Monte Carlo full waveform inversion of tomographic crosshole

Monte Carlo full waveform inversion of tomographic crosshole data using complex geostatistical apriori informationKnud S. Cordua*, Thomas M. Hansen, and Klaus Mosegaard, Technical University of Denmark, Department ofInformatics and Mathematical ModellingSummaryThis paper presents a Monte Carlo full waveform inversionstrategy based on a Bayesian formulation of the inverseproblem. Existing full waveform inversion strategies oftenrelies on a migration based approach, which suffers fromlack of uncertainty estimates. Using a Bayesian approach,the solution to the inverse problem is formulated as an aposteriori probability density. We demonstrate that samplesfrom the solution to the full waveform inverse problem canbe obtained using the extended Metropolis algorithm inconjunction with complex a priori information. The a prioriinformation is described by a training image using ageostatistical algorithm. A posteriori samples from thesolution to the inverse problem provide a means ofobtaining resolution analysis of the solution. The suggestedstrategy is tested on synthetic crosshole full waveformground penetrating radar (GPR) data, but is equally wellapplicable to seismic waveform data. The forward problemis solved using finite-difference time-domain calculationsof Maxwell’s equations. To our knowledge this is the firstexample of performing full waveform inversion using theextend Metropolis Algorithm and, in this way, provide anuncertainty estimate of a tomographic full waveforminverse problem.IntroductionGround penetrating radar (GPR) crosshole tomography is apopular method used to obtain tomographic images of nearsurfacegeological structures and geophysical parameters.The crosshole GPR experiment involves a transmittingradar antenna (20 MHz – 1 GHz, (see Reynolds, 1997))lowered into a borehole and a receiving antenna placed inan adjacent borehole. The boreholes are typically separatedby a distance of 5m – 20m and are 10 m – 20 m deep. Thetransmitting antenna is kept fixed in one borehole while thesignal is recorded at multiple locations in the oppositeborehole. This procedure is repeated for multipletransmitter positions until the inter borehole section hasbeen covered with measurements. For a correspondingseismic experiment the source pulse frequency is typicallyin the order of 500Hz – 1000 Hz (e.g. Paasche et al., 2006;Belina et al., 2009).Ray-based inversion of first arrival travel times andamplitudes of GPR or seismic data provide estimates of thesignal velocity and attenuation distribution of the interboreholeregion. Unfortunately, ray-theory is based on ahigh-frequency approximation and accounts only for asmall part of the information content of the full waveformsignal. Therefore, tomographic images obtained using raytheoryare limited in resolution.Ernst et al. (2007a,b) introduced and applied an algorithmfor inversion of tomographic crosshole GPR data. Thiscode was modified by Belina et al. (2009) and applied toseismic waveform data. Ernst et al. (2007a,b) and Belina etal. (2009) demonstrated that sub-wavelength features canbe resolved by properly modelling the wave propagationand, hence, utilizing the full information content of thewaveform data. The algorithm introduced by Ernst et al.(2007a,b) relies on the migration-based full waveforminversion concept introduced by Tarantola (1984). Thismethod is desirable because it, in relatively few iterations,is able to infer geophysical parameters in order to minimisea misfit function between the observed and modelledwaveforms. However, the migration based approach hassome limitations: Firstly, the method depends on a goodinitial guess of the large scale features of the solution inorder to ensure convergence. Secondly, when using thismethod in a tomographic crosshole setup, a dense receiverand transmitter coverage is needed in order to avoidnumerical artefacts along the boreholes. Finally, the methodis based on an optimization algorithm and does not provideany uncertainty estimate of the solution.Here, we formulate the GPR crosshole problem as aprobabilistic, Bayesian, full waveform inverse problem. Weuse the extended Metropolis Algorithm (Mosegaard andTarantola, 1995) to obtain samples of the a posteriori pdf.In this way, not only a single estimate, but multiplerealizations of tomographic images which all honour thedata uncertainties and a priori information are provided.This, in turn, provides a means of performing resolutionanalysis of the solution (Mosegaard, 1998). The a prioriinformation about the problem is controlled through ageostatistical algorithm. This approach allow us toincorporate complex a priori information described by both2-point and multiple-point statistics. In this study we limitthe example only to concern a priori information describedby a training image (i.e. multiple-point statistics). Wedemonstrate how geostatistical formulated a prioriinformation serves as a guide in the initial part of theinversion procedure. In this way convergence of thesuggested full-waveform inversion algorithm becomesindependent of the initial model.SEG Expanded abstracts

Monte Carlo full waveform inversionMethodologyConsider that the subsurface can be represented by adiscrete set of model parameters, m , and that a data set,d , of indirect observations of the model parameters isprovided. The model parameters describe some physicalproperties of the subsurface that influences the dataobservations. Hence, the forward relation between themodel parameters (i.e. the model) and the data observationscan be expressed as (e.g. Tarantola, 2005):d = g( m ), (1)where g is a linear or non-linear mapping operator whichoften relies on a physical law. Here, the forward relation inequation (1) is given as a finite-difference time-domainsolution of Maxwell’s equations. However, any numericalwave propagation modelling strategy for GPR or seismicsignals can be applied. The inverse problem is to inferinformation about the model parameters based on a set ofobservations, a priori information about the model, and theforward relation between the model and the dataobservations.In a Bayesian formulation the solution to the inverseproblem is given as an a posteriori probability density,which can be formulated as (e.g. Tarantola, 2005):σ ( m) = kρ( m) L( m ), (2)MMwhere k is a normalization constant, ρM( m)is the a prioriprobability density, and L( m)is the likelihood function.ρ ( ) Mm describes the probability that the model satisfiesthe a priori information. L( m)describes how well themodelled data explains the observed data given a datauncertainty. Hence, the a posteriori probability densitydescribes the probability that a certain model is a solutionto the inverse problem.A highly nonlinear inverse problem refers to the case wherethe a priori probability density is far from being Gaussianor the forward relation between the model and data are farfrom being linear. In the case of full waveform inversionthe forward relation is expected to be highly nonlinear.Moreover, the a priori information described by a trainingimage is highly non-Gaussian.The extended Metropolis algorithm is a versatile toolwhich, in particular, is useful to obtain samples fromsolutions to non-linear inverse problems using arbitrarilycomplex a priori information. The minimum requirement ofthe algorithm is; 1) a “black box” algorithm that is able tosample the a priori probability density and, 2) a “black box”algorithm that is able to compute the likelihood for a givenset of model parameters. The flowchart of the extendedMetropolis algorithm is as follows: 1) The a priori samplerproposes a sample, mpropose, from the a priori probabilitydensity, which is a perturbation of a previous acceptedmodel, m accept. 2) The proposed sample is accepted with theprobability (known as the Metropolis rule):Paccept⎛ L( m ) ⎞propose= min 1,⎜L( accept) ⎟⎝ m ⎠3) If the proposed model is accepted, mproposethe a posteriori probability andOtherwisempropose(3)is a sample ofmproposebecomes maccept.is rejected. 4) The procedure iscontinued until a desirable number of models have beenaccepted.In this study the algorithm that provides the a prioriinformation is the Single Normal Equation SIMulation(snesim) algorithm, which is a fast geostatistical algorithmthat produces samples (conditional or unconditional) froman a priori probability density defined by a training imagefor a relatively low number of categorical values (Strebelle,2002). Hansen et al. (2008) suggest a strategy termedperturbed simulation, which is capable of producingperturbations of spatial distributions using geostatisticalalgorithms. Thus, perturbed simulation serves as a “blackbox” that produces samples of a priori probability densitiesdescribed by both two-point and multiple-point statistics.The flow of this algorithm is as follows: 1) An initialunconditional sample of the a priori probability density(here defined by a training image) is provided. 2) A subareaof the sample is randomly chosen. 3) The model parameterswithin this area are set to unknown. 4) The unknown modelparameters are resimulated conditional to the rest of themodel parameters using a geostatistical algorithm (heresnesim) and a perturbation is obtained. 5) This procedure isrepeated in order to obtain multiple samples of the a prioriprobability density.The size of the perturbation area governs the exploratorynature of the Metropolis algorithm. The size of theperturbation area is chosen subjectively. In the extreme casewhere the area covers the entire model the outcome of theperturbed simulation algorithm is uncorrelated to theprevious model. Contrary, if the area only constitutes asingle model parameter, the perturbed model is highlycorrelated with the initial model. According to themetropolis rule a small perturbation area results in aproposed model that is more probable of being accepted ascompared to a proposed model obtained using a largerperturbation area. Therefore, the perturbation area shouldSEG Expanded abstracts

Monte Carlo full waveform inversionbe chosen carefully in order to ensure an efficientalgorithm. Gelman et al. (1996) found that the acceptancerate should be around 23% for high-dimensionaldistributions. For large acceptance rates the algorithm isexploring the a posteriori probability density too slowly. Onthe other hand, for smaller acceptance rates too manycomputationally expensive trials are performed. Therefore,we suggest to automatically change the size of theperturbation area while running the algorithm such that acertain acceptance rate is maintained. A constantacceptance rate results in a larger perturbation area in theburn-in period than in the subsequent sampling period.This effect is beneficial because the algorithm needs toperform large perturbations in the initial part in order tofind models of large probability and, hence, producerepresentative samples of the a posteriori probabilitydensity.Finally, the likelihood function is defined as a Gaussiandistribution:⎛ 1⎞L( ) kexp ( g( ) d )/ σ⎝⎠Ni i 2m = ⎜− ∑ m −obs ⎟, (4)2 i = 1where g( m) i represents the amplitude of the individualsample points of all the simulated waveforms obtainedithrough equation (1) (i.e. the FDTD algorithm) and ared obsthe sample points of the observed waveform data. σ is thestandard deviation of the expected amplitude uncertainty ofthe waveform data.Results and discussionFigure 1 shows a training image that mimic a matrix of claywith embedded channels of unconsolidated sand.Electromagnetic signals in near surface sediments aresensitive to the dielectric permittivity and the electricalconductivity of the materials. In this study we limitourselves only to consider the influence of the dielectricpermittivity, which is primarily governing the phasevelocity of the signal. Water saturation of clay is often highcompared to sandy deposits. Therefore, the dielectricpermittivity of the clay is set to a relative dielectricpermittivity of εr≈ 4,57 (0,14m/ns) and the permittivity ofthe sand channels is set to εr≈ 2,75 (0,18m/ns) (e.g. Toppet al., 1980). Figure 2 (left) is the synthetic reference to beconsidered and is, at the same time, an unconditionalsample of the training image obtained using snesim. Theelectrical conductivity is set to a constant value of 3 mS/mand is, in the following, assumed known.A full waveform synthetic data set is calculated using theFDTD algorithm. A Ricker wavelet with a centralfrequency of 100 MHz is used as source pulse. The sourcepulse is assumed known during the inversion. Thetransmitter and receiver positions are separated by 2 m and0.25 m, respectively (see figure 2 left). Data acquired with atransmitter-receiver angle larger than 45 degrees fromhorizontal are omitted since, in practice, these data areviolated by effects of wave guiding in the boreholes (cf.Peterson, 2001). This leads to a total of 248 dataobservations (i.e. recorded waveforms).Depth [m]05101520250 5 10 15 20 25Distance [m]Figure 1. Training image which mimic sandy channelstructures embedded in a matrix of clay deposits.Depth [m]0246810Reference model120 2 4Distance [m]Distance [m]Figure 2. Left) Synthetic reference model. Black asterisksshow transmitter positions and the yellow dots showreceiver positions. Right) The initial model used as inputfor the inversion.0Initial guess120 2 4SEG Expanded abstractsNoise is subsequently added to the data by performing arandom phase shifting of the synthetic waveforms. Thephase shift is normal distributed with zero mean and astandard deviation of 0.4 ns since this is a typicalmagnitude found in GPR travel time data (e.g. Looms et al,in press). The phase shift results in an amplitude2468104.64.44.243.83.63.43.232.82.6Relative dielectric permittivity ( ε/ε 0)4.64.44.243.83.63.43.232.82.6Relative dielectric permittivity ( ε/ε 0)

Monte Carlo full waveform inversion3uncertainty with a standard deviation of 10 − . Accordingly,the standard deviation of the data uncertainties, σ , is set to3. The standard deviation of the amplitude noise isindicated by the red error bar in figure 3 and compared withtwo waveforms recorded at 0 degrees (short offset / highamplitude) and 45 degrees transmitter-receiver angles (longoffset / low amplitude), respectively.1Recorded at short offset0.5Recorded at long offset0-0.5-110 − 100 150 200 250 300 350 400Normalized amplitudeSample no.Figure 3. Dashed line is a waveform recorded at shorttransmitter-receiver offset (0 degrees). Solid line is awaveform recorded at long offset (45 degrees). The heightof the red errorbar indicates 2 times the standard deviationof the noise added to the data.Depth [m]05100 2 405100 2 405100 2 4Distance [m]05100 2 405100 2 4Figure 4. Five statistically independent samples from the aposteriori probability density of the full waveform inverseproblem.The initial model used for the Metropolis algorithm ischosen as an unconditional sample of the training imageusing a different random seed than for the reference model(see figure 2 right). Burn-in was reached after 2000accepted models. Hereafter samples accepted by theMetropolis rule are representative samples of the aposteriori probability density. Figure 4 shows the 2000 th ,6000 th , 1000 th , 14000 th , and 18000 th accepted sample usinga priori information defined by the training image (figure 1)and Gaussian data uncertainty (equation 4). Only a slightdeviation between the individual samples is seen, whichindicates little a posteriori model uncertainty. Figure 5shows the a posteriori mean and variance based on 18000samples from the a posteriori probability density after theburn-in period. From the a posteriori variance it is seen thatthe overall structures of the model is recovered (varianceclose to or equal to zero) whereas the higher variancesalong the edges between the clay and sand deposits aresubjected to uncertainty. A comparison between thereference model (figure 2 left) and the mean of the samples(figure 4 right) confirms that the waveform inversion is ableto recover the sand structures very well. Moreover, itshould be noted that the algorithm is able to reach theseresults even though it is initiated in a model uncorrelatedwith the reference model (compare figure 2 right and left).Depth [m]0246810Mean120 2 4Distance [m]4.64.44.243.83.63.43.232.82.6( ε/ε 0)0246810Variance120 2 4Distance [m]Figure 5. Mean (left) and variance (right) of 18000 samplesdrawn from the a posteriori probability density of thesolution to the full waveform inverse problem.The suggested Monte Carlo inversion strategy is preferablein that it allows for arbitrary antennae geometry.Furthermore, complex a priori inversion can be includedusing the perturbed simulation algorithm. Finally, a fulldata covariance matrix can be included in order to accountfor correlated data errors, which are often present intomographic inverse problems (e.g. Maurer and Musil,2004; Cordua et al., 2008, 2009). However, note that thisexhaustive sampling strategy needs substantially morecomputationally expensive forward calculations comparedto the traditional migration based approach.ConclusionsWe have demonstrated the potential of producing samplesof the solution to a tomographic full waveform inverseproblem using the extended Metropolis algorithm withcomplex a priori information. The methodology provides ameans of evaluating the a posteriori uncertainty, which isnot provided using optimisation based strategies for fullwaveform inversion. Moreover, the present approach isrobust with regard to the initial guess of the solution and thetransmitter-receiver density. Finally, the extendedMetropolis Algorithm is flexible regarding the choice of apriori information and specification of data uncertainties.SEG Expanded abstracts0.80.70.60.50.40.30.20.10( ε 2 /ε 02 )

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