appraisal analysis in geophysical inverse problem: tool ...€¦ · outline of the talk •...

Appraisal Analysis in Geophysical Inverse Problem:

Tool for Image Interpretation and Survey Design

Partha S. Routh

Boise State University

Collaborators:

Doug Oldenburg, University British Columbia

Tim Johnson, Greg Oldenborger & Carlyle Miller, Boise State University

Imaging from Wave Propagation Workshop, IMA, Minneapolis,

October 17-21 2005

Image Interpretation Problem

• What scale of structure we can believe?

• What features can be resolved?

• How much of this picture has come from prior information?Sources Receivers

Slowness (1/Velocity) Map

Survey Design Problem

• Can we improve the model by designing a new survey?

• Where should we place our sources/receivers with logistical constraints?

• How can we improve signal-to-noise ratio?

Sources Receivers

Basic Goal of Interpretation & Survey Design

Interpretation: We do the experiment and try to find the “right” support volume from the data.

Survey Design: We define the support volume and design the “right” experiment for enhanced resolution.

Physics

Model Perfect DataIdeal World

Imaging/Inversion

Survey DesignPhysics

Incomplete Data+

ErrorsModel

Real World

Imaging/Inversion Prior InformationModel Estimate

Outline of the Talk

• Linearized Appraisal Analysis: (Backus & Gilbert, 1970) Point Spread Function (PSF)Averaging Kernel

• PSF for survey design

• Nonlinear Appraisal Analysis and image interpretation

• Nonlinear PSF• Region of Data Influence Analysis (Oldenburg & Li,1999)• Funnel Function Analysis (Oldenburg, 1983)• Nonlinear Resolution with Inverse Scattering Approach

(Snieder, 1991)

• Conclusions and a Recipe

Outline of the Talk




• Nonlinear PSF• Determining the region of data influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach


Linear Problem

Forward Problem: ε+= Gmd obs

Inverse Problem:

20

2)()(min mmWdGmW m

obsd −+−= βφ

)()(ˆ 01 mWWdWWGWWGWWGm m

Tm

obsd

Td

Tm

Tmd

Td

T ββ ++= −

Solution:

Where:

emRIRmm +−+= 0)(ˆ

)()(

)()(1

1

εβ

β

dTd

Tm

Tmd

Td

Td

Td

Tm

Tmd

Td

T

WWGWWGWWGe

GWWGWWGWWGR−

−

+=

+=

R: Resolution Matrix

Appraisal AnalysisWhat can we tell about the model?

Model EstimateModel

Limitation from ResolutionOperator

Bias from prior Information

Errors in Data

Linear Resolution Matrix

)()( 1 GWWGWWGWWGR dTd

Tm

Tmd

Td

T −+= β

Rows of R are Averaging Functions

Columns of R are Point Spread Functions

Nonlinear Problem

Forward Problem: ε+= )(mFd obs

20

2)())((min mmmWdmmFW m

obsd −++−+= δβδφInverse Problem:

)(

))ˆ()((ˆ)(

0mmWW

mPmFdWWJmWWJWWJ

nmTm

nobs

dTd

Tnm

Tmnd

Td

Tn

−−

−−=+

β

δδβ

Solution:

εδδ

εδ

+++=

++=

)()(

)(

mQmJmFd

mmFd

nnobs

nobs Linearization Error

011 ))ˆ()((ˆ mWWAmPmQWWJAmRm m

Tmnd

Td

Tnnn δβεδδδδ −− ++−+=

Linearized Resolution Matrix

Nonlinear Problem: Linearized Resolution

)()())()(( 1nnd

Tdn

Tnm

Tmnnd

Tdn

Tnn mJWWmJWWmJWWmJR −+= β

Physics:

Model

Acquisition Parameters

Discretization

Data Noise

Prior Information

Regularization

)()())()(( 1111nndn

Tnmnndn

Tnn mJCmJCmJCmJR −−−− += β

Bayesian Formulation

Linear Problem

( ) jjj xmxgd ε+= )(),(

0.5

,

Kernel Model Data

=

kernels: =)(xg j Damped Sine & Cosine functions

Experiment: cosine (j = 0,17) sine (j = 1,7)

Linear Problem

Resolution Matrix

True Model

Inverted Model

Point Spread FunctionsR

esol

utio

n de

ceas

es w

ith d

epth

Point Spread Functions

• Kernels (# and type)• Data accuracy

• Details of model objective function

• Reference model

Survey

Prior information

Routh et. al (2005); Routh et. al, (in prep)

Outline of the Talk




• Nonlinear PSF• Determining the region of data influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach


PSF for comparing between two surveys

Model From survey-1

11 Kernels

Model From survey-2

24 Kernels

Exp #1Exp #2

PSF

Survey Optimization using PSF

• Physics

• Data noise

• Prior information

• Regularization

• Model discretization

Why PSF?


“ Given a region of interest where target is most likely to occur, can we design survey to provide enhanced resolution? ”

sources receivers

Previous works: Maurer & Boerner, 1998; Curtis, 1999a, 1999b, van denBerg et. al, 2003, Routh et. al, 2005.

Methodology

• We pose the inverse problem by maximizing a resolution measure i.e. point spread function (PSF).

•

• Very Nonlinear Constrained Optimization Problem:

Newton strategy based on primal interior point.Global optimization using simulated annealing.

PSF Delta-Like

Survey Optimization


( ) maxmin2 s.t.)(min ξξξξ ≤≤∆− kkk pW

ParametersSurvey:ξ

-2

Survey Optimization using PSF: Newton Method

( )

∑

∑

=

=

−−

−−∆−=

N

k k

kk

N

k k

kkkk pW

1max

min

1max

2

log

1log)()(

ξξξλ

ξξλξξψMinimize:

[ ]( )

−

−∆=

eepW

YXJW kkkk

λλ

ξδξ

λλ

)(Solve:

δξαξξ +=+ )()1( nnUpdate:

Tomography Example

Inverted Model

Raypaths

Tomography Example: Raypaths

After (Newton)

After (SA)

Before

Tomography Example

PSF (Original Survey) : Region I

PSF (SA)PSF (Newton)

Tomography Example

PSF (Original Survey): Region II

PSF (Newton) PSF (SA)

Tomography Example : BG Averaging Functions

AVG (Newton) AVG (SA)

Averaging Function before survey optimization

Tomography Example: Inverted Models

After: Newton After: SA

Nonlinear Example: Controlled Source Electromagnetic Survey Design

TX

Ex

Ey

Hx

Hy

HzFrequency Sounding

HiE ωµ−=×∇

sJEH +=×∇ σGoverning Equations:

( ) ( )( ) ( ) ( )ωωω

ωω

ZYX

YX

HHHEE

,,,Data:

Region of Interest (ROI)

Depth (m)

Con

duct

ivity

(S/m

)• What frequencies provide better resolution at the region of

interest? with constraints 100001.0 ≤≤ f

Goal of Survey Design

Survey Design: Nonlinear CSEM Problem

• Before Survey Optimization: Frequencies used: 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192 Hz

Survey Design: Nonlinear CSEM Problem

• After Survey Optimization: Frequencies are: 0.88,2.6,16.3,63,73,81,129,190,229,279,287,352,372,1396,2251 Hz

More Survey Design Questions…

• Which fields (E or H ) and what components are better to acquire?

• Which fields can be combined to give better resolution at region of interest?

• Methodology: Compare the PSF from different multi-component surveys at the region of interest.

PSF for Multi-Component DataExHyHz ExHy Ex

Model

PSF

PSF in ROI

PSF for Multi-Component DataHy Hz Rho-Phi

Model

PSF

PSF in ROI

Outline of the Talk

• Linearized Appraisal Analysis: Backus-Gilbert Theory (Backus & Gilbert, 1970)

Point Spread Function (PSF)Averaging Kernel

• Point Spread Function for survey design


• Nonlinear Point Spread Function• Region of Data Influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach


Back to Point Spread Function

kdTd

Tm

Tmd

Td

Tk

kk

GWWGWWGWWGp

Rp

∆+=

∆=

− )()( 1β

Inversion

Delta-Fn PSF

kpk∆

PSF as Model Construction Process

For a single PSF

kdTd

Tkm

Tmd

Td

T GWWGpWWGWWG ∆=+ )()( β

PSF construction as an optimization problem

( ) ( ) 22min kkdkkm GGpWpW ∆−+∆−β

[ ]

∆=

0

kdk

m

d GWp

WGW

β

Development of Nonlinear PSF

( ) ( ) 22min kkdkkm GGpWpW ∆−+∆−β

Linear Operator

( ) ( ) 22 )()(min kkdkkm FpFWpW ∆−+∆−β

Nonlinear Operator

Routh et. al, 2005 (in prep)

Nonlinear Example

jdxxmxg

jjed ε+∫=

− )()(

• To compare linear and nonlinear PSF chose a impulse response function.

• Solve the nonlinear inverse problem and obtain the model. This is nonlinear PSF

• Generate the linearized resolution matrix and extract the linearized PSF for the kth cell.

kxm δ=)(

)(ˆ)( xmxp nonlink =

klink Rxp δ=)(

Comparison of Nonlinear PSF & Linear PSF

Outline of the Talk




• Nonlinear PSF• Region of Data Influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach


Region of Data Influence

• Invert the data with two different reference model

• Basic idea: region not sensitive to data will recover the reference model

• Compute the RDI

• Region NOT sensitive to data will indicate RDI 1

refref mm 21 ,

refref mmmm

RDI12

12

−

−=

refref mmmm 2211 , →→

22)())((min refm

obsd mmWdmFW −+−= βφ

Oldenburg and Li, 1999

Region of Data Influence

RDI Map

Inversion Ref=1.0

Inversion Ref=1.5

Region of Data Influence Inversion Ref=1.0

RDI Map Inversion Ref=1.5

Region of Data Influence: DC Resistivity ExampleData Resistivity Model

Observed

Predicted

Outline of the Talk







Funnel Function Approach

• Consider computing average value within a window

• Window is our support volume

• Can we compute unique average value?

• Can we compute upper & lower bound of the average value?

20∆

−x20∆

+x0x

∆1

)(xm

Oldenburg, 1983; Backus & Gilbert, 1970a,b,c

Funnel Function Approach

• Backus-Gilbert Inference theory provides the general mechanism.

( )

( ))(),,()(

)(),,()(

00

00

xmxxpxe

xmxxgxd

=

=

Inference theory prediction kernel

Oldenburg, 1983

Funnel Function cont…

• What are our choices for ?),( 0xxp

( )( )

B

A

xmxmxxBxexxBxxp

xmxmxxAxexxAxxpxmxexxxxp

)()(),,()(),(),(

)()(),,()(),(),()()()(),(

00000

00000

0000

==⇒=

==⇒=

=⇒−=

∆∆

δ

Oldenburg, 1983)(),( 00 xxxxA −→ δ

• Approach of Backus & Gilbert (1970):

• express inference prediction kernel as linear combination of kernels

• Make prediction kernel close to delta function

∑=j

jj xgxxxA )()(),( 00 β

Funnel Function Cont…• In funnel function we chose the support volume

• Recognize that unique average cannot be obtained since

• Determine the bounds on the average value by solving two optimization problem (max & min) subject to fitting the data. (We use interior point Gauss-Newton method to solve).

),(),( 00 xxBxxP ∆=

∑≠∆j

jj xgxxxB )()(),( 00 α

Bxm )( 0

Oldenburg, 1983; Routh et. al. 2005 (in prep)

Funnel Function Cont…

Funnel Functions for Tomography Problem

∆

Brm )( 0

Actual Average

2.1)(05.0)( 0 ≤≤ xmxm UPPER

2)(05.0)( 0 ≤≤ xmxm UPPER

2)(05.0)( 0 ≤≤ xmxm LOWER

2.1)(05.0)( 0 ≤≤ xmxm LOWER

Routh et. al, (in prep)

Outline of the Talk







Nonlinear Appraisal Using Inverse Scattering

• 3D Equation for Electric Field ( ) :

• Perturbation E Field

• Scattering Equation

• For sparse data

• The scattering matrix is given by

)()(22SrrSEkE −=+∇ δω

∫ ′′′′−= vdrErrrGirE )()(),()( δσωµδ

( ) 01QEQIE −−=δ

( ) 01QEQIPE OBS −−=δ

kj

kjCrrGiQ jjjkkj

==

≠−=

0

),(ωµ

GCQ =

0=⋅∇ E

Forward Scattering

Single Scattering: Johnson, T. C., Routh, P. S., and Knoll, M.D., 2005, Fresnel volume georadarattenuation-difference tomography,Geophysical Journal International, 162, 9-24.

Multiple Scattering: Routh, P. S., and Johnson, T.C, 2005, Multiple scattering formalism in 3D georadarproblem, SEG expanded abstracts, pg 1-4.

Nonlinear Appraisal Using Inverse Scattering

( )( ) ( )+++=+++=

+++=−−− CGCGECEGCGCEGCGCCGCGGCCGGCGE

EGCCGCGCGCGEOBS

OBS

~~~~~~~~~~~~~~~)(~~~

10

10

10

02

δ

δ

....321 +++= CCCC

OBS

OBS

OBSOBS

OBSOBS

EGECECCwhere

EGGGCCGG

EGGCGECGGEGGEGGCGGGCGG

EGGCGGEGC

δ

δ

δδ

δδ

11010

111

1

1101

110

111

11

1

11

11

~&ˆ~̂....~~~

~~~~~~~~~~

~~~~~̂

−−

−−

−−−−−−−

−−−

==

+

−+

+−=

( ) HH GIGGG ~~~~ 11 −− += β PGGNote =~:

• Series expression for the perturbation data

• Expand scattering strength

• Finally, we obtain expression of estimated scattering strength

• One choice for the inversion operator

Lax, 1951; Weglein et. al., 2003; Cheney & Bonneau, 2004; Malcolm & de Hoop, 2005

Nonlinear Appraisal Using Inverse Scattering• Substitute

• Writing terms upto second order

++== −−− CGCEGCGEandEGEC obsobs ~~~~~~ 10

1101 δδ

HOTCGGGCGGEGGCGCEGGCGGC +−+= −−−−−−− ~~~~~~~~~~~~~~~~̂111

011

011

GGRWriting ~~ 1−=

HOTCGRCRRECGCRECRC +−+= −− ~~~~~~̂1

01

0

0== oncontributinonlinearorderhigherseeweIRIf

CCor

CGCECGCECC~~̂

,

~~~~~~̂1

01

0

=

+−+= −−

cancel

Conclusions• Geophysical Data are “inadequate, insufficient and noisy.”

Appraisal analysis provides a means to address non-uniqueness in solutions.

• Point Spread Function is particularly useful measure since it is connected to all of the components that affect resolution: physics, prior information, data noise, regularization, model discretization.

• We developed methodology to examine survey design problem.

• Examined funnel functions to get an estimate of bounds for a given support volume.

• Region of data influence method provided a measure where physics have sampled the Earth.

• We arrived at a formalism for nonlinear appraisal using inverse scattering approach, needs more work.

Recipe

2.1)(05.0)( 0 ≤≤ xmxm UPPER

2)(05.0)( 0 ≤≤ xmxm UPPER

2)(05.0)( 0 ≤≤ xmxm LOWER

2.1)(05.0)( 0 ≤≤ xmxm LOWER

Det

erm

ine

the

RD

I

Compute Funnel functions

Determine RDI using two prior

Determine the Resolution in Region of Interest with PSF

Acknowledgements

• Margaret Cheney, Frank Natterer and Bill Symes for organizing and inviting to present in the Imaging from wave propagation workshop at IMA.

• IMA for financial support to present the talk.

• Thanks to Benjamin White of ExxonMobil Research for pointing out the commuting errors that helped in correcting the formulation for the nonlinear resolution using inverse scattering.

• EPA grant X970085-01-0

• NSF-Epscor grant EPS0132626

• Inland Northwest Research Alliance

appraisal analysis in geophysical inverse problem: tool ...€¦ · outline of the talk •...

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