1 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 ob- servations (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 con- sidered. 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 distin- guished. 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 spheri- cal 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 quasige- oid 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 aforemen- tioned “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 be- comes more intricate whenever similar variations are taken into account because Stokes’ the- ory is based on potential theory where explicit time dependence is not compatible with har- monic 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 consid- ered 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 geo- potential, 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 4 5 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 ob- servation 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 po- tential 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 spe- cific 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.