to account for localised deformation events or changed through later geodetic adjustments. These updates can be applied in the supplementary displacement parameter S, if required.

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Figure 6.19 Static RF to Kinematic RF transformation with no associated deformation model

In the case illustrated in Figure 6.19 the interframe transformation epoch tAB = t0. There is no propagation in the static frame so:

DXA =0 In the example shown, it is not possible to propagate within the target kinematic frame as there is no associated deformation model. If the location is on a stable tectonic plate with no seismic deformation and a suitable time-dependent transformation (e.g., a 14 parameter transformation based on a plate motion model) is used then the transformation will be accurate.

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Figure 6.20 Static RF to Kinematic RF transformation using a deformation model

In the case illustrated in Figure 6.20 the interframe transformation epoch tAB = t0. There is no propagation in the static frame so:

DXA =0

In the example shown, tB > tAB so the propagation sign mB = 1 (forward propagation).

nnn DXD= + C + P + S (from 6.12) B(ttBAB® )ååå B B B ttBAB®®® tt BAB tt BAB so the simplified schema is expressed using Eq. 6.14 is used: XXX=+fm[] (D ) (from 6.14) BABAt (AB) B B

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6.4.8 Static RF to semi-kinematic RF

This transformation is used to relate pre-earthquake data defined in a static RF (e.g., legacy geodetic or cadastral control) to positioning in a semi-kinematic RF.

Figure 6.21 Transformation from a static RF to a semi-kinematic RF

In the case illustrated in Figure 6.21 the interframe transformation epoch tAB = t0. There is no propagation within a static RF so:

DXA =0 (from 6.11)

In the example shown, tB > tAB so the propagation sign mB = 1 (forward propagation).

nnn (from 6.18) DXCPSBBBB=ååå + + ttBAB®®® tt BAB tt BAB

If the epoch in the target RF is earlier than a seismic displacement event that updates the semi-kinematic RF then some care is required. If the transformation is required to the current version of the semi- kinematic frame and not the stale coordinate before the most recent deformation event then Eq. 6.19 should be used:

nnn (6.19) DXCPSBBBB=ååå + + ttMAX®®® AB ttMAX AB ttMAX AB where tMAX is the greater of tA or tB. So the simplified schema is expressed using Eq. 6.14 is: XXX=+fm[] (D ) (from 6.14) BABAt (AB) B B

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6.4.9 Static RF to Static RF

This is the simplest case transformation between two static RF using a conformal transformation or distortion model between the RF (Figure 6.22).

Figure 6.22 Conventional transformation between two static RF

As the transformation is invariant with time, no epoch is required and so the simplest form of the schema is used using Eq. 6.24 which is a straightforward conformal or grid transformation:

XX= f [] (6.24) BABAt (AB) In practice, the supplementary term (S) may be used for static transformations where localised updates or readjustments have occurred.

6.5 Management and evolution of time-dependent transformations

In plate boundary regions where earthquakes are a regular occurrence it is evident that epoch fixed RF have a limited lifespan due to increased strain arising from both interseismic and seismic deformation. Whenever the strain exceeds the appropriate threshold, described in Chapter 3, the RF should be redefined at a more current epoch and a transformation grid estimated to enable transformation between the old and new RF at a defined reference epoch. An increasingly large number of seismic events that require coseismic and postseismic displacement models over time also becomes difficult to manage.

It is prudent then to realise a current static or semi-kinematic RF at regular intervals with a new interframe displacement grid at the interframe transformation epoch. The displacement grid is a

129 combination of both interseismic, coseismic and postseismic displacements between the two reference epochs that define the RF and their interframe transformation.

Displacement within a RF can be monitored at each CORS station that realises the RF. This can be achieved by comparing the reference epoch coordinates computed using Eq. 6.17 or Eq. 6.21 with the reference epoch coordinates for the station. Whenever the difference between the two exceeds a specified tolerance an alert can be raised. This would usually happen after an earthquake, or commencement of a slow-slip event. In the absence of any of these episodic events, the interseismic velocity model would need to be verified and updated by analysis of the CORS time-series. Repeat observations over a dense network of passive geodetic monuments can also be used to verify and improve the precision of the deformation models in current use.

An Absolute Deformation Model (ADM) can only improve as more CORS come online and the network of passive geodetic monuments (stable ground marks not continuously occupied by a space geodesy sensor) with known site velocities expands, especially in tectonically active regions characterised by complex deformation. Campaign-style GNSS measurements over a dense network of stable passive geodetic monuments in a deforming zone allow for high resolution modelling of the deformation field. Together with longer time-series for CORS and passive stations, these observations result in improved definitions of intraplate deformation and fault models. Moreover, they can help identify microplates and crustal blocks where networks are sparse. Ongoing refinements of an ADM and associated patch models can mitigate the need for regular updates of a locally-used reference epoch. Monitoring of the performance of the model can identify when actual deformation differs from the modelled deformation outside specified tolerance limits. In these instances the ADM can be redefined if the site velocity is incorrect, or alternatively the patch model can be updated to accommodate unmodelled deformation.

6.6 Legal considerations with use of kinematic coordinates

By necessity, coordinates in a kinematic RF require an epoch to fix a moving position in time within the kinematic RF. Coordinates used for a legal definition of position often do not have datum epoch metadata as a static RF has traditionally been assumed. Nevertheless, a geodetic datum and projection have always been required as essential metadata in a legal definition in order for the position described to be unique and traceable. Many cadastres still have a legal coordinate definition quite low in the hierarchy of a legal definition of a boundary, with original survey boundary marks and occupations taking precedence over any coordinate system. Where cadastral or other coordinates have legal importance if used in the context of a semi-kinematic or kinematic RF, then a legal definition of a coordinate requires considerably more metadata to uniquely identify the position. This information includes: (1) epoch of the coordinate in the reference frame, (2) the site velocity and (3) the version of the reference frame taking into account seismic displacement updates or readjustments. The fundamental principle is that a legal coordinate can be directly traceable to a RF, even if the RF is kinematic in nature.

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6.7 Height transformation schema

The schema is designed to handle displacements in three dimensions so it is worth noting a few variations in its application with regards to heights. As shown and justified in more detail in previous chapters, vertical deformation should be handled differently from horizontal deformation in some spatial applications. For some engineering and hydrology applications, secular changes in elevation should not be ignored or regressed to a different epoch. In other words, the ellipsoid height and derived orthometric heights should be completely kinematic. For example, if a region is subsiding due to mining activity, or ground water changes, then fixing the elevations of survey control is not advisable for affected projects that include infrastructure that is outside the area of subsidence. Another example is flood modelling in areas of rapid uplift due to GIA. Again, these elevations need to be treated in a kinematic manner and not fixed to a certain reference epoch. As most geodetic transformations require ellipsoidal height data, some care is required with any transformed height that is used.

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Chapter 7 Case Studies

Three case studies from regions with differing tectonic settings are described to show how the LRF development and complex time-dependent transformation schema described in this thesis can be applied in practice. The first case study is the development of a plate fixed LRF for the Australian continent. The Australian continent is a tectonically stable land mass with geodetic strain rates typically less than 0.1E-9 yr-1. The second case study is from near Seddon in the South Island of New Zealand, a location which has been subject to a number of significant and well recorded earthquakes since 2013. The third case study is from the Gisborne region of the North Island of New Zealand, a location characterised by rapid and complex episodic slow-slip events near the boundary between the Pacific and Australian Plates.

7.1 Australian Case Study

The Australian continent forms the stable portion of the interior of the Australian Plate. With the exception of significant but isolated intraplate earthquakes e.g., Newcastle (Mw 5.6, 28th December 1989),

Tennant Creek (Mw 6.6, 26th January 1988) and Meckering (Mw 6.5, 14th October 1968) (Leonard et al., 2007) no significant intraplate deformation between stable geodetic monuments fixed to bedrock across the continent has been observed within the interseismic period. But since the year 2000, large regional earthquakes along the margins of the Australian Plate have resulted in ~4 millimetre coseismic and postseismic displacements across much of the Australian Plate (Tregoning et al., 2013). A large portion of the Australian continent is covered in regolith which overlies crustal bedrock at depths of up to 200 m (Wilford et al., 2016) in the lower sections of the Murray-Darling Basin. Many large aquifers also exist within the continental interior. Geodetic stations located on regolith and sedimentary basins where groundwater abstraction, coal seam gas extraction and underground mining is occurring are also subject to observable deformation relative to the stable underlying plate.

Chapter 2 shows how a stable plate motion model (PMM) can be estimated by inversion of interseismic site velocities estimated for bedrock fixed space geodetic sensors located within the stable portion of a tectonic plate. For most geodetic applications within stable plate regions, a PMM alone is sufficient for time-dependent transformations and propagation of coordinates between different epochs. In this case study a PMM is developed for the Australian continent. Following the approach described in Chapter 2, a residual velocity model incorporating regolith deformation can then be developed. The PMM can then be used to estimate a base interseismic velocity model grid of ITRF velocities that can be augmented with localised deformation displacements and used for geodetic applications.

7.1.1 Development of a stable Australian PMM

In this case study, a stable Australian Plate Reference Frame (SAPRF2014) has been developed using the approach described in Chapter 2. Geoscience Australia have published the latest IGb08 (GPS realisation of

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ITRF2008) (Rebischung et al., 2012) set-of-station coordinate (SSC) solution (Geoscience Australia, 2014a) and associated SINEX file for the Asia-Pacific Reference Frame (APREF) encompassing the extent of the Australian continent (Geoscience Australia, 2014b). GPS data for all continuous GPS (CORS) sites forming the APREF network were processed using the Bernese GPS software Version 5.0 (Dach et al., 2007) and the ITRF site velocities for all stations in the network were estimated using the CATREF software (Altamimi et al., 2004). Known coseismic and equipment change offsets were isolated from the velocity estimation and a power-law noise model applied to estimate more realistic station velocity uncertainties from the APREF GPS time-series. A selection of 46 AuScope and Australian Regional GNSS Network (ARGN) stations (Figure 7.1) was used for the inversion of the Euler Pole of the Australian Plate that fitted the following criteria:

• Station is located within the Australian continent and Tasmania • Antenna mounts are directly anchored to cratonic bedrock using reinforced concrete pillars with rooftop, tower and regolith sites excluded (e.g., ADE1, PERT, BUR2 and MOBS) • The ITRF site velocity (horizontal component) uncertainty is less than 0.45 mm yr-1 • Well-distributed selection of stations over the Australian continental landmass

The mean horizontal velocity uncertainty of the 46 stations is 0.4 mm yr-1 with a uniform standard deviation of 0.04 mm yr-1, hence no weighting strategy was applied to the inversion. Figure 7.2 shows the ITRF site velocities of the selected network.

Figure 7.1 ARGN and AuScope CORS selection used to estimate a stable Australia PMM

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Figure 7.2 ITRF site velocities for selected ARGN and AuScope stations

The ITRF2008 Euler Pole (rad yr-1) for the Australian Plate was estimated by inversion of the 46 site velocities using Eq. 2.2 yielding:

p éù7.2905E-9 Ω = êú5.7479E-9 AustPlate êú ëûêú5.8807E-9

The standard errors of the rotation rates using Eq. 2.8 are:

s = 3.652E-11 rad yr-1 s = 4.147E-11 rad yr-1 s = 4.512E-11 rad yr-1 wz wy wx

The equivalent Euler Pole rates are estimated using Eq. 2.3:

! -1 ! ! wp = 0.630 Ma fp = 32.35 lp = 45.17

The SAPRF2014 Euler Pole is closely aligned with the published ITRF2008 Euler Pole for the Australian Plate which was estimated from a sparser network of 19 sites forming a subset of the 46 sites used in this study (Altamimi et al., 2012). Site velocities estimated using the ITRF2008 PMM differ by 0.3 mm yr-1 from velocities estimated from SAPRF2014.

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The equivalent SAPRF2014 to ITRF2008 transformation parameters and uncertainties were computed using Eq. 2.5 yielding:

-1 -1 -1 r!x =-1.5038E-3 arcsec yr r!y =-1.1856E-3 arcsec yr r!z =-1.2130E-3 arcsec yr

s = 9.31E-6 arcsec yr-1 s = 8.55E-6 arcsec yr-1 s = 7.53E-6 arcsec yr-1 r!x r!y r!z

The rotation rates are multiplied by the epoch-difference (Dt) to compute the rotation parameters between ITRF2008 and SAPRF2014 at different epochs.

The rotation rate parameters can be used in a 14 parameter transformation model with zeros for all other parameters. The velocity residuals were then computed using Eq. 2.7 and are shown in Figure 7.3. The velocity residuals are largely within the uncertainty of the site velocities used for the inversion and this indicates that the Australian continent is currently stable at the level of uncertainty of the observations during periods of interseismic stability. Large regional plate boundary earthquakes result in observable deformation within the Australian continent at ~< 5mm. Tregoning et al., (2013) show agreement between observed seismic deformation (both coseismic and postseismic) and modelling. As the uncertainties currently exceed any interseismic deformation signal no residual DM has been developed for SAPRF2014. By c.2022 many of the AuScope stations (Figure 7.1, green circles) used for the inversion of a refined SAPRF will have a sufficiently long time-series to improve the uncertainties of the site velocities and better quantify the magnitude of any intraplate deformation.

Figure 7.3 Estimated velocity residuals for stable Australian Plate. The uncertainty of the velocities (0.4 mm yr-1) are generally greater than the velocity residuals which suggests that intraplate deformation is currently insignificant during interseismic periods.

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7.1.2 Application of SAPRF2014 in practice

SAPRF2014 can be used as a basis for representation of spatial data in Australia as coordinates of bedrock fixed within the SAPRF2014 will change by less than 0.4 mm yr-1 in the absence of any seismic deformation (local or large regional earthquakes). Kinematic ITRF2008 coordinates can be transformed to SAPRF2014 coordinates by a 14 parameter transformation, or more simply a four parameter transformation that defines the three rotation rates and a difference in epoch. The reference epoch for SAPRF2014 can be arbitrary, however the model can be used to propagate stable plate coordinates into the future for the purpose of datum modernisation, e.g., GDA2020 and ATRF. Defining a reference epoch is beneficial to support data integration and surveying until spatial software improvements can handle kinematic ITRF coordinates in a robust and assured fashion. A fixed reference epoch is also important for interframe transformations. SAPRF2014 at a defined epoch e.g., 2020.0 could be described as SAPRF2014(2020.0) in order to clearly show the reference epoch for frame coordinates and velocities.

7.1.3 Development of an Australian site velocity grid

SAPRF2014 can be used to generate a 1 degree resolution grid 2D site velocity (topocentric East and North velocities in mm yr-1) grid model over continental Australia. The tectonic stability of the Australian Plate supports this grid resolution, however in areas of significant localised deformation (e.g., underground mining and coal seam gas areas) and urban areas a higher resolution grid could be generated if required. There is currently insufficient duration of CORS data to estimate intraplate deformation of any significance beyond the 0.3 mm yr-1 uncertainty.

Plate motion models do not describe vertical deformation, so the 2D grid can be augmented with a topocentric vertical (Up) velocity grid to create a 3D interseismic velocity grid. Interseismic vertical motion at bedrock sites is not apparent within the Australian continent beyond 0.05 mm yr-1 on geological timescales (Braun et al., 2009). Sites located on regolith however, can be expected to show non-zero velocities on annual and decadal timescales depending upon hydrological conditions. The postseismic response from far-field plate boundary earthquakes can also affect the vertical velocity on longer period timescales.

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7.2 New Zealand case study 1 – multiple coseismic events

A case study location in New Zealand has been chosen to test the application of the schema presented in this thesis. The study uses a working deformation model in conjunction with actual CORS GPS data which is used to simulate point positioning at different epochs and also to test the application of the schema. The case study location is Cape Campbell located about 30 km SE of Blenheim on the South Island in the Marlborough region. This location has been subject to a number of significant earthquakes between 2013 and 2016 resulting in complex coseismic and postseismic displacement and so is an ideal location to test the schema.

7.2.1 Tectonic setting of Cape Campbell

On 14th November 2016 a large earthquake sequence initiated by a Mw7.8 mainshock ruptured a series of faults near Kaikōura, New Zealand (Figure 7.4). The affected location is within the active plate boundary zone between the Pacific and Australian Plates.

Figure 7.4. Kaikōura earthquake, November 2016, tectonic setting (left) and observed displacements (right) (from Hamling et al., 2017). Vectors show horizontal coseismic displacement and blue and red shading indicates vertical coseismic displacement. CORS station CMBL is used to demonstrate the application of the schema in this case study.

The coseismic displacements resulting from the earthquake sequence were very significant and complex. In addition, sites in the northern portion of the affected area near Seddon and Cape Campbell were also displaced by an earlier doublet of large earthquakes, a Mw6.5 earthquake on 21st July 2013 in Cook Strait and a Mw6.6 on 16th August 2013 at Lake Grassmere. These events were large enough to also warrant coseismic displacement models (implemented as reverse patches) to be estimated to update NZGD2000.

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LINZ in conjunction with GNS New Zealand have developed high-resolution deformation models in the affected area using a combination of InSAR, slip dislocation modelling and GPS data (Crook and Donnelly, 2017; LINZ, 2019; Hamling et al., 2017). The deformation models include an ITRF96 interseismic velocity model used with NZGD2000 and multi-resolution coseismic and postseismic patch models (displacement grids) for the 2013 doublet and 2016 earthquakes.

A site within this study area is used to test the schema using scenarios described in detail for different schema variations in the previous chapter. The estimation is compared with actual GNSS observations and tabulated data for the location which is also a CORS station (CMBL) forming part of the NZ Geonet monitoring network. The purpose of the study is to test the application and logic flow of the schema and not necessarily to test the accuracy of the current NZGD2000 deformation model, as (CMBL) may have been used in the generation of the deformation models implemented by LINZ.

7.2.2 GNSS point positioning test data

Actual GNSS observation data for Geonet CORS station CMBL were used to evaluate the schema. Rinex GNSS observation files for two epochs either side of the deformation sequence epochs were chosen. Rinex files (cmbl0010.08o) for Day of Year 001, 2008 (epoch 2008.0 or 1st January 2008) and (cmbl0600.19o) for Day of Year 060, 2019 (epoch 2019.164 or 1st March 2019) and antenna metadata were obtained from the Geonet ftp server at ftp://ftp.geonet.org.nz/gnss/rinex. The data were used to simulate a typical GNSS point positioning observing session in an Earth-fixed kinematic RF such as ITRF2014.

The Rinex GNSS observation files were submitted to both NRCan-PPP (NRCan, 2019) and AusPOS (GA, 2019a) post-processing services to estimate a sub-centimetre precise position at each measurement epoch in ITRF2014. The files were also submitted to the PositioNZ-PP online processing service (LINZ, 2020) to compare the results from different online processing services. PositioNZ-PP still uses the ITRF2008 RF and the ITRF2008 solutions were transformed to ITRF2014 using the IERS 14 parameter transformation (IERS, 2019b). The comparative coordinates are shown for both epochs in Tables 7.1 and 7.2. Likewise, the 2008 data NRCan solution was reported in terms of ITRF2005 and was transformed to ITRF2014 using the IERS 14 parameter transformation. The submission was cross-checked with the site log file to ensure the correct antenna type and antenna reference point (ARP) height were entered and correctly formatted in the RINEX file header. The ITRF2014 coordinate estimations and uncertainties were estimated by the services as shown in Tables 7.1 and 7.2. The three solutions cross-validate the different online GNSS post-processing services and techniques used by each service and a weighted mean solution was estimated together with respective PU at 95% confidence. The PositioNZ-PP processing reports also include both ITRF96 (transformed using the LINZ transformation) and NZGD2000 coordinates derived using the deformation model. These are also shown in the table for comparison with the schema case study. Tabulated NZGD2000 coordinates (pre and post earthquake sequence) for CMBL

138 were also extracted from the LINZ geodetic database. The post-earthquake listed position was transformed to NZGD49 using the LINZ online transformation tool which uses the NTv2 grid transformation. The listed coordinates in the NZGD2000 and NZGD49 datums are listed in Table 7.3. The online transformation tool applies the grid transformation to the latest (post-earthquake corrected) version of NZGD2000 which is v20180701 (1st July 2018). This infers that the superseded NZGD49 static datum is implicitly updated to maintain consistency with NZGD2000 updates. To estimate NZGD49 coordinates pre-earthquake, the grid transformation would be applied to the pre-earthquake NZGD2000 coordinates.

[email protected] ellipsoidal coordinates PU at 95% CI (m) CMBL Latitude Longitude ellipsoid height E/Long. N/Lat. h AusPOS -41° 44’ 56.55666” 174° 12’ 49.70157” 256.073 m 0.004 0.003 0.008 NRCan1 -41° 44’ 56.55670” 174° 12’ 49.70168” 256.078 m 0.006 0.004 0.012 PosNZ-PP2 -41° 44' 56.55663" 174° 12' 49.70164" 256.071 m 0.001 0.001 0.005 Mean -41° 44’ 56.55666” 174° 12' 49.70163" 256.073 m 0.003 0.002 0.007 Mean -41.74904352 174.21380601 (in decimal degree format)

[email protected] Cartesian coordinates UTM zone 60S grid coordinates CMBL X (m) Y (m) Z (m) E (m) N (m) AusPOS -4741514.320 480471.186 -4225019.186 268339.131 5374334.405 NRCan1 -4741514.323 480471.184 -4225019.190 268339.133 5374334.404 PosNZ-PP2 -4741514.319 480471.184 -4225019.184 268339.132 5374334.406 Mean -4741514.320 480471.184 -4225019.186 268339.132 5374334.405 1 NRCan position transformed from ITRF2005 to ITRF2014 using IERS Transformation. 2 PositioNZ-PP position transformed from ITRF2008 to ITRF2014 using IERS Transformation. Reported coordinates in these frames are shown below.

[email protected] Cartesian coordinates UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h NRCan -4741514.326 480471.185 -4225019.199 268339.132 5374334.400 256.086

[email protected] Cartesian coordinates UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741514.318 480471.186 -4225019.181 268339.130 5374334.407 256.068

[email protected] Cartesian coordinates UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741514.319 480471.171 -4225019.204 268339.146 5374334.390 256.083

NZGD2000 (v20180701) Cartesian coords. UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741516.730 480469.785 -4225018.428 268340.700 5374336.525 257.252

Table 7.1 Pre-earthquake sequence observed coordinates at epoch 2008.0 for test point CMBL

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[email protected] ellipsoidal coordinates PU at 95% CI (m) CMBL Latitude Longitude ellipsoid height E/Long. N/Lat. h AusPOS -41° 44’ 56.46280” 174° 12’ 49.74801” 257.272 m 0.005 0.003 0.008 NRCan -41° 44' 56.46266" 174° 12' 49.74804" 257.280 m 0.004 0.002 0.008 PosNZ-PP2 -41° 44' 56.46276" 174° 12' 49.74786" 257.280 m 0.001 0.001 0.005 Mean -41° 44' 56.46274" 174° 12' 49.74797" 257.278 m 0.005 0.003 0.008 Mean -41.74901743 174.21381888 (in decimal degree format)

[email protected] Cartesian coordinates UTM zone 60S grid coordinates CMBL X (m) Y (m) Z (m) E (m) N (m) AusPOS -4741517.237 480470.403 -4225017.823 268340.109 5374337.335 NRCan -4741517.246 480470.403 -4225017.826 268340.110 5374337.339 PosNZ-PP2 -4741517.242 480470.405 -4225017.839 268340.108 5374337.334 Mean -4741517.243 480470.404 -4225017.827 268340.109 5374337.337 2 PositioNZ-PP position transformed from ITRF2008 to ITRF2014 using IERS Transformation. Reported coordinates in ITRF2008 frame are shown below.

[email protected] Cartesian coordinates UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741517.242 480470.407 -4225017.827 268340.106 5374337.336 257.278

[email protected] Cartesian coordinates UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741517.233 480470.377 -4225017.857 268340.136 5374337.307 257.289

NZGD2000 (v20180701) Cartesian coords. UTM zone 60S coords. ellipsoid ht. (m) CMBL X (m) Y (m) Z (m) E (m) N (m) h PosNZ-PP -4741516.772 480469.806 -4225018.435 268340.682 5374336.549 257.289

Table 7.2 Post-earthquake sequence observed coordinates at epoch 2019.167 for test point CMBL.

NZGD2000 (04-Oct-2010) ellipsoidal coordinates PU at 95% CI (m) CMBL Latitude Longitude ellipsoid Height E/Long. N/Lat. h GDB -41° 44’ 56.56492” 174° 12’ 49.71021” 256.109 m 0.08 0.08 0.35 GDB -41.74904581 174.21380839 (in decimal degree format) NZGD2000 (04-Oct-2010) Cartesian coordinates UTM zone 60S grid coordinates X (m) Y (m) Z (m) E (m) N (m) GDB -4741514.198 480470.973 -4225019.400 268339.338 5374334.157

NZGD2000 (v20180701) ellipsoidal coordinates PU at 95% CI (m) CMBL Latitude Longitude ellipsoid height E/Long. N/Lat. Ht GDB -41° 44’ 56.48898” 174° 12’ 49.77112” 257.273 m 0.05 0.05 0.25 GDB -41.74902472 174.21382531 (in decimal degree format) NZGD2000 (v20180701) Cartesian coordinates UTM zone 60S grid coordinates X (m) Y (m) Z (m) E (m) N (m) GDB -4741516.756 480469.818 -4225018.427 268340.669 5374336.544

NZGD49 (from v20180701) ellipsoidal coordinates CMBL Latitude Longitude ellipsoid height GDB -41° 45’ 02.65673” 174° 12’ 49.12585” 245.332 m GDB -41.75073798 174.21364607 (in decimal degree format) Table 7.3 Tabulated NZGD2000 and derived NZGD49 data for test point CMBL from LINZ geodetic database (GDB), 2019. PU is indicative based on the order assigned to the station accuracy.

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7.2.3 Estimation of displacements from published models

The current version of the NZGD2000 deformation model (version 20180701, LINZ, 2019) updated in December 2018 (http://apps.linz.govt.nz/ftp/geodetic/nzgd2000_deformation_20180701_full.zip) was used in this case study to estimate interseismic velocities, coseismic and postseismic displacements for testing the schema. The NZGD2000 deformation model is defined in terms of ITRF96 consistent with the original realisation of NZGD2000 ([email protected]). The model is composed of different sub-model grids for each component and earthquake event of the deformation field. The four nodes of each model grid relevant to the test point CMBL are shown in the Tables 7.4 to 7.12 below.

The NZGD2000 deformation model does not currently include uncertainties for the modelled interseismic velocities, coseismic and postseismic displacements. In order to validate the schema, realistic uncertainties for model values have been synthetically estimated using the sample variance of the four data values for the deformation grid cell. In practice, a more robust estimate of uncertainty derived from the modelling technique used to generate the deformation model would be used.

7.2.4 Estimation of the interseismic velocity

The interseismic velocity grid sub-model is described by the csv file in the NZGD2000 deformation model: grid_igns2011_nz.csv. The file format is a regular 0.1° grid of nodes in decimal ellipsoidal coordinate format. For each node, the modelled ITRF96 topocentric horizontal ( E! and N! ) velocities are tabulated (Table 7.4). Vertical (Up) velocities and are not included in the model and are assumed to be zero, however realistic values and uncertainties are included here for the purpose of testing the full schema. The nodes and location of CMBL are shown in Figure 7.5.

Node coordinates ITRF96 Topocentric velocities (m yr-1) Velocity uncertainties (m yr-1) 1s Node Longitude Latitude E! N! U! σE! σN! σU! SW 174.2 -41.8 -0.0235 0.0326 0.0000 0.0015 0.0005 0.0015 SE 174.3 -41.8 -0.0243 0.0322 0.0000 0.0015 0.0005 0.0015 NW 174.2 -41.7 -0.0217 0.0327 0.0000 0.0015 0.0005 0.0015 NE 174.3 -41.7 -0.0227 0.0324 0.0000 0.0015 0.0005 0.0015 Table 7.4 ITRF96 Interseismic velocity model node data used for test point CMBL (LINZ, 2019). The vertical velocity is set to 0.000 as a default value assuming that no secular vertical motion is occurring at the site within the bounds of the vertical motion uncertainty. Realistic uncertainties have been simulated based on the model variances within the grid cell.

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Figure 7.5 Location of test point CMBL and interseismic velocity model data

The interseismic topocentric velocity (in terms of ITRF96) for CMBL and uncertainties were then estimated by bilinear interpolation of the model using Eqs. 5.9 to 5.11 as follows: E! =-0.0227m yr-1 N! = 0.0326m yr-1 U! = 0.0000m yr-1

s E! = 0.0015 m yr-1 s N! = 0.0005m yr-1 sU! = 0.0015m yr-1

Using Eqs. 6.1 and 6.2 the estimated forward secular displacement and propagated uncertainty in ITRF96 between epoch 2008.0 and 2019.164 in topocentric format is: DED() =(2019.164- 2008)×- 0.0227 = -0.253 m sDED() =0.017 m (2008.000® 2019.164) (2008.000® 2019.164)

DND()(2008.000® 2019.164) =(2019.164- 2008)× 0.0326 = 0.364 m sDND()(2008.000® 2019.164) =0.006 m

DUD()(2008.000® 2019.164) =(2019.164- 2008)× 0.000 = 0.000 m sDUD()(2008.000® 2019.164) =0.017m

7.2.5 Estimation of coseismic displacements for each significant earthquake

The highest resolution coseismic displacement models (patches) were used to estimate the coseismic displacement at CMBL as a result of three significant earthquakes impacting the site.

For the 21st July 2013 Cook Strait earthquake (seismic event 1), 3D sub model grid_cs_20130721_L3.csv was used to estimate the topocentric coseismic displacements for that event (Table 7.5).

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Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(C)1 DN(C)1 DU(C)1 sDE(C)1 sDN(C)1 sDU(C)3 SW 174.212500 -41.7500000 -0.005 -0.028 0.006 0.001 0.003 0.001 SE 174.221875 -41.7500000 -0.006 -0.031 0.007 0.001 0.003 0.001 NW 174.212500 -41.7421875 -0.004 -0.026 0.005 0.001 0.003 0.001 NE 174.221875 -41.7421875 -0.005 -0.029 0.006 0.001 0.003 0.001 Table 7.5 21st July 2013 Cook Strait earthquake coseismic displacement model (patch) node data used for test point CMBL (LINZ, 2019). Realistic uncertainties have been simulated.

The coseismic displacements and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEC()1 =- 0.005m DNC()1 =- 0.028m DUC()1 = 0.006m

sDEC()1 = 0.001m sDNC()1 = 0.003m sDUC()1 = 0.001m

For the 16th August 2013 Lake Grassmere earthquake (coseismic event 2), 3D sub model grid_lg_20130816_L4.csv was used to estimate topocentric coseismic displacements for that event (Table 7.6).

Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(C)2 DN(C)2 DU(C)2 sDE(C)2 sDN(C)2 sDU(C)2 SW 174.2125000 -41.75000000 -0.193 0.002 -0.076 0.007 0.002 0.005 SE 174.2148438 -41.75000000 -0.195 0.004 -0.075 0.007 0.002 0.005 NW 174.2125000 -41.74804688 -0.199 0.004 -0.079 0.007 0.002 0.005 NE 174.2148438 -41.74804688 -0.200 0.006 -0.079 0.007 0.002 0.005 Table 7.6 16th August 2013 Lake Grassmere earthquake coseismic displacement model (patch) node data used for test point CMBL (LINZ, 2019)

The coseismic displacements and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEC()2 =- 0.197m DNC()2 = 0.004m DUC()2 =- 0.077m

sDEC()2 = 0.007m sDNC()2 = 0.002m sDUC()2 = 0.005m

For the 14th November 2016 Kaikōura earthquake sequence, sub models grid_ka_20161114co_H_L4_R_19.csv (horizontal) and grid_ka_20161114co_V_L4_16.csv (vertical) were used to estimate initial topocentric coseismic displacements for that event (Table 7.7).

Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(C)30 DN(C)30 DU(C)30 sDE(C)30 sDN(C)30 sDU(C)30 SW 174.2125000 -41.75000000 1.312 2.274 1.031 0.02 0.02 0.02 SE 174.2148438 -41.75000000 1.348 2.311 1.051 0.02 0.02 0.02 NW 174.2125000 -41.74804688 1.255 2.214 0.997 0.02 0.02 0.02 NE 174.2148438 -41.74804688 1.289 2.249 1.018 0.02 0.02 0.02 Table 7.7 14th November 2016 Kaikōura earthquake sequence initial coseismic displacement model (patch) node data used for test point CMBL (LINZ, 2019)

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The initial coseismic displacements and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEC( )= 1.303m DNC( )= 2.264m DUC( )= 1.025m 30 30 30 sDEC( )= 0.020m sDNC( )= 0.020m sDUC( )= 0.020m 30 30 30

Following the initial release of the initial coseismic displacement model for the Kaikōura earthquake, two incremental updates were made to the model. The first correction (horizontal only) from grid_hor_refinement.csv is shown in Table 7.8.

Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(C)31 DN(C)31 DU(C)31 sDE(C)31 sDN(C)31 sDU(C)31 SW 174.10 -41.750 0 0 0 0.002 0.005 0.002 SE 174.25 -41.750 0 0 0 0.002 0.005 0.002 NW 174.10 -41.625 0.027 -0.033 0 0.002 0.005 0.002 NE 174.25 -41.625 0.011 -0.018 0 0.002 0.005 0.002 Table 7.8 coseismic displacement model (patch) 1st correction node data used for test point CMBL (LINZ, 2019)

The first coseismic displacement correction and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEC( )= 0.000m DNC( )= 0.000m DUC( )= 0.000m 31 31 31 sDEC( )= 0.002m sDNC( )= 0.005m sDUC( )= 0.002m 31 31 31 The second correction from grid_hor_refinement2.csv and grid_vrt_refinement2.csv are shown in Table 7.9.

Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(C)32 DN(C)32 DU(C)32 sDE(C)32 sDN(C)32 sDU(C)32 SW 174.200 -41.760 0.252 -0.044 0.078 0.01 0.02 0.01 SE 174.220 -41.760 0.216 -0.026 0.078 0.01 0.02 0.01 NW 174.200 -41.745 0.248 0.005 0.066 0.01 0.02 0.01 NE 174.220 -41.745 0.216 0.023 0.050 0.01 0.02 0.01 Table 7.9 coseismic displacement model (patch) 2nd correction node data used for test point CMBL (LINZ, 2019)

The second coseismic displacement correction and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEC( )= 0.226 m DNC( )= 0.004m DUC( )= 0.061m 32 32 32 sDEC( )= 0.01m sDNC( )= 0.02m sDUC( )= 0.01m 32 32 32

The overall coseismic displacement for the Kaikōura earthquake was then estimated by summation of the initial model with the two corrections as follows:

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DEC()3 = 1.529m DNC()3 = 2.268m DUC()3 = 1.086m The uncertainties were estimated using standard error propagation as follows:

sDEC()3 = 0.022m sDNC()3 = 0.029m sDUC()3 = 0.022m

The total coseismic displacement (in a topocentric frame) for all events between the test data epochs was then estimated by summation of the coseismic displacement from the three events as follows:

= (-0.005) + (-0.197) + (1.529) = 1.327 m å DEC()=DEC()123+D EC ()+D EC () 2008£C< 2019.164

= (-0.028) + (0.004) + (2.268) = 2.244 m å DNC()=DNC()123+D NC ()+D NC () 2008£C< 2019.164

å DUC()=DUC()123+D UC ()+D UC () = (0.006) + (-0.077) + (1.086) = 1.015 m 2008£C< 2019.164

The associated topocentric uncertainties are: s åDEC( )= (0.001)222 + (0.007) + (0.022) = 0.023m s åDNC( )= (0.003)222 + (0.002) + (0.029) = 0.029m s åDUC( )= (0.001)222 + (0.005) + (0.022) = 0.023m

7.2.6 Estimation of postseismic displacements

The NZGD2000 deformation model only includes postseismic deformation for the Kaikōura earthquake sequence for the study area. The distributed postseismic displacement models used are simplified and constitute just two linear ramp functions (ramped displacement corrections between two defined epochs) to model the postseismic deformation over a period of three months after the mainshock. The postseismic correction is either a full displacement for epochs after the defined postseismic period, or a linear proportional displacement if the epoch falls within the postseismic period. The displacement factor is linear from 0 at the start of the period to 1 at the end of the period. The functional model used by LINZ for estimating postseismic displacement is documented in the deformation model distribution (LINZ, 2019). The ramp function approach can be improved using the approach described in Chapter 6 to describe postseismic deformation using a combination of logarithmic and exponential parameters (amplitude and decay time) and velocity corrections where this data is available. The first ramp function using sub model grids grid_ka_20161114pe1_H_L3_R_01.csv (horizontal) and grid_ka_20161114pe1_V_L3_01.csv (vertical) defines postseismic deformation between 14th November 2016 and 14th December 2016 and is shown below in Table 7.10.

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Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(P)31 DN(P)31 DU(P)31 sDE(P)31 sDN(P)31 sDU(P)31 SW 174.212500 -41.7500000 0.072 0.062 0.117 0.01 0.02 0.03 SE 174.221875 -41.7500000 0.076 0.080 0.105 0.01 0.02 0.03 NW 174.212500 -41.7421875 0.097 0.101 0.047 0.01 0.02 0.03 NE 174.221875 -41.7421875 0.069 0.084 0.097 0.01 0.02 0.03 Table 7.10 14th November 2016 Kaikōura earthquake sequence (event 3) 1st postseismic displacement model (patch) data used for test point CMBL (LINZ, 2019)

The first postseismic displacement correction and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEP( )= 0.075m DNP( )= 0.069 m DUP( )= 0.108m 31 31 31 sDEP( )= 0.01m sDNP( )= 0.02m sDUP( )= 0.03m 31 31 31

The second ramp function using sub model grids grid_ka_20161114pe3_H_L3_R_01.csv (horizontal) and grid_ka_20161114pe3_V_L3_01.csv (vertical) defines postseismic deformation between 14th December 2016 and 14th February 2017 and is shown below in Table 7.11.

Node coordinates Topocentric displacements (m) Uncertainties (m) 1s Node Longitude Latitude DE(P)32 DN(P)32 DU(P)32 sDE(P)32 sDN(P)32 sDU(P)32 SW 174.212500 -41.7500000 0.038 0.027 0.048 0.004 0.005 0.01 SE 174.221875 -41.7500000 0.037 0.033 0.050 0.004 0.005 0.01 NW 174.212500 -41.7421875 0.046 0.042 0.024 0.004 0.005 0.01 NE 174.221875 -41.7421875 0.034 0.037 0.046 0.004 0.005 0.01 Table 7.11 14th November 2016 Kaikōura earthquake sequence (event 3) 2nd postseismic displacement model (patch) data used for test point CMBL (LINZ, 2019)

The second postseismic displacement correction and uncertainties (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DEP( )= 0.039 m DNP( )= 0.030m DUP( )= 0.046m 32 32 32 sDEP( )= 0.004m sDNP( )= 0.005m sDUP( )= 0.01m 32 32 33 The overall postseismic displacement (in a topocentric system) for the Kaikōura earthquake was then estimated by summation of the two ramp displacements as follows:

The summation of the modelled postseismic forward displacements and uncertainties in ITRF96 between epochs 2008.0 and 2019.164 is: = 0.114 m m å DEP()=DEP()3 s åDEP()= 0.011 2008£P< 2019.164

= 0.099 m m å DNP()=DNP()3 s åDNP()= 0.021 2008£P< 2019.164

= 0.154 m m å DUP()=DUP()3 s åDUP()= 0.032 2008£P< 2019.164

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7.2.7 Supplementary corrections to the deformation model

The magnitude of the deformation from the Kaikōura earthquake sequence was sufficiently large that an area greater than that covered by the coseismic displacement model used to define permanent changes to NZGD2000 coordinates was deformed. Rather than apply permanent changes to the established NZGD2000 network in these outer regions, a model correction patch (also referred to as a forward patch) is applied to coordinates after the earthquake. The supplementary correction has been estimated from the NZGD deformation model components as shown in Table 7.12. No uncertainties are assigned to these corrections.

grid_ka_20161114co grid_ka_20161114pe1_ grid_ka_20161114pe3_ Node coordinates _H_L2_F.csv H_L2_F.csv H_L2_F.csv Node Longitude Latitude DE(S)31 DN(S)31 DE(S)32 DN(S)32 DE(S)33 DN(S)33 SW 174.21250 -41.75000 -0.100 0.131 0.007 0.010 0.007 0.009 SE 174.25000 -41.75000 -0.109 0.134 0.007 0.010 0.006 0.009 NW 174.21250 -41.71875 -0.086 0.139 0.007 0.010 0.007 0.009 NE 174.25000 -41.71875 -0.094 0.142 0.007 0.011 0.007 0.009 Table 7.12 14th November 2016 Kaikōura earthquake sequence (event 3) supplementary topocentric displacement model (forward patch) data used for test point CMBL (LINZ, 2019)

The supplementary correction (in terms of ITRF96) for this event were then estimated by bilinear interpolation of the model as follows:

DES( )=- 0.100 m DNS( )= 0.131m 31 31 DES( )= 0.007m DNS( )= 0.010m 32 32 DES( )= 0.007 m DNS( )= 0.009 m 33 33 DES()=- 0.086 DNS( )= 0.150 å 3 å 3

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7.2.8 Interframe transformation parameters

The IERS 14 parameter transformation defined between ITRF2014 and ITRF96 (IERS, 2019b) could be used in this case study, however in New Zealand a different transformation between ITRF2014 and ITRF96 (Tables 7.13 and 7.14) has been mandated (LINZ, 2017). Using Eq. 3.5 the ITRF2014 to ITRF96 (LINZ) parameters for the epochs 2008.000 and 2019.164 used in the case study are estimated and shown in the table.

tx ty tz rx ry rz s t0 (mm) (mm) (mm) (mas) (mas) (mas) (ppb) (year) 6.4 3.99 -14.27 -0.16508 0.26897 0.11984 1.08901 2000.0

t!x t!y t!z r!x r!y r!z s! rates (mm yr-1) (mm yr-1) (mm yr-1) (mas yr-1) (mas yr-1) (mas yr-1) (ppb yr-1) 0.79 -0.6 -1.44 -0.01347 0.01514 0.01973 -0.07201 2000.0

tx ty tz rx ry rz s tA (mm) (mm) (mm) (mas) (mas) (mas) (ppb) (year) 21.5 -7.5 -41.9 -0.423 0.559 0.498 -0.29 2019.164 12.7 -0.8 -25.8 -0.273 0.391 0.278 0.51 2008.000 Table 7.13 ITRF2014 to ITRF96 (New Zealand) PV notation transformation parameters (LINZ, 2017) at t0 and at the case study RF transformation epochs 2019.164 and 2008.000

tx ty tz rx ry rz s t0 (mm) (mm) (mm) (mas) (mas) (mas) (ppb) (year) 6.4 3.99 -14.27 0.16508 -0.26897 -0.11984 1.08901 2000.0

t!x t!y t!z r!x r!y r!z s! rates (mm yr-1) (mm yr-1) (mm yr-1) (mas yr-1) (mas yr-1) (mas yr-1) (ppb yr-1) 0.79 -0.6 -1.44 0.01347 -0.01514 -0.01973 -0.07201 2000.0

tx ty tz rx ry rz s tA (mm) (mm) (mm) (mas) (mas) (mas) (ppb) (year) 21.5 -7.5 -41.9 0.423 -0.559 -0.498 -0.29 2019.164 12.7 -0.8 -25.8 0.273 -0.391 -0.278 0.51 2008.000 Table 7.14 ITRF2014 to ITRF96 (New Zealand) CF notation transformation parameters (LINZ, 2017) at t0 and at the case study RF transformation epochs 2019.164 and 2008.000

7.2.9 Summary of test position data and model estimation for use in schema test case studies

The CMBL test position data (X) and deformation model component estimations in geodetic Cartesian format above are summarised below together with their topocentric uncertainties for use in the schema test case studies.

The ITRF2014 positions from Tables 7.1 and 7.2 are described in vector form together with their uncertainties represented as 1s for use in the schema accordingly.

The test position geocentric Cartesian coordinates, equivalent UTM zone 60S grid coordinates, ellipsoid height and associated 1s uncertainties in topocentric format at epoch 2008.0 are:

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éù-4741514.320 X = êú480471.184 ITRF2014(2008.000) êú êú-4225019.186 ëûCARTESIAN

XITRF2014(2008.000) = (268339.132,5374334.406,256.073)UTM60 S & h

σXITRF2014(2008.000) = (0.0015,0.0010,0.0035)TOPOCENTRIC

The test position geocentric Cartesian coordinates, equivalent UTM zone 60S grid coordinates, ellipsoid height and associated 1s uncertainties in topocentric format at epoch 2019.164 are:

éù-4741517.243 X = êú480470.404 ITRF2014(2019.164) êú êú-4225017.827 ëûCARTESIAN

XITRF2014(2019.164) = (268340.109,5374337.337,257.278)UTM60 S & h

σXITRF2014(2019.164) = (0.0025,0.0015,0.0040)TOPOCENTRIC

The interseismic velocity and uncertainty from Section 7.2.4 is indicated as follows in both topocentric vector and Cartesian form (using Eq. 5.7): éù-0.0227 éù-0.0194 D! = êú0.0326 D! = êú0.0248 ITRF96 êú ITRF96 êú êú0.0000 êú0.0242 ëûTOPOCENTRIC ëûCARTESIAN ! σDITRF96 = (0.0015,0.0005,0.0015)TOPOCENTRIC

Similarly, the derived displacements and uncertainties for the different epoch intervals used in the case studies are shown as follows: éù-0.253 éù-0.216 D = êú0.364 D = êú0.276 ITRF96(2008.0® 2019.164) êú ITRF96(2008.000® 2019.164) êú êú0.000 êú0.271 ëûTOPOCENTRIC ëûCARTESIAN

σDITRF96(2008.000® 2019.164) = (0.017,0.006,0.017)TOPOCENTRIC

éù-0.435 éù-0.372 D = êú0.625 D = êú0.475 ITRF96(2000.000® 2019.164) êú ITRF96(2000.000® 2019.164) êú êú0.000 êú0.465 ëûTOPOCENTRIC ëûCARTESIAN σ0D = .029,0.010,0.029 ITRF96(2000.000® 2019.164) ( )TOPOCENTRIC

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éù-0.182 éù-0.155 D = êú0.261 D = êú0.199 ITRF96(2000.000® 2008.000) êú ITRF96(2000.000® 2008.000) êú êú0.000 êú0.194 ëûTOPOCENTRIC ëûCARTESIAN σ0D = .012,0.004,0.012 ITRF96(2000.000® 2008.000) ( )TOPOCENTRIC

The summation of coseismic displacement between epochs 2008.000 and 2019.164 and associated uncertainty from Section 7.2.5 is indicated as follows in both topocentric vector and Cartesian form (using Eq. 5.7):

éù1.327 éù-2.379 C = êú2.244 C =êú-1.093 å ITRF96(2008.000® 2019.164) êú å ITRF96(2008.000® 2019.164) êú êú1.015 êú0.993 ëûTOPOCENTRIC ëûCARTESIAN σ0C = .023,0.029,0.023 å ITRF96(2000.000® 2008.000) ( )TOPOCENTRIC

The summation of postseismic displacement between epochs 2008.000 and 2019.164 and associated uncertainty from Section 7.2.6 is indicated as follows in both topocentric vector and Cartesian form (using Eq. 5.7):

éù0.114 éù-0.192 P = êú0.099 P =êú-0.095 å ITRF96(2008.000® 2019.164) êú å ITRF96(2008.000® 2019.164) êú êú0.154 êú-0.029 ëûTOPOCENTRIC ëûCARTESIAN σ0P = .011,0.021,0.032 å ITRF96(2000.000® 2008.000) ( )TOPOCENTRIC

The supplementary displacement (forward patch) between epochs 2008.000 and 2019.164 from Section 7.2.7 is indicated as follows in both topocentric vector and Cartesian form (using Eq. 5.7). No uncertainties are assigned to this component in the case study since the forward patch conceptually reduces the uncertainty.

éù-0.086 S = êú0.150 å ITRF96(2008.000® 2019.164) êú êú0.000 ëûTOPOCENTRIC éù-0.091 S = êú0.096 å ITRF96(2008.000® 2019.164) êú êú0.112 ëûCARTESIAN

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The transformation parameters between ITRF2014 and ITRF96 (LINZ) at epoch 2019.164 (Table 7.14) are converted to metres and radians for use in the Bursa-Wolf (CF notation) seven parameter transformation (Eq. 3.3) using Eq. 7.1:

éùXX é0.0215 ù é 1-- 2.414E 9 2.710E - 9ùéù êú ê ú ê úêú êúYY= ê-0.0075 ú+ (1 + (--- 2.9E 10))ê 2.414E 9 1 2.051E- 9úêú × êúZZ ê--0.0419 ú ê2.710E- 9- 2.051E- 9 1 úêú ëûITRF96(2019.164) ë û ë ûëûITRF2014(2019.164)

(7.1)

The transformation parameters between ITRF2014 and ITRF96 (LINZ) computed at epoch 2008.0 (from Table 7.14) are also converted to metres and radians for use in the Bursa-Wolf (CF notation) 7- parameter transformation (Eq. 3.3) for schema testing using Eq. 7.2:

éùXX é0.0127 ù é 1-- 1.348E 9 1.896E - 9ùéù êú ê ú ê úêú êúYY= ê-0.0008 ú+ (1 + (5.1E-- 10))ê 1.348E 9 1 1.324E- 9úêú × êúZZ ê--0.0258 ú ê1.896E- 9- 1.324E- 9 1 úêú ëûITRF96(2008.000) ë û ë ûëûITRF2014(2008.000)

(7.2)

The schema described in Chapter 6 can be tested for different scenarios using the sample data interpolated from the NZGD2000 deformation model summarised in 7.2.9.

7.2.10 Kinematic RF (with no DM) to Kinematic RF (with a defined DM)

This test involves the transformation of CMBL from ITRF2014 (an Earth-fixed kinematic RF with no explicitly described deformation model) at epoch 2019.164 (tA) to ITRF96 at epoch 2008.0 (tB). In New Zealand, ITRF96 can be propagated to a different epoch using the NZGD2000 deformation model which is defined in terms of ITRF96. The specific schema example for this case is described in 6.4.1 and as illustrated in Figure 6.8.

The position XA and uncertainty in the source RF (RFA) (ITRF2014 at epoch 2019.184) are described as follows (from sample data summarised in 7.2.9):

éù-4741517.243 X = êú480470.404 σX = (0.0025,0.0015,0.0040) A êú A TOPOCENTRIC êú-4225017.827 ëûCARTESIAN

Following the schema logic flow described in Section 6.3.1:

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The interframe (ITRF2014 to ITRF96) transformation epoch tAB is determined using the case testing criteria described in Section 6.3.1 and Table 6.1 as:

tAB = 2019.164

As ttAB= A there is no intraframe propagation within RFA so:

DXA =0 (from 6.11) The interframe transformation operation f []X from ITRF2014 at epoch 2019.164 to ITRF96 at ABt (AB) A epoch 2019.164 is then computed using Eq. 7.1 as follows:

éùX é-4741517.232 ù êúY ê480470.376 ú êú= ê ú êúZ ê-4225017.855 ú ëûITRF96(2019.164) ë û The propagation in the target frame (ITRF96 between epochs 2008.000 and 2019.164) is then estimated using Eq. 6.12 in Cartesian form:

éùéùéùéùéù-----0.216 2.379 0.192 0.091 2.878 DX =êúêúêúêúêú0.276 +- 1.093+- 0.095+ 0.096 =- 0.816 (from 6.12) B êúêúêúêúêú êúêúêúêúêú0.271 0.993- 0.029 0.112 1.347 ëûëûëûëûëûCARTESIAN

In the example shown, tB < tAB so the propagation sign mB = -1 (reverse propagation in the target RF). The simplified schema transformation is expressed using Eq. 6.14 as follows:

éùéùéù---4741517.232 2.878 4741514.354 X =+êúêúêú480470.376 (- 1)×- 0.816= 480471.192 B êúêúêú ëûëûëûêúêúêú--4225017.855 1.347 4225019.202

The topocentric uncertainties are expressed using Eq. 6.13 as follows:

s E =+++++++=0.002522 0 0 0 0 0.017 0.0232 0.0112 0.031 B s N =+++++++=0.001522 0 0 0 0 0.006 0.0292 0.0212 0.036 B sU =+++++++=0.00422 0 0 0 0 0.017 0.0232 0.0322 0.043 B

The transformed position in the target frame (ITRF96 at epoch 2008.000) and estimated uncertainty at 95% CI (~2s) are therefore:

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éù-4741514.354 X = êú480471.192 ITRF96(2008.000) êú êú-4225019.202 ëûCARTESIAN

XITRF96(2008.000) = (268339.127,5374334.416,256.109)UTM60 S & h

2σXITRF96(2008.000) = (0.062,0.072,0.086)TOPOCENTRIC

The schema estimation using the NZGD2000 deformation model can then be compared with the actual observation at the target epoch.

Using the published deformation model and schema the transformed position is:

XITRF96(2008.000) = (268339.127,5374334.416,256.109)UTM60 S & h which is compared with the observed position at that epoch (from Table 2.1):

XITRF96(2008.000) = (268339.146,5374334.390,256.083)UTM60 S & h

The observed minus modelled residuals (uE, uN, uh) are:

υX = (0.019,-- 0.026, 0.026)UTM60 S & h

The residuals are well within the 95% CI bounds estimated from the model, suggesting that the simulated model uncertainties may be overly pessimistic.

7.2.11 Kinematic RF (with no DM) to a semi-kinematic RF

This test involves the transformation of CMBL from ITRF2014 (an Earth-fixed kinematic RF with no explicitly described deformation model) at epoch 2019.164 (tA) to the latest version of NZGD2000

(v20180701) (tB). In New Zealand, the NZGD2000 deformation model which is defined in ITRF96 is used to propagate ITRF96 coordinates between epochs. The specific schema example for this case is described in Section 6.4.2 and as illustrated in Figure 6.11.

The position XA and uncertainty in the source RF (RFA) (ITRF2014 at epoch 2019.184) are described as follows (from sample data summarised in Section 7.2.9):

éù-4741517.243 X = êú480470.404 σX = (0.0025,0.0015,0.0040) A êú A TOPOCENTRIC êú-4225017.827 ëûCARTESIAN

Following the schema logic flow described in Section 6.3.1:

The interframe (ITRF2014 to NZGD2000 v20180701) transformation epoch tAB is determined using the case testing criteria described in Section 6.3.1 and Table 6.1 as:

tAB = 2019.164

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As ttAB= A there is no intraframe propagation within RFA so:

DXA =0 (from 6.11)

The interframe transformation operation f []X from ITRF2014 at epoch 2019.164 to ITRF96 at ABt (AB) A epoch 2019.164 is then computed using Eq. 7.1 as follows:

éùX é-4741517.232 ù êúY ê480470.376 ú êú= ê ú êúZ ê-4225017.855 ú ëûITRF96(2019.164) ë û

The propagation in the target frame uses Eq. 6.17 in Cartesian form where the interseismic displacement (D) is estimated from the NZGD2000 reference epoch (2000.0) together with a supplementary forward patch correction:

éùéùéù---0.372 0.091 0.463 DX =êúêúêú0.475 + 0.096 = 0.571 (from Eq. 6.17) B êúêúêú êúêúêú0.465 0.112 0.577 ëûëûëûCARTESIAN

tB < tAB so the propagation sign mB = -1 (reverse propagation in the target RF). The simplified schema transformation is expressed using Eq. 6.14 as follows:

éùéùéù---4741517.232 0.463 4741516.769 X =+êúêúêú480470.376 (- 1)× 0.571= 480469.805 B êúêúêú ëûëûëûêúêúêú--4225017.855 0.577 4225018.432

The topocentric uncertainties are expressed using Eq. 6.13 as follows:

s E =+++++++=0.002522 0 0 0 0 0.029 0 0 0.029 B s N =+++++++=0.001522 0 0 0 0 0.010 0 0 0.010 B sU =+++++++=0.00422 0 0 0 0 0.029 0 0 0.029 B

The transformed position in the target frame (NZGD2000 v20180701) and estimated uncertainty at 95% CI (~2s) are therefore:

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éù-4741516.769 X = êú480469.805 NZGD2000(v20180701) êú êú-4225018.432 ëûCARTESIAN

XNZGD2000(v20180701) = (268340.683,5374336.549,257.285)UTM60 S & h

2σXNZGD2000(v20180701) = (0.058,0.020,0.058)TOPOCENTRIC

This position agrees within 3 mm of the PositioNZ-PP transformation in Table 7.2. The schema estimation can also be compared with the tabulated data in the target frame. Using the published deformation model and schema the transformed position is:

XNZGD2000(v20180701) = (268340.683,5374336.549,257.285)UTM60 S & h which is compared with the tabulated position (from Table 7.3):

XNZGD2000(v20180701) = (268340.669,5374336.544,257.273)UTM60 S & h

The tabulated minus modelled residuals (uE, uN, uh) are:

υ(0.014,0.005,0.012)X =--- UTM60 S & h

The residuals are well within the 95% CI bounds estimated from the model, suggesting that the simulated model uncertainties may be overly pessimistic. The residuals are also well within the uncertainty limits specified for the tabulated data for CMBL.

The schema can also be used to estimate coordinates of the target semi-kinematic RF pre-earthquake sequence. In this example, NZGD2000 coordinates as originally realised (2000.0) are required and so all seismic displacements are required to be modelled. Eq. 6.16 is used for the target RF propagation as follows:

éùéùéùéùéù-----0.372 2.379 0.192 0.091 3.034 DX =êúêúêúêúêú0.475 +- 1.093+- 0.095+ 0.096 =- 0.617 B êúêúêúêúêú êúêúêúêúêú0.465 0.993- 0.029 0.112 1.541 ëûëûëûëûëûCARTESIAN

The epoch of the target RF predates the tB < tAB so the propagation sign mB = -1 (reverse propagation in the target RF). The simplified schema transformation is expressed using Eq. 6.14 as follows:

éùéùéù---4741517.232 3.034 4741514.198 X =+êúêúêú480470.376 (- 1)×- 0.617= 480470.993 B êúêúêú ëûëûëûêúêúêú--4225017.855 1.541 4225019.396

The topocentric uncertainties are expressed using Eq. 6.13 as follows:

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s E =+++++++=0.002522 0 0 0 0 0.029 0.0232 0.0112 0.039 B s N =+++++++=0.001522 0 0 0 0 0.010 0.0292 0.0212 0.037 B sU =+++++++=0.00422 0 0 0 0 0.029 0.0232 0.0322 0.049 B

The transformed position in the target frame (NZGD2000 pre-earthquake) and estimated uncertainty at 95% CI (~2s) are therefore: éù-4741514.198 X = êú480470.993 NZGD2000(2000.0) êú êú-4225019.396 ëûCARTESIAN

XNZGD2000(2000.000) = (268339.318,53743344.160,256.108)UTM60 S & h

2σXNZGD2000(2000.000) = (0.078,0.074,0.098)TOPOCENTRIC

The schema estimation can also be compared with the tabulated data in the target frame. Using the published deformation model and schema the transformed position is:

XNZGD2000(2000.000) = (268339.318,53743344.160,256.108)UTM60 S & h which is compared with the tabulated position defined on 4th October 2010 (20101004) (from Table 7.3):

XNZGD2000(2000.000) = (268339.338,53743344.157,256.109)UTM60 S & h

The tabulated minus modelled residuals (uE, uN, uh) are:

υX = (0.020,- 0.003,0.001)UTM60 S & h

The residuals are well within the 95% CI uncertainty limits specified for the tabulated data for CMBL before the earthquake sequence.

7.2.12 Kinematic RF (with DM) to a static RF

This test involves the transformation of CMBL from ITRF96 (with a defined DM in New Zealand) at epoch

2019.164 (tA) to the latest superseded static RF used in New Zealand, NZGD49 (tB). The specific schema example for this case is described in Section 6.4.3 as illustrated in Figure 6.14. In this example, the position in the target static RF is one which predates any surface deformation.

The position XA and uncertainty in the source RF (RFA) (ITRF96 at epoch 2019.184) are described as follows (from sample data summarised in Section 7.2.9):

éù-4741517.232 X = êú480470.376 σX = (0.0025,0.0015,0.0040) A êú A TOPOCENTRIC êú-4225017.855 ëûCARTESIAN

Following the schema logic flow described in Section 6.3.1:

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The interframe (ITRF96 to NZGD49) transformation epoch tAB is determined using the case testing criteria described in Section 6.3.1 and Table 6.1 as:

tAB = 2000.000 The propagation in the source frame uses Eq. 6.11 in Cartesian form where the interseismic, coseismic, and postseismic are estimated from the NZGD49 and ITRF96 transformation reference epoch (2000.0) together with a supplementary forward patch correction:

éùéùéùéùéù-----0.372 2.379 0.192 0.091 3.034 êúêúêúêúêú DX =0.475 +- 1.093+- 0.095+ 0.096 =- 0.617 (from Eq. 6.11) A êúêúêúêúêú êúêúêúêúêú0.465 0.993- 0.029 0.112 1.541 ëûëûëûëûëûCARTESIAN tA > tAB so the propagation sign mA = -1 (reverse propagation in the source RF). The propagated coordinate in the source RF (ITRF96 at epoch 2000.0) is then estimated using: XX=+m ()D X A(tAB ) A A A éùéùéù---4741517.232 3.034 4741514.198 X =+êúêúêú480470.376 (- 1)×- 0.617= 480470.993 A(tAB ) êúêúêú ëûëûëûêúêúêú--4225017.855 1.541 4225019.396

The propagated ITRF96 coordinate at epoch 2000.000 is identical to the NZGD2000 coordinate since NZGD2000 is realised by ITRF96 at epoch 2000.000: éù-4741514.198 XX==êú480470.993 ITRF96(2000.000) NZGD2000(2000.000) êú êú-4225019.396 ëûCARTESIAN

XNZGD2000(2000.000) = (268339.318,53743344.160,256.108)UTM60 S & h

The interframe transformation operation fm[()]XX+D from ITRF96 at epoch 2000.000 can ABt (AB) A A A then use the NTv2 NZGD2000 to NZGD49 grid transformation which estimates the following transformed coordinates in the target frame (there is no propagation in this frame):

XNZGD49(2000.000) =(- 41.75075904,174.21362891,244.1665)flh Post-earthquake realisation of NZGD49 can be done by only applying the interseismic and supplementary displacements in the source RF as follows: éùéùéù---0.372 0.091 0.463 êúêúêú DX =0.475 + 0.096 = 0.571 A êúêúêú êúêúêú0.465 0.112 0.577 ëûëûëûCARTESIAN

157 tA > tAB so the propagation sign mA = -1 (reverse propagation in the source RF). The propagated coordinate in the source RF (ITRF96 at epoch 2000.0 corrected for subsequent seismic deformation) is estimated using:

XX=+m ()D X A(tAB ) A A A éùéùéù---4741517.232 0.463 4741516.769 X =+êúêúêú480470.376 (- 1)× 0.571= 480469.805 A(tAB ) êúêúêú ëûëûëûêúêúêú--4225017.855 0.577 4225018.432

The propagated ITRF96 coordinate at epoch 2000.000 corrected for later earthquakes is also identical to the NZGD2000(v20180701) coordinate: éù-4741516.769 X = êú480469.805 NZGD2000(v20180701) êú êú-4225018.432 ëûCARTESIAN

XNZGD2000(v20180701) = (268340.683,53743336.549,257.285)UTM60 S & h

The interframe transformation operation fm[()]XX+D from ITRF96 at epoch 2000.000 ABt (AB) A A A corrected for earthquakes can then also use the NTv2 NZGD2000 to NZGD49 grid transformation which estimates the following transformed coordinates in the target frame (there is no propagation in this frame):

XNZGD49(v20180701) =(- 41.75073795,174.21364624,245.344)flh

The position is compared with the 4th October 2010 tabulated NZGD2000 position transformed to NZGD49 using the NTv2 grid transformation as follows:

XNZGD49(2000.000) =(- 41.75073798,174.21364607,245.332)flh

The tabulated minus modelled residuals (uE, uN, uh) are:

υ(0.014,0.004,0.012)X =--- TOPOCENTRIC

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7.3 New Zealand case study 2 – multiple SSEs

7.3.1 Tectonic setting of Gisborne

The Gisborne region of the North Island of New Zealand lies within the foreland of the Hikurangi subduction margin along the Pacific and Australian Plate boundary zone (Figure 7.6). Deformation in the region is characterised by frequent slow-slip events (SSEs) which result from subduction of the Pacific Plate beneath the Australian Plate along the Hikurangi Trough (Wallace and Beavan, 2010; Wallace and Eberhart-Phillips, 2013; Wallace et al., 2018). GNS Science, New Zealand, has established a GNSS CORS network (GeoNet) in this region to actively monitor deformation and CORS GISB located NW of Gisborne has been selected for this case study (Figure 7.7). GISB is also a zero-order geodetic station that forms part of the NZGD2000 fiducial network. The station has been operating continuously since 2002 and has recorded displacement from a large number of SSEs. For this reason the station has been chosen as a case study for separation of cumulative SSE displacement and interseismic motion prior to use in the schema. The current NZGD2000 Deformation Model does not account for SSEs and site velocities interpolated from the model conflate both secular interseismic and episodic SSE displacements.

Figure 7.6 Location of 2nd New Zealand case study (background image: www.niwa.co.nz)

159

Figure 7.7 GeoNet – CORS station GISB (image: www.linz.govt.nz)

This case study shows how improvement in positioning accuracy can be achieved by modelling SSE separately from interseismic motion using the cumulative patch model approach described in Chapter 4.

7.3.2 GISB time-series analysis

Time-series data for GISB were obtained from the GNS Geonet time-series web server (GNS, 2020). The GNS Geonet time-series data are topocentric East and North displacements in terms of ITRF2008, however NZGD2000 and its associated deformation model are defined in terms of ITRF96. A selection of GISB RINEX files widely spaced over the observation history were downloaded from the GNS data server and processed using the LINZ PositioNZ-PP online processing service. The output ITRF2008 and ITRF96 coordinates enabled the topocentric ITRF2008 time-series to be fitted to these frames with a precision of 2 mm. The topocentric time-series of GISB in terms of ITRF96 was then generated for analysis in this case study. The time-series clearly shows episodic SSE deformation as well as the trending interseismic site velocity (Figure 7.8).

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Figure 7.8 GPS time-series for GISB CORS showing SSEs and trending interseismic motion. The North time-series is shown in red, the East time-series is shown in blue and the epochs of SSE are indicated with purple arrows. (GNS GeoNET data portal https://fits.geonet.org.nz/ )

7.3.3 Estimation of GISB interseismic velocity from the NZGD2000 deformation model

The first part of the case study is to analyse the performance of the current NZGD2000 deformation model (LINZ, 2019) in such a challenging deformation setting. The interseismic velocity grid sub-model is described by the csv file: grid_igns2011_nz.csv. The modelled topocentric horizontal ( E! and N! ) velocities around GISB are tabulated (Table 7.15). In this case study, vertical velocities and displacements are not considered as they are not currently estimated in the NZGD2000 deformation model.

Node coordinates ITRF96 Topocentric velocities (m yr-1) Node Longitude Latitude E! N! SW 177.8 -38.7 0.001842 0.019837 SE 177.9 -38.7 0.000754 0.018977 NW 177.8 -38.6 0.002614 0.019788 NE 177.9 -38.6 0.001498 0.019227 Table 7.15 Interseismic velocity model node data used for GISB (LINZ, 2018)

The interseismic topocentric velocity (in terms of ITRF96) for GISB was then estimated by bilinear interpolation of the model using Eqs. 5.9 to 5.11 as follows:

E! = 0.0014m yr-1 N! = 0.0192m yr-1 The LINZ geodetic database (https://www.geodesy.linz.govt.nz/gdb/) was accessed to obtain the current NZGD2000 coordinate information for zero order station GISB. These are shown in Table 7.16.

ellipsoidal coordinates GISB Latitude Longitude Ellipsoid Height NZGD2000 ([email protected]) 38° 38' 07.21293" S 177° 53' 09.72572" E 87.177 m Table 7.16 GISB tabulated NZGD2000 coordinates (v20180701 definition)

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A plot of the comparison between observed displacements with respect to the deformation model is shown in Fig. 7.9.

Figure 7.9 Comparison between predicted NZGD2000 deformation model and observed displacements. Residuals are shown in an ITRF96 aligned topocentric frame with the predicted velocity signal removed.

The plots show good agreement at the start of the observation timeseries, however the effects of differential interseismic motion and episodic SSE are evident as the residuals increase with time approaching 30 mm by 2020 with a mean residual of 13 mm E and 11 mm N over the observation timespan.

7.3.4 Deconvolution of the GISB timeseries into interseismic and SSE components

The next step was to deconvolve the GISB timeseries into trending interseismic and SSE components. The episodic nature of SSE makes automation of the process difficult. In this case study the trending interseismic velocity was tuned visually to optimise the "stepness" of the time-series with flat interseismic motion and inclined SSE/coseismic displacement (Figure 7.10). The optimal tuning interseismic velocities were estimated as:

E! =-0.0080m yr-1 N! = 0.0215m yr-1

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The cumulative SSE displacement in the format used in the schema was estimated as:

å DEC()= 0.196 m å DNC()=- 0.067m 2000® 2019.548 2000® 2019.548

Figure 7.10 Topocentric timeseries of GISB displacement with the interseismic displacement signal minimised

The observed SSE displacements have a recurrence interval of approximately ~ 2 years with mean displacements of 0.018 m E and -0.006 m N associated with each event taking place over a time span of up to 0.08 yr (1 month).

7.3.5 Forward modelling using deconvolved SSE timeseries

At present the magnitude of the SSE is absorbed into the allowable tolerance of the NZGD2000 deformation model and the PU assigned to NZGD2000 geodetic stations. While these tolerances may be acceptable for current positioning requirements, the SSE displacements are not modelled separately from the overall interseismic displacement signal and are implicitly included. In the future, more stringent tolerance requirements for real-time positioning precision may require deconvolution of time-series

163 affected by SSE into interseismic and coseismic displacements or postseismic functions for each event. The cumulative displacement can be used in the schema approach described in Chapter 6 for intraframe propagations within SSE affected regions. The benefit of this approach is that interseismic velocities can be used for CORS monitoring of SSE with any residual displacements (observed minus modelled) likely to be attributable to an incipient SSE event.

The coseismic and postseismic displacement grid models could potentially include these cumulative SSE displacements that can be interpolated. This approach would benefit from a denser CORS network, or campaign-style occupation network in the SSE affected regions. In addition, Insar and other remote sensing techniques could be used to make higher resolution models of SSE surface displacements.

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Chapter 8 Discussion and Recommendations

This thesis has described a logic framework for a time-dependent transformation schema especially suited to tectonically active or GIA affected regions, but which can also be applied to tectonically stable regions. The schema fills a vital gap in current geodetic transformation workflows and pipelines for enabling spatial data alignment over time as well as the need for GNSS positioning and spatial data to be properly aligned in context. The case studies used to demonstrate and test the schema show the significant improvement in precision that can be attained over existing and somewhat simplistic transformation strategies.

Application of a robust time-dependent schema to handle complex deformations is essential for contemporary geodetic applications. The main driver for this is the rapid uptake of precision GNSS positioning by a diverse spectrum of users many of whom have limited knowledge of geodesy or geodynamics. The significant potential benefits arising from this positioning accuracy are at risk of being compromised by a lack of functionality within spatial data software to handle this precision in the context of a highly dynamic Earth.

Geodesy and geophysical studies focussing on plate tectonics and Earth deformation are significantly advanced and fully harness the improvements in space geodetic positioning methods such as GNSS. From a scientific perspective it seems eminently logical that geodetic reference frames in practice should account for the multitude of deformations that affect the Earth’s surface. So, it seems somewhat paradoxical that there is still a gap in the application of these major scientific advances within spatial data management and surveying.

Time dependent geodetic transformations are still a novelty in many GIS packages and positioning systems. High-end users of reference frames such as civil project engineers should be reliant on the surveying and spatial professions to advise on the application of reference frames on their projects. In reality though, misunderstandings and failure to consider RF differences are impacting on the integrity of many projects. This arises from misalignment of field surveys and designs. In developed countries where there is usually a robust management of geodetic reference frames in practice, the problem is overcome by well managed geodetic infrastructure (including geodetic information), accessible GNSS data streams and well distributed surveying directions and guidelines. In many developing countries where this infrastructure is not as well established, kinematic global reference frames are often used in a somewhat ad hoc manner. This leads to significant issues with misalignments between positioning, design and setting out. It can also lead to issues with definition of cadastral boundaries by coordinates using unlocalised GNSS positioning. The problem is exacerbated by the limited geodetic knowledge of survey practitioners in many developing countries.

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Another emerging issue is how to correctly align and visualise the exponentially increasing volume of spatial data being captured by technologies such as laser scanning and UAVs. Sensors for these products are often controlled by GNSS, but the actual data collected is static. In other words, the data collected does not intrinsically contain deformation information. The problems arise when this data are used, often many years later in conjunction with GNSS positioning and where the data are not transformed to the later epoch, or vice versa. Similar problems arise if later point cloud data are merged with earlier point clouds and the misalignment is not identified or identified but ignored. Urban data models such as integrated BIM are intrinsically static in nature. The frame for these emerging products does not currently handle RF kinematics and displacement.

There is also a great effort made by many geodetic agencies and jurisdictions to modernise national geodetic datums to ensure contemporary alignment with global reference frames used intrinsically by positioning services such as GNSS. The very sound argument used to justify these datum modernisation programs is that mass-market access to precise positioning requires underlying spatial data to align with the GNSS position. Examples include navigation for autonomous vehicles, precision instrument landings for aircraft, pilotless manoeuvring of vessels in port areas, automated mining, UAV delivery services, precision agriculture and so on. These strong drivers for change are motivated primarily by safety and efficiency considerations and the need for simplicity in geodetic transformations. Other drivers include monitoring pressing global challenges such as sea level rise and land subsidence in low lying cities using a global reference frame. Compatibility of spatial data across national borders is also improved if national datums are aligned closely with a global reference frame. At a regional level these initiatives have already been well advanced with EUREF and SIRGAS, for example.

The underlying requirement for these aims to become a reality is a robust strategy for time-dependent transformations between any reference frames used for both positioning and spatial data management. Implementation of time dependent transformations within GIS is still in its infancy. To some extent this is due to a lag in standards agencies and geodetic registries to properly define and handle time dependence in a robust way. Efforts to date have concentrated on conformal time-dependent transformations such as a 14 parameter transformation. These have been applied in some post-processing services such as AusPOS, NRCan-PPP and OPUS. These approaches work well in stable tectonic plate settings where earthquakes are rare and plate boundary deformations are very localised. In tectonically active countries, development of time-dependent transformation tools and deformation models has been achieved independently at a jurisdictional level (e.g. LINZ, New Zealand). The structure and format of the models used is different in every case. The lack of a global standard or approach makes implementation of time-dependent transformations within GIS and positioning problematic.

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Chapter 2 reviewed the impact of Earth deformation on RF and positioning, highlighting the differences between stable plate regions and plate boundary regions noting the different approaches required for handling transformations in these different settings. In generally stable plate settings such as Australia, Europe (excepting Scandinavia, Southern Italy and Greece), Eastern USA, Brazil, Africa (excepting the Atlas region and Rift Valley), Southern India and most Pacific Island states, a plate motion model is sufficient to handle most of the observed displacements. In other regions that include active plate boundaries and/or GIA, a grid model of interseismic velocities is recommended. Such countries and regions would include: Japan, Taiwan, The Philippines, Indonesia, Papua New Guinea, New Zealand, Chile, Peru, Mexico, California, Canada, Alaska, Iceland, Fennoscandia, Southern Italy, Greece, Turkey, Israel, Caucasus, Iran, Afghanistan, Nepal, China and Myanmar. In addition to an interseismic velocity model, coseismic and postseismic models would be required for affected regions after each significant earthquake to update existing RF to account for the earthquake deformation. A strategy for the development of a plate fixed RF and transformation to a global frame was also outlined. This approach has already been adopted in Australia with the implementation of GDA2020 and ATRF.

Chapter 3 examined positioning and dimensional tolerance standards for different geodetic applications, considering the effects of deformation on positioning. The impact of deformation was analysed in more detail in Chapter 4 to consider the time validity of reference frames particularly in deforming zones. In plate boundary zones with very high interseismic strain rates such as New Zealand and Papua New Guinea, RF updates for the reference epoch are required at least every 20 years. In other earthquake prone areas, updates are required after a number of significant earthquakes to recalibrate or redefine the RF. The benefit of this update is that all deformation (both interseismic and seismic) can be conflated into a single transformation grid with a commensurate reset in the propagated uncertainty. Chapter 5 recommended a standard format and metadata requirements for time-dependent transformations. These recommendations were developed in Chapter 6 into a full schema that can accommodate the most complex deformation settings.

Finally, in Chapter 7 case studies for diverse tectonic settings in Australia and New Zealand were presented to demonstrate the application of the schema and PMM and the improvements in precision over existing methods.

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