Electromagnetic Imaging and Simulated Annealing - CiteSeerX

JOURNAL OF GEOPHYSICAL

RESEARCH, VOL. 96, NO. B5, PAGES 8057-8067, MAY 10, 1991

ElectromagneticImaging and Simulated Annealing DOMINIQUE GIBERT Ecole Nationale S•pdrie•re des Mines de Paris JEAN VIRIEUX lnstit•t

de Gdodynamiq•e, CNRS, Valbonne, France

In contrast with acousticalimaging methods,for which the wave field is dominated by propagationeffects,electromagneticimaging of conductivemedia suffersfrom the diffusive behavior of the electromagneticfield. An important questionto addresswhen working toward the achievementof electromagneticimaging concernsthe possibility of resolving the diffusion damping. Exact inversionwill be looking at the solvability of the integral equation relating a diffusivefield to its dual wavefield. This equation is ill posed because its Laplace-likekernel makesthe inverseproblem of finding the dual wave field a notori-

ouslydifficult (both numericallyand mathematically)one. Stochasticinversionis another alternative based on least squaresfitting. In this inverse problem approach, extracting

thewavefieldisstilla relatively instable process, although theL2 misfitfunction fordata without noisepresentsa global minimum. The simulated annealing overcomesthis instability for parameterizationof this problemdesignedas follows. The unknownwave field is expected to be a sequenceof impulsive functions. The number of impulsive functions can be determined by using a statistical criterion, called AIC, which comes from the Prony technique. The simulated annealing is applied to the positions of the reflections, while the amplitudes, which are not taken as parameters, are obtained by linear fitting. The simulated annealing method provesto be efficient even in the presenceof noise. Furthermore, this nonlinear numerical inversion furnishes statistical quantities which allows an estimation of the resolution. Simple synthetic examples illustrate the performance of the inversion, while a synthetic finite element example showsthe final pseudo-seismicsection to be processedby standard seismicmigration techniques. INTRODUCTION

been applied only to homogeneous media with analytFor typical Earth conductivitiesthe electromagneticical Green functions. This "reverse" propagation reresponseis mainly controlled by a diffusion process quiresa processoppositeto the diffusionwhich might broadeningthe signaland makingit quitedifferentfrom be consideredas an unstable procedure. The link between seismicand electromagneticphea seismicsignal. Obtaining the electromagneticresponseof an arbitrary conductivemedium is in itself nomena is emphasizedwith the integral transformaan already difficult task investigatedwith approaches tion betweena fictitious wave field and the electromagas different as integral equations,in both time domain netic field. This transformation has been known for a [SanFilipoandHobmann,1985]andfrequency domain long time [Filippi and Frisch, 1969; Lavrent'evet al., [Wannamaker et al., 1984],or as finite difference for- 1980].An illustrationof the forwardproblemhasbeen mulations[OrisiaglioandHobmann,1984].Recoveringgivenrecentlyby Lee et al. [1989]:they first compute conductiveinformationis an even more difficult oper- the fictitious wave field responseto a given conductive ation investigated with limitedsuccess [Weidelt,1972, medium and transform it into an electromagneticre1975; Tripp et al., 1984; Barnett, 1984; Smith, 1988; sponse, which is a stable procedure. Although they apply finite differencecodes, they suggestthe use of Tarits, 1989]. Although a mathematical parallelismbetweenelec- efficientseismicmodelingtools as ray tracing programs tromagneticand seismicsignalsin layered media has for computingthe responseof a conductivemedium. been underlinedpreviously[Weldell, 1972; Kuneiz, A more ambitious target is extracting the fictitious 1972; Lee et aL, 1987; Levy et aL, 1988; Lee et at, wave field from the electromagneticresponse.The en1989],seismicforwardmodelingtechniques, as well as countered difficulties are similar to the ones met in the the differentseismicmigrationandinverseformulations, Prony'smethodor in exponentialfittings[McDonough havenot met a greatsuccess in electromagnetism. The and Huggins,1968].Constructionof an exactinversion and stability, methodof ZhdanovandFrenkel[1983],whichproposedoperatorleadsto problemsof convergence the "reverse"propagationof residusfrom the observers the analysisof which is beyond the scopeof this paper. into the mediumwidelyusedin seismicmigration,has How can we transform this ill posed inverse problem into a better posedproblem with a tractable solution? Copyright 1991 by the American GeophysicalUnion. Paper number 91JB00278.

0148-0227/ 91/ 91JB-00278505.00

From the statistical point of view a stable solution will

be the stochasticinverse[Tarantola,1987].How to obtain this solution is the questionwe want to addressin this paper. 8057

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GIBERT AND VIRIEUX: ELECTROMAGNETIC IMAGING AND SIMULATED ANNEALING

Once obtained, the fictitious wave field can be processedby seismicmigration methods, opening to us the possibilityof reconstruction of conductivityinside a given medium. After a brief descriptionof electromagneticpropagation in conductivemedia and the integral transformation we set up the inverseproblem and describethe difficulties we must solve. This leads us to preprocess our data in order to find our parameterization. Then, the solutionis constructedby simulatedannealing. With the help of one-dimensional examples,we discussthe efficiencyof this technique,the stoppingcriterion, and the structureof the final solution. Finally, two-dimensional mediumgeometryis investigatedwith inversionof synthetic finite element data.

associatedwith smooth variations. We look after sharp discontinuitiesof conductivity, following an approach

commonin seismicmigration problem. Moreover,we assumethat the responsehas been deconvolvedof the sourcefunction. The excitation function is a plane wave propagatingvertically.The associated waveresponse is an impulsivesignalfor eachreflection, N

U(q)=

u.(q - q.),

(6)

whenweneglectresponse beyondcriticaldistancewhich is a well-verifiedassumptionexceptfor very steepreflections. Insertingthis expression into equation(4), one componentof the electricfield can be written:

DIFFUSION of ELECTROMAGNETIC RESPONSE IN CONDUCTIVE MEDIA

N

-

If we assumethe displacementcurrent is negligible,

(7)

the electric field E satisfiesthe diffusion equation,

0E(r, t)

v xv x

= S(r,t)

wherer, equalsqn 2. In the time domainwe obtaina (1) similar expression,

where •r is the electric conductivityof the medium. p is

the magneticpermeabilitywhichis takenconstant.The sourceterm S is causal.FollowingLeeet ai. [1989],we introduce a fictitious wave field U and a source term F

related by the followingrelation:

V x V x U(r, q)+ pa(r)

0:•U(r,q) Oq:•

We want to estimate the time shifts r. and the ampli-

tudesUn, from a givenset of M measureddata E(rm) or E(w,•). BecauseE(w,•) hasa simplerform, we se-

= F(r, q)

lect it for our numerical investigations.The number of

(2) events is denoted by N.

From our different numerical tests we can argue that where q is an independent variable with the dimension the selection of the number N of events is very imof square root of time. The corresponding"seismic" portant for a convergencetoward reasonablesolutions. velocityturnsout to be 1/•. An integralrelation Similar conclusionshave been previously stated for the

betweenE(t) and U(q) (seeLeeet ai. [1989]for refer- Prony'smethod applied in time seriesanalysis[Van BiaricumandMittra, 1978]wherean erroneous number ences),

E(t) - I

(3)

of eventsleadsto unphysicaltime shifts and amphtudes of reflections. In other words, the number N of reflections, i.e., the number of basic functions, must be also estimated

introducesa kernelcorresponding to the diffusivegreen function of three-dimensionalmedium. The spatial dependenceof E and U has been left implicit. This transformation involvesonly the time t and the variable q and is independent of r. The source terms S and F are related by the same expression.One can write the

expression (3) in the spectraldomainundera simpler

from the data.

In a moretheoreticalframework[Hadamard,1932] this problem is ill posedbecausevaluesof someparameters are not continuous

e-•i• qU(q)dq

of the data.

Let us

assumetwo models with one reflection of given amplitude defined by:

E1(w)- U1e-vq•¾,

_

form,

E(w)-

functions

(4)

(o)

The distance between these two models can be defined

alreadyusedby Weldelf[1972].We selectthe following

as the usual modulus

square root of • 1

V•- •(,-!-1),

(5) For any valuesof amplitudesU• and U2 we can find ar-

bitrary timesrl and r2 makingthis distanceassmallas with an exponentially damped kernel, making the in- we want. From the practical point of view this means

verseproblemdifficultto solve.FollowingKunetz[1972] that a wide rangeof modelsgivesidenticalimagesin and Levy et ai. [1988],we assumethat only reflections the data space. Fortunately,this difficultyis partially contributeto U, and we neglectthe slowretrodiffusion overcomeby the observedfrequencyrangeand the max-

GIBERT

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IMAGING

AND SIMULATED

ANNEALING

8059

imumdepththat we consider.This oftengivesa stable verseproblemsunderconstraints.Let us underlinethat solution.in any case,we can increasethe stability by amplitudesare not involved in the nonlinear inverse the verypowerfulconstraintI U, I< 1. This constraint, problem as unknowns: they are computedas quantiwhich states that reflectionshave a smaller amplitude ties once the nonlinear parameters, i.e., the reflection than the emitted incidentwave,makesthe problemwell positions,are selected. What makes this inverse problem ill posed? Let us behavingwith a globalminimumfor the misfitfunction, as we shall see. considerthe misfit function, hereafter also referred to

as the energyfunction,with L2 norm definedby

ESTIMATION or TnS NUMSE•t OF I•FLECTIONS

M

Selectingthe numberN of eventsis a very impor$ - 1/2m----1 E (Eob0(•,•) - E(•,•,r•,...,r,))2 (15) tant step and makesthe problemmore stable. Levy et al. [1988]haveproposed on a similarsubjectto reduce the nonlinearproblemof finding amplitudesU, where we sum over M frequenciessampled in the oband times v, to a linear problemin amplitudeswhen served frequency range. For a zero-mean Gaussian the time vn of eachevent is fixed. The solutionis regu- white noisewith varianceaa, the energyfunctionmay larizedby reducingthe numberof significanteventsby readily be convertedto a probability densityfunction mean of a minimization of the L• norm of the solution

vector(seealsoOldenburg [1990]).

•P-•e-s/•

We preferan alternativeapproachwhichwe describe now. Let us defineone componentof the reflectedfield by:

(16)

where C is a normalizingconstant. For each assumed positionof the reflectionswe computethe amplitude. The misfit function is deducedfrom the amplitude mis-

N

E(•,•)- E U.e-¾'•.

(m--1,...,M), (11) fit.

For a givenmodelwith threereflectionsat positions q = 0.067., 0.139, and 0.204, let us plot the misfit

in sucha way that the M frequenciesare sampledwith function which is a three-dimensional function in this a regular step in the squareroot of frequency,which particularcase. We considerslicesof this functionas means that

with The reflected

•,• = m6y+ •o.

field can now be written: N

E(ym) -- Em- E UnoZn Z•',

shownin Figure1, and obvioussymmetries are related to permutationsof the three events.For example,the (12) globalminimumis locatedat sixdifferentplaces.When the third event is at q - 0.204, i.e., betweenthe first and secondslicestarting from the top of the Figure 1, we have two symmetrical positionscomingfrom permutation of the first two events. Two other positions

(13)of the globalminimumare alsofoundnearthe fourth

slicefrom the top of Figure I whenthe third eventis at q = 0.139.Finally,weobservetwootherpositions when with the third eventis at q - 0.067 near the sixth slicefrom 0Z, = e- ,½•-Z,o, the top. Althoughthe globalminimumis well defined, the procedureto reachit by gradientmethodscan be quite slow,becausewe haveto turn aroundapparent The expression (13) is verysimilarof the Pronymodel potentialbarriers:startingin the sixth slicefrom the [e.g.,Kay andMarpie,1981]widelyappliedfor spectral top, we do not converge to the globalminimumof this analysisof transient signalsin the terminology of sig- sliceif we needfor that to go througha situationwith nal processing.N is the order of the model. Among two nearbyreflections.When an unwantedreflection many ways to evaluate the order of Prony's model, we is near a true one, the inverseprocedurebalancesbehave found the spectral analysisof the data covariance tween these two reflectionsfor interpreting the single matrix introducedby VanBlaricumandMittra [1978]a reflectionand givesa misfit functionhigherthan for a very efficientmethod, whichwe presentin the appendix. singleevent,creatinga small barrier not seenin the Let us repeat that the decouplingbetweenthe num- Figure 1. The gameis to go to the globalminimum ber of reflectionsand the reflection positions provides usinga ratherlongpath preventing the situationof two us with a stable and efficient inversionapproach, while nearbyreflections. Moreover,the main valley of the inverting simultaneouslythesequantitiesis a more un- globalminimumpresentsa rather fiat bottomwhich stable procedure. can be very sensitiveto noiseperturbation: another reasonto avoidusinggradientmethods.The two sides SIMULATED ANNEALING of the valley havevery steepslopeswhich can be seen, Once we know how many reflectionsare expected, we for example,on local variationsperpendicularto slices must locate them: this is our nonlinear inverseproblem. near the fourth slice from the top and, consequently, In order to estimate the misfit function we need the amany discretization of the costfunctionmusttakecareof plitude U• which is computed by standard techniques apparentlocalminimacomingfrom the sampling.How [Lawsonand Hanson,1974]of linear least squaresin- to escapefrom thesedifficulties?

Z.- e-'••".

(14)

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GIBERT AND VIRIEUX:

ELECTROMAGNETIC

IMAGING AND SIMULATED ANNEALING

accepted models obeys the Boltzmann law

reflection 1:q"time" (s'•

reflection 'time" O.O5 0.1 0_15 02 o•

7•B - IZ e-s/T,

(18)

.02

where Z is a normalizingpartition function. The simi-

laritiesbetweenthe probabilitydistributions 7• of (16) and 7•B of (18) allowsfor a bridgebetweensimulated annealingand inverseproblemtheory [Mosegaardand Tarantola,1991]. In fact, the temperatureT and the

noisevariancea2 playthe samerole,anddecreasing the temperatureduring the coolingscheduleis equivalent to gradually enhancingthe influenceof the data

(i.e.,increasing thesignal-to-noise ratio)whenselecting

'"':• ....... ß above :• tO'• .:• to" •ms to-' • 10-' • m-' •

below

Fig. 1. Misfit function for three reflections. Slices are plotted in grey scale. Variations perpendicularto slicesare shown on two backgroundplanes. Symmetry of the figurescomesfrom permutationsbetweenthe three reflections: the global minimum is found above the secondslice from the top, above the fourth one, and near the sixth one. For each of these slices, two positionsof the global minimum are obtained by permutation of the two other events. The path performed by a gradient method from any arbitrary point in the cube is a rather complex line which must avoid a situation with two reflectionsnearly at the same position.

the modelsin the Metropolisloop. In practice,the Metropolisalgorithm is not efficient at low temperaturewherethe rejectionprobabilitylevel of most perturbationsis high, and it can be replaced by a faster procedureknown as the heat bath method

[Creulz,1980]andappliedby Rolhman[1986].Foreach of the k possiblevaluesof the currentlyconsideredtime shift vn (keepingall the other time shift parameters

fixed) one computesthe relativeprobabilityof acceptance by the followingexpression, e-Sk/T

7•A(*n,k) -- Zk=l K e-Sk/T '

(19)

for eachk between1 and K. The denominatorof (19) may be viewed as a marginal partition function. From this probabilitydistributionone makesa randomguess. This guessis alwaystaken and givesthe new value for vn. We must repeat this procedurefor every reflection. Whether simulated annealing is efficient or not strongly dependson the possibility of the processto jump from one particular acceptable model to another

The simulated annealingtechnique,originally pro- one. When the temperatureis high (i.e., when the posedby Kirkpatricket al. [1983]and Cerny[1985],is singaLto-noise ratio is assumedlow), almosteveryparthe answer we select compared to a pure Monte-Carlo approach. Although this techniquerequires the solution of the forward problem many times, it has proven

ticular model is likely to be accepted. However, when

its efficiency in manysituations(seevanLaarhovenand Aarls [1987]for a review and Rothman[1985, 1986], [1989], [1990] geophysical applications).Simulatedannealingis a two-

belongingto a relatively small subsetof the model set, remain with a significantlikelihood of acceptance.For such low temperatures, and due to the local nature of the attempted transitions, it may be difficult, in practice, to tunnel from an acceptable model to another

loop procedure which is a compromisebetween local convergentmethods and Monte-Carlo methods. The

the temperaturedecreases (i.e., whenthe signal-to-noise ratio is assumedlargerand larger),only a few models,

one since the number of worse and intermediate

mod-

first (inner)loopisthe Metropolisalgorithmwhichcon- els to visit may be large [Hajek, 1988]. Obviously,this sistsof randomlyperturbingthe modelparametersand acceptingthe new model with a probability

number changesaccordingto the manner in which we perform our model perturbations. It has been reported previouslythat the tunnelingeffectis sometimeseasier 7•M-- min(1; e-aS/T), (17) when the perturbation involvesseveral model parame[Ettelaieand Moore,1987;HeyndwhereAS is the changein the misfit function due to the ters simultaneously model perturbation, and T is a parameter called, for his- erickz and De Raedt, 1988]. Our numericalattempts torical reasons,the temperature. The probability 7)M is showedus that these more sophisticatedperturbation suchthat whenthe changeAS is positive(i.e., the new schemeshave no effectsin our problem. Consequently, modelis worse)the perturbationis not systematically at eachiteration, we haveperturbeda singleparameter, rejected.The second(outer)loopof simulatedanneal- namely, the position of one reflection. ing is the coolingschedulewhichconsistsof decreasing In summary,our algorithm proceedsin the following the temperaturewhile loophagover the Metropolis al- steps: selectionof the number of eventsN as explained gorithm. It canbe shownthat for an infinitenumberof in the previous section and computation of the accepattempted perturbationsthe energydistribution of the tanceprobability(equation(19)) usedfor the drawing

GIBERT AND VIRIEUX:

ELECTROMAGNETIC

of a new value of the selectedposition. When we have consideredevery reflection,we have performed a single iteration. The simulated annealing repeats this iteration while the temperature is decreasedin an adequate

IMAGING AND SIMULATED ANNEALING

8061

TABLE 1. Model Description

Conductivity,s/m

Layer thickness,m

0.01

300.

0.10

100.

0.02

100.

0.01

150.

0.10

oo

way.

COOLING SCHEDULE

Good control over the convergenceparameter T is the difficulty of the simulatedannealing. By a too slow cooling,we can makethe solutionvery expensive,while a too quick cooling might trap us into an unwanted

solution[Kirkpatricket al., 1983; Kirkpatrick,1984]. equal to the noise variance,

In other words, the convergencerate toward a minimum increaseswhen the coolingrate increases,but the T. oi,•= sd(noise) 2, (23) probability to be in the global minimum diminishes. Therefore, we must have an adequateslow cooling,and might be goodsincefor this particularchoicethe proba-

until now, despitetheoreticalattempts [Hajek, 1988], only experiences in differentpracticalsituations[Rolhman, 1985]provideboundsto this cooling.Nullon and $alamon[1988] and Andresenet al. [1988]proposean adaptative annealingschedulefor global optimization problems.

bility distributions 7• (equation(16)) and7•, (equation (18)) are identical,and the modelspaceis sampledaccordingto the a posterioridistribution7• [Mosegaard and Tarantola,1991]. By decreasingthe temperature below this level it can be shown that each parameter convergestowardsthe maximum-likelihoodsolution

In our case, we have selectedthe strategy proposed [Mosegaardand Tarantola,1991]. This encourages us to continuethe coolinguntil the followingfinal temperconstantthermodynamic speedA. We must verify that ature: Tnoise Tnoise the averagedenergyat iteration n + 1 is below the aver< Tii.., < •. 10 (24) 100 aged energyat the iteration n by A times the standard deviation,denotedsd(), of the energyat iteration n. FinM]y, increasingin one step the temperature to Getting realisticestimatesof thesestatistical quantities will allow the computation of the conditional probabilimpliesthat the energydistributionat a given iteration ity distributions 7•A (equation(19)) whichare our final must be sampled accurately before changingthe tem- solution and give an insight upon the location error of the reflection. perature. We have the followingformula,

by Huanget al. [1986]wherethe coolingis madeat a

< $(n + 1) >=< $(n) >-•

sd($(n)),

SIMPLE EXAMPLE FOR TESTING SIMULATED ANNEALING

(•0)

where A is chosenin order to guaranteean appreciable The selected example is a one-dimensionalmedium overlap between the energy distributions at iterations with four reflections, described in Table 1. The posin and n-9 1. The greater this overlap, the larger the tions of primary reflectionsare convertedin travel time probability to escapefrom local minima. The cooling r and variable q in Table 2. The amplitudes of refleclaw• tions are alsogivenin Table 2. We computeelectromag-

T(n+ 1)- T(n)e-•T(")/'d(s(")),

(21) netic responses using(3) at 100 frequencies between1 Hz and 1.5 kHz.

We add a Gaussian

white

noise with

is deduced[Huanget al., 1986]. The choiceof the pa- zero mean to this synthetic data. We construct three rameter A is not crucial and we have taken A - 0.75,

a rather high value comparedto those(~ 0.1) often chosenby mostauthors.In practice,the thermal quasiequilibriumat a giventemperatureis quicklyreachedin accordance with the resultsobtainedby Creutz[1980],

data setswith signal-to-noiseratio of 30 dB, 20 dB and

10 dB (Figure 2). The spectralanalysisof data covariancematrices using the Akaike's information crite-

rion-AIC- (seeTable 3) showsus that threereflections can be located for the first two data sets, while the

and 10 iterationsat constanttemperatureare sufficient presenceof important noisein the last one allows only to give goodestimatesof averageenergyand standard two reflections to be located. The statistical criterion is deviation. constructed for a maximum of 15 expected reflections Finally, we must specifythe initial temperature and (= N'), a numbersignificantly higherthan the a priori the stoppingtemperatureof the coolingprocedure.The selectednumberof reflections(N), alwayslowerthan initial temperaturemustbe high enoughto allownearly

any transitions(i.e., the probabilitydistribution7•Aof (19) is almostuniform). The temperatureis taken as the averagedenergyover 100 randomconfigurations incremented

of the standard

Events

deviation:

= < s(,) > + sd(S(,)).

TABLE 2. Positions and Amplitudes of Events

(22)

1 2

This choiceallows more than 80% of possibletransitions,as suggested by Kirkpatrick[1984].The selection of the final temperature is lessobvious. A temperature

3

4

AmplitudeU -0.52

r, ms

q, V•

4.524

0.067

0.28

19.088

0.139

0.11

28.860

0.170

41.420

0.204

-0.31

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GIBERT

AND VIRIEUX:

ELECTROMAGNETIC

IMAGING

AND SIMULATED

TABLE

3. AIC

ANNEALING

Value for an a Priori

Number

of Events

Signal-to-Noiseratio, dB Events 30

-0.25

-3

-2

-1

0

20

10

1

1388

345

289

2

331

321

277

299

3

318

315

4

334

331

322

5

361

359

348

6

382

380

366

Bold numbers give the AIC selectionfor the number of events.

1og10[period (s)]

The amplitudesare alsowell defined(Table4). Let us note that the two intermediate

events have the same

sign in amplitudeand are not resolvedby our procedure. The estimatedamplitude of the secondreflection is nearly the sum of these two intermediate real reflec-

tions. Once the positionsare given, we computethe conditional probabilityat Tnoise(Figure3). This probability wouldleadto a possibleanalysisof the final resolutionwhichis not performedin this study.Intuitively, one can seein Figure 3 the decreasein resolutionwith

(b)

depth. It is interestingto follow the evolution of the event

, -3

locations whenweproceedto the cooling(Figure4). In -2

-1

0

log]o[period (s)]

the samefigurewe give the temperatureevolution,the decrease of the averagedenergy< $ >. One canseethe hierarchywherethe mostsuperficialreflectionis first located, the seconddeeperone, and so on. This evolution is the one we obtain in all the numerical simulations

we undertake. The energyvariation are best seenwith

the computationof the derivativeA < $ >/AT, which is analogousto a specificheat. The decreaseof the energyis associated with a better organizedsystem,which is often related,if not always,to the interfaceposition.

o.o-

;> •

_ -

r = ?.. -0.25 -

snr

(=

= 30 dB

•30. 75-

-3

-2

-1

0

0.5-

-

log]o[period (s)] Fig. 2. Synthetic data obtained by convolution. Three signal-to-noise ratiosare defined:(a) a ratio of

30dB•variance a2= 10-5),(b)a ratioof20dB(vari-

ance•r• = 10-4) and (c) a ratio of 10 dB (variance o.2_ 10-3).

0.0 0.0

, , , , I , 0.05

, , i , , i' , 0.1

0.15

02

q "time" (i •=)

Fig. 3. For a signal-to-noiseratio of 30 dB, conditional probabilities associatedto each of the inverted six events. This is necessaryin order to have a signifi- events. Flags indicate final positionsof the reflections

cantnumberof termsin the sumand productof (A9) obtainedat theendofcooling schedule (T = Tlinalfrom (seethe appendix). equation(24)). For eachreflection,conditionalproba-

For the high signal-to-noiseratio of 30 dB, the re- bilities have been computedby varying the positionof solvablethree reflectionsgivenby the criterionAIC are the considered event(seeequation(19)) whilekeeping well located(Figure3) whenTlinal equalsTnoise/10. the remainingreflectionsat their final positions.

GIBERT AND VIRIEUX: ELECTROMAGNETIC

IMAGING AND SIMULATED ANNEALING

8063

TABLE 4. Amplitude Estimation of Events After Simulated Annealing

Events

True Amplitude

1

Figure 3

-0.520

-0.520

2

0.280

3

O. 110

4

ß

'

ß

-0.272

ß

..

•

ß

ß

ee•

ße ß

''

ßß '

...

ß'

ß

:..-'''.

ß.

'

ß'""'""

.

.

•')

e. ß. ß '

15 0

•

'

.' .,..

ß

.

ß,,. ,,. --

.

ß

.

0.05

-.,..

...

ß

ß

.-'

ß

'. .ß,.. ß .

0.0

.

5). This is an a posterioriproofof theimportance of detectingthe numberN of eventsand givesa justification we have used.

has a very small amplitude which can be used to reject

.

" 0

-

the two deepesteventsboth in locations(Figure7) and in amplitudes(Table 3). We emphasise that this event

ß

ß

-

-0.375

Of course, when the noise increases,events are much more difficult to locate. Figure 6 presentsthree located pulsesat the final temperatureT, oise/10with a strong degradationof resolution: only the first event is perfectly resolved.Introducing an extra event with respect to the AIC still givesa nonlocatedevent which perturbs

O' ßß ß.

-

for the AIC

ß

-

O. 451

0.463

On the bottom of Figure 4, the specificheat presents three bumpsrelated to the temperatureinterval where an interfacestarts to freezeat a given position. When four reflectionsare a priori sought, the unresolvedevent, whichis the secondone with a small amplitude(Table4), perturbsthe positions of the threeother reflectionsand decreases the globalresolution(Figure

ß

e• e•e

.'.

-0.828

-0.525

-0.288

-0.291

Figure 8

0.001

0.376

0.381

-

Figure 7

-0.525

-0.009

.

ee

,-.0.1

e

Figure 6

-0.520

0.353

-0.310

ß .

Figure 5

it.

25

50

75

For a noisy environment as the final example, the

100

AIC

allows two reflections.

One can see that

the first

one is welllocated, but the second one is located in a wrong position near the first one at the final tempera-

0.0

ture (Figure8). The wholestrategyfailsfor this signal to-noise ratio, and the secondreflection interprets only

noisein the data (Figure2c). Two-DIMENSIONAL

T

-5.0

25

50

75

100

REALISTIC

EXAMPLE

In this last section we would like to test the performanceof simulated annealingon synthetic data obtained by an entirely different forward modeling than

the integralequation(3). We haveuseda finite element

T = T•. snr

'-' 0.75•

= 30

(= 10-') dB

-

,.•

-

0

.

•' 0.5: Ill

cl 0 •

[ [ [ [ I I i [ [ I ill

25

50

II

75

ill

I I I I [ I

100

iterationcountof coolingschedule

Fig. 4. Evolution of reflectionpositionson the top panelduringthe cooling.Intermediatepanelshowsthe prescribed temperature evolution and the decreasing energy. Variations of the energy are better seenwith the specificheat coefficienton the bottom panel.

-

-

,• 0.?..5I:1

-

o.ol...; ..... .... ..... .... ......... ..... .... 0.0

0.05

0.1

0.15

0.2

q "time" (sw)

Fig. 5.

Same as Figure 3 for a priori four reflections.

Please note the unlocalized

event.

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GIBERT

AND VIRIEUX:

ELECTROMAGNETIC

IMAGING

AND SIMULATED

ANNEALING

L04-•

-•

-

T=T•..(=104

-

snr

= 20

)

T=

dB

ant=

= 0.75-_ ß•

(= lo 10 dB

0.5: .

•

-

o

:

...,i

• o25:

"• : 0.0

..... ........ ,...

, [ [

..,.,"

] [ [ [ I [[

0.0

0.05

] [ I • [ 0.1

1 [ ] ] [ I [ 0.15

02

1, 80.Ol 0.0

q "time" (s•")

I

T"'"[[ [ I [ ] ' ' I [

0.05

OA

0.15

02

q "Ume"

Fig. 6. Same as for Figure 3 but with a signal-to- Fig. 8. Sameasfor Figure3 but for a priori two events and a signal-to-noiseratio of 10 dB. One reflection is correctly located, while the secondis at a wrong position and fits somenoisein the signal.

noise ratio of 20 dB. Three reflections are deduced from the criterion AIC.

program[Wannamaker et ai., 1985]fora simplerealistic medium.This modelis designedsuchthat the deepest and third interface is fiat with an important constrast of the resistivity. Above this interface a more complex pattern of interfacesis consideredfirst for perturbing the diffusionprocessand for checkingthe performance of the simulated annealing. Moreover, resistivity con-

trastsare smallerthan for the deepestinterface(Figure 9). We neglectdisplacement currentasusuallydonein the usual magnetotelluricapproximation,and we consider transversemagneticmode for our experiment. We use 63 traces with only 50 frequenciesbetween 1 Hz and 1.5 kHz with the regular sampling of the

squareroot of frequency(equation(12)). An extra Gaussianwhite noiseis addedwith a signal-to-noiseratio of 30 dB.

During the coolingfor each trace, the parameter A is equal to 0.75, while 10 iterations are performed at each temperature before the selectionof the next position. The final temperatureis taken as Tnoise/10.The criterion AIC

turns out to select between three and six

events, although only two reflectionsare correctly located' the final conditional probabilities of these extra events are rather

The conditional probability is shownfor each trace in an equivalentseismicsectionin Figure 9. We emphasize that the electromagneticsourceis a vertical plane wave exciting the medium at the free surface. In other words, the pseudo-seismic sectionis the one expected after a CMP stack before migration. The simulated annealing has the capacity to detect two interfaces: The top interface is the best localized, while the secondis slightly blurred out. This secondinterface, which is fiat in the real medium, has a slopein good correlation with the resistivity aboveit. The complexity on the right of the model is entirely unresolved and does not even degradethe solution: interfacesare too near each other with a too small jump of resistivity. Sometimes,reflectionsare mislocated with a spread conditionalprobability: a posteriori analysismight require the restarting of the coolingschedule,and an improved picture would be obtained. We have not undertaken this strategy and have preferred to present straight resultsfrom the inversion.

fiat.

horizontal position (m)

L0-

T= snr

(= lo = 20 dB

•

= 0.75•

-

,,•

-

o

p- 600 tim

.

0.5:

•

-

I:l

-

0

-

...,i

p - 200 tim

ßo 025-Cl 0.0

Fig. 7. Same as for Figure 3 but for a priori four events and a signal-to-noiseratio of 20 dB. Pleasenote the unlocalized event and the unwanted perturbation of

Fig. 9. The two-dimensionalmodel on the top with a deep fiat interface. From TM signalsat the free surface, we deducea pseudoseismic sectionwhichis equivalent to a CMP stack beforemigration. More accurate readability of the event position is obtained by plotting the conditional probability at the temperature

the latest

Wnoise.

0.0

0.05

0.1

OA5

02

q "time" (•)

reflection.

GIBERT AND VIRIEUX: ELECTROMAGNETIC

IMAGING AND SIMULATED ANNEALING

8065

horizontal position (m) 250

500

750

1000

1250

1500

.[;.•.''' ............................... '''' •'.......... ';,,-'; ':;;;;.-'.'' ............ !:;'..-..;;•;;;-'.;,.•,-;L.4.-;.-':. ':...... 3:.:.:!:.:./

......... '"'•". •••••$'-.'-•g•'"::::• ''"'•-'-'-'-"'-"'''" •----• 'ß' ..' ,

.'--'-.'-•. ..::-••••.-'..:.-• '---'""'-'-'-"-"" -'-'-'-'""•-" '- -.

................ """ '"'"" ""'"' '""••"•'"'"'"'""•:':'"•••:•'-'::'.::• ....... '.................... .'-':-'5::.'

• •

•.....•...:..:•..•.:••:•••.•...:..,: :'-'••:•'.::-'.'-'.'::::-'.',' .................................. •. '•'":• ................. '''" Fig. 9.

o.eo 0.4o - o.6o

::..'..'.'• 020 - 0.40 -'-'A::•BELOW

(continued)

The conversionfrom pseudotimeto depth will require with œ- 0,...,N and j - N + 1,...,M. By summing the use of migration techniquesused in seismicdata the N terms on the LHS, we obtain the expression, analysis. The discussionof this transformation is beN N N yondthe scopeof this paper. Poststackmigrationmight be appliedto this pseudo-seismic sectionand, hopefully, E atE. i-t - n=l E UnoZn E atZJ•-t' (A3) the deepest interface will becomefiat. l=0 •=0 CONCLUSION

We have shown that diffusiveelectromagneticsignals can be convertedto impulsivepseudo-wavesignalsby a procedure similar to deconvolution. This procedure is efficiently performed by simulatedannealing. We have studied the different stepsin order to achievethis transformation and have definedthe parameterscontrolling the cooling. Except for the very noisyexample,we succeedin locating reflections,as well as in estimating their amplitudes. This successenables us to propose the different migrations of seismicexploration in order to construct the contrast of conductivity inside the medium. This will be the purposeof a future work. Each trace is inverted independentlyfrom the neighboring ones. A promisingstrategy for future work will include

the coherence

between

traces to accelerate

with (j = N + 1,..., M). The polynomialon the RHS is evaluated at one of its roots Z, and is consequently equal to zero. Thus the linear system N

E atEi-t= 0

(A4)

with a0 - 1 andj - (N + 1,..., M), holdsand canbe written in the compactform: œa - 0, where we have introduced the matrix œ,

œ=

E1-..Et½+l ] :

EM-N

the

'..

ß

."

EM

,

(A5)

searchfor the globalminimumfor eachtrace. This technique will improve the pseudo-seismicsection we have andthe vectorat = (a•v,.-., a0). As usual,we reduce to a squaresystemby multiplying(A4) with the conobtained in the previousparagraph.

jugated transposedmatrix œaof œ. If we divide also

(A4) by the numberof unknownparameters N + 1, we

APPENDIX

The spectral analysisof the data covariancematrix

introduced by Van Blaricumand Mittra [1978]turns out to be a very efficientmethodfor determiningthe

obtain the data covariancematrix 7• and the equivalent system,

'R,a= 0.

(A6)

number of reflections.

This expression showsus that a is an eigenvector of the matrix 7• with aneigenvalue equalto zero. The rank of matrix 7• is lower or equal to the numberof events (A1) N. Of course,in presenceof noisydata, eigenvalues

Let us definethe polynomialP(Z) of total degreeN with roots N

N

P(Z)- H (Z- Z,)- E atZN-t' n=l

are not identically zeroes: we must select a threshold

underwhichwe decidewith a givencriterionthat the

with a0 equalto 1. Multiplying(13) by at, we obtain eigenvalueis almost zero. For white noisewith variance

the followingexpression:

•ra, thecovariance matrixhasthefollowing formwith I as the identity matrix,

N

- y] n=l

oZZ'. ,

7•' = 7•+ o'•I,

(A7)

8066

GIBERT AND VIRIEUX:

ELECTROMAGNETIC

from which one can obtain:

7•'a -- •'2a

(AS)

whichstatesthat the thresholdcanbe takenasa2 [Van Blaricum and Mittra, 1978;Kay and Marpie, 1981]. As statistical test of decision, we use the Akaike's test se-

lected by Waz and Kailath [1985]for a very similar problem. We compute a data covariancematrix for a numberof reflectionN' higher than the expectednumber N. The true number of reflections is the one which makes the Akaike's information criterion-AICminimum.'

IMAGING

AND SIMULATED

ANNEALING

Huang, M.D., F. Romeo, and A.L. Sangiovanni-Vincentelh, An efficient generM cooling schedule for simulated anneahng, Proc. IEEE Int. conf. Computer-Aided Design, Santa Clara, 381-384, 1986. Kay, S. M., and S. L. Marpie, Jr., Spectrum anMysis: a modern perspective, Proc. IEEE, 69, 1380-1419, 1981. Kirkpatrick, S., Optimization by simulated annealing: Quantitative studies, J. Statist. Phys., 3•, 975-986, 1984. Kirkpatrick, S., C. D. Gelart, and M.P. Vecchi, Optimization by simulated anneMing, Science, œœ0,671-680, 1983. Kunetz, G., Processing and interpretation of magnetotelluric soundings,Geophysics,37, 1005-1021, 1972. Landa, E., W. Beydoun, and A. Tarantola, Velocity model estimation from prestack waveforms: Optimization by simulated anneMing, Geophysics,54, 984-990, 1989. Lavrent'ev, M. M., V. G. Romanov, and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis

AIC(N)--2Log[ 1-Ii:•v+• ]Uv'-•V)(M-•V') _ N½ +

- :v),

(in Aussian),Nauka, Moscow,1980.

(4s) Lawson, C. L., and R. L. Hanson, Solving Least Squares wherethe Ai are the eigenvaluesof 7•' arrangedin deProblems,Prentice-HaH,EnglewoodCliffs, N.J., 1974. creasingorder. We found this criterion very successful Lee, K. H., G. Liu, and H. F. Morrison, A new approach in obtaining the number of reflectionsin even relatively to modeling the electromagneticresponseof conductive noisy signals. For a more detailed descriptionof this media, Geophysics,15•, 1180-1192, 1989.

criterionwe referthe readerto Wa• andKailalh [1985]. Lee, S., G. A. McMechan, and C. L. Aiken, Phase-fieldimaging: The electromagneticequivMent of seismicmigration, Acknowledgments.Many thanks to D. W. Oldenburg for Geophysics,5œ,1678-693, 1987. making available his preprints and for helpful comments. This paper benefited from stimulating discussionswith C. Levy, S., D. W. Oldenburg, and J. Wang, Subsurfaceimaging using magnetotelluric data, Geophysics,53, 104-117, Hubaris, V. Jouanne, M. Menvielle, P. Podvin, M. Roussig1988. nol, J. T. Smith, D. Sornette, and P. Tarits. Very construcMcDonough, R.N., and W.H. Huggins, Best least-squares tive reviews were made by K. Mosegaard and A. Tarantola. representationof signMs by exponentials, IEEE Trans. This research was supported by the sponsorsof CGGM de Autom. Control, A C-13, 408-412, 1968. l'EcoledesMines(CFP, CGG, IFP, SNEAP).

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