Monte Carlo full waveform inversion of tomographic crosshole

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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)
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Monte Carlo full waveform inversion of tomographic crosshole

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