Regional Gravity field modeling as multi-resolution representation estimated from the combination of heterogeneous data sets

Transcription

1 Regional Gravity field modeling as multi-resolution representation estimated from the combination of heterogeneous data sets Verena Lieb 1, Klaus Börger 2, Wolfgang Bosch 1, Johannes Bouman 1, Kirsten Buße 1, Denise Dettmering 1, Barbara Görres 3, Martin Fuchs 1, Christoph Haberkorn 1, Wilhelm F. Kersten 3, Sabine Kirsch 1, Gerhard Ressler 1, Michael G. Schmidt 1, Christian Schwatke 1, and Florian Seitz 1 (1) DGFI German Geodetic Research Institute, Munich, Germany, Centre of Geodetic Earth System Research (CGE) (2) German Space Situational Awareness Centre (GSSAC), Uedem, Germany (3) Bundeswehr Geoinformation Centre (BGIC), Euskirchen, Germany

2 2 Motivation Satellite gravimetry Task: Software developement for the flexible computation of regional gravity and geoid models from Altimetry airborne gravimetry Difficulty: How to combine data sets with different distribution, resolution, and accuracy? terrestrial gravimetry heterogeneous data sets. Aim: Construction of precise height systems in operational areas.

3 3 Approach Regional gravity modeling using radial basis functions (e.g. Blackman). gravity observations (e.g. airborne, terrestrial) functionals of the Earth s gravity field (e.g. gravity anomalies Dg) ם target area observations x computation grid mgal

4 4 Observations GOCE GG altimetry, airborne, sea ground, terrestrial measurements sensitive to different frequency bands defined by resolution levels j limited by maximum degree L j in a series expansion related to the spatial resolution r on the Earth s surface j [level] L j [deg] r [km] frequency satellite gravimetry degree l altimetry mean spatial resolution: 10 km airborne + terrestrial gravimetry

5 5 Combination of data sets Q ΔF x = d J,q b J x, x q q=1 Q = d J,q q=1 L J l=0 2l + 1 4π R r l+1 B l P l cos θ GOCO03s Observation 1 up to d/o Functional F Combination of GRACE, GOCE, SLR, technique Modified basis function b J GRACE ΔV GOCE V zz (e.g.) Altimetry N = SSH - DOT Terrestrial, air-, shipborne, bathymetry measurements Dg δg Estimation of unknown scaling coefficients d J by using an extended Gauß-Markov model and VCE (rigorous combination at one level j).

6 6 MRR Multi-Resolution Representation: Computing a target signal F J from a smoothed version F j and a number of detail signals G j L = 255 j = 8 G 9 L = 511 j = 9 G 10 L = 1023 j = 10 G 11 L = 2047 J = 11 Computation of detail signals G j from different observations related on their maximum spectral content at the specific frequency band of G j mgal

7 7 Relative weighting Observation j = 8 (L = 255) j = 9 (L = 511) j = 10 (L = 1023) j = 11 (L = 2047) GOCE V xx GOCE V xy GOCE V xz GOCE V yy GOCE V yz GOCE V zz ERS-1e ERS-1f Jason 1 GM Envisat EM Cryosat RADS Airb. North Sea Airb. Baltic Sea Terrestrial Data Bathymetry Prior information GOCO03s d/o

8 8 Relative weighting Observation j = 8 (L = 255) j = 9 (L = 511) j = 10 (L = 1023) j = 11 (L = 2047) GOCE V xx GOCE V xy GOCE V xz GOCE V yy GOCE V yz GOCE V zz ERS-1e ERS-1f Jason 1 GM Envisat EM Cryosat RADS Airb. North Sea Airb. Baltic Sea Terrestrial Data Bathymetry Prior information GOCO03s d/o Criteria high sensitivity

9 9 Relative weighting Observation j = 8 (L = 255) j = 9 (L = 511) j = 10 (L = 1023) j = 11 (L = 2047) GOCE V xx 1 GOCE V xy 10-4 GOCE V xz 10-1 GOCE V yy 1 GOCE V yz 10-4 GOCE V zz 1 ERS-1e ERS-1f Jason 1 GM Envisat EM Cryosat RADS Airb. North Sea 1 1 Airb. Baltic Sea Terrestrial Data 1 1 Bathymetry 10-1 Prior information GOCO03s d/o Criteria high sensitivity no correlations spatial distribution (prior information not sufficient)

10 10 Coefficients coefficients d mgal std. dev. σ 11 mgal selected coefficients (78%) mgal 0 2 mgal j = 11 (78%) j = 10 (66%) j = 9 (39%) j = 8 (59%) filled up with mean values. detected outliers depending on: observation type -> data gaps dimension of computation grid

11 11 Summation of detail signals MRR: Dg 11, MRR mgal mgal Dg 11 : σ 11 : mgal mgal ΔDg 11 mgal Largest standard deviations in less observed regions! Dg 11, rig ΔDg, mean +/- std: / mgal Largest differences at data gaps close to the borderline. Improvement: MRR-solution contains optimized spectral information in all frequency domains (contribution of GOCE)!

12 12 Outlook & Summary Outlook improving selection of input data choosing prior information with higher spectral content (e.g. topographic models) considering correlations between detail signals (e.g. introducing a filter matrix) improving outlier detection validation with real data further study areas Criteria high sensitivity no correlations spatial distribution (prior information not sufficient) Summary Rig. j = 11 MRR combination up to j = 11 + less unknowns to estimate - larger number of unknowns - relative weighting of obs. at highest level + relative weighting of obs. at each level + spectral information in all frequency bands + improved handling of data gaps stabilized solution Exploiting the highest degree of information out of each data set. The authors want to thank the BKG (Bundesamt für Kartographie und Geodäsie, Leipzig, Germany) for providing us the high-resolution terrestrial and airborne gravimetry data sets.

13 Appendix

14 14 Comparison with EGM2008 Dg 11, final mgal Difference Dg 11, MRR EGM2008 (j = 11, l = 2023, Blackman smoothed) ΔDg, mean +/- std: / mgal Largest differences at data gaps. Differences up to +/- 10 mgal in western parts (new data set?) and in the Baltic Sea (missing airborne data in EGM?). Difference Dg 11, rig EGM2008 ΔDg, mean +/- std: / mgal Larger differences (especially in western parts). Missing spectral information in mid and low frequency domains. ΔDg 11, MRR ΔDg 11, rig mgal mgal

15 15 Software Specifications Study area: longitude latitude L 11 = 2047 (J = 11) depending on (spectral/spatial) resolution of input data Computation grid: Reuter depending on resolution level J # grid points = # unknowns Q Background model: GOCO03s 1 up to d/o 127 further serves as prior information due to rank deficiency problems 1 Combination of GRACE, GOCE, SLR,

16 16 Software Output Computation of the target signal up to max. level J = 11 Synthesis: Blackman scaling funct. Output: different functionals of the Earth s gravity field (e.g. gravity anomalies Dg)

17 17 Coefficients j = 8 j = 9 j = 10 j = 11 j = 12

18 18 Analysis N ΔF x = d J,q b J+1 x, x q q=1 = N q=1 L J l=0 2l + 1 4π d R J,qΦ J+1,l r l+1 P l cos ψ Observation equation for one observation: Deterministic part ΔF x + e x = b T J+1 x, x q d J IN: ΔF observation e measurement error b J+1 (Nx1) vector of basis functions OUT: d J (Nx1) vector of scaling coefficients Stochastic part D ΔF k = σ 2 1 k P k IN: ΔF k vector of observations P k weighting matrix of observations OUT: σ k variance components (VCs) Estimation of unknown scaling coefficients d J Introduction of additional observations μ d μ d + e d = d with D μ d = σ 2 1 d P d μ d prior information avoiding singularity problems rank deficiencies (in general number of grid points too large)

19 19 Modelling approach Analysis Extented Gauß-Markov model for several observation techniques: y = V 1 μ d = y 1 μ d vector of observations y 1 y K μ d + e 1 e K ed = (Nxn k ) T b J+1,1 T b J+1,k I d J (Nx1) matrix of scal. functions vector of scal. coefficients D y 1 μ d = σ k 2 σ d 2 P Σ d k = 1 K n k various observation techniques number of observations from technik k Note: the measurements y are treated as independent observations, i. e. without correlations. Soving the normal equations (by iteratively determined VCs) results in d J. Extracting the erroneous observations y and applying the law of error propagation then results in the variance covariance matrix D d J.

20 20 Detail signal j = 11 G 11 G 11 min max [mgal] mean [mgal] +/- std. [mgal] Dg σ large standard deviations in areas of replaced mean-coefficients -> outlier detection Dg 11, red σ 11