What Type of Geoid Is Needed for Oceanic Applications and What Can Geodesy Deliver?

What type of geoid is needed for oceanic applications and what can geodesy deliver?

by E. Groten, Darmstadt e-mail: [email protected]

Abstract: Interdisciplinary application of geoids as reference surfaces for geophysical and oceanographic purposes is affected by a variety of imprecise definitions and conse- quent uncertainties. This is still all the more true in connection with time variable geoids as soon as secular changes of mean sea level (MSL) as well as long-period and long-term variations play a role. From a mathematical as well as from a physical viewpoint the combination of gravimetric, satellite (GRACE, GOCE as well as GPS and GLONASS), levelling, tide gauge, altimetric and solid earth tide data poses sev- eral unresolved problems. This is also true for shipborne and airborne gravimetry on sea. One principal problem arises due to the fact that in geodesy mainly relative observations (not absolute data) are available. Absolute gravimetry is one of the few exeptions. On the other hand, ocean circulation models are dominated by gradient computation so that relative data (up to an unknown constant) are sufficient in many cases. With GRACE and GOCE data, which will soon be available, the situation im- proves substantially. Nevertheless, an identification of oceanographic needs and clearer and unique definitions of geodetic quantities and parameters may now be helpful. Mainly global definitions and practical implementations of the geoid are still far apart from uniqueness and from desired accuracy, respectively; this affects also the quasigeoid which coincides with the geoid on the ocean. The main impact of better gravity data at sea will be due to the possibility to better separate stationary from transient ocean circulation as well as its small scale from larger scales phe- nomena.

1. Introduction

Using assimilation techniques C. Wunsch and M. Gaposchkin were among the first to interre- late ocean circulation and geoid in least-squares approaches where a set of parameters is op- timized and the associated residuals and the errors, which can now be checked and verified by satellite altimetry, are presently being used to stepwise evaluate improved models and also take into account temporal variations of geoids and circulation models at the same time. So this parameter estimation process quite well elucidates the interrelations between dynamic topography and geoid or geopotential. As Topex-Poseidon (T/P) and similar altimetry data deliver averages of SST (= sea surface topography) over 10 days or so, GRACE could give additional information in the future on regional short-term variability, typically in the Antarc- tic and arctic area. Until now mostly the EGM 96 geoid model served as the base for such investigations. Dealiasing plays a specific role where short-term variability with time is considered. Here focus is on geodetic aspects of geoid computation.

Particularly with respect to GRACE it is important to note that it is possible to determine now temporal changes of the geoid, at least for low and medium degree harmonic expansion, more precisely than the geoid itself. So transient and stationary geoids should be clearly distinguished.

Recently, the geoid became more popular as a reference surface for a variety of geophysical and mathematical or physical considerations. As a level surface it fulfills a series of require- 2 ments related to level surfaces of the geopotential. As a particular level surface quite close to mean sea level (MSL) it is easily evaluated, at least in spherical approximation, from gravity anomalies, ∆g, and related quantities in terms of a perturbation equation. The convolution type Stokes’ integral can be expressed in terms of a low-pass filter (n-1)-1 applied to a spherical harmonics expansion ∆g = ∑ ∆gn . For higher approximation, however, the evaluation n becomes more “intricate”, as ellipsoidal and terrain corrections have to be taken into account. Based on the well known Runge Theorem related integral equation or similar solutions were designed by Molodensky, Bjerhammar, Moritz, Arnold etc. The differences between quasigeoid and geoid need no further explanation if we confine ourselves to the ocean because both surfaces are basically identical there. However, as the aforementioned filter function tends to infinity for n = 1 most evaluations ignore the terms for n < 2 which leads to a constant term No which is basically the difference between absolute and relative geoids. If, besides ∆g, also altimetric data are used at sea, they may either be used to evaluate the difference between the sea surface, and after appropriate reduction, between mean sea level (MSL) and the geoid. Or we ignore that difference (of the order of d < 1.5 m) and solve an “improperly posed problem” (in Hadamard’s sense) and determine ∆gn from altimetric observations using the inverse (high-pass) filter function (n-1). With GPS-data at hand we may also replace the aforementioned “free boundary value problem” by a fixed boundary value problem where (n-1) is re- placed by (n+1) whenever GPS positioning is accurate enough. In coastal areas where sea surface noise becomes relevant we may combine tide gauge data together with GPS- positioning in order to bridge oceanic and continental sections of the geoid. The situation becomes more intricate whenever similar variations are taken into account because Stokes’ theory is based on potential theory where explicit time dependence is not compatible with harmonic or even conservative fields of force so that tidal reductions and similar operations need to be applied to eliminate such dependences. On the other hand, such problems can be avoided by step-wise constant functions so that time dependent ∆g and geoids can be considered in detail.

Whenever oceanic circulation models are derived from gradient observations and related par- tial differential equations, such relative geoids, as mentioned before, will be sufficient. Details of the earth’s gravity field on ocean circulation modeling are found in (Dombrowsky et al., 1999).

However, problems may come up, if the aforementioned step-functions are not compatible with the applied tidal corrections and vice versa. In principle, this is a matter of convention but such conventions must be applied to make the data compatible with each other. For long tide gauge records simple averaging techniques may be sufficient but for short tidal records and for altimetric observations careful considerations may be appropriate. If, for instance, secular mean sea level variations of the order of a few millimeters are considered for the latter type of observations, systematic deviations could arise. For details see (Fenoglio-Marc and Groten, 2002).

2. Absolute geoid

In some cases the actual location of the geoid and the associated numerical value of the geopotential, W°, are needed. In that case also the aforementioned terms for n = 0 and n = 1 have to be considered. But in geocentric reference systems all terms for n = 1 disappear so that only the zero degree term is of interest. Depending on the selection of the appropriate (for linear approximation) approximation figure (sphere, ellipsoid etc.) correction terms need to be taken 3

6 into account which make things intricate. Also in view of long-periodic variations of MSL (such as El Nino) more sophisticated mathematics and detailed considerations of physical effects are necessary (Marcus et al., 2001).

In Fig. 1 those types of observations are summarized which are usually being applied to Ver- tical Datum and geoid evaluation. It is important to realize that, depending on the type of observation and data processing, the observations are transformed from “observation space” (i.e. reality) to stationary “model spaces”. As far as tidal corrections are concerned, three different model spaces are to be distinguished.

a) zero tide, b) mean tide and c) tide-free, as mentioned before. The reason for using 3 models is due to the fact that, even after eliminat- ing time-dependent tides, permanent tides can only be handled in those different models if (1) we want to stay as close as possible to reality and (2) take all non-harmonic parts of tidal potential out, in order to be able to apply potential theory in terms of harmonic functions. For non-experts the transition from one to the other models is not always easily understood. But comparison and compatibility are only possible within one model space. Also non-tidal time dependences have to be considered in an analogous way. One good example is time-varying ocean bottom pressure variations as considered in detail by Wuensch et al. (2001).

Another aspect of data reductions is due to the fact that national height systems are basically independent of each other. Thus the evaluation of gravity anomalies is reduced to different “zero-levels” which leads to systematic offsets in geoid computation ∆g → N where N is the geoid height, i.e. the separation of the ellipsoid from the geoid. All what is true for one specific epoch holds, of course, also for repeat observations. Optimal data treatment is possible whenever data are related to the same reference frame, as, e.g., in case of the TIGA project. This is illustrated in Fig. 2. The transition from one national height system to any other or to a common continental system is illustrated in Fig. 3 (courtesy BKG, Dr. Ihde) for the European System. The pioneer attempt to evaluate a global unified height system based on the joint use of altimetric, gravimetric, levelling and positioning (GPS, GLONASS etc.) data is illustrated in (Bursa et al., 2002).

In the absence of strong currents and dominant circulation, mean sea level reflects to a large extent the underlying ocean-bottom topography and gravitational fields so that its variations are close to those of the associated geoid. There the inverse application of Stokes’ integral to altimetric data leads to mean gravity anomalies which can be used to validate shipborne gravimetry which, in pre-GPS-times, was jeopardized by long-period distortions of gravimetric records. Using a modified Stokes approach for that purpose Belikov, Fenoglio-Marc and Gro- ten have reduced the effect of distortions in classical shipborne data in the Mediterranean sea (Belikov, 1994). Also from that viewpoint the advent of data from GRACE and GOCE will imply a substantial improvement of the possibilities to model the gravity and potential field substantially.

3. Temporal changes of the geopotential and of the geoid

There are well known short-term changes of the sea surface. Typical cases are tidal variations. Also temporal changes of the mean sea level (MSL) have been investigated. Whenever ther- mal expansion is dominant, the corresponding variations of the geopotential are quite small. If we can ignore mass exchange between atmosphere and ocean, e.g. in connection with El Nino 7

or similar effects, the zero term W0 in the expansion of the geopotential in terms of spherical harmonics will remain constant where

∞ W = W0 + ∑Wn n=1 with

GM W = , 0 r

GM = terrestrial gravitational constant and r = geocentric radius vector. Meanwhile, a num- ber of geophysical effects, such as sea bottom pressure variations, have been investigated and associated changes in the geopotential W were determined.

For areas of post-glacial rebound, such as Fennoscandia (Mäkinen et al., 2000), also the geoidal variations have been studied. The main experience of temporal changes of the geoid related to long-term subsidence and uplift of the earth’s surface originates from Fennoscandia where vertical displacements of the geoid were found to amount to about one tenth of the associated displacements of the earth’s surface. This ratio is, of course, related to post-glacial uplift and the structure of the lithosphere in that area. At sea, due to different density relations and a large steric contribution the corresponding relationship should be much less than one tenth. Such effects lead to variations of the shape of the geoid as well as of the (constant)potential W0 at the geoid. Within the “Fennoscandian Uplift” project thus temporal (secular) variations

dW0/dt ≡ W& 0 have been studied.

In general, the changes of the geoid are, of course, much smaller than those of the sea surface or of mean sea level. Bursa (Bursa et al. 2002) has introduced the quantity R(o) = GM/W0 which is independent of underlying tidal models, contrary to ellipsoidal models. As R(o) has metric dimension it, more or less, represents the volume of the earth and is a useful reference parameter in dealing with global sea level uplift or subsidence. Annual variations of R(o) are less than 1 mm within 1993-2000. Bursa’s solution and his associated determination of a global vertical datum was critically reviewed by Sacerdote and Sanso (2001). The mathematical difficulties inherent in the determination of W0 were discussed by Lehmann (2000).

Consequently, also W& 0 may be considered as a global quantity representing associated uplift or subsidence of mean sea level; see Fig. 4. However, it should be kept in mind that this time derivative of the potential at the geoid is less variable than the sea level or any other part of the earth's surface and may, consequently, denote the variations of the geopotential reference with time. Remember that W 0( r ) is less sensitive to errors in its numerical determination than the location, r , of the geoid itself.

In general, it is expected that with the present dedicated gravity satellite missions, such as GRACE, GOCE etc, as well as the new altimetric missions, in combination with terrestrial projects of COST and TIGA type (combining tide gauge, terrestrial GPS and space-borne GPS altimetry) physical geodesy will soon be able to provide much more accurate geoids for 8 9 coastal and pelagic ocean areas. Consequently, geodesy may then (in a few years) provide substantially better dynamic reference surfaces (geoids) and systems (unified vertical datums). For geodynamic purposes W& 0 may be more important than W0 itself! It is a measure of stabil- ity. Short and medium-term variations of the earth’s gravity field have been investigated in numerous papers; see, e.g., (Cazenave et al. 1999, Dong et al. 1996, Gruber et al. 2000, Wahr et al 1998, Wünsch et al. 2001). Observations, in terms of spherical harmonics coefficients, of associated effects will be available from monthly analysis of the gravity field obtained from LEO-type satellites.

R. Britt (2002) reports on recent long-term variations of the second degree zonal harmonic which may be attributed to large-scale ocean mass shift, such as in Pacific Decadal Oscilla- tion, occurring since 1998 and detected by satellite techniques in W2, leading to W& 2 (Cox and Chao, 2002). However, the length of the GRACE campaign is relatively short (∼5 years) for observations of such changes.

From terrestrial relative gravimetry in tectonic active areas it is known that regional gravity variations of 30 microgal are typical for medium terms such as one year. This can also be considered as an upper limit for regular cases. Higashi (1995) used superconducting gravime- ters for that purpose. Also absolute gravimetry can be used (van Dam, 1999; Ming and Hager, 2001).

With low harmonic variations of the geopotential as deduced from GRACE and high harmonics derived from GOCE, together with “in situ” data, we may obtain absolute ocean currents (Zahel and Schroeter, 2002) as a base for determining surface as well as deep circulation models of mass and heat exchange; whereas the long-wave part of ocean circulation and the short-wave part of the geopotential (dealiasing) may be improved, based on monthly GRACE and averaged long-term GOCE data sets. In addition, ocean bottom pressure data deduced from time-varying gravity field models could be used to separate steric from non-steric surface effects in altimetry data. Similar to tidal modeling procedures interdependences between circulation and tides as well as those between circulation and ocean bottom topography may be better modeled, also in view of non-linearities. Moreover, gravity field information may be applied to improved ocean circulation modeling which enables the separation of stationary from time-variable circulation as well as the investigation of small-scale stationary circulation related to climatic modeling. With the aforementioned inclusion of ocean floor pressure data in circulation models barotropic and barocline parts may be separated for improved modeling of ocean mass and heat transport. The role of the separation of MSL from the geoid in 3D- modeling of ocean circulation, besides the tilt of MSL w.r.t. the geoid, still appears unclear, mainly in view of absolute current modeling, as mentioned above.

CONCLUSIONS

For tide gauge stations equipped with permanent or temporary “repeat” GPS stations geocentric locations and their temporal variations can be determined with sufficient accuracy where the temporal changes of the geocenter, due to tidal and atmospheric effects, are assumed to amount to one centimeter or less. Using leveling, tide gauge stations may be connected to the associated local or national height datum and related regional (relative) geoids. As the geopotential cannot be directly observed those relative geoids differ from the actual geoid (with 0 geopotential W ) by a constant N0 which is different for each local geoid, determined by Stokes formula or a similar approach (e.g., collocation). The various vertical height datums can be interconnected by solving an “improperly posed” mixed boundary value problem (al- 10 timetric-gravimetric BVP) which is, at present, possible with accuracy of less than one or two decimeters yielding a unified global vertical datum related to an (absolute) geoid of geopotential W0. In this way orthometric heights and their temporal variations for all tide gauge stations can be determined. In view of a deformable earth, secular variations may become of interest so that, over longer intervals ∆t, also temporal variations dW0/dt ≡ W& 0 (t) and temporal changes of the geoid have to be determined. With results from GRACE and GOCE becom- ing soon available the aforementioned accuracy of a few decimeters may increase. New altimetric satellites together with terrestrial projects such as COST, GLOSS etc. may substantially contribute to improved solutions of the BVP. In so far, the present situation may soon change substantially, giving way to determining precise reference surfaces (geoid etc.) and reference systems for oceanic investigations. Bruns’ well known formula can be used to determine the separations of various levels surfaces from their potential differences divided by normal gravity γ. More precise gravity information could also substantially help to explain more accurately the deviations of eustatic from steric ocean surface data in order to replace present reduction procedures based on surface temperatures.

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