c.c.tscherning, university of copenhagen, denmark. developments in the implementation and use of...

C.C.Tscherning, University of Copenhagen, Denmark

.

Developments

in the implementation and use of

Least-Squares Collocation.

IAG Scientific Assembly, Potsdam, Sept. 2013

LEAST-SQUARES COLLOCATION

Developed 1960+ by H.Moritz and T.Krarup for gravity field modelling.

Optimal (best) linear approximation method in reproducing kernel Hilbert-Space of harmonic functions or in an equivalent stochastic process. Minimum norm solution

All types of Data (linearized) can be used and ”predicted”

Errors can be estimated, Parameters may be solved for.

Convergence of method proven for increasing number of data in ideal situation.


LEAST-SQUARES COLLOCATION OPTIMAL

LSC optimal so it gives best results using less data.

Example with reduced point masses (RPM, a Radial basis-function) and LSC used in GOCINA test area with GOCE data.

Gravity anomaly residuals (mgal)

Height anomaly differences (m)

From (Herceg et al., 2012)


EGM08 GOCE Dir R3

RPM RPM all

LSC

Mean 0.38 0.28 -0.12 0.36 -0.68

St.devia. 20.04 11.89 11.97 15.95 10.87

Mean 0.02 0.01 0.00 0.02 -0.03

St.devia 0.58 0.52 0.52 0.59 0.50

LEAST-SQUARES COLLOCATION – Problems.

Hilbert space of harmonic functions outside a (Bjerhammar) sphere needed in order to make the reproducing kernels or covariance functions computable. But data at poles inside sphere !


LEAST-SQUARES COLLOCATION – Solution.

Use coefficients of an EGM as observations and change to spherical approximation. Predict equivalent data at poles at 20 km altitude.Equivalent to Remove/restore of EGM.Side effect: Data variance decreases and data are decorrelated !


LEAST-SQUARES COLLOCATION BASIC EQUATION


Construction of approximation to anomalous potential, from data with error ,

If no error,

LEAST-SQUARES COLLOCATION SOLUTION OF EQUATIONS

As many equations as observations (+parameters)

Multi-processing enables large systems to be Cholesky reduced.

Results from 2.40 GHz Intel® Computer


N 37971 22464 22464

Processors 22 22 4

Time (s) 440 136 391

GLOBAL LEAST-SQUARES COLLOCATION

GOCE TRF Tzz data used in approximate equal-area grid with up-ward continued gravity data at the poles. Results for coefficient


Model N, Number of observations

Estimate Error-estimate

EGM96 0.111 0.036

EGM2008 0.100 0.012

GOCE TIM2 10000000 0.105 0.015

LSC grid 42219 0.120 0.054

LSC grid 164212 0.106 0.028

LSC grid ~650000 ? 0.014

GLOBAL LEAST-SQUARES COLLOCATION

GOCE TRF Tzz data used in approximate equial-area grid with up-ward continued gravity data at the poles compared to GO_DIR-r2.


LSC COVARIANCE FUNCTIONS

Empirically estimated functions represented by Reproducing kernels assures positive definitness of .

Kernel estimated in 1974 still valid globally.


LSC COVARIANCE FUNCTIONS

Locally empirically estimated functions may be represented by reproducing kernels using basic model from 1974. Model is not optimal since it is isotropic.

Recent development Reguzzoni & Gatti: Anisotropic covariance modelling based on locally adapted coefficient variances in gravity field estimation.

Shows improvements compared to isotropic models !!


Local LSC COVARIANCE FUNCTIONS

Locally empirically functions are generally estimated from gravity anomaly data. But if not available or the associated histogram is not uniform, GOCE gravity gradient data may be used.

Disagreement between the analytic models determined from the data, blue: red: , green analytic from gravity. (Arabelos et al., 2013)


LSC Software

First general program written in Algol, 1972.

General FORTRAN program written in 1974 for 3D LSC, geocol, now geocol19, by C.C.Tscherning et al..

in 1986 2D LSC, gpcol, by R.Forsberg.

Both programs available for scientific or teaching purpose free of charge.

Programs also developed at TUGraz, POLIMI, UHannover.


LSC ERROR ESTIMATES + IMPROVEMENTS

Show:

(1) Sometimes quality of data

(2) Influence of different data types

(3) But may be improved knowing local signal standard deviation or (rms)

(4) Only show location of data


Example of local improvement of error estimates


Gravity anomalies from GOCE Tzz & EGM2008 to 512. (ITG-Grave2010c to 36 subtracted everywhere), units: mgal.

Differences and LSC error estimates


Differences gravity fromGOCE Tzz- EGM2008 to 512 and LSC error estimates.

Tzz RMS in 1 deg. blocks and scaled error estimates


Tzz RMS (E.U.) and and scaled error estimates (mgal).

Global scaling using Tzz RMS in 1 deg. Blocks.


GOCE Tzz (-ITG-GRACE2010c to 36) RMS (E.U)

Global scaling of LSC error estimates in 1x1 deg blocks.


Scaled error-estimates, mgal.

Conclusion

(1) LSC not anymore restricted due to large number of observations if multiprocessing can be used.

(2) Analytic ellipsoidal or an-isotropic kernels under development.

(3) Software available in GRAVSOFT package.

(4) The scaling of LSC derived error-estimates improves the error estimates, so that the variation of the error due to changing local signal standard deviation is seen.

Thanks to those who have contributed to the development:

D.Arabelos, M.Reguzzoni, F.Sansò, R.Barzaghi, R.H.Rapp, P.Holota, H.Sünkel, P.Knudsen, R.Forsberg, M.Veicherts, B.Sørensen, G.Moreaux,


c.c.tscherning, university of copenhagen, denmark. developments in the implementation and use of...

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