A fundamental inverse problem in geosciences

Transcription

1 A fndamental inverse problem in geosciences Predict the vales of a spatial random field (SRF) sing a set of observed vales of the same and/or other SRFs. y i L i ( ) + v, i,..., n i ( P)? L i () : linear fnctionals : vi random noise E { ( P)} m( P) trend of the nknown field E{( ( P) m( P)) ( ( Q) mq ( ))} ( P, Q) V fnction

2 Setting the stage yi Li ( ) + vi si + vi, i,..., n s( Pi ) Axiliary hypotheses: ) E { ( P )} 0, E { s } 0, E { s } 0 0 i y s+ v ) sv 0 + y s v 3) s [ i, j] L Lj ( Pi, P j c i T ( ) L i ( P, P i P s[ i] ) ) Signal V model + V propagation law 4) v [ i, j] E{ vivj} σ vivj Noise V model

3 Optimal generalized interpolation yi Li ( ) + vi si + vi, i,..., n s( Pi ) y s+ v Linear nbiased prediction of the nknown field with minimm mean sqared error ˆ( P) T c ( P ) ( + ) s s v y ˆ s ( s + v ) y Least-sqares collocation (LS) Kriging Wiener-Kolmogorov (WK) theory

4 Hidden characteristics of the LS soltion The signal prediction algorithm is not affected by the spatial distribtion of the prediction points (the LS estimate at P is the same, regardless of the nmber and/or the geometry of other prediction points). ˆ( P) ˆ( Q) T c ( P ) ( + ) T c ( Q ) ( + ) s s s s v v y y R P Data points y s + v i i i ˆ ( R)... ˆ s ( s + v ) y Q Prediction points

5 Hidden characteristics of the LS soltion ˆ s ( s + v ) y Model-denial occrs throgh the LS estimation procedre, in the sense that: [ i, j] ( Pi, P j ˆ s s+ v) T ( ) s e ˆ ˆ e The ato-covariance strctre of the estimated SRF is different from the ato-covariance strctre of the corresponding tre SRF (field smoothing).

6 De to the fact that: ˆ e - Smoothing effect in the field estimate ˆ ( P). - Field images based on LS are nsitable for applications where the spatial signal variability is a key element for scientific inference. - Signal details not inherent or proven by the actal data (i.e. artifacts) are not prodced, bt.. - Inability to reprodce even the empirically determined V fnction model ( P, Q).

7 Example ( P, Q) 750 mgal o ( τ PQ ) o τ + ( ) a ξ 9.5 km Spatial resoltion: km

8 Smoothed image that does represent realistically the spatial variability of the nderlying field

9 Histogram of the tre signal realization Signal realization Max Min Mean 0.70 σ Histogram of the LS signal soltion theoretical σ 7.39 mgal û LS soltion Max 4.6 Min Mean.95 σ 5.39

10 Preliminary conclsions Signal predictions from LS shold be listed, not mapped! LS is the optimal prediction method for localized signal recovery (minimm pointwise MSE). LS is nsitable for spatial field mapping spatial field mapping (fails to reprodce the model-based spatiostatistical variability of the nknown field). The final reslt of LS denies its fndamental bilding component (i.e. the V fnction of the nderlying nknown signal)!

11 A revised formlation Apply a post-processing processing correction algorithm (de-smoothing transformation) on the LS soltion sch that: ˆ R{ˆ} ) The V strctre of the SRF is preserved ˆ ) The estimation error remains small in some sense e ˆ i.e. minimize some fnctional of e

12 R ˆ ˆ A linear revised formlation Linear transformation of the LS soltion where R is a filtering matrix that needs to be T ) ( ) ( ˆ R I R I e e + determined according to some optimality criteria Note that: e ˆ e ˆ LS estimation error (LS &R-filtering) estimation error

13 Filtering optimization Determine the transformation matrix R ˆ Rˆ e ˆ sch that T ˆ Rˆ R V-matching constraint sbject to the optimal prediction criterion e e + ( I R) ˆ ( I R) T From LS δ e trace δ e minimm

14 The optimal matrix R Solving the previos optimization problem, we get the reslt: ) ( ˆ / / / / / R ˆ ˆ ˆ ˆ ) ( / / / / / R - Symmetric - Positive definite or eqivalently

15 Alternative forms for the optimal filtering matrix R Using the matrix identity: / ) ( / STS S ST we obtain the eqivalent compact expressions for the matrix R R ) ( ˆ / expressions for the matrix R ˆ ˆ ) ( R / ) ( ˆ ˆ / R ) ( ˆ / R

16 Prediction error o-variance matrix of the prediction error e e + ( I R) ˆ ( I R ) T If all eigenvales of the filtering matrix R * are larger than one, i.e. R I + δr positive definite matrix then it can be shown that trace ( e ) trace( e) * not necessarily optimal..

17 Test (noise filtering) (τ ) o τ + ( ) a o 0 mgal Spatial resoltion: km σ v 35 mgal (white noise) ξ 7 km

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19 theoretical σ 4.83 mgal û û Signal realization Max Min Mean σ 4.96 LS soltion Max 7.4 Min -9.0 Mean 0.74 σ 0.9 V-adaptive soltion Max 4.47 Min Mean 0.73 σ 4.64

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21 ˆ ˆ LS estimation errors Max 8.70 Min Mean 0.78 σ 9.88 V-adaptive estimation errors Max 3.98 Min Mean 0.77 σ.0

22 Test (noise filtering + spatial interpolation) ( ) o τ Spatial resoltion: km τ + ( ) a σv 5 mgal (white noise) o 0 mgal ξ 8 km

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24 theoretical σ 4.83 mgal û û Signal realization Max Min Mean -.9 σ 4.94 LS soltion Max 3.04 Min Mean 0.67 σ 8.85 V-adaptive soltion Max 3.48 Min Mean 0.94 σ.8

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26 ˆ ˆ LS estimation errors Max 39.0 Min Mean.86 σ.58 V-adaptive estimation errors Max 34.5 Min Mean.3 σ.8

27 onclsions LS provides optimal local accracy, yet poor global spatial rendering in SRF prediction problems. New prediction approach based on optimal desmoothing throgh V-matching filtering; i.e. reprodce the signal V fnction with minimm loss in the MSE accracy Identify relevant application areas where global spatial accracy is an important isse (preserving non-stationary patterns, spatial field mapping, ) Are there any fndamental isses hidden on the trade-off between local and global accracy in SRF estimation from discrete data?

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