Toolbox for Analysis of Flexural Isostasy (TAFI)—A MATLAB Toolbox for Modeling Flexural Deformation of the Lithosphere GEOSPHERE; V

Software Contribution

GEOSPHERE Toolbox for Analysis of Flexural Isostasy (TAFI)—A MATLAB toolbox for modeling flexural deformation of the lithosphere GEOSPHERE; v. 13, no. 5 Sumant Jha, Dennis L. Harry, and Derek L. Schutt Department of Geosciences, Colorado State University, 1482 Campus Delivery, Fort Collins, Colorado 80523, USA doi:10.1130/GES01421.1

7 figures; 5 tables; 2 supplemental files ABSTRACT typically involving subsidence in the vicinity of the load (assuming the load represents an addition of weight to the lithosphere) and uplift in surrounding CORRESPONDENCE: sumant.jha@colostate.edu This paper describes a new interactive toolbox for the MATLAB computing areas (the peripheral “bulge”) (Fig. 1). The magnitude and wavelength of the platform called Toolbox for Analysis of Flexural Isostasy (TAFI). TAFI supports flexural deflection depends on the magnitude of the load and the strength of CITATION: Jha, S., Harry, D.L., and Schutt, D.L., two- and three-dimensional (2-D and 3-D) modeling of flexural subsidence the lithosphere, with greater strength resulting in more broadly distributed 2017, Toolbox for Analysis of Flexural Isostasy (TAFI)—A MATLAB toolbox for modeling flexural and uplift of the lithosphere in response to vertical tectonic loading. Flexural (larger-wavelength) deformation. Flexure results in variations in the regional deformation of the lithosphere: Geosphere, v. 13, deformation is approximated as bending of a thin elastic plate overlying an gravity field due to deformation of density interfaces within the lithosphere. no. 5, p. 1555–1565, doi:10.1130/GES01421.1. inviscid fluid asthenosphere. The associated gravity anomaly is calculated by Such density interfaces may include the crust-mantle boundary, the interface summing the anomalies produced by flexure of each density interface within between the sedimentary basin fill and underlying rocks, the seafloor in marine Received 26 August 2016 the lithosphere, using Parker’s algorithm. TAFI includes MATLAB functions areas, and other significant intra-crustal interfaces. The relationships between Revision received 9 March 2017 Accepted 5 July 2017 provided as m-files (also called script files) to calculate the Green’s functions flexural deformation and the gravity field are dependent upon the partition- Final version posted 11 September 2017 for flexure of an elastic plate subjected to point or line loads, and functions ing of the load between surface (topography) and subsurface (buried density to calculate the analytical solution for harmonic loads. Numerical solutions anomalies) components (Karner and Watts, 1983; Forsyth, 1985; Macario et al., for flexure due to non-impulsive 2-D or 3-D loads are computed by convolv- 1995; McKenzie, 2003). ing the appropriate Green’s function with a spatially discretized load function read from a user-supplied file. TAFI uses MATLAB’s intrinsic functions for all x computations and does not require any other specialized toolbox, functions, b x or libraries except those distributed with TAFI. The modeling functions within 0 X TAFI can be called from the MATLAB command line, from within user-written Peripheral Bulge z1 programs, or from a graphical user interface (GUI) provided with TAFI. The GUI w z facilitates interactive flexural modeling and easy comparison of the model to Flexural Moat b 2 gravity observations and to data constraining flexural subsidence and uplift. ρi x Q0 ma w ρc

INTRODUCTION ρm

OLD G In this paper we present the Toolbox for Analysis of Flexural Isostasy (TAFI). TAFI is a suite of tools for the MATLAB computing platform that supports forward modeling of flexural subsidence and uplift of the lithosphere in response Z to tectonic loading. TAFI also calculates the gravity anomaly due to flexure of Figure 1. Flexure of an elastic plate under an applied line load Q0. OPEN ACCESS the lithosphere, and allows for easy comparison of modeled deflection and Variables characterizing the shape of the basin are the distance be- gravity curves with observations to facilitate a rapid interactive search for the tween the load position to the crest of the flexural bulge,x b; the distance from the load to the point of zero deflection, ; the maxi- best-fitting flexural model. x0 mum deflection of the basin,w max; and the amplitude of peripheral

Vertical tectonic loads emplaced on the lithosphere may be in the form uplift, wb. Density interfaces are present at the surface and base of of static surface loads such as topography, static subsurface loads due to lat- the plate, at depths z1 and z2 (relative to user defined datum) prior eral variations in lithosphere or asthenosphere density, or dynamic subsur- to flexural bending, respectively. The density of the infill, crust, and mantle are ρi, ρc, and ρm respectively. Additional optional density This paper is published under the terms of the face loads created by tractions at the base of the lithosphere imposed by flow interfaces within the plate (not shown) can be used when computing CC‑BY license. in the underlying mantle. These tectonic loads result in vertical deformation, the gravitational response (see text).

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1555 by guest on 28 September 2021 Software Contribution

The flexural behavior of the Earth’s lithosphere can be approximated with The Green’s function for a line load is a 2-D flexural profile (in thex -z plane) a model in which the lithosphere is treated as a thin elastic plate overlying an across a basin of infinite length, normalized by the magnitude of the load. In inviscid fluid asthenosphere (Watts, 2001). Analytical solutions describing the TAFI these are referred to as 2-D plate models, with the basin shape described flexural bending of homogeneous infinite and semi-infinite thin elastic plates along a profile measured perpendicular to the trend of the line load. TAFI subjected to point, line, or harmonic loads have been widely used in flexural computes the flexural deformation for spatially distributed 2-D loads (loads isostatic analysis (Hertz, 1884; Nadai, 1963; Walcott, 1970). These elastic plate with finite width and infinite along-strike extent) by convolving the line load models have been shown to match the long-wavelength (>25 km) bathymet- Green’s function with a load discretized along a profile trending perpendic- ric features and the gravity field near most major oceanic tectonic features, ular to strike (the x-direction). Greens functions for both infinite (unbroken) including ocean trenches, ridges, island chains, transform faults, and hotspot plates and semi-infinite (broken) plates are provided in TAFI. For 2-D models, swells (Turcotte, 1979; Watts et al., 1980; Bodine et al., 1981; Dahlen, 1981). On TAFI also provides an analytical solution for harmonic loads, in which the load continents, elastic plate models have also been applied to analyze isostasy magnitude varies sinusoidally in the x-z plane and is of infinite extent in the in continental rifts, mountain belts, and sedimentary basins (e.g., Karner and y-direction. Watts, 1983; Watts et al., 1982; Chase and Wallace, 1988; Forsyth, 1985; McNutt TAFI computes the gravity field resulting from the flexed plate usingParker’s et al., 1988; Flemings and Jordan, 1989; Weissel and Karner, 1989; Stern and method (Parker, 1973). This allows the user to specify a layered density struc- ten Brink, 1989; ten Brink et al., 1997). Elastic plate models have also been used ture for the lithosphere, with the gravitational attraction of each flexed layer to assess glacial isostatic rebound (e.g., Stern et al., 2005) and to analyze isos- summed to compute the total gravity anomaly. Examples of density interfaces tasy on scales spanning continents and ocean basins (Banks et al., 1977; Watts that contribute strongly to the gravity anomaly are the crust-mantle boundary, and Burov, 2003; McKenzie and Bowin, 1976; Watts, 1978; McNutt and Menard, the interface between the sedimentary basin fill and underlying crust, and in 1982; Lowry and Smith, 1994; Djomani et al., 1995; McKenzie and Fairhead, marine environments, the seafloor. 1997; Petit and Ebinger, 2000; Flück et al., 2003; Crosby and McKenzie, 2009). The MATLAB geodynamic modeling functions provided in TAFI to calcu- TAFI provides an easy-to-use interactive tool to model isostatic deforma- late plate flexure and the corresponding gravity anomaly are similar to those tion of the lithosphere as flexural bending of an elastic plate. TAFI consists of a available in other software packages. Examples include Lithflex1 and Lithflex2 suite of MATLAB scripts and functions, coded as m-files, which facilitate rapid (http://csdms .colorado .edu /wiki /Geodynamic _models; Slingerland et al., 1994); interactive forward modeling of flexural deformation of a thin elastic litho Flex2D (http://www .ux .uis .no /~nestor /work /programs .html); MECAIR (http:// sphere overlying an inviscid asthenosphere. Only intrinsic MATLAB functions aconcagua.geol.usu.edu/~arlowry/code_release.html); gFlex, (https://github are used, providing portability across operating systems without need of spe- .com/awickert /gFlex; Wickert, 2016), and MATLAB routines by Fredrik Simons cial libraries or packages other than MATLAB itself. TAFI computes analytical (http://geoweb .princeton .edu /people /simons /software .html). The primary Green’s functions that represent the flexural response of a thin elastic plate feature that distinguishes TAFI from these other software packages is TAFI’s subjected to vertical impulsive (line or point) loads. The flexural response for easy-to-use graphical user interface (GUI), which facilitates interactive flexural non-impulsive loads can be obtained by convolving the Green’s functions with modeling through easy modification of model parameters and instantaneous discretized functions representing loads that are distributed either in a radially visual comparison of model results to data. The GUI provides a powerful in- symmetric pattern (simulating, for example, a seamount), along linear profiles structional tool, allowing users to interactively explore how changes in plate (simulating loading by a thrust belt or seamount chain), or arbitrarily on a hori boundary conditions, plate strength, load position and magnitude, and super- zontal (x-y) plane (topography). position of loads affect flexural deformation. TAFI’s GUI is designed to serve The Green’s function for a point load represents a two-dimensional (2-D) as a template to which additional isostatic analysis tools, such as solutions for radial cross section in the r-z plane (r is the radial coordinate position) across a different plate rheologies (e.g., viscous, viscoelastic, or visco-plastic), may be radially symmetric flexural basin, normalized by the magnitude of the load. In added by users familiar with MATLAB coding. TAFI, these are referred to as axisymmetric plate models, with the basin shape described along a profile measured radially away from the point load. The point load Green’s function can be convolved with a spatially distributed load to com- METHODS pute the flexural response for axisymmetric loads of finite radius and for loads that are distributed arbitrarily in the horizontal (x and y) directions. In the first in- Calculating Flexural Deformation stance, the user supplies a load discretized as a function of radial distance, with TAFI returning a flexural profile across a similarly axisymmetric basin. In the TAFI models the lithosphere and asthenosphere system as a thin elastic second instance, the user supplies a load discretized on an x-y grid. In this case, plate of constant flexural rigidity overlying an inviscid substrate. Flexure is the TAFI returns a grid representing the three-dimensional (3-D; x, y, and z) shape of result of a vertical load (pressure) applied to the plate. Horizontal forces are the basin. Accordingly, TAFI refers to such models as 3-D plate models. neglected, as these have a relatively small effect on the elastic flexural profile

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1556 by guest on 28 September 2021 Software Contribution

3 under reasonable geological conditions (Caldwell et al., 1976; Turcotte, 1979). α −x /α x wx()= Q0 ecos semi-innite plate, line load; (5) The equations describing the vertical deflection of the plate in 2-D Cartesian 4D α and radial coordinates are: α2 r wr()= Q0 kei innite plate, point load, (6) 4 2παD Dw∇+()xy,,∆ρgwx()yq= a ()xy, ; (1)

where a is the flexural parameter, which depends on the flexural rigidity and  d 2 1 d  2 D + ww()rr+=∆ρg () q ()r , (2) density structure of the plate (Table 1), Q0 is the load magnitude, and kei is the  drr 22dr  a zeroth-order Kelvin function (Hertz, 1884; Nadai, 1963). For the special case of

Q0 = 1, Equations 4–6 are impulse responses (Green’s functions), which can be where w is the vertical deflection of the plate, ∇4 is the fourth derivative gra- convolved with load functions to obtain the flexural deformation of the plate

dient operator, qa is the vertical load applied to the plate, Dr is the density under geologically realistic spatially distributed (non-impulsive) loads. For a contrast between the mantle underlying the plate and the material filling the distributed 2-D sinusoidal surface (topographic) load on an infinite plate, the basin (usually either water or sedimentary rocks), x, y and r are Cartesian and solution to Equation 1 is: radial coordinate positions (respectively), g is gravitational acceleration, and D Q x is the flexural rigidity of the lithosphere (Hertz, 1884; Nadai, 1963).D depends wx()= 0 sin2π , (7) 4 λ on the thickness of the elastic plate (Te), the plates’ Young’s modulus (E ), and  2π ()ρρmc− +D   Poisson’s ratio (n):  λ 

3 ETe D = . (3) where r is the density of mantle, r is the density of the rocks forming the 12()1− ν2 m c topography and filling the basin (usually the crystalline upper crust), and l is Analytical solutions of Equations 1 and 2 for a line load and infinite (con- the wavelength of the load (Turcotte, 1979).

tinuous) plate geometry, a line load and semi-infinite (broken) plate geometry, The maximum deflection of the plate w( max), amplitude of the peripheral

and a point load and infinite plate geometry are: uplift (wb), distance from the load to the crest of the peripheral uplift (xb), and

3 the point of zero deflection x( 0) between the basin and peripheral uplift are α −x /α  xx wx()=+Q0 ecossin  innite plate, line load; (4) attributes commonly used to compare flexural models to observations (Fig. 1). 8D  αα

TTABLEABLE 1. OUTPUT PPARAMETERSARAMETERS CHARACTERIZINGCHARACTERIZING THE FLEXURAL BABASINSIN SHAPE ZeZeroro crcrossingossing distance FlexuralFlexural bulgebulge positionposition FlexuralFlexural parameterparameter PePeripheralripheral bulgebulge amplitude

Isostatic model (x0) (xb) (aα) (wb)

14/ 14/ 3 Elastic halfhalf-space-space,, π 3π  4D  Q0α α α −000.06767 2-D line load* () 2-D line load* 2 4 ()ρρmi− g 4D 14/ 14/ 3 Elastic continuous plate,plate, 3π  4D  Q0α α παpa −000.043432 2-D line load* () 2-D line load* 4 ()ρρmi− g 8D

14/ #  D  2 α rb Elastic continuous plate,plate,   Q kei b # ––– – ETe Q0 kei point load ()ρρmi− +  2πD α  R2 

14/ −Q 14/ * − 0 Elastic continuous plate,plate, λ λ  D  4 2π 2-D sinusoidal load*  g  π 2‑D sinusoidal load* 2 4 ρi g ()ρρmc−+gD   λ 

Note:Note: 22‑D—t-D—twowo‑dimensional;-dimensional; aα—flexural—flexural parameter;parameter; D—flexural—flexural rigidity;rigidity; Q0—load magnitude; rρm—mantle densitdensity;y; rρi—infill—infill densitdensity;y; rb—radial position of flflexuralexural bulge;bulge; lλ—sinusoidal load wavwavelength;elength; E—Young’s—Young’s modulus; Te—elastic thicthickness;kness; R—Ear—Earth’sth’s radiusradius;; g—acceleration due to gravitgravity;y; keikei—zeroth—zeroth orderorder of KelvinKelvin function. – (dash)—closed foformrm solutions not known.known. ParametersParameters areare determineddetermined by searchsearch of solution vectorvector w. *Turcotte*Turcotte and SchubertSchubert (2002). #BrBrotchieotchie and SilSilvestervester (1(1969).969).

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1557 by guest on 28 September 2021 Software Contribution

These attributes, which describe the shape and width of the basin, depend PROGRAM DESIGN AND COMPUTATIONAL APPROACH 1 8 on the flexural parameter, a, and are obtained in closed form for the line and 2 harmonic load models (Table 1). For the point load model, these values are TAFI is structured hierarchically, with an upper-level script that provides a determined by searching the calculated deflection curve for the minimum, GUI to lower-level functions that perform the geodynamic calculations, visuali maximum, and zero value nearest to the load. zation, and input-output operations (Fig. 2; Table 2). The appearance of the GUI 3 is defined in the file “TAFI.fig” (Fig.1 3 ). This is a MATLAB encoded file that de- 6 4 Calculating the Gravity Anomaly fines the elements available from within the GUI. TAFI.fig also associates the 5 GUI elements with actions and variables used in TAFI for modeling and visual- Flexure of density interfaces within the lithosphere produces a gravity ization. The actions invoked by user interaction with the GUI elements are pre- 7 anomaly that is dependent upon the density contrast across the interface, the scribed in the elements’ respective callback functions, which are coded in the mean depth of the interface, and the shape of the flexural deformation. The file “TAFI.m”. The callback functions call TAFI lower-level functions that perform 1If viewing Figure 3 in the full-text article, please visit http://doi .org /10 .1130/GES01421 .S1 to download the gravity anomaly produced by flexure of a single density interface is calculated computational, data input-output, or visualization tasks and pass variable values PDF and, using Adobe Acrobat or Reader, interact with using Parker’s method (Parker, 1973): obtained from the GUI elements (e.g., the flexural rigidity) to these functions. its layers. n−1 The geodynamic and gravity modeling functions are MATLAB implementa- ∞ ||k n F()∆∆gz=−2πρG exp()− ||kr0 F[]h (). (8) tions of Equations 4–7 and Equation 8, respectively. These functions can also ∑n=1 Supplementary Lile to Jha et al., 2017, Toolbox for Analysis of Flexural Isostasy (TAFI) n! –A Matlab® toolbox for modeling Llexural deformation of the lithosphere. be called from the command line, allowing access from outside of the GUI or

This supplementary Lile describes TAFI's directory structure, GUI use and input Lile formats. Here, F(Dg) is the Fourier transform of the gravity anomaly, r is the position from within user-written programs (Table 3). The MATLAB implementation of vector, k is the wavenumber, G is the universal gravitational constant, Dr is the Equations 4–6 to obtain Green’s functions (by setting Q = 1) for the flexural TAFI DIRECTORY STRUCTURE 0

TAFI is organized into a main directory, “TAFI”, which contains the Liles deLining the density contrast across the interface, F(h) is the Fourier transform of the de- response of an elastic plate to impulse (line or point) loads is straightforward,

GUI ("TAFI.Lig" and "TAFI.m") and a README Lile describing TAFI’s installation and use trended flexural relief on the interface (prescribed by the plate flexure), andz 0 as is the implementation of Equation 7 to describe harmonic loading. Flexure (manuscript Figure 2, Table 2). The main TAFI directory also contains nine subdirectories. is the mean depth of the interface. Inverse Fourier transforming F(Dg) provides due to non-unit impulse loads is obtained by scaling the Green’s function by a The subdirectory "GUI_Functions” contains Liles accessed by the GUI that deLine icons, the the gravity field produced by topography on the buried interface. The gravita- user-specified load magnitude. For distributed 2-D loads, the flexural response default parameter values that populate the GUI when it is opened (deLined in the Lile

GUI_Functions/Defaults/TAFI_Defaults.m), and functions needed to enable or disable GUI tional effect of multiple density interfaces within the lithosphere is obtained by is calculated by convolving the user-supplied discretized load function with the elements depending on the plate and load geometry combinations. The subdirectories summing the effect of each interface. semi-infinite or continuous plate Green’s functions for line loads (Equations “Geodynamic_Functions” and “Gravity_Functions” contain the functions used to calculate 4–5). For 3-D distributed loads, the flexural response is calculated by convolving the Llexure and gravity proLiles (2-D) or Lields (3-D). The “Load_Functions” subdirectory

contains functions to read and re-sample the user-provided discretized load, and functions Upper-level script the user-specified load matrix with the point load Green’s function (Equation to convolve the uniformly sampled load function with the Green’s Function. The resampling 6). MATLAB’s intrinsic convolution functions require that the load and Green’s Graphical User Interface and convolution functions can be called from the Matlab command line, as well as from functions be discretized at similar uniform intervals. For 2-D models, the load Lower-level functions is resampled to the Green’s function interval (specified by the user) prior to 2Supplemental File 1. TAFI’s installation, directory structure, use and properties of GUI elements, default 1 Main convolving. For 3-D loads, the Green’s function is calculated at the grid spacing 6 behaviors, input-output file formats, and other oper- 2 used in the user-supplied load function, and no resampling is required. ating instructions. Please visit http://doi .org /10 .1130 Supplemental TAFI.m TAFI.fig To compute the gravity anomaly, TAFI requires the Fourier transform of the /GES01421.S2 or the full-text article on www.gsapubs graphics (GUI control, data I/O, GUI .org to view Supplemental File 1. utilities deflection in the wavenumber domain (Equation 8). Estimation of the gravity graphics, geodynamic appearance and gravity callbacks) field in the wavenumber domain via the series summation in Equation 8 is then straightforward. The gravity field in the wavenumber domain is then inverse Fourier transformed to yield the gravity field as a function of position. Four terms of the series in Equation 8 are used, as empirical testing showed 3 that this is sufficient to reproduce the calculated gravity anomaly to within 10–9 Geodynamic 5 mGal in realistic modeling scenarios. functions 4 Data I/O calculate deflection Gravity functions calculate FLEXURE AND GRAVITY MODELING USING TAFI gravity

TAFI’s installation, directory structure, use and properties of GUI elements, Figure 2. Hierarchical structure of Toolbox for Analysis of Flexural Isostasy (TAFI). Upper-level script default behaviors, input-output file formats, and other operating instructions 3Supplemental File 2. TAFI m-files. Please visit http:// (TAFI.m) provides a graphical user interface (GUI; TAFI.fig) to lower-level functions that perform the geo- 2 doi.org/10.1130/GES01421.S3 or the full-text article dynamic and gravity calculations and to data and model input-output (I/O) utilities. Supplemental graphic are described in Supplemental File 1 . The TAFI m-files are in Supplemental on www.gsapubs.org to view Supplemental File 2. utilities are used to plot model results. Numbers correspond to TAFI’s MATLAB functions (Table 2). File 23. Here, we briefly describe TAFI’s functionality. All operations described

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1558 by guest on 28 September 2021 Software Contribution

TTABLEABLE 2. TTOOLBOXOOLBOX FOR ANALANALYSISYSIS OF FLEXURAL ISOISOSTASYSTASY (T(TAFI)AFI) DIRECTDIRECTORYORY SSTRUCTURETRUCTURE TToolboxoolbox ffilesiles DescrDescriptioniption 1T1 TAFI.mAFI.m Main TTAFIAFI ffile,ile, which contrcontrolsols intinteractioneraction with GUI objects 2T2 TAFI.figAFI.fig TTAFIAFI GUI ffileile with slidersliders,s, edit bboxes,oxes, bbuttons,uttons, drdropdownopdown menusmenus,, and plotsplots.. AAccessedccessed frfromom TTAFI.m.AFI.m. SubdirSubdirectoryectorySSecondecond-le-levelvel subdirsubdirectoryectoryDDescriptionescription 5E5 Examplexample DDataata (NGDC Aleutian TTrenchrench data BathBathymetricymetric data in kilomekilometers.ters. used fforor the case study in this paper) FrFreeee-air-air gravitgravityy data in mGalmGal.. 3G3 Geodynamiceodynamic FFunctionsunctions (functions ttoo calculatcalculatee Geodynamic Utilities FFunctionsunctions ttoo dedefinefine defdefaultault constants and calculatcalculatee the flflexuralexural wwavelengthavelength (Equations 4–6 in ttext).ext). thethe fflexurallexural rresponseesponse of an elastic platplate)e) GrGreen’seen’s FFunctionunction CContinuous2D_flex.m,ontinuous2D_flex.m, CContinuous3D_flex.m,ontinuous3D_flex.m, Halfspace2D_Halfspace2D_flex.m,flex.m, CContinuous3Dgrid_flex.montinuous3Dgrid_flex.m Sinusoidal HarHarmonic2D_flex.mmonic2D_flex.m 4G4 Gravityravity FFunctionsunctions (functions ttoo calculatcalculatee the gravity_Cgravity_Callback.m—Interactsallback.m—Interacts with GUI and parparkg.mkg.m or parparkg_3D.mkg_3D.m to calculatcalculatee the gragravityvity anomaanomaly.ly. gravity anomalanomalyy caused bbyy fflexure)lexure) parparkg.mkg.m and parparkg_3D.m—Useskg_3D.m—Uses PParker’sarker’s (1(1973)973) method ttoo calculatcalculatee the gragravityvity anomaanomaly.ly. 6 GUI FFunctionsunctions (functions ttoo contrcontrolol GUI DefDefaultault BacBackupskupsBBackupackup of TTAFIAFI defdefaultault ffiles.iles. element behabehaviorvior and appearance) GUI FFunctionunction Utilities FFunctionsunctions ttoo set slider defdefaultault ranges and vvaluesalues and enabenablele or disadisableble GUI elements and set the TTAFIAFI eenvironment.nvironment. 3 Load FFunctionsunctions (functions ttoo imporimport,t, ininterpolate,terpolate, CoConvolutionnvolution FFunctionsunctions ttoo coconvolvenvolve the imporimportedted load with GrGreen’seen’s function. and conconvolvevolve spatiallspatiallyy distrdistributedibuted loads with ImporImportt LoadLoad Load ffileile imporimportt utilitutilityy GUI widget and corrcorrespondingesponding MAMATLABTLAB ffileile which imporimportsts distrdistributedibuted load vvectorector (2-D)(2-D) and GrGreen’seen’s functions) ImporImportt GGridrid grgrid (3‑D).id (3-D). Load FFunctionunction Utilities UtilitUtilityy functions ttoo reread,ad, ininterpolate,terpolate, and rremoveemove the imporimportedted distrdistributedibuted loadload..

5O5 Outpututput PParametersarameters outputparam.m—Foutputparam.m—Functionunction to calculacalculatete output parametparametersers αa,, xbb,, x00,, wmamaxx,, andand wbb Plot FFunctionunction TTAFIPlot2D.mAFIPlot2D.m and TTAFIPlot3D.m—FunctionsAFIPlot3D.m—Functions used to plot fflexurelexure and gragravityvity rresponseesponse and data constraintsconstraints.. Manual (Help ffileile fforor TTAFI)AFI) FleFlexurexure Manual FFilesiles TToolboxoolbox fforor AAnalysisnalysis of FleFlexuralxural IsostasIsostasyy (T(TAFI)AFI) .doc and .pdf NoNote:te: NumberNumberss in the ffirstirst column corrcorrespondespond ttoo bboxesoxes in FFigureigure 2. AAbbreviations:bbreviations: NGDC—NationalNGDC—National GeopGeophysicalhysical DaDatata CCenter;enter; 2-D—t2-D—twowo dimdimensional;ensional; 3-D—th3-D—threeree dimensional; GUI—graphical user ininterface.terface.

PParameters:arameters: αa—f—flexurallexural parametparameter;er; xbb—f—flexurallexural bbulgeulge position; x00—z—zeroero crcrossingossing position; wmamaxx—maximum flflexuralexural amplitudeamplitude;; wbb—per—peripheralipheral bbulgeulge amplitudeamplitude..

below can be done outside of TAFI’s GUI by invoking TAFI’s functions from the instead of the load magnitude is passed to the corresponding geodynamic func- command line as described in Supplemental File 1 [footnote 2]. The GUI is tion. In the GUI, the user can vary these variables with sliders and edit boxes and designed to allow the user to vary model parameters and observe the results see the change in the computed flexural curve displayed in real time. Using the immediately using sliders, edit boxes, and dropdown menus, thus facilitating TAFI GUI, the user can also interactively change the load position and scale the interactive fitting of flexural models to data. magnitude of distributed loads. TAFI uses default values for g (9.8 m/s2), Earth’s Flexure and gravity modeling in TAFI requires passing model parameters radius (R, 6370 km), E (8.5 × 1010 N/m2), and Poisson’s ratio (0.25). These values as arguments to TAFI’s geodynamic modeling functions, which return cal- can be modified by users as described in Supplemental File 1 (footnote 2). culated deflection and gravity fields. The model parameters are specified in The elastic plate is required to have at least two density interfaces, one at the function invocation when using TAFI from the MATLAB command line. the top of the plate and another at the base of the plate. These density inter- When using the TAFI GUI, the model parameters are specified with sliders, edit faces represent the boundaries between the basin fill and underlying crust, boxes, and dropdown menus (Fig. 3 [see footnote 1]). The flexure and gravity and between the crust and mantle (Fig. 1). The mantle and infill densities curves are displayed in the GUI plot window and are updated interactively are required to calculate the flexural parameter (a) (Table 1), which is used when the model parameters are changed. to compute the flexural deflection of the lithosphere for non-harmonic loads To build a flexure model in TAFI, the user must specify the plate type, load (Equations 4–6). The density contrasts between these layers and the depth geometry, flexural rigidity, load magnitude, load position, and density contrast of interfaces are also used to calculate the gravity anomaly associated with between the mantle and basin fill. The functional form of the flexural response flexure of the lithosphere (Equation 8). When using the TAFI GUI, the density of the elastic plate for 2-D and 3-D flexure models is prescribed by the plate interfaces are defined by specifying the basin fill, crust, and mantle densities. geometry (infinite or semi-infinite plate) and the load geometry (selecting from When invoking TAFI from the MATLAB command line, the interfaces are deline, point, 2-D harmonic, 2-D distributed, axisymmetric distributed, or 3-D dis- fined by specifying the density contrasts between the layers and their depths. tributed load geometries; Table 4). The remaining model parameters are defined Additional density interfaces, representing lithological boundaries within the by the user and passed to TAFI’s geodynamic functions. In the case of distributed lithosphere, can be specified when accessing TAFI from the command line. loads (2-D or 3-D), load magnitudes are provided by the user as a MATLAB col- These additional density interfaces are used to compute the gravity anomaly, umn vector (2-D) or matrix (3-D). For sinusoidal loads, the load wavelength (l) but are not used in the flexural deformation calculation.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1559 by guest on 28 September 2021 Software Contribution

Figure 3. Toolbox for Analysis of Flexural Isostasy (TAFI) graphical user interface (GUI). When viewed in the PDF of this paper using Adobe Acrobat or Reader, clicking on green circles activates a dynamic figure demonstrating GUI features. These include (1) available plate geometry (pulldown menu); (2) available load geometry (pulldown menu); (3) changing the flexural rigidity (slider); and (4) changing the load magnitude (slider). Clicking on the red circle resets the figure to its initial display. Other model parameters controlled with sliders and text boxes are the load position, load wavelength for harmonic loads, and plate density structure (infill density, crust density, mantle density, and the respective interface depths). Plot appearance is controlled with “Xmin”, “Xmax”, and “Spacing” in the “Plot Parameters” panel. Flexure or gravity data can be imported using the “Data Import Utility”, which prompts for an input file name. The data are plotted by clicking the “Plot Data” button. X, Y, and Z buttons and edit box in the “Data Shift” panel allow data to be shifted horizontally and vertically to adjust the model fit. Buttons to interact with the

imported data and plots are also provided. The flexural parameter (α [indicated as Alpha in the TAFI GUI]), zero crossing distance (x0 [X0 in the GUI]), flexural bulge position x( b [Xb in the GUI]),

the maximum flexural depth w( max [Wmax in the GUI]) and the amplitude of peripheral bulge (wb [Wb in the GUI]) are displayed in the “Outputs” panel. Flexure and gravity models and imported data are shown in the plot panel.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1560 by guest on 28 September 2021 Software Contribution

TTABLEABLE 3. LISLISTT OF TOOLBOXTOOLBOX FOR ANALANALYSISYSIS OF FLEXURAL ISOISOSTASYSTASY (T(TAFI)AFI) FUNCTIONS CCALLABLEALLABLE FRFROMOM THE MAMATLABTLAB WWORKSPACEORKSPACE FunctionFunction ArgumentsArguments DescriptionDescription

flexparam.mflexparam.m D, gγ, g,, g, E, Te, R, load geometrgeometry,y, plateplate geometrygeometryFFunctionunction to calculacalculatete the flflexuralexural parameterparameter fforor a givgivenen load and plaplatete geometrgeometry.y. CContinuous2D_flex.montinuous2D_flex.m αa, D, x GrGreen’seen’s function forfor infiniteinfinite plateplate and line impulse load.load. Continuous3D_flex.mContinuous3D_flex.m aα, D, x Green’sGreen’s function forfor infiniteinfinite plateplate and point impulse load. Continuous3Dgrid_flex.mContinuous3Dgrid_flex.m aα, D, x, y Green’sGreen’s function forfor infiniteinfinite plateplate and 33-D-D distributeddistributed load.load. Halfspace2D_flex.mHalfspace2D_flex.m aα, D, x Green’sGreen’s function forfor semisemi-inf-infiniteinite plateplate and line impulse load.load.

Harmonic2D_flex.mHarmonic2D_flex.m x, lλ, Q0, g, g, , gγ, D FunctionFunction to calculatecalculate flexuralflexural responseresponse forfor infiniteinfinite plateplate and 22-D-D sinusoidal load.load. parkg.mparkg.m w, Dr∆ρ, z, D∆x FunctionFunction to calculatecalculate the gravitygravity fieldfield due toto flexedflexed elastic plate.plate. Dr∆ρ and z can be vectorsvectors toto definedefine multiple interfaces.interfaces. parkg_3D.mparkg_3D.m nx, ny, D∆x, D∆y, z, Dr∆ρ, w FunctionFunction to calculatecalculate the gravitygravity fieldfield due toto flexedflexed elastic plateplate forfor a 33-D-D distributeddistributed load. Loadconv2D.mLoadconv2D.m w, Q, load geometry,geometry, D∆x, x FunctionFunction to convolveconvolve Green’sGreen’s function with 22-D-D distributeddistributed load.load. Loadconv3D.mLoadconv3D.m w, load scale,scale, 33-D-D load matrix,matrix, nx, ny, D∆x, D∆y FunctionFunction to convolveconvolve Green’sGreen’s function with 33-D-D distributeddistributed load.load. Note:Note: 22-D—t-D—twowo dimensional; 33-D—thr-D—threeee dimensional; D—flexural—flexural rigidity;rigidity; gγ—density—density contrast betweenbetween infillinfill and mantle;mantle; g—acceleration due to gravity;gravity; E—Young’s—Young’s modulus;

Te—elastic thickness;thickness; R—Ear—Earth’sth’s radius; x—arra—arrayy of position along the x--axis;axis; y—array—array of position along the y--axisaxis (for(for load grids);grids); lλ—sinusoidal load wavelength;wavelength; Q0—v—verticalertical load magnitude; Dr∆ρ—array—array containing density contrast betbetweenween successivesuccessive lalayersyers (inf(infillill –cr–crust,ust, crcrust–mantle);ust–mantle); z—arra—arrayy containing depth of successivesuccessive densitydensity ininterfacesterfaces along zz-axis;-axis; w—array—array containing verticalvertical flexuralflexural deflectiondeflection valuesvalues of plate;plate; D∆x—interval—interval spacing along x-x-axis forfor a spatiallspatiallyy distributeddistributed line and point loads as well as load grids; D∆y— intervalinterval spacing along y--axisaxis forfor spatiallyspatially distributeddistributed load grids; nx, ny—number of grid nodes along x- and y- axis respectivelyrespectively forfor a spatiallyspatially distributeddistributed load grid;grid; Q—distributed—distributed load magnitude array;array; load scale—scaling parameterparameter forfor distributeddistributed loads.loads.

USING TAFI—ALEUTIAN TRENCH 2-D CASE STUDY We chose a profile that crosses the trench ~45 km to the west of the “Aleu- tian 8” profile of Levitt and Sandwell (1995) (Fig. 4). Bathymetric and free-air TAFI can be used to explore plate flexural behavior without loading data. gravity data along this profile were downloaded from the U.S. National Geo- However, the primary purpose of TAFI is to facilitate forward modeling to de- physical Data Center (NGDC). We prepared the data for use in TAFI by remov- termine the flexural parameters that best fit the observed vertical deflection ing the mean and regional tilt to correct for elevation changes not associated and gravity data. To demonstrate the TAFI workflow, we model the flexural with flexure (primarily, changes in seafloor depth due to plate age). The cor- deformation of the lithosphere along a 2-D profile crossing the Aleutian Trench rected flexural and gravity profiles were saved in text (.txt) format (format is (Fig. 4). The Aleutian Trench extends ~2900 km from the Gulf of Alaska (USA) described in Supplemental File 1 [footnote 2]). to the Kamchatka Peninsula (Russia) and is the plate boundary where the Launching TAFI from the GUI shows the default infinite plate model. Pacific plate is being subducted under the North American plate. A number of Dropdown menus are used to change the plate geometry and load shape studies have shown that the seafloor bathymetry near the Aleutian Trench can from the default settings to semi-infinite and 2-D line load respectively. Fol- be fit with a 2-D semi-infinite elastic plate model subjected to a line load (Cald- lowing Levitt and Sandwell (1995), the TAFI model is then modified to change well et al., 1976; Levitt and Sandwell, 1995; Watts, 2001; Bry and White, 2007). the infill, crust, and mantle densities and the depths to their associated inter-

TTABLEABLE 4. TTOOLBOXOOLBOX FOR ANALANALYSISYSIS OF FLEXURAL ISOISOSTASYSTASY (TAFI)(TAFI) GEODYNAMICGEODYNAMIC FUNCTIONS FOR DIFFERENT PLATEPLATE AND LOLOADAD GEOMETRGEOMETRYY COMBINACOMBINATIONSTIONS PlatPlatee geometrgeometryyLLoadoad geometgeometryry Geodynamic functiofunctionn InfInfiniteinite22-D-D impulse load CoContinuous2D_flex.mntinuous2D_flex.m 3-D3-D impulse load CoContinuous3D_flex.mntinuous3D_flex.m PePeriodicriodic loadinloadinggHHarmonic2D_flex.marmonic2D_flex.m 2-D2-D distdistributedributed load Continuous2D_flex.mContinuous2D_flex.m DistrDistributedibuted axisymmetricaxisymmetric load Continous3D_flex.mContinous3D_flex.m 3-D3-D distdistributedributed load Continuous3Dgrid_flex.mContinuous3Dgrid_flex.m SemiSemi-inf-infiniteinite22--DD impulse load Halfspace2D_flex.mHalfspace2D_flex.m 22-D-D distributeddistributed load Halfspace2D_flex.mHalfspace2D_flex.m Note:Note: 22-D—t-D—twowo dimensional; 33-D—th-D—threeree dimensional.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1561 by guest on 28 September 2021 Software Contribution

o N ALASKA NORTH AMERICAN 60 N PLATE BERING SEA GULF OF ALASKA

C H A N E L E R E M U T P T I A N Figure 4. Location of the Aleutian Trench E R o flexural profile (red line). Black lines are O 50 N R profiles from Levitt and Sandwell (1995).

N JUAN DE FUCA

C PLATE

N 0 1000 km 40oN 160oE 170 oE 180oE 170oW 160oW 150oW 140oW

faces to 1000 kg/m3, 2750 kg/m3, 3200 kg/m3, 7000 m, and 12,000 m respec- harmonic loads were verified by comparing TAFI solutions with gravity models tively. TAFI’s plot window displays both the initial and modified flexure and calculated from the flexural admittance function (Watts, 2001). TAFI’s solutions gravity curves. for distributed 3-D loads were verified by computing 3-D models with (1) a To constrain the flexure and gravity modeling of the Aleutian Trench by point load located in the center of the load grid, and (2) a line load crossing curve fitting in TAFI, the Aleutian bathymetry and gravity data (saved earlier) the load grid. Radial deflection and gravity profiles extending away from the are imported into TAFI (Fig. 5) (data import process and file format are de- point load and striking perpendicular to the line load were verified to match scribed in Supplemental File 1 [footnote 2]). Curve fitting involves adjusting with TAFI’s comparable axisymmetric point load and 2-D line load solutions, the modeled flexure and gravity curves to fit the imported data. This is done by respectively (Fig. 7). revising the flexural rigidity, load magnitude, and load position through their respective sliders and edit boxes in TAFI’s GUI. The model curves are updated each time a parameter value is changed, with the new model curves added to SUMMARY the existing plot. The curve-fitting process is terminated when the modeled curves fit the data reasonably well (Fig. 6). The final flexural model parameters The Toolbox for Analysis of Flexural Isostasy (TAFI) consists of MATLAB and attributes for the best-fitting Aleutian Trench TAFI model (Fig. 6) are found functions that compute the flexural response of an infinite or semi-infinite to lie within the range of values estimated by Levitt and Sandwell (1995) and thin elastic plate overlying an inviscid substrate and subjected to line, point, Bry and White (2007) (Table 5). and 2-D periodic loads. TAFI also provides functions to convolve the flexural Green’s functions with non-impulsive loads, allowing computation of deflection under spatially distributed 2-D or 3-D loads, and provides functions to CODE VERIFICATION compute the Bouguer or free-air gravity fields associated with plate flexure. The geodynamic functions in TAFI can be called from the MATLAB user inter- The flexural responses for 2-D and 3-D line and point loads (Equations 4–6) face or within user-written MATLAB code, or accessed through TAFI’s graphical and 2-D sinusoidal loads (Equation 7) were verified by comparing the results user interface (GUI). The GUI facilitates interactive flexural modeling and com- obtained from TAFI with solutions computed in Microsoft Excel and Generic parison of the model to gravity observations and data constraining flexural Mapping Tools (Wessel et al., 2013), and by comparing the zero crossing, max- subsidence and uplift. TAFI uses only MATLAB intrinsic functions and func- imum deflection, and flexural bulge position and amplitude with analytical tions distributed as m-files with TAFI, and does not require additional tool- solutions where available (Table 1). Gravity solutions for the line, point, and boxes or libraries.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1562 by guest on 28 September 2021 Software Contribution

100 Figure 5. Gravity and flexure profiles for 200 the default semi-infinite line load model. 100 50 Bathymetry and free-air gravity along the 100 200 300 400 500 600 700 Aleutian Trench profile (Fig. 4) are shown 0 100 200 300 400 500 600 700 for comparison. 0 –100 Distance (km) Distance (km) –200 Gravity data –50 Gravity data –300 Inial model Inial model Test models

Gravity (mGal) –100

Gravity (mGal) –400 Best ﬁt model –500 –150 –600 –200 –700

–5 –4 –4 –2 –3 Distance (km) 100 200 300 400 500 600 700 –2 0

–1 Figure 6. Curve-fitting process of Toolbox for Analysis of Flexural Isostasy (TAFI), 2 Bathymetric data 0 showing initial gravity and flexure model, 100 200 300 400 500 600 700 Aleutian Trench gravity and bathymetry, Inial model 1 Distance (km) and the gravity and flexure curves for 4 Test models

Flexural deflecon (km) Bathymetric data the best-fitting model (Table 1). The test Flexural deflecon (km) Best fit model 2 Inial model curves were created by changing the rigidity and load sliders in the graphical user 6 3 interface.

TABLETABLE 5. BESBESTT--FITFIT MODEL PPARAMETERSARAMETERS FOR ALEUTIAN TRENCH Input parameparameterterVValuealue LeLevittvitt and SaSandwellndwell (1995)*(1995)*BBryry and WhiteWhite (2007)† FleFlexuralxural rigidityrigidity (D) (N-m(N-m)) 2.89 × 1023 2.18 × 1022–3.23 × 1024 6.48 × 1020–4.55 × 1023 3 § § InfInfillill densitdensityy (ρri) (k(kg/mg/m ) 1000 1000 1000 3 CrCrustalustal density (rρc) (k(kg/mg/m ) 2750 ––– – 3 Mantle densitdensityy (ρrm) (k(kg/mg/m ) 3200 3200 33003300** # Depth ttoo ttopop of platplatee (z1) (m(m)) 7000 ––– – # CrCrustust-mantle-mantle boundarboundaryy depth (z2) (m)(m) 12,000 ––– – 13 Load magnitude (Q0) (N/m)(N/m) 1.16 × 10 ––– –

Output rresultesult VaValuelueLLevittevitt and SaSandwellndwell (1995)*(1995)*BBryry and WhiteWhite (2007)† FlexuralFlexural parameterparameter (aα) (km)(km) 84.2384.23331.7–110.61.7–110.6 18.41–94.7918.41–94.79

FleFlexuralxural bulgebulge position (xb) (km(km)) 198.46 ––– –

ZerZeroo crcrossingossing (x0) (km(km)) 132.31 ––– –

Maximum flexuralflexural amplitude (wmaxmax) (km)(km) 5.98 ––– – Elastic thicthicknesskness (km(km)) 33.933.911224.3–51.94.3–51.944.5–40.5–40 NoNote:te: ResultsResults of LeLevittvitt and SandwellSandwell (1995)(1995) and BryBry and WhiteWhite (2007) Aleutian TrenchTrench models shownshown forfor comparison.comparison. DashesDashes indicateindicate parameters/resultsparameters/results not givengiven in those papepapers.rs. *Fle*Flexuralxural rigidityrigidity range forfor the Aleutian TrenchTrench calculatcalculateded based on the avaverageerage density contrast (2200 kg/mkg/m3) and minimum and maximum valuesvalues of the flexuralflexural parametparameterer frfromom the LeLevittvitt and SSandwellandwell (1995)(1995) paper.paper. †FleFlexuralxural rigidityrigidity and fflexurallexural parameterparameter range calculatedcalculated based on the range of elastic thicknessthickness of the Aleutian plateplate and gigivenven Young’sYoung’s modulus (8 × 101010 N/m2) and PPoisson’soisson’s ratio (0.25) values.values. §InfInfillill densitdensityy calculatedcalculated frfromom the aaverageverage density contrast and mantle densitdensity.y. # z1 and z2 indicatindicatee depths belobeloww sea-lesea-levelvel datum prpriorior ttoo flflexuralexural bending.bending.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1563 by guest on 28 September 2021 Software Contribution

3-D Flexure Model 3-D Gravity Model

50 m) 0.5 (k 0 l) 0 Ga on –0.5 c

(m –50

–1.0 y ﬂe it –100 de

–1.5 av

Gr al –2.0 –150 ur 150 150 ex 250 100 250 Fl 10000 200 Di 200 150 stan 50 150 Dist 50 50 100 ce 100 ance 0 (km) (km) 0 50 m) (k 0 ance 0 ce (k m) Dist Distan

Comparison of 2-D with 3-D ﬂexure model Comparison of 2-D with 3-D gravity model

20 –0.2 20 40 60 80 20 40 60 80 100 0 0 l)

m) 0.2 Distance (km)

Distance (km) Ga

(k –20 0.4 (m on 0.6 y –40 it c

0.8 av –60 ﬂe 1.0 Gr de 2-D ﬂexure –80

al 1.2

ur 1.4 3-D ﬂexure –100 2-D gravity ex 3-D gravity

Fl 1.6 –120 1.8

Figure 7. Toolbox for Analysis of Flexural Isostasy (TAFI) three-dimensional numerical model showing a solution for a line load. The load is located along the left (x = 0 km) edge of the model and has a magnitude of 109 N/m. Other model parameters are as in Figure 3. Top row: Deflection (left) and gravity field (right). Bottom row: Comparison of TAFI’s two-dimensional (2‑D) analytical line load solution (solid lines) with the three-dimensional (3-D) numerical solution (circles). The profile location is indicated by solid black lines in the figures in the top row. The mismatch between the analytical and numerical solutions near x = 0 is due to an approximation of the Kelvin’s function used in the axi symmetric Green’s function. The Kelvin function returns “infinity” at radial coordinater = 0. To overcome this issue, r is assigned a small value (10–18) instead of 0.

ACKNOWLEDGMENTS Brotchie, J.F., and Silvester, R., 1969, On crustal flexure: Journal of Geophysical Research, v. 74, p. 5240–5252, doi:10.1029/JB074i022p05240. We would like to thank geophysics graduate and undergraduate students of the Department of Geo- Bry, M., and White, N., 2007, Reappraising elastic thickness variation at oceanic trenches: Journal sciences, Warner College of Natural Resources, Colorado State University, who reviewed the tool- of Geophysical Research, v. 112, B08414, doi:10 .1029/2005JB004190. box and provided constructive comments. We thank Dr. Andrew Wickert, University of Minnesota, Caldwell, J.G., Haxby, W.F., Karig, D.E., and Turcotte, D.L., 1976, On the applicability of a universal whose reviews made this paper better. We also thank the science editor of Geosphere, Dr. Raymond elastic trench profile: Earth and Planetary Science Letters, v. 31, p. 239–246, doi:10.1016/0012 M. Russo, for his helpful suggestions. This material is based upon work supported by the National -821X(76)90215-6. Science Foundation under grants 1043700 and 1358664, and under Cooperative Agreement 0342484 Chase, C.G., and Wallace, T.C., 1988, Flexural isostasy and uplift of the Sierra Nevada of Califor- through subawards administered and issued by the ANDRILL Science Management Office at the nia: Journal of Geophysical Research, v. 93, p. 2795–2802, doi:10.1029/JB093iB04p02795. University of Nebraska–Lincoln, as part of the ANDRILL U.S. Science Support Program. Crosby, A.G., and McKenzie, D., 2009, An analysis of young ocean depth, gravity and global residual topography: Geophysical Journal International, v. 178, p. 1198–1219, doi:10.1111/j .1365-246X.2009.04224.x. Dahlen, F.A., 1981, Isostasy and ambient state of stress in the oceanic lithosphere: Journal of REFERENCES CITED Geophysical Research, v. 86, p. 7801–7807, doi:10.1029/JB086iB09p07801. Banks, R.J., Parker, R.L., and Huestis, S.P., 1977, Isostatic compensation on a continental scale: Djomani, Y.H.P., Nnange, J.M., Diament, M., Ebinger, C.J., and Fairhead, J.D., 1995, Effec- Local versus regional mechanisms: Geophysical Journal of the Royal Astronomical Society, tive elastic thickness and crustal thickness variations in west central Africa inferred from v. 51, p. 431–452, doi:10.1111/j.1365-246X.1977.tb06927.x. gravity data: Journal of Geophysical Research, v. 100, p. 22,047–22,070, doi:10.1029 Bodine, J.H., Steckler, M.S., and Watts, A.B., 1981, Observations of flexure and the rheology of /95JB01149. the oceanic lithosphere: Journal of Geophysical Research, v. 86, p. 3695–3707, doi:10.1029 Flemings, P.B., and Jordan, T.E., 1989, A synthetic stratigraphic model of foreland basin develop- /JB086iB05p03695. ment: Journal of Geophysical Research, v. 94, p. 3851–3866, doi:10.1029/JB094iB04p03851.

GEOSPHERE | Volume 13 | Number 5 Jha et al. | Toolbox for Analysis of Flexural Isostasy (TAFI) Downloaded from http://pubs.geoscienceworld.org/gsa/geosphere/article-pdf/13/5/1555/3995597/1555.pdf 1564 by guest on 28 September 2021 Software Contribution

Flück, P., Hyndman, R.D., and Lowe, C., 2003, Effective elastic thickness Te of the lithosphere in Slingerland, R.L., Furlong, K., and Harbaugh, J., 1994, Simulating Clastic Sedimentary Basins: western Canada: Journal of Geophysical Research, v. 108, 2430, doi:10.1029/2002JB002201. Englewood Cliffs, New Jersey, Prentice Hall, 352 p. Forsyth, D.W., 1985, Subsurface loading and estimates of the flexural rigidity of continen- Stern, T.A., and ten Brink, U.S., 1989, Flexural uplift of the Transantarctic Mountains: Journal of tal lithosphere: Journal of Geophysical Research, v. 90, p. 12,623–12,632, doi:10 .1029 Geophysical Research, v. 94, p. 10,315–10,330, doi:10.1029/JB094iB08p10315. /JB090iB14p12623. Stern, T.A., Baxter, A.K., and Baxter, P.J., 2005, Isostatic rebound due to glacial erosion within the Hertz, H., 1884, On the equilibrium of floating elastic plates: Wiedemann’s Annals, v. 258, p. 449– Transantarctic Mountains: Geology, v. 33, p. 221–224, doi:10.1130/G21068.1. 455, doi:10.1002/andp.18842580711. ten Brink, U.S., Hackney, R.I., Bannister, S., Stern, T.A., and Makovsky, Y., 1997, Uplift of the Karner, G.D., and Watts, A.B., 1983, Gravity anomalies and flexure of lithosphere at the mountain Transantarctic Mountains and the bedrock under the East Antarctic ice cap: Journal of Geo- ranges: Journal of Geophysical Research, v. 88, p. 10,449–10,477, doi:10 .1029 /JB088iB12p10449 . physical Research, v. 102, p. 27,603–27,621, doi:10.1029/97JB02483. Levitt, D.A., and Sandwell, D.T., 1995, Lithospheric bending at subduction zones based on depth Turcotte, D.L., 1979, Flexure: Advances in Geophysics, v. 21, p. 51–86, doi:10.1016/S0065 -2687 soundings and satellite gravity: Journal of Geophysical Research, v. 100, p. 379–400, doi:10 (08)60261-7. .1029/94JB02468. Turcotte, D.L., and Schubert, G., 2002, Geodynamics: New York, Cambridge University Press, Lowry, A.R., and Smith, R.B., 1994, Flexural rigidity of the Basin and Range–Colorado Plateau– 456 p., doi:10.1017/CBO9780511807442. Rocky Mountain transition from coherence analysis of gravity and topography: Journal of Walcott, R.I., 1970, Flexural rigidity, thickness, and viscosity of the lithosphere: Journal of Geo- Geophysical Research, v. 99, p. 20,123–20,140, doi:10.1029/94JB00960. physical Research, v. 75, p. 3941–3954, doi:10.1029/JB075i020p03941. Macario, A., Malinverno, A., Haxby, W.F., 1995, On the robustness of elastic thickness estimates Watts, A.B., 1978, An analysis of isostasy in the world’s oceans 1. Hawaiian-Emperor seamount obtained using the coherence method: Journal of Geophysical Research, v. 100, p. 15,163– chain: Journal of Geophysical Research, v. 83, p. 5989–6004, doi:10.1029/JB083iB12p05989. 15,172, doi:10.1029/95JB00980. Watts, A.B., 2001, Isostasy and Flexure of the Lithosphere: Cambridge, UK, Cambridge Univer-

McKenzie, D.P., 2003, Estimating Te in the presence of internal loads: Journal of Geophysical sity Press, 458 p. Research, v. 108, 2438, doi:10.1029/2002JB001766. Watts, A.B., and Burov, E.B., 2003, Lithospheric strength and its relationship to the elastic and McKenzie, D.P., and Bowin, C.O., 1976, The relationship between bathymetry and gravity in seismogenic thickness: Earth and Planetary Science Letters, v. 213, p. 113–131, doi:10 .1016 the Atlantic Ocean: Journal of Geophysical Research, v. 81, p. 1903–1915, doi:10.1029 /S0012-821X(03)00289-9. /JB081i011p01903. Watts, A.B., Bodine, J.H., and Steckler, M.S., 1980, Observation of flexure and the state of stress McKenzie, D.P., and Fairhead, D., 1997, Estimates of the effective elastic thickness of the conti- in oceanic lithosphere: Journal of Geophysical Research, v. 85, p. 6369–6376, doi:10 .1029 nental lithosphere from the Bouguer and free air gravity anomalies: Journal of Geophysical /JB085iB11p06369. Research, v. 102, p. 27,523–27,552, doi:10.1029/97JB02481. Watts, A.B., Karner, G.D., and Steckler, M.S., 1982, Lithospheric flexure and the evolution of McNutt, M.K., and Menard, H.W., 1982, Constraints on yield strength in the oceanic lithosphere sedimentary basins, in Kent, P.S., Bott, M.H.P., McKenzie, D.P., and Williams, C.A., eds., The derived from observations of flexure: Geophysical Journal of the Royal Astronomical Soci- Evolution of Sedimentary Basins: Philosophical Transactions of the Royal Society of London ety, v. 71, p. 363–394, doi:10.1111/j.1365-246X.1982.tb05994.x. A, v. 305, p. 249–281, doi:10.1098/rsta.1982.0036. McNutt, M.K., Diament, M., and Kogan, M.G., 1988, Variations of elastic plate thickness at continental Weissel, J.K., and Karner, G.D., 1989, Flexural uplift of rift flanks due to mechanical unloading of thrust belts: Journal of Geophysical Research, v. 93, p. 8825–8838, doi:10.1029/JB093iB08p08825. the lithosphere during extension: Journal of Geophysical Research, v. 94, p. 13,919–13,950, Nadai, A., 1963, Theory of Flow and Fracture of Solids: New York, McGraw-Hill, 705 p. doi:10.1029/JB094iB10p13919. Parker, R.L., 1973, The rapid calculation of potential anomalies: Geophysical Journal of the Royal Wessel, P., Smith, W.H.F., Scharroo, R., Luis, J.F., and Wobbe, F., 2013, Generic Mapping Tools: Astronomical Society, v. 31, p. 447–455, doi:10.1111/j.1365-246X.1973.tb06513.x. Improved version released: Eos (Transactions, American Geophysical Union), v. 94, Petit, C., and Ebinger, C., 2000, Flexure and mechanical behavior of cratonic lithosphere: Grav- p. 409–410. ity models of the East African and Baikal rifts: Journal of Geophysical Research, v. 105, Wickert, A.D., 2016, Open-source modular solutions for flexural isostasy: gFlex v1.0: Geoscien- p. 19,151–19,162, doi:10.1029/2000JB900101. tific Model Development, v. 9, p. 997–1017, doi:10.5194/gmd-9-997-2016.