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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
• Linearized Appraisal Analysis: (Backus & Gilbert, 1970) Point Spread Function (PSF)Averaging Kernel
• PSF for survey design
• Nonlinear Appraisal Analysis and image interpretation
• Nonlinear PSF• Determining the region of data influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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
• Linearized Appraisal Analysis: (Backus & Gilbert, 1970) Point Spread Function (PSF)Averaging Kernel
• PSF for survey design
• Nonlinear Appraisal Analysis and image interpretation
• Nonlinear PSF• Determining the region of data influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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?
Survey Optimization using 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
Survey Optimization using PSF
( ) 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 Appraisal Analysis and image interpretation
• Nonlinear Point Spread Function• Region of Data Influence• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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
• 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• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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
• Linearized Appraisal Analysis: Backus-Gilbert Theory (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• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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
• Linearized Appraisal Analysis: Backus-Gilbert Theory (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• Funnel Function Analysis (upper and lower bounds)• Nonlinear Resolution with Inverse Scattering Approach
• Conclusions and a Recipe
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