2. Monte Carlo Gravity Field Simulator.
In order to be able to assess the improvement of the accuracy of the various quantities expected from the GOCE mission, experiments were conducted using the simulated coefficient error estimates for GOCE computed by P.Visser (personal communication, 1998) as well as the error estimates of the coefficients of the EGM96 geopotential model (Lemoine et al., 1996). In Figure 2.1 the error degree variances from the EGM96 error model to degree 360 and from the GOCE error model to degree 250 are shown. From this Figure it is clear that at degree l = 200 the error degree variance of the GOCE error model remains below 0.02x10-5 ms-2 while the corresponding error of the EGM96 has almost reached a value equal to 0.26x10-5 ms-2 (more than 10 times larger).
The error spectrum has been used to derive the observation requirements for GOCE, see ESA(999, Table 4.1).
Figure 2.1: Error degree variances according to EGM96 and GOCE error models
The error of estimation of a gravity field quantity (like a point or mean geoid height) from space-measurement like those planned for GOCE should depend on the magnitude of the gravity field variation in the vicinity of the quantity being estimated: The number of measurements needed for the estimation of a point or mean value is higher in an area with high gravity field variation than in an area with low gravity field variation.
However, the error estimates available from EGM96 depend on the distribution of the data used to determine the coefficients, and not on the local gravity variation. The error estimates of GOCE seems only to depend on latitude (Rummel et al., 1995). The error estimates
reflect the existence of the polar gaps, for a near plar sun-syncroneous orbit, and the fact that the orbits becomes closer when the latitude increases numerically.
In the least squares collocation procedure the problem is solved by using a local covariance function, where the low-degree variances are equal to scaled error degree variances, obtained from the spherical harmonic expansion used in the remove-restore procedure (see e.g. Knudsen, 1987). The scale factor is larger than 1 if the local gravity field varies more than the global and it is less than 1 if we are in a smooth region. This has worked quite satisfactory, except that it seems that the errors in the low harmonics have been too low, when all error degree variances are multiplied with the same constant. Probably, a function dependent on the degree instead of a constant should have been used.
We therefore decided to use a similar procedure for the error variances for the spherical harmonic model coefficient error estimates. The scale factor is computed as the regional gravity variance divided by the global gravity variance as computed from the coefficients of the used spherical harmonic model. It is used then either as a constant scale factor for all degrees or a linear factor being equal to 1 at degree 0 and equal to the scale factor at the maximal degree used.
There is however a problem. If the data in the spherical harmonic expansion have been computed from mean values of blocks with size larger than 180/nmax, where nmax is the
maximum degree of expansion, then the local variation is erroneous. It should therefore be compared to the variation of the topography: If the topography has a large variation, then an error in the gravity data may have been detected.
The standard deviation of the gravity versus the standard deviation of the topography is shown in Figure 2.2. For the computation of the standard deviation of gravity, the GPM98A geopotential model (Wenzel, 1998) was used from degree 61 to degree 1800. Gravity anomalies were computed on a 5 x 5 grid on 3o equal areas. Then, the standard deviation for each 3o block was computed from the 25 point values. The same procedure was followed for the computation of the standard deviation of the topography. In this case the model used was the spherical harmonic model GTM3A, complete to degree and order 1800, based on the expansion of ETOPO5 (Wenzel, personal communication 1997). In Figure 2.2 there are many points showing that it is possible to have erroneous values due to errors either in gravity or in the topography.
These ideas have been implemented in a FORTRAN program HARMEXG. For a given error degree-variance model, pertubations of the spherical harmonic coefficients are calculated so that they for a given degree have mean value zero and variance equal to the error degree variance of this degree. Values in a grid or in a number of discrete points may be computed. Typically between 20 and 50 different sets of values are computed. The values may be pertubations of the geoid, of the gravity anomaly or the deflections of the vertical. A scaling procedure may be selected according to the ideas described above (no scaling, constant scaling or linear scaling).
The program was initially used to produce grids in 4 regions: Mediterranean, Himalaya, Antarctic and Indian ocean. Images of the geoid and gravity grids for EGM96 error degree variances, and for the three types of COCE error degree variances were produced. Here are only shown the 8 grids for the Mediterranean Region and the topography (Fig. 2.3 - Fig. 2.11). All the other figures can be accessed through the project home-page, http://www.gfy.ku.dk/~cct/task2.htm . In the figures the latitude effect is clearly seen. The difference between EGM96 errors and GOCE errors are also striking.