Satellite altimetry, since its inception with Skylab, Geos 3 and Seasat in the 1970's, has had enduring influence on both Geodesy and Physical Oceanography, the metric study of the Earth and its marine environment from space (Kirwan et al., 1983; Douglas and Cheney, 1990; Cheney, 1995). To cite just one example of the former, the detailed mapping of the mean ocean surface it provides enables a complimentary inferential map to be drawn of the ocean bottom almost everywhere, extending the limited reach of historical bathymetry (e.g., Smith, 1998). Thus the altimetric sea surface is computed as a residual of the altimeter height from an estimated satellite trajectory. It happens that this bottom-mapping ability depends happily on the circumstance that the high satellite orbit carrying the altimeter, which is affected by the same gravitational field as the ocean surface which the instrument also senses, naturally acts to filter from the altimeter signal poorly known short wavelength effects on the trajectory. Thus these components of the resulting mean altimeter surface can be said to arise almost entirely from the severe density contrast at the bottom of the ocean.
Indeed though not essential to the bottom mapping application, the longer wavelengths of the altimetric mean sea surface which carries most of the power of the oceanographic sea surface topography due to currents are also largely undistorted by their strong orbit components because we already know (from extensive tracking of satellite orbits by other instruments) these broader parts of the gravitational field.
Broadly speaking these filter principals motivated the extensive use of the altimetry in the 1970's and 1980's (the latter from the Geosat orbit) to aid in the determination of both low and high degree terms in the field from averages of the mean surface or its gradient (e.g., Rapp, 1983). On the oceanographic side, the improved knowledge of the broad components of the geoid (the gravitational surface of the quiet ocean) from satellite tracking has, with the accompanying highly accurate knowledge of the Topex/Poseidon (T/P) orbit, permitted these same components of the mean sea topography (above the geoid) to finally emerge from the relative obscurity of earlier satellite altimeter estimates (e.g., Mather et al., 1978; Engelis, 1986; Nerem et al., 1994).
It has long been recognized of course that these (simple, residual) applications of satellite altimetry are only approximations and that the long-term averaged altimeter height residuals carry signals not just from the geoid or the "stationary" ocean topography but from a host of other influences on the orbit, in particular errors in the so-called accurate geopotential used for its computation, and at the ocean's surface from the unsteady component of the currents (time variable topography).
In fact, almost since the birth of satellite altimetry one of the goals of Satellite Geodesists has been to resolve all geodetic as well as oceanographic signals in the altimeter height of an orbit in a joint solution (Wunsch and Gaposchkin, 1980; Wagner, 1986, Engelis, 1988; Nerem et al., 1990). The rigorous solution is not only complex but at present limited to low spatial resolution (;SPMgt;1500 km) because of the accuracy limitations of satellite tracking instruments to sense the damped effects of the short scale geopotential on the orbit in order to distinguish its source geoid components in the altimetry from the same scale components in the sea topography which has negligible effect on the trajectory (Wagner, 1986).
Currently, most applications of altimeter data in oceanography either avoid the geopotential altogether, seeking only time-varying effects from collinear differences in Exact Repeat Missions (ERM), or employ statistical estimates of the orbit plus geoid error in assimilating the direct heights into ocean models (e.g., Nerem et al., 1994).
Our purpose here was to try and improve just the geopotential from the altimetry with as little influence from uncertain surface effects, oceanic and geoidal.
This dictates the use of altimetry at track crossovers (sea height differences) as a special filter of the complete detailed geoid (and mean sea topography) signal as well, passing only the geopotential-orbit information and undesired but hopefully smaller short or long term ocean signals (e.g., tides and other ocean dynamics), with due care for the time between crossover passes.
We deal with two kinds of the crossovers: between the ascending (A) and descending (D) track of one orbit, single-satellite crossovers (SSCs), and between tracks of two satellites, dual-satellite crossovers (DSCs). Of the DSCs there are 4 types: AA, DD, AD, and DA depending on the pass sense for each satellite at the crossover location (see KlokoŸn¡k et al., 1995).
Crossovers have been exploited previously in orbit-geopotenial modeling by Nerem et al., 1990 and Shum et al., 1994 [using Geosat SSCs and direct altimetry in conjunction with a joint geoid-ocean model],, Scharroo and Visser, 1998 [using SSCs in a refinement of the Ers1 orbit and geopotential model, and testing DSCs] and Moore et al., 1998 [using SSCs for Ers1 and DSCs between Ers1 and T/P to assess the validity of a joint geoid-ocean model using direct Ers1 altimetry and other tracking data].
As stated above our goal was to exploit, short of a joint solution, the maximum amount of altimeter information that could be ascribed principally to geopotential error in all three principal satellite altimeter orbits (thus crossovers); first to see what that information implied about the accuracy of current geopotential orbit models, second to see if we could improve those models with our data at the same time showing that our improvement was genuine and not just empirical.
For this purpose we chose as observations the differences of altimetric sea heights at geographic cross-over locations from passes either near contemporaneous or in the same month. Such measurements would contain no geoid or "stationary" and presumably only small seasonal ocean signals to confuse a purely satellite-geopotential analysis (Shum et al.,1990; KlokoŸn¡k et al., 1995, 1999). In addition, we preprocess our data by averaging it into small geographic bins to take advantage of Rosborough's (1986) geographic representation of the dominant (circular) orbit-geopotential effects.
The first part of our study was published a year ago (KlokoŸn¡k et al., 1999). We concluded that the substantial spectrum of orbit errors predicted for the trajectory model used on the extensively tracked altimeter satellites Geosat, Ers1 and T/P, Jgm3 (Tapley et al., 1996), are indeed realized in these cross over observations (none of which were used in developing Jgm3). An important finding was that while for the most part the errors found agreed well with the predictions, a fair number of low order terms were anomalous especially when the observations were from DSCs over multi-year gaps. [Recall, DSCs sense the geographically correlated parts of the radial orbit error due to the geopotential unobserved by SSCs, adding independent strength to the elucidation of the radial error from crossover altimetry; see formulae in KlokoŸn¡k et al., 1995). We also observed that the agreement (between errors and predictions) were generally good for each of the three SSCs for Geosat, Ers1 and T/P. These facts led us to the conclusion that non-stationary (or interannual) oceanography might have affected the multi-year DSC data. But the comparisons then were only with a statistical prediction (e.g., KlokoŸn¡k et al., 1998, 2000). Here we aim to invert our data and examine residuals to prove the nature of the anomalies.
In the following we first discuss the sources of our SSC and DSC measures and
the formulation of their inverse. Then we give results of the inversions
(adjusted coefficients, residuals and their statistics), and finally we
give other evidence from independent data which increase our confidence that
our solutions and their interpretation are valid.