A fundamental inverse problem in geosciences

Predict the values of a spatial random field

(SRF) using a set of observed values of the

same and/or other SRFs.

nivuLy iii ,...,1 , )( =+=

)()}({ PmPuE =

),())}()(( ))()({( QPCQmQuPmPuE u=−−

?)( =Pu

trend of the unknown field

CV function

:)(uLi linear functionals :iv random noise

Setting the stage �

nivsvuLy iii

iPs

ii ,...,1 , )(

)(

=+=+=���

0s === }{ , 0}{ , 0)}({ EsEPuE i

vsy +=

Auxiliary hypotheses:

1) 0s === }{ , 0}{ , 0)}({ EsEPuE i

0Csv= vsy CCC +=

),( ],[ jiuji PPCLL ji=sC

1)

jvivjiji vvE σ }{ ],[ ==vC

),( ][T

)( iui PPCLiPu =sc

2)

3)

CV propagation law

Noise CV model

Signal CV model+

4)

Optimal generalized interpolation

1T −+=

Linear unbiased prediction of the unknown field

with minimum mean squared error

nivsvuLy iii

iPs

ii ,...,1 , )(

)(

=+=+=��� vsy +=

yCCc vss 1)( )(ˆ

T)(

−+= PuPu

− Least-squares collocation (LSC)

− Kriging

− Wiener-Kolmogorov (WK) theory

yCCCu vsus 1)( ˆ −

+=

“Hidden” characteristics of the LSC solution

The signal prediction algorithm is not affected

by the spatial distribution of the prediction points

(the LSC estimate at P is the same, regardless of the

number and/or the geometry of other prediction points).

yCCc vss 1)( )(ˆ

T)(

−+= QuQu

yCCc vss 1)( )(ˆ T

)(−

+= PuPu

... )(ˆ =Ru

Data points

Prediction points

iii vsy +=

number and/or the geometry of other prediction points).

yCCCu vsus 1)( ˆ −

+=

P

Q

R

“Hidden” characteristics of the LSC solution

Model-denial occurs through the LSC

estimation procedure, in the sense that:

yCCCu vsus 1)( ˆ −

+=

uusvsusu CCCCCC )( T1ˆ ≠+=

−

The auto-covariance structure of the estimated

SRF is different from the auto-covariance structure

of the corresponding true SRF (field smoothing).

),( ],[ jiuji PPC=uC

euu CCC −= ˆ

uue −= ˆ

Due to the fact that:

- Smoothing effectSmoothing effect in the field estimate

ueuu CCCC ˆ ≠−=

- Field “images” based on LSC are unsuitable for

).(ˆ Pu

- Field “images” based on LSC are unsuitable for applications where the spatial signal variabilityspatial signal variability is a key element for scientific inference.

- Signal details not inherent or proven by the actual data (i.e. artifacts) are notnot produced, but ..

).,( QPCu

- Inability to reproduce even the empirically determined CV function model

Example

2

)(1

C )( ),(

a

QPCQPC ouu ττ

+

=−==

2mgal 750 C =o

km 9.5 ξ = Spatial resolution: 2 km

Smoothed image that does

represent realistically the spatial

variability of the underlying field

Histogram of the LSC signal solution

-80 -60 -40 -20 0 20 40 60 80 1000

10

20

30

40

50

60

70

80Histogram of the true signal realization

Signal realization

Max = 65.30

Min = -76.85

Mean = 0.70

σ = 22.71

u

-80 -60 -40 -20 0 20 40 60 80 1000

10

20

30

40

50

60

70Histogram of the LSC signal solution

LSC solution

Max = 42.62

Min = -49.86

Mean = 2.95

σ = 15.39

theoretical σ = 27.39 mgal

ˆ u

Preliminary conclusions

− Signal predictions from LSC should be listed, not mappednot mapped!

− LSC is the optimal prediction method for localized signal recoverylocalized signal recovery (minimum pointwise MSE).

− The final result of LSC denies its fundamental building component (i.e. the CV function of the underlying unknown signal)!

− LSC is unsuitable for spatial field mappingspatial field mapping(fails to reproduce the model-based spatio-statistical variability of the unknown field).

A revised formulation

}ˆ{ ˆ uu ℜ=′

Apply a postApply a post--processing “correction” processing “correction”

algorithm (dealgorithm (de--smoothing transformation) smoothing transformation)

on the LSC solutionon the LSC solution

such that:

1) The CV structure of the SRF is preserved

uu CC ˆ =′

2) The estimation error remains small in some sense

eC ′of functional some minimize i.e.

uue −′=′ ˆ

uRu ˆ ˆ ⋅=′

A linear revised formulation

Linear transformation of the LSC solution

where R is a filtering matrix that needs to be

determined according to some optimality criteria

T)()( ˆ RICRICC uee −−+=′

determined according to some optimality criteria

Note that:

uue −′=′ ˆ

uue −= ˆ LSC estimation error

(LSC &R-filtering) estimation error

uRu ˆ ˆ =′

Filtering optimization

Determine the transformation matrix R

such that

T

uue −′=′ ˆ

subject to the optimal prediction criterion

uuu CRCRC T ˆˆ ==′ CV-matching constraint

T)()( ˆ RICRICC uee −−+=′

eC ′δ minimum δ =′eCtraceFrom LSC

The optimal matrix R

Solving the previous optimization problem,

we get the result:

2121212121 )( ˆ/////uuuuu CCCCCR −

=

2121212121ˆˆˆˆ )( ///// −−

= uuuuu CCCCCR

- Symmetric

- Positive definite

or equivalently

Alternative forms for the optimal filtering matrix R

Using the matrix identity: 21112/1 )( /−−= STSSST

we obtain the equivalent compact expressions for the matrix R

uuu CCCR 21)( ˆ/−

=

expressions for the matrix R

121ˆˆ )( −

=uuu CCCR

/

211 )( ˆˆ/

uuuCCCR

−=

21)( ˆ/−

= uuu CCCR

Prediction error

Co-variance matrix of the prediction error

If all eigenvalues of the filtering matrix R*

T)()( ˆ RICRICC uee −−+=′

If all eigenvalues of the filtering matrix R*

are larger than one, i.e.

)(2 )( ee CC tracetrace ×≤′

δRIR +=

then it can be shown that

positive definite matrix

* not necessarily optimal..

Test 1 (noise filtering)

2

)(1

C )(

a

C ou ττ

+

=

2mgal 220 C =o km 7 ξ =

Spatial resolution: 2 km

noise) (white mgal 35 σ =v

Signal realization

Max = 45.33

Min = -42.88

Mean = -0.04

σ = 14.96

LSC solution

Max = 27.42

Min = -29.10

ˆ u

u

theoretical σ = 14.83 mgal

Min = -29.10

Mean = 0.74

σ = 10.29

CV-adaptive solution

Max = 41.47

Min = -42.48

Mean = 0.73

σ = 14.64

ˆ u ′

LSC

estimation errors

Max = 28.70

Min = -31.94

Mean = 0.78

σ = 9.88

ˆ uu −

CV-adaptive

estimation errors

Max = 31.98

Min = -32.05

Mean = 0.77

σ = 11.01

ˆ uu −′

Test 2(noise filtering + spatial interpolation)

2C 220 mgalo = ξ 8 km=

Spatial resolution: 2 km

(white noise)σ 5 mgal v =2

)(1

C )(

a

C ou ττ

+

=

Max = 35.76

Min = -40.09

Mean = -1.19

σ = 14.94

LSC solution

Max = 31.04

Min = -24.38

ˆ u

uSignal realization

Min = -24.38

Mean = 0.67

σ = 8.85

Max = 32.48

Min = -28.56

Mean = 0.94

σ= 11.18

ˆ u ′

theoretical σ = 14.83 mgal

CV-adaptive solution

LSC estimation

errors

Max = 39.01

Min = -35.59

Mean = 1.86

σ = 11.58

ˆ uu −

CV-adaptive

estimation errors

Max = 34.25

Min = -36.92

Mean = 2.13

σ = 11.81

ˆ uu −′

Conclusions

− LSC provides optimal local accuracy, yet poor

global spatial rendering in SRF prediction problems.

− New prediction approach based on optimal de-

smoothing through CV-matching filtering;

i.e. reproduce the signal CV function with minimum i.e. reproduce the signal CV function with minimum i.e. reproduce the signal CV function with minimum i.e. reproduce the signal CV function with minimum loss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracyloss in the MSE accuracy

− Are there any “fundamental” issues hidden on the

trade-off between local and global accuracy in SRF

estimation from discrete data?

− Identify relevant application areas where global

spatial accuracy is an important issue (preserving

non-stationary patterns, spatial field mapping,F)