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ProbabilisticProbabilistic Geoacoustic Geoacoustic InversionInversion
SACLANT Undersea Research Centre, La Spezia, Italy&
University of Victoria, Victoria B.C. Canada
145th ASA, Nashville TN, April 28 – May 2 2003
Stan DossoStan Dosso
Introduction
l Determining geoacoustic model parameters (wave speeds, attenuations, density, porosity etc) from acoustic data (full-field, reflectivity, ambient noise etc)is a nonlinear inverse problem ? non-unique
l Complete solution:Ø Treat data according to their uncertaintiesØ Include existing (prior) knowledgeØ Provide parameter estimates, uncertainties,
inter-relationships
l Probabilistic Inversion (Tarantola et al ) provides rigorous & general approach
Outline
l Probabilistic Inversion:Ø Define Likelihood, Prior, Posterior Probability Density (PPD)Ø Parameter estimates: optimizing PPDØ Parameter uncertainties / relations: integrating PPD
? Importance Sampling? Markov Chain Monte Carlo (Gibbs Sampler)
l Examples:Ø IT2001 Benchmark geoacoustic inversionØ Matched-field inversion Ø Reflectivity inversion Ø Source localization with environmental uncertainty
Probabilistic Inversion
l Combine data & priorinformation to definePosterior Probability Density (PPD)
l PPD quantifies modelprobability over M-Dparameter space
Data
Prior
PPD
PPD
l PPD quantifies knowledge of model parameters due to resolving power of data and prior information ? not a function of the inversion algorithm
l Global minimum-misfit model is a property of PPD but global min does not generally correspond to “true” model and is not the primary goal
truetrue
Example: Local/Global Optima
l 2-D marginal probability distributions for inversion of noisy synthetic reflectivity data
l True solution (cross) not in global optimum, not at local optimum
l Note strong inter-parameter correlations
PPD: Bayes Theorem
l Bayes Theorem:
l Likelihood: data uncertainty distribution, interpreted as function of m (for measured d). Typically
l Prior: existing knowledge of m
)]d,m(exp[)m|d( EP −∝
)m()m|d()d()d|m( PPPP =
PPD PriorLikelihood
Data misfit
PPD Definition
l Bayes Theorem:
l PPD:
)m()]d,m(exp[)d|m( PEP −∝
'm)]d,'m([exp
)]d,m(exp[)d|m(d
Pφ
φ
−
−=
∫M
)m(log)d,m()d,m( e PE −=φgeneralized misfit
? Interpret M-dimensional distribution?
Parameter Estimates
l M-D PPD interpreted in terms of properties defining parameter estimates & uncertainties
l Parameter estimates:Ø Maximum A Posteriori (MAP) model
Ø Posterior mean model
}{ )d|m(m PmaxMAP Arg=
'm)d|'m('mm dP∫=><
MAP and Mean
l For uni-modal, symmetric distributions MAP & Mean coincide
l Choice problem-dependent:Ø MAP is most probableØ Mean has smallest variance
l Understanding uncertainty distribution preferable to simply estimating parameters
MAPMean
MeanMAP
Mean
MAP
PPD Optimization
l MAP estimate requires maximizing PPD (minimizing misfit )
l Nonlinear problems can have many local minima and preclude gradient-based minimization
l Global Search methods:Ø Genetic Algorithms (Gerstoft)
Ø Simulated Annealing (Collins; Dosso & Chapman; Knobles)
Ø Hybrid Inversion (Gerstoft; Fallat & Dosso; Musil & Chapman )
Etc…
)m(φ
Parameter Uncertainties
l Marginal Probability Distribution:
Ø Reduces the M-D PPD to M 1-D parameter probability distributions by integrating out M –1 parameters
Ø Joint (2-D) marginals defined similarly
m')d|m'()'()d|( dPmmmP iii −= ∫ δ
Marginal Distributions
l Marginal DistributionsIntegrates M–1 parameters ? rigorous, quantitative
uncertainty distribution
l Misfit “Slice” (sensitivity)Holds M–1 parameters fixed ? approx qualitative uncertainty
potentially misleading
1-D Marginals 1-D Slices)( 1mφ
F(m
2 )
Credibility Intervals
l ß % Credibility Interval: interval containing ß % of the area of the marginal distribution
l Highest Probability Density (HPD) credibility interval:interval of minimum width containing ß % of area
ß %
Covariance / Correlation
l Covariance Matrix:
Ø Diagonal terms ? Parameter Variances (stnd dev)2
Ø Off-diag terms ? Inter-parameter Covariances
l Correlation Matrix: Normalize to quantify parameter inter-relationships
m')d|m'()ˆ'()ˆ'( dPmmmm jjiiij −−= ∫C
–1 < Rij< +1 correlationbetween mi & mj
jjiiijij CCCR /=
PPD Integration
l Marginals, covariance, etc require integrating PPD
l For nonlinear problems, numerical integration isrequired using Importance Sampling and Markov Chain Monte Carlo methods
'm)d|'m()'m( dPfI ∫=
Importance Sampling
l Monte Carlo method based on preferentially sampling regions of parameter space where integrand is large
Consider drawing Q models mi from g(m)
)m()d|m()m(1
'm)'m()'m(
)d|'m()'m(
1 i
ii
gPf
Q
dgg
PfI
Q
i∑
=≈
= ∫
? How to choose g(m)?
Gibbs Sampler (GS)
l Metropolis GS samples from Gibbs distribution
by accepting perturbations to m if uniform r.v. ? on [0,1]
m']/)m'([exp]/)m(exp[
)m(dT
TPG
φφ
−−
=∫
]/exp[ Tφξ ∆−<
? Simulated Annealing applies GS as T? 0 to min
? PPD P(m|d) is a Gibbs distribution sampled at T=1
φ
GS in Importance Sampling
l GS at T =1 in Importance Sampling: g(m) = P(m|d)
l Speed-up: adaptive perturbations, re-parameterization
l Convergence: monitor several independent samples
(Integration ~ 5–10 X slower than fast optimization)
)m(1
)m(
)d|m()m(111
ifQg
Pf
QI
Q
ii
iiQ
i∑=
=≈ ∑=
? efficient, unbiased PPD integration
Parameter Rotation
l Searching along parameter axes inefficient for correlated parameters ? rotate to principle axes parameter space by diagonalizing covariance (Collins & Fishman; Perkins; Neilsen & Knobles etc)
m2
m1
m2 m1
‘‘
Likelihood Function
l Likelihood function P(d|m) expresses data uncertainty (error, noise) distribution
l Uncertainties include measurement & theory errors ? often not well known
l Proceed with reasonable assumptions, e.g. Gaussian distribution with unknown stnd dev sØ ML estimate for s (Gerstoft & Mecklenbräuker)Ø Include s as unknown in inversion (Michalopoulou)
? Check assumptions after inversion!
Matched-Field Inversion
l Matched-field methods typically match spatial acoustic fields using incomplete source spectral information
l Consider acoustic field data on N-sensor array at F freqsFor uncorrelated complex Gaussian errors:
? Can’t compute df (m) – df for unknown spectrum
][ 222
1
/|d)m(d|exp)m|d( ffff
F
f
NP σσπ −−= −
=
− ∏
Incoherent Processing
l For unknown source spectrum
l Maximizing P(d|m) over Af & ?f
FfA f
i
fff ,...,1)m(de)m(d =→
θ
2
2|d|)d,m(1)m( ][
1 f
ffBE
F
f σ−∑
== Bf = normalized
Bartlett match
? Incoherent sum of Bartlett mismatch weighted by SNR? Coherent likelihood processors etc obtained similarly
Prior Distribution
l Expresses existing knowledge of m (subjective)
Ø Lower / upper bounds:
Ø M-dimensional Gaussian:
∞−
≤≤=
+−
otherwise
if0)m(P loge
iii mmm
2/]m̂m[]m̂m[)m(P log 1e −−∝ −
MT C
Etc…
Linearized Inversion
l Analytic results for Linear/Gaussian case? Linearization?
l Linearized inversion based on local functional derivativesCan fail:Ø Parameter estimates may converge to local minimumØ Uncertainties characterize single minimumØ Derivatives may not characterize nonlinear uncertainty
l Matched-field inversion is strongly nonlinear due tomodal interference ? consider:Ø Invert modal wave-numbers (Frisk & Becker; Rajan)
Ø Invert modal dispersion curves (Potty & Miller)
Ø Inversion via modal decomposition (Neilsen & Westwood)
Etc…
Benchmark Testcase
l Matched-field inversion for IT2001 Workshopbenchmark test case
l Blind Test: Inversioncarried out with no knowledge of solution
l Parameterization:seabed represented as L layers
Model Parameterization? What is L?
water
Number of Layers
l Seek the minimum number of layers consistent with the resolving power of data
l Examine misfit of optimal model vsnumber of layers(PE prop model)
? L = 3 layers resolved Layers
Mis
fit
Marginal Distributions
l MAP estimate with one standard deviation uncertainties compared to true solution ? note error in prior bounds for a
Sediment Profile
MAPTrue
Error in prior
Matched-field Inversion
l PROSIM ’97 experiment (Nielsen et al, SACLANTCEN)
l Acoustic data on verticalline array (VLA) due to towed source
VLA
Acoustic Data
l 300–800 Hz LFM “chirp” signal on 64-element VLA
l Effective SNR of5–10 dB (includes theory error)
Parameterization
l Model parameterized as 3 geoacoustic, 5 geometric unknowns
l Adiabatic normal modeprop model (300, 400,500, 600 Hz)
l Data errors assumedcorrelated over scale of significant modes
Marginal PPDs
l Marginals for 4 km range compared to results of 16 inversions of independent data at 2–6 km
Synthetic Marginals
l Compare to marginalsfor synthetic data with noise of assumed statistics
Correlation Matrix
l Water-depth / layer thickness negatively correlated (– 0.9)
l Layer thickness / layer speed positively correlated (+0.7)
Rows of Correlation Matrix
Joint Marginals
l 2-D uncertainties illustrate parameter correlations
Reflection Inversion
l Inversion of towed-array Reflection-coefficient data
l Two sites along seismicline in Baltic (hard till and soft mud inclusion)
Ref
lect
ion
Coe
ffGrazing Angle (o)
Can different geoacousticproperties be resolved?
Marginal PPDs
l Marginal distributions quantify geoacoustic differences resolved by data
Mud Till
High-Resolution Reflectivity
l High-resolution bottom loss in Straits of Sicily using towed source & fixed receiver (Holland, SACLANTCEN)
l Low-velocity silty-clay produces Angle of Intromission
BL Inversion
l Inversion carried out with ML estimate for data stnd dev s , and by including s in inversion
l High-resolution data define VP , ?, aP , VS(not aS )
ML
Inv
Check: Data Statistics
l Assumed Gaussian errors ? data residuals[d–d(m)]/s should be Gaussian(0,1) and uncorrelated
Residuals Auto-correlation
l
Localization withEnvironmental Uncertainty
l Probabilistic inversion can incorporate environmental uncertainty in source localization
l Example: Consider benefit of geoacoustic inversionto localization
V1
V2
V3
VS
Vb
Geoacoustic Inversion
l Geoacoustic inversion (50, 100, 200 Hz SNR=10 dB) ? Use PPD as prior for source localization
(95% HPD intervals & M-D Gaussian)
Joint Marginals
l Joint marginalsshow correlated parameters V2 V3
V3
VS
VS
rh VS
Probabilistic Localization
l Joint marginals in (r, z) by integrating unknown environmental parameters (100 Hz, SNR=5 dB)
l PA = probability within ± (200, 5) m in (r, z)
Geoacoustic Inv PPD as Prior
Known /UnknownEnvironment
Summary
l Probabilistic inversion provides general approach:Ø Parameter estimates (MAP & mean)Ø Parameter uncertainties (marginal probability, variances,
credibility intervals)Ø Parameter inter-relationships (correlations, joint marginals)
l Explicitly treats data uncertainties & prior info
l Natural framework for transferring uncertainties in inversion
References
Contributions by the author to this field include:
S.E. Dosso et al., Estimation of ocean-bottom properties by matched-field inversion of acoustic field data, IEEE J. Oceanic Eng. 18, 232–239 (1993).
M.R. Fallat & S.E. Dosso, Geoacoustic inversion for the Workshop 97 testcases using simulated annealing, J. Comp. Acoust. 6, 29–44 (1998).
M.R. Fallat & S.E. Dosso, Geoacoustic inversion via local, global, and hybrid algorithgms, J. Acoust. Soc. Am. 105, 3219–3230 (1999).
M.R. Fallat, P.L. Nielsen & S.E. Dosso, Hybrid inversion of broadband Mediterranean Sea data, J. Acoust. Soc. Am. 107, 1967–1977 (2000).
S.E. Dosso, M.J. Wilmut & A.S. Lapinski, An adaptive hybrid algorithm for geoacoustic inversion, IEEE J. Oceanic Eng. 26, 324–336 (2001).
S.E. Dosso, Quantifying uncertainty in geoacoustic inversion I: A fast Gibbs sampler approach, J. Acoust. Soc. Am. 111, 129–142 (2002).
S.E. Dosso & P.L. Nielsen, Quantifying uncertainty in geoacoustic inversion II: Application to a broadband shallow-water experiment, J. Acoust. Soc. Am. 111, 143–159 (2002).
S.E. Dosso & M.J. Wilmut, Quantifying data information content in geoacoustic inversion, IEEE J. Oceanic Eng. 27, 296–304 (2002).
S.E. Dosso & M.J. Wilmut, Effects of incoherent and coherent source spectral information in geoacoustic inversion, J. Acoust. Soc. Am. 112, 1390–1400 (2002).
S.E. Dosso, Benchmarking range-dependent propagation modeling in matched-field inversion,J. Comp. Acoust. 10, 231–242 (2002).
S.E. Dosso, Environmental uncertainty in ocean acoustic source localization, Inverse Problems 19, 419–431 (2003).
M. Riedel & S.E. Dosso, Uncertainty estimation for AVO inversion, At press: Geophysics (2003).A.S. Lapinski & S.E. Dosso, Bayesian inversion for the Inversion Techniques 2001 Workshop, At press: IEEE J.
Oceanic Eng. (2003).S.E. Dosso & C.W. Holland, Geoacoustic uncertainties from seabed reflection data, Submitted to: J. Acoust. Soc.
Am. (2003).
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