Document: GOCE_s2.doc Date: 2001.05.08, Author: C.C.Tscherning.

1.2.2. Slice 2. High level processing architecture.

Module Description.

In the following input to slice 2 relates to EGG-C tasks 3, 5, 6 and 7 and are associated with the use of the space-wise approach only.

**Task 3. Aid to preprocessing**.

Title: (SW 3.1). Parameters for regional covariance functions.

Abstract: Ground gravity data from equal-area blocks covering the Earth and from selected calibration areas are used to estimate parameters of an analytic expression of the covariance function.

Input: (1) A-priori spherical harmonic model. (2) 5’ mean free-air anomalies, globally, (3) point free-air anomalies in calibration areas, (4) digital terrain models and after launch: (5) GOCE SST and SGG data.

Constants: Gravity in calibration area must be in IGSN71 and associated heights must be in a well defined datum.

Structure:

- The contribution from the a-priory global model is subtracted from the gravity data.
- In the calibration areas the residual topographic effect is calculated and subtracted.
- In each block or calibration area the empirical covariance function is estimated.

(d) The parameters of an analytic representation are determined.

Output: File with area boundaries and parameters of associated analytic covariance function.

Status: GRAVSOFT programs GEOCOL, TC, EMPCOV and COVFIT will execute tasks (a) – (d). The optimal size of the regions and the location of the calibration areas have not been decided. Data to be used for calibration must be certified.

Critical items: Before launch some areas lack ground data to be used in the covariance estimation. After launch the satellite data can be used for this purpose.

Title: (SW 3.2). Calibration of GOCE data and the associated error-estimates.

Abstract: GOCE data may be calibrated using ground data. This will be done in a unified procedure using least-squares collocation with parameters which will represent contingent biases and drifts.

Input: Gravity data from selected calibration areas reduced with respect to a global spherical harmonic model and topographic data, see SW 5.3.1. Covariance function parameters (see SW 3.1). Before launch: simulated GOCE data with simulated bias and drift errors. After launch: obviously real data.

Constants: All data must be given in the same reference system.

Structure:

- Identification of calibration areas. Selection of ground data.
- Selection and "reduction" of the data (see SW 5.3.1).
- The ground data are used to determine upward continued data, which are compared with the GOCE data. The spectrum (or covariance function) of differences between the computed and the observed data is used to calibrate the error-estimates.
- The data are used to determine an approximation of the anomalous gravity field for the area using simultaneously ground and GOCE data. As a part of this, parameters and their error-estimates may be determined.
- If calibration factors are significantly different from zero, they should be applied on the level 1b data.

Output:

Sets of calibration parameters, including contingent noise scale factors.

Status:

The GEOCOL program has been prepared as a prototype for carrying out the task, and used in the CIGAR projects. The program works in spherical approximation. Updates must be made.

Topographic reduction of GOCE data types are not fully implemented in the GRAVSOFT program TC.

Since the character of calibration factors as yet have not been specified, only determination of bias and tilt (linear drifts) have been tested. Contacts with industry are needed to identify the character of the possible calibration parameters (non-linear ??):

Critical Items: Availability of validated data in a selected calibration area. Update needed for TC in order to be used to reduce GOCE data for topographic effects. However, if a sufficiently "smooth" area is used for calibration, this may not be needed.

Title (SW 3.3). Gross-error check.

Abstract: GOCE data are predicted from ground data and from other GOCE data located in an area around the data to be checked. The difference will be compared with the estimate of the error of prediction. The data will be flagged if the difference is larger than 3 times the error of prediction.

Input: Global digital terrain model. Ground data (if available) and GOCE data reduced with respect to a spherical harmonic model as well as topography (see SW 5.3.1). Covariance functions (see SW 3.1).

Constants: all data in the same reference system.

Structure:

- Reduced GOCE data are predicted from ground data and from other (reduced) GOCE data located in an area around the data to be checked.
- The difference will be compared with the estimate of the error of prediction.
- The data will be flagged if the difference is larger than 3 times the error of prediction.

Output: notification of flagged data. Statistics of flagged data.

Status: The procedure has been tested in spherical approximation using only one SGG data-type using a dedicated program GEOGRIDX. How SST data can be tested using this program is an open question.

Several data-types can be used to test individual quantities using the more resource-demanding program GEOCOL.

Critical items: Implementation in the program GEOGRIDX the possibility of using simultaneously different quantities.

Further tests needed of software. The program uses isotropic covariance functions. This has as a consequence that data (e.g. in mountains) which vary strongly may be flagged erroneously.

Task 5.3. Gravity model space-wise.

Title: (SW 5.3.1) Computation of anomalous quantities corresponding to the GOCE observables.

Abstract: Computation of the differences between the observations and the contribution from the a-priori spherical harmonic model and contingent time-varying quantities.

Input: Coefficients of the (currently best) spherical harmonic model. Models for time-varying phenomena. Calibrated GOCE level 1b data, including attitude data, see SW 3.2.

Constants: a-priori coefficients, models for time-variations.

Structure: The spherical harmonic model is used to calculate the values corresponding to the GOCE observables in their proper reference frame. The models for time variations are used in a similar manner. Differences are formed. The data are then used as input for gross-error detection see SW 3.3, and contingently flagged.

The task will be repeated using updated spherical harmonic models.

Output: Anomalous quantities (in GRAVSOFT format).

Status: The GRAVSOFT program GEOCOL is able to perform this task as a prototype. The program is able to handle data which are along-track weighted mean values.

Critical items: The prototype treats SGG data as if they contained all gravity field harmonics. How this problem should be solved is still an open question. SST data may be treated as potential differences using the state-vector. But information is lost when converting from single velocity vector components to the absolute velocity. Furthermore there is the problem of how non-eliminated non-inertial forces should be treated is. Will the function of thrusters eliminate these forces ??

Title: (SW 5.3.2) Computation of gridded values at mean satellite altitude.

Abstract: Least-squares collocation (GEOCOL) is used to predict gridded values of potential (from SST) differences and of the two independent derivatives T_{zz} and T_{yy}-T_{xx} in an earth-fixed frame, with z-axis in the direction to the centre of the Earth. All the values at the same parallel will be associated with a fixed distance to the centre of the Earth. A grid will be constructed for each region which has an individually defined covariance function, cf. SW 3.1.

Input: Anomalous quantities, see SW 5.3.1. Parameters for analytic covariance function, cf. SW 3.1.

Constants: n/a

Structure:

- Data (anomalous quantities) from a specific region is selected.
- An approximation of the anomalous potential for the region is constructed.
- Values of potential differences, and derivatives T
_{zz}and T_{yy}-T_{xx}in an earth-fixed frame at a fixed radial distance (e.g. on a sphere) are computed. - Their error-estimates are computed simultaneously.

Output: Grids on GRAVSOFT format. Regional solution to be used in Task 9.

Status: A prototype GEOCOL is designed to solve this task. It works presently in spherical approximation. Examples of use are included in the CIGAR reports (Arabelos and Tscherning, 1993) and furthermore in (Arabelos and Tscherning, 1990, 1993). Further tests are especially needed in combining SST and SGG data.

The program may use data which have errors which are correlated along-track.

Critical Items: Test of combination of SSG and SST data.

Title: (SW 5.3.3) Computation of corrections to the a-priori spherical harmonic coefficients from gridded data.

Abstract: Corrections to the a-priori spherical harmonic coefficients as well as their error estimates will be determined using the method of fast spherical collocation (Sanso and Tscherning, 2001) from data gridded equidistantly in longitude. The data must have assigned the same noise variance for each parallel. T_{zz} and potential difference data can be used.

Input: Global covariance function parameters. Output from SW 5.3.2.

Constants: n/a.

Structure:

- The parameters of a global covariance function are determined, using the GRAVSOFT programs EMPCOV and COVFIT.
- The data computed regionally are used as input to a new program SPHGRID which implements Fast Spherical Collocation.
- Corrections to spherical harmonic coefficients and the error-estimates of these coefficients are determined. Error covariances are determined.

Output: Updated spherical harmonic coefficients and error-estimates.

Status: The prototype program SPHGRID has been tested to degree and order 720 using simulated T_{zz} point data.

Critical items: The real data will be correlated. A method to take this into account is not available at present.

Title (SW 5.3.4) Computation of corrections to the a-priori spherical harmonic coefficients from non-gridded data.

Abstract: A global approximation to the anomalous potential is constructed using selected data using GEOCOL. Sparse matrix techniques will be used together with the conjugent gradient method (cgm) for solving the normal equations.

Input: Global digital terrain model. Global covariance function parameters. (see SW 5.3.3). Selected level 1b data reduced using the spherical harmonic coefficients produced in SW 5.3.3. Probably a subset of T_{zz }data in the orbital frame must be used

Constants: n/a.

Structure:

- Data are selected so that the largest data density is in areas with the largest gravity field variation.
- Normal equations with sparse matrices are established using software written by G.Moreaux (Moreaux, 2001).
- The equations are solved using the cgm method.

Output: Approximation to the anomalous potential. Estimated spherical harmonic coefficients. Global grids of gravity anomalies and geoid heights evaluated at the Earth’s surface.

Status: Tests with 41000 observations using a full matrix has been made (Tscherning, 2001). Software to compute finite covariances has been developed. Prototypes exist. The software has not yet been tested on satellite data.

Critical items: How will the finite covariance functions perform with data scattered in altitude ? How will the cgm method work for large, global datasets ?

Task 6. Solution evaluation. (This description is common for all products produced by Tasks 6 and 9).

Title: 6.1. Evaluation of sets of coefficients of spherical harmonic coefficients.

Abstract: The sets of spherical harmonic coefficients and their error-covariances are evaluated. This will be done by using the coefficients for orbit prediction, comparing derived gravity and geoid values with ground data and comparing the results of ocean current velocities derived from altimetry with observed velocities.

Input: Different sets of estimated spherical harmonic coefficients and their error-covariance matrices. Satellite tracking data from selected satellites. Ground data: gravity anomalies, gravity disturbances, deflections of the vertical, height-anomalies from GPS and levelling. Satellite radar altimetry. Ocean current data.

Constants: Datum transformation constants. Reference system constants.

Structure.

- Calculation of orbits for selected low-flying satellites and comparing with tracking data.
- Comparing gravity, deflection of the vertical and geoid data derived from the coefficients with ground data using interpolation.
- Evaluating differences, taking into account the variance-covariances of the estimated coefficients.
- Select a final product to be adopted by ESA as a final GOCE only solution.

Output: recommendation to ESA about best set of coefficents.

Status: The process has been tested when selecting the EGM96 model. An international group took part in the evaluation, and a similar group will have to be established for GOCE. This will have to be negotiated with IAG, represented by IgeS.

Critical items: A method to evaluate the error-covariances is lacking.

Title: (6.2). Evaluation of grids of mean gravity anomalies and geoid heights.

Abstract: Grids of e.g. 0.5 degree mean gravity and geoid heights not computed from spherical harmonic models are evaluated by comparing interpolated values with observed mean values on the ground.

Input: Grids of gravity anomalies and height anomalies at terrain level as well as the error estimate of the values.

Mean values of ground data. Mean values of geoid heights computed using oceanographic data (satellite radar altimetry and in-situ current, salinity and temperature data).

Constants: datum and reference system parameters.

Structure: The gridded values are compared the observed mean data values, and the mean and standard deviation of the differences are calculated. The differences are compared using the estimates of the errors of the two data-types.

Output: Estimates of the agreement between the observed and the calculated values. Possible scale-factor correction of error estimates.

Critical items: Correct estimates of the errors of the ground data are missing. Mean height anomalies do not exist at present. Mean values of oceanographic "geoid" heights do not exist.

9. Regional solutions.

Here we specialize to the use of LSC.

Title: (SW-LSC 9.1) Computation of regional solutions using LSC.

Abstract: LSC is used to compute regional approximations to the anomalous potential in a number of equal-area blocks covering the area which also is covered by GOCE. Each block must have an overlap with the neighbouring block.

Input: Regional covariance function models (see SW 3.1). Reduced GOCE data checked for gross-errors and systematic errors (see SW 5.3.1, 3.2 and 3.3). It is feasible to treat up to 200000 observations simultaneously.

Constants: covariance function parameters.

Structure:

- The program GEOCOL is run with the given input parameters and data.
- The regional solution is used to compute values (and their error-estimates) gridded in longitude at the mean orbit altitude, see SW 5.3.2.
- The solution is used to compute detailled grids of gravity disturbances, gravity anomalies and height anomalies at ground level. Error estimates are also computed.

Output: gridded values at satellite altitude and at ground level.

Critical Items: Since an isotropic covariance function is used, the error-estimates will not reflect the changes in the gravity field variation inside the area. A topographic reduction may help here.

**Module interaction** is described in the following figure.

References:

Arabelos,D. and C.C.Tscherning: Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT. Bulletin Geodesique, Vol. 64, pp. 363-382, 1990.

Arabelos, D. and C.C.Tscherning: Regional recovery of the gravity field from SGG and SST/GPS data using collocation. In: Study of the gravity field determination using gradiometry and GPS, Phase 1,

Final report ESA Contract 9877/92/F/FL, April 1993.

Arabelos, D. and C.C.Tscherning: Regional recovery of the gravity field from SGG and Gravity Vector data using collocation. J.Geophys. Res., Vol. 100, No. B11, pp. 22009-22015, 1995.

Moreaux, G.: Some preconditioners of harmonic spherical spline problems. Inverse Problems, Vol. 17, pp. 157-177, 2001.

Sanso', F. and C.C.Tscherning: Fast spherical collocation. Paper prepared for IAG2001, Budapest, Sept. 2001.

Tscherning, C.C.: Computation of spherical harmonic coefficients and their error estimates using Least Squares Collocation. J. of Geodesy, Vol. 75, pp. 14-18, 2001.