Probabilistic Engineering Mechanics 39 (2015) 56–68

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Probabilistic Engineering Mechanics

http://d0266-89

n CorrE-m

zenon@

journal homepage: www.elsevier.com/locate/probengmech

Varying dimensional Bayesian acoustic waveform inversion for 1Dsemi-infinite heterogeneous media

S. Esmailzadeh a, Z. Medina-Cetina a,n , J.W. Kang b, L.F. Kallivokas c

a Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USAb Department of Civil Engineering, Hongik University, Seoul, South Koreac Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, Austin, TX 78712, USA

a r t i c l e i n f o

Article history:Received 21 January 2014Received in revised form16 October 2014Accepted 5 November 2014Available online 13 November 2014

Keywords:Probabilistic calibrationBayesian inversionBayesian model selectionBayesian partition modelingReversible jump MCMCWave equationPerfectly matched layers (PML)

x.doi.org/10.1016/j.probengmech.2014.11.00220/& 2014 Elsevier Ltd. All rights reserved.

esponding author.ail addresses: saba61@tamu.edu (S. Esmailzadtamu.edu (Z. Medina-Cetina).

a b s t r a c t

This paper introduces a methodology to infer the spatial variation of the acoustic characteristics of a 1Dvertical elastic heterogeneous earth model via a Bayesian calibration approach, given a prescribed se-quence of loading and the corresponding time history response registered at the ground level. This in-volves solving an inverse problem that maps the ground seismic response onto a random profile of theground stratigraphy (i.e. a 1D continuous spatial random field). From a Bayesian point of view, the so-lution to an inverse problem is fully characterized by a posterior density function of the forward modelrandom parameters, which explicitly overcomes the solution's non-uniqueness. This subsurface earthmodel is parameterized using a Bayesian partition model, where the number of soil layers, the location ofthe layers' interfaces, and their corresponding mechanical characteristics are defined as random vari-ables. The partition model approach to an inverse problem is closely related to a Bayesian model se-lection problem, where the likely dimensionality of the inverse problem (number of unknowns) is in-ferred conditioned on the experimental observations. The main benefit of the proposed approach is thatthe explicit regularization of the inverted profile by global damping procedures is not required. A Re-versible Jump Markov Chain Monte Carlo (RJMCMC) algorithm is used to sample the target posterior ofvarying dimension, dependent on the number of layers. A synthetic case study is provided to indicate theapplicability of the proposed methodology.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

A subsurface earth model is composed of complex geophysicalformations, which embodies a wide range of physical and me-chanical heterogeneities. The aim of probabilistic inverse modelingis to reconstruct the random field structure of these subsurfaceproperties, while accounting for various sources of uncertaintystemming from ground seismic geologic measurement errors,aleatory formations, and limited theoretical understanding aboutunderground wave propagation.

In practice, one of the main goals of geophysical investigationsis to identify the main geomorphological features of an unknownmedium, meaning the spatial location and concentration of geo-logical features such as the transition between materials, dis-continuities and material concentrations [34]. In the case of avertical 1D profile, this requires the definition of the location of

eh),

the sharp transitions between layer interfaces (i.e. material prop-erties), and the characterization of its corresponding mechanicalproperties.

In a horizontally stratified earth model, prior to making aninference about the likely variation of the elastic parameterswithin the geological layers, an assumption must be made con-cerning the number of layers in a certain depth range of interest.This assumption defines the dimensionality (i.e., the number ofunknowns) of the inverse problem. In reality, however, such in-formation is rarely available for the dimension and definition ofthe parameter space to be fixed. Consequently, fixing the numberof layers based on an incorrect assumption results in an erroneoussubsurface characterization.

To relax the hypothesis about the subsurface structure or spa-tial layering of the media's mechanical parameters, before theforward model is calibrated, it is proposed to define the number oflayers, their locations, and their corresponding mechanical para-meters, as random variables. From a Bayesian perspective, this setup is closely associated with probabilistic model selection, where acollection of models with varying number of parameters are

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 57

presented for inversion, and the task is to select the models thatmost likely describe the experimental observations.

To illustrate the applicability of the proposed probabilistic ca-libration method, a one dimensional horizontally stratified med-ium is presented in terms of a Bayesian partition model [8]. Thepartition model divides the unknown material field into a numberof non-overlapping regions, where each region represents a soillayer. Formulating the inverse medium problem in terms of apartition model may help reduce the dimensionality of the para-meter space. Hence, regularizing the solution through specificprior distributions, which bears smoothness constraints (in aBayesian inversion framework [45,12,20]), or regularization terms(in deterministic optimization problems [44,36,15]), is precluded.

A generalization of the simulation-based Markov Chain MonteCarlo methods, the so-called reversible jump [18], is used tosample the posterior distribution of varying dimensionality. In thissetting, the Markov Chain is capable of undergoing dimensionchanges while moving among a number of candidate models. Thekey aspect of the reversible jump algorithm is the introduction ofsome auxiliary random variables to equalize the dimensionality ofthe parameter space across models. A series of one-to-one de-terministic functions are defined to perform dimension matchingsuch that the detailed balance condition is satisfied. Balance con-dition is the necessary condition for a Markov Chain to converge tothe target density [7].

Since the introduction of Bayesian inference methods to thegeophysical community, this has received a great deal of attentionin a variety of applications [14,16,17,45,41,43]. However, a limitednumber of studies have addressed the subsurface parameter esti-mation as a model selection problem, many of which resort toapproximate methods to fulfill the model determination [10,13].The varying dimensional formulation was first introduced to thegeophysics literature by Malinverno [31] in a 1D-DC resistivitysounding inversion, and later implemented in a number of geo-physical probing inverse problems [39,11,1,35].

The major impact of utilizing probabilistic calibration via aBayesian approach is the systematic exploration of all combina-tions of the model parameters through a transparent definition ofthe impact of the participating uncertainty sources. During suchexhaustive parameter exploration, a probability metric is definedto assess the likelihood of selecting sets of parameters that serveto approximate the experimental observations (likelihood); butalso a probability density is defined to reflect the degree ofknowledge on the model parameters (prior) before the modelinversion. The combination of these two states of knowledgeabout the model of interest yields the following benefits: a tran-sition from deterministic to probabilistic model parameters, as-sessment of the type and degree of correlation between the modelparameters (e.g. linear or non-linear), measurement of the impactof the varying experimental observations (e.g. the effect of thenumber of observations on the prediction of confidence levels),assessment of the model performance, and most importantly, thatamong a number of competing models to choose from, selectingthe best model which can describe the process that generated theobservations. Key applications of the probabilistic subsurfaceimaging include integrated site investigation, since the recovery ofgeophysical mechanical parameters allows enhanced geomecha-nical characterization. [33,34]. The varying parameter dimension-ality is formulated through a Bayesian inversion, to populate likelyconfigurations of a heterogeneous elastic medium occupying asemi-infinite domain.

A key defining characteristic of full waveform inversion is thenumerical solution of the equations of motion. The governingforward physics consist of a 1D transient scalar acoustic wavepropagation, where in order to model the semi-infinite extent ofthe physical domain, a perfectly matched layer (PML) is introduced

at the truncation boundary to emulate the infiniteness of the earthstructure [22]. A displacement–stress mixed finite element schemeis used for the numerical solution of the PML-augmented wavePDE.

2. Bayesian approach to inverse problems

An inverse problem is described as the process of estimatingsome characteristics or parameters of a physical system from a setof directly measurable responses of the system (observations). Thevector of model parameters θ, and the vector of observablequantities dobs are mapped through a forward model. The forwardmodel operator G relies on a physical theory to predict the out-come of a possible experiment, or in other words to approximatethe reality: θ≈ Gd ( )obs , or

θ ϵ= +Gd ( ) (1)obs

where ϵ is the tradeoff component which quantifies the deviationbetween model predictions and experimental observations. Thisrandom term contains both theoretical and measurement errors(assuming that the forward model is an unbiased estimate to thetrue physical process). Explicit distinction, however, could bemade between model and observational errors in a full un-certainty quantification framework (UQ) [32].

The basis of the proposed UQ framework is found on the de-finition of a “true process” vector d, which in general representsvalues of observable variables (in this case displacement timehistory response of earth at the surface level). Notice that in atypical geomechanical processes, d is not known. However, if thetrue process is assumed to be random, d can be defined as a vectorof random variables. On the other hand, what the modeler candetermine are the following: (1) a vector of physical observationsdobs, and (2) a vector of model predictions dpred (prescribed at thesame control points in space and time). dpred represents a vector ofpredictions stemming from the forward model G, conditioned on avector of control parameters θ. dpred could deviate from the trueprocess (d) as a result of the model not fully capturing the un-derlying physics, due, for example to the fact that either the gov-erning PDE is an inadequate idealization of the true process, in-itial/boundary conditions are insufficiently modeled, or due to thedeficiency of the computational scheme or lack of resolution of thenumerical solver.

Having denoted physical random deviations between d anddobs (observation error), and between d and dpred (model error) byδdobs and δdpred respectively (δ = −d d dobs obs and δ = −d dpred

θ= − Gd d ( )pred ), the observed data is defined as = +d dobs pred

ϵδ δ+ = +d d dobs pred pred .In general, the error components δdobs and δdpred are not

identifiable, meaning that several different combinations of valuescould be equally consistent with the observed data. However, thisdoes not mean that all the possible values are equally likely [28].For example, error trends that significantly deviate from zero mostlikely imply either a bias in the model or a mis-calibration of thedata acquisition instrument. The Bayesian method provides a basisfor quantifying a priori and a posteriori measures of plausibility ofeach type of error [27]. In this study, the model discrepancy termδdpred vanishes, since the data is synthesized by perturbing themodel output. Therefore, the error component can be defined by asingle uncertainty metric as shown in Eq. (1). Notice that this latterformulation is valid also when the model predictions are unbiasedalong the domain of interest (where d is defined). That is, whenthe probabilistic expectation δ δ− =d d[ ] 0obs pred [32].

In a Bayesian approach to inverse problems, a prior distributionθp ( ) is incorporated in estimating each model unknown, which

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–6858

quantifies the initial uncertainty about the parameter. Ideally, thisdensity limits the space of plausible parameters by giving higherprobability to those which can help us to describe the system'sresponse more accurately. The objective of the inversion is to findthe posterior distribution θ|p d( )obs , built to fully describe themodel parameters in terms of a density function, given the datadobs is observed. The Bayes theorem in this context is defined as

∫θ

θ θθ θ θ

| =||Θ

pp p

p pd

dd

( )( ) ( )( ) ( ) d (2)

obsobs

obs

where the likelihood function θ|p d( )obs assess the probability thatthe observed realization dobs is produced by the model parametersθ. Under the customary assumption that the random error com-ponents ϵ = ϵ … ϵ( , , )n

T1 are such that ϵ σ∼ 0 I( , )n

2 , with In beingan n�n identity matrix (i.e., uncertainty associated with the datais normal with mean zero and standard deviation s and datapoints are independent of each other), the likelihood function isfound with reference to a multivariate normal density:

θ

θ θ

π| =

| |

× − − −−⎡⎣⎢⎢

⎤⎦⎥⎥

p

G G

dC

d C d

( )1

[(2 ) ]

exp12

( ( ) ) ( ( ) )(3)

obsn

d

obsT

d obs

1/2

1

where n is the number of observations and σ=C Id n2 is the cov-

ariance of the error term. The quantity in the denominator of Eq.(2) (the probability of observing the data dobs) is a normalizingconstant, such that the posterior is integrated to one.

3. Forward model

This section introduces the forward model used in the modelinversion. The forward physics describes seismic vertical propa-gation of compressional waves within a horizontally stratifiedsemi-infinite elastic earth, when the media is subjected to a uni-form excitation p t( ) over the surface. This problem can be treatedas a one dimensional problem along the depth direction. In acomputational setting, a major issue associated with this geo-acoustic inverse problem is to model the semi-infinite physicaldomain. In order to arrive at a computationally finite region themedium must be truncated at some depth. If the truncatedboundary is fixed or inadequately modeled, the propagating wavesare (partially) reflected in the domain, and distort the invertedprofile [23].

To address the issue, a perfectly matched layer (PML) approachis adopted, and a PML buffer zone is introduced at the truncationinterface [22]. The PML enforces the rapid decay of the wavemotion within the buffer zone, with ideally no reflection back intothe domain. Fig. 1 illustrates the schematic representation of the

Fig. 1. Schematic presentation of the 1D problem. (a) Original semi-infinite soilmedia. (b) PML truncated domain.

problem. We refer to the original work for the extensive deriva-tions of the model [22], however, for the sake of completeness weonly include the governing wave equation: find ν z t( , ) and ∑ z t( , )

such that

ν ν

ν

∂∂

+ ∂∂

−∂∑

∂=

∈ ∈

∂∑∂

+ ∑ − ∂∂ ∂

=

∈ ∈

z t

tc z g z

z tt

z tz

z L t T

z tt

c z g z z t c zz t

z t

z L t T

( , )( ) ( )

( , ) ( , )0,

for (0, ), (0, ],

( , )( ) ( ) ( , ) ( )

( , )0,

for (0, ), (0, ], (4)

t

t

2

2

22

subject to

ν

νν

=

∑ =

=∂∂

=

∑ =

L t

t p t

z

tz

z

( , ) 0

(0, ) ( )

( , 0) 0

( , 0) 0

( , 0) 0 (5)

t

where ν is the normalized displacement with respect to the soil'sdensity ρ (i.e., ν ρ= u) and ∑ denotes the stress. g z( ) is an at-tenuation function which accounts for the artificial dissipation ofthe wave motion within the buffer zone, and c z( ) is the 1D soilcompressional wave velocity random field which is the inverseproblem parameter. Eq. (4) presents the displacement ν( )–stress∑( ) mixed equations governing wave propagation in a PMLtruncated one dimensional domain. The reader is cautioned notto confuse the standard deviation ∑ appeared in Section 2 withstress denoted by ∑ (i.e. upright Greek letter).

4. Bayesian partition models

As described in the preceding section, our geo-acoustic inverseproblem identifies the spatially dependent coefficient of a PMLaugmented wave equation c z( ), given the probed medium's re-sponse to a known excitation. This describes a functional inverseproblem where the unknown quantity is a function of the spatialcoordinate. Hence, in our Bayesian probabilistic setup, the inverseproblem parameter comprises a real-valued random field c z( ) (ofinfinite dimensionality), which assigns a probability densityfunction to the subsurface property of interest at each point in thespatial domain. In order to arrive at a computationally feasibleproblem, this random field (and the forward model) must be ap-proximated by its discretized version. Hence the velocity field isapproximated with an N-dimensional joint probability density

… |p c z c z d( ( ), , ( ) )N N obs1 1 , with N being the number of grid blocks inthe domain.

One way of treating the problem is to assign a prior to eachrandom variable = …c cc ( , , )N

T1 , and directly apply the Bayesian

formulation to form the posterior density of |c dobs, and implementMCMC methods to explore the resulting very high-dimensionalposterior density. Although MCMC methods converge to the pos-terior by definition as the number of samples grows, in such highdimensional, highly correlated target density configurations, slowchain mixing and serious lack of convergence may arise, whichrender the whole inversion procedure almost computationallyimpractical.

Instead of exploring the value of zc( ) at each of the N gridblocks, we introduce a varying dimensional Bayesian model toparameterize the velocity random field. We opt for a Bayesianpartition model [9,19] representation of the material field, as-suming that c(z) takes the form of a step function (Fig. 2). This

Fig. 2. Partition model presentation of the 1D velocity random field.

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 59

setting is well suited for parameterizing a stratified earth modelwith sharp transition between materials. The Bayesian partitionmodel can be defined by

=c z Zc( ) (6)

=

≤ < ≤ ⋯ >

≤ < ≤ ⋯ >⋮ ⋮ ⋱ ⋮≤ < ≤ ⋯ >

−

−

−

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

I z z I z z z I z z

I z z I z z z I z z

I z z I z z z I z z

Z

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) (7)

k

k

N N Nk

(1)1 1

(1)2

(1)1

(2)1 1

(2)2

(2)1

( )1 1

( )2

( )1

where Z is called the basis matrix, each column of which forms abasis function. This formulation states that the true k-layer soil ismade up of a linear combination of these basis functions and thecorresponding coefficients (c). = …c cc ( , , )k

T1 hold the value of

partition weights (i.e. wave velocity at each layer), and I (. ) is theindicator function which assumes the value one, if its argument istrue, and zero otherwise. The vector … −z z( , , )k1 1 denotes the −k 1change point locations (position of the layer interfaces), where k isunknown number of partitions (layers). …z z, , N(1) ( ) are the co-ordinates of N prespecified grid points, which conveniently coin-cide with the forward model discretization mesh. Fig. 2 shows thepartition model presentation of a 1D velocity random field. Bypartitioning the velocity field the reduced dimension parametervector will be = =

−k c z( , { } , { } )i ik

i ik

1 11 .

5. Bayesian model selection

The proposed Bayesian partition model is categorized within aspecial class of models namely variable dimension models. Avariable dimension model is defined as a model whose number ofunknowns is an unknown itself [9]. A Bayesian variable dimensionmodel is composed of a set of plausible models ={ }k k

K1, each

reflecting a hypothesis about the data. Having K such competingmodels, it is desired to find the model that best describes theobservations. Here, k represents a k-layer subsurface structure.

This definition by nature pertains to the spatial case of a Bayesianmodel selection problems, where the competing models belong tothe same family, with differing number of parameters, namelynested models [38].

A variable dimension model can be formulated as an extensionto the standard Bayesian inference (see Section 2), where a priordistribution is assigned on the model indicator k, which impliesextending the prior modeling from parameters to models.Θ Θ= ⋃ ×∈ k{ }k k is the parameter space associated with the setof models k, where denotes the space of model indicators.Having defined priors πk on the indicator parameter k (beingconsidered now part of the parameters), and the parameter sub-space Θk, by virtue of Bayes's theorem

∫θ

θ θθ θ θ

ππ

| =| |

∑ | |Θ∈p

p p

p pd

dd

( , )( , ) ( )

( , ) ( ) d (8)k k obs

obs k k k k k

k k obs k k k k kk

where θk is a set of parameters specific to model k, andθ|p d( , )k kobs and θ |p ( )k k are the likelihood function and the

prior density of the model specific parameters θk, respectively,given k is the true model. Bayes factors in the Bayesian modelselection context offer a thorough criterion to pairwise compar-ison of members in { }k . The relative plausibility of model i ver-sus model j, having observed dobs, is determined by the Bayesfactor given by

=||

BFpp

dd

[ : ]( )( ) (9)

i jobs i

obs j

where |p d( )obs i is the marginal likelihood of data under modeli, which is the normalizing constant of the posterior density (see

the denominator in Eq. (8)), and is given by

∫ θ θ θ| = | |Θ

p p pd d( ) ( , ) ( ) d(10)obs i obs i i i i i

i

The above quantity (equation (10)) is the basis for the Bayesianmethod's natural penalty against complex models, also known asOccam's razor [8]. The Bayesian embodiment of Occam's razor isbriefly explained in Appendix A. Non-Bayesian model selectionprocedures [2,42,25] rely on comparing (penalized) maximumlikelihoods of the competing models [46]. Such criterion breaksdown when a set of nested models is to be compared, and over-fitting becomes a serious problem. Since a more flexible model isable to describe the data better, it gives rise to a higher likelihoodmeasure, and therefore is favored in a likelihood-based test, whileit performs poorly in terms of prediction. The Bayesian method'sbuilt-in penalty against overly complex models provides a robusttool to treat varying dimensional problems.

6. Prior elicitation

The first step in a Bayesian inference analysis setting is tospecify prior densities to the model parameters θ (given the modelrepresentation ( ) is chosen). The prior distribution θp ( ) is basi-cally a tool to summarize the initially available information on theprocess, and to quantify the uncertainty associated with this in-formation. Selecting standard vague or non-informative priors isfavored in this case in order to base the inference only on theexperimental observations.

A number of techniques are currently available for constructingsuch standard priors [26]. The use of non-informative priors,however, is rather delicate for varying dimensional model settings,since the majority of non-informative priors are improper. Im-proper priors are only defined up to a proportionality constant(i.e., not integrable) [26]. In general, improper priors cannot beassigned to model specific parameters in Bayesian model selection

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–6860

problems, as the choice of the arbitrary normalizing constant willinfluence the Bayes factor (Eq. (9)). Notice that the Bayes factor is amultiple of this normalizing constant (Eqs. (9) and (10)). Propervague priors (proper priors with large dispersion parameter) alsodo not address the difficulty, for they give rise to the so-calledJeffreys–Lindley paradox [29,24]. The Jeffreys–Lindley paradox is aproblem related to the stability of the Bayes factor, which causesthe simplest model (which might be a very poor reflection of thedata) to always be favored by the Bayes factor.

We address these concerns in our choice of priors. In particular,we propose the use of a hierarchical Bayes approach to model thelack of information on the parameters of the prior distribution, byintroducing a second level of prior distributions on these para-meters. Hence we refrain from using improper priors, yet avoidingany subjective input to the inference by introducing ungroundinformative priors. The posterior kernel (of varying dimension)according to the Bayes rule is

θ θ θ| ∝ |p k p k p kd d( , ) ( , ) ( , ) (11)k obs obs k k

where θk is the parameter vector associated to the k soil layermodel. dobs denotes the ×n 1 vector of normalized displacementresponse, recorded at the soil surface. Introducing the second layerof hierarchy will lead to

θ θ θ

θ σ σ λ λ

| ∝ | |

∝ | | | |

p k p k p k p k

p k p v k p k p p v p k p

d d

d c z

( , ) ( , ) ( ) ( )

( , ) ( , , ) ( ) ( ) ( ) ( ) ( ) (12)

k obs obs k k

obs k2 2

For the ease of notation, we define vectors θP and θH whichcontain the model specific parameters and the global hyperpara-meters, respectively. The global parameters are unknowns whichbear on parameters common to all models. Thusθ = × − ×c z( , )P k

Tk

T T1 ( 1) 1 , θ σ λ= v( , , )H

T2 , θ θ θ= ( , )k PT

HT T , and θ θ= k( , )k

T T .Superscript T denotes transposition. The definitions of the priorsare

σ σ| ∼ ×v k vc c Ilog , , ( , ) (13a)k k2

02

σ α δ∼ ( , ) (13b)20 0

ζ ηη

Γ ζ∼ =

ζζ η− −v a v e( , )

( ) (13c)v

0 00

0

( 1) ( )0

0 0

| ∝−

−⎛⎝⎜

⎞⎠⎟p k T

kz( )

1 (13d)

1

λλ

λ| ∼∑ ! − !

= …=

−k

i kk K

1

( / ) ( 1), 1, ,

(13e)iK i

k

1

1

λ ι κ∼ a ( , ) (13f)0 0

We, a priori, assume that the velocity field within each layer ispopulated from a log-normal distribution. Hence, the log-velocityfield has a multi-variate Gaussian prior density (Eq. (13a)). Thisassumption ensures that velocity is a positive-valued random filed.We further suppose that …c c, , k1 are a priori independent. Thecorrelation structure of the layered elastic properties will be re-constructed a posteriori (if there exists any). c0 is set to 200 m/s,meaning that before the inversion, the media is assumed to behomogeneous.

The hyper parameters s2 and v are the noise variance and theprecision parameter respectively. We opt for a broadly non-in-formative priors for these parameters (α δ= = 0.010 0 andζ η= = 0.010 0 ), (Eqs. (13b) and (13c)). Since the inference is highlysensitive to the choice of v, it is important to avoid fixing thisparameter [8]. By considering it as a random variable we elevate

the robustness of the method against poor choices of v. There is norestriction in using improper priors for the global parameters(which are common among all the models) [3], since in Bayesfactor calculation (Eq. (9)) the arbitrary constant of proportionalitycancels out. Notice that the Bayes factor is a multiple of the priornormalizing constant. Hence, the problem with the arbitraryproportionality constant, which brings about Lindley's paradox, isremoved.

z is the position vector of the −k 1 layer interfaces. |p kz( ) (Eq.(13d)) reflects the prior assumptions about the position of thematerial interfaces. We define an underlying grid of T points(which coincides with the finite element discretization of thephysical domain). This prior suggests that given a k layer model isthe true process, and there are T candidate nodes to locate −k 1interfaces, any combination of … −z z( , , )k1 1 is equally likely. Weassign a hierarchical truncated Poisson prior on λ|k , with K beingthe maximum number of layers in partitioning (Eq. (13e)). Thissetting controls the prior weights given to over-parameterizedmodels. λ is a hyperparameter to be elicited from the data. Anatural choice of prior on this parameter is a flat Gamma dis-tribution ι κ= =( 0.010 0 ). The defined prior θp k( , )k does not placean explicit penalty on the model complexity. However, as statedearlier, the marginal likelihood contains a built-in penalty on themodel dimension, which strongly depends on the prior variance v

of the coefficients c [8].

7. Posterior computation

A customary burden of using Bayes factors (Eq. (9)) is thecomputation of, oftentimes, high dimensional marginal likelihoodintegrals (Eq. (10)). To circumvent this difficulty, one may resort toalternative solutions such as Monte Carlo simulation basedmethods (e.g., pseudo-priors [4,5]), or asymptotic approximationto Bayes factors (e.g., Schwartz's criteria also known as BIC) [42].The latter is widely used in a variety of applications includinggeophysical modeling (e.g., see [13,10,45]) due to the ease of itsimplementation. BIC provides a first-order approximation to thelogarithm of the Bayes factor as the sample size grows. The ap-plicability of the approximation, however, is restricted to modelswith regular likelihoods, and i.i.d. data structures. Also the methodcalls for the derivation of maximum likelihood estimates for theparameters of all models, which is an unfavorable fact when K islarge. Monte Carlo methods provide essentially the only accuratemean of inferring the posterior which does not depend on theknowledge of the proportionality. Markov Chain Monte Carlo(MCMC) is an iterative stochastic method designed to generaterandom samples from the posterior kernel. A generalization ofMCMC, the so-called Reversible Jump MCMC (RJMCMC), was in-troduced by Green [18] to generate random samples which aredistributed according to a varying dimensional posterior. The keyaspect of the reversible jump algorithm is the introduction ofsome auxiliary random variables to equalize the dimensionality ofthe parameter space across models in order for the Markov chainto be able to move among different dimensions. RJMCMC has re-cently become increasingly popular in geophysical inversion as arobust tool for subsurface modeling. A detailed introduction togeophysical transdimensional Bayesian inversion can be found inSambridge et al. [40].

7.1. Reversible jump MCMC implementation as birth–death process

In this section we extract the details involved in the RJMCMCalgorithm designed for our specific inversion setup. Suppose wewant to draw random samples from varying dimensional target

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 61

distribution θ |p k d( , )k obs (Eq. (11)), where the sequence of randomsamples constructs a Markov chain.

In order to traverse the varying dimensional posterior surface,we iteratively perform four types of transitions: Birth (B), Death(D), Move (M), and Perturb (P). Different search strategies havebeen designed depending on the application (e.g., see the originalwork by Green [18], and Denison et al. [6]). As long as the algo-rithm satisfies the balance condition, and the acceptance ratioremains computationally efficient, we assume that our approachoffers a flexible design.

Let us suppose that at the s( ) th step the chain is at θ θk , ,sP

sH

s( ) ( ) ( )

(denoting number of layers, model specific parameters= … = … −c c z zc z( , , ) , ( , , )s s

ks T s s

ks T( )

1( ) ( ) ( )

1( )

1( ) , and hyper-parameters

σ λv, ,s s s2( ) ( ) ( ) respectively). The possible transitions are the follow-ing: (B) add an intersection at a random location with probabilityp

kB( )s( ). (D) Delete a randomly chosen intersection with probability

pk

D( )s( ) . (M) Swap a randomly chosen intersection for a randomly

chosen available node in with probability pk

M( )s( ) , where is the

set of candidate node locations, and T is the size of the set(| | = T ). (P) Perturb velocity of a randomly chosen layer withprobability p

kP( )s( ). Where + + + = ∀p p p p k1,

kB

kD

kM

kP s( ) ( ) ( ) ( ) ( )

s s s s( ) ( ) ( ) ( ) . Notice

that (B) and (D) involve dimension changes in θPs( ) , while (M), and

(P), proposes moves within the current dimension, hence thelatter proceeds similar to regular Metropolis–Hastings algorithm[37]. Below is the definition of each transition:

�

Fig.solut

Birth: * = +k k 1s( ) .With probability = *p q

kB

k k( )

,s s( ) ( ) , a Birth move is proposed, and alayer interface i is added at an available random grid location(Fig. 3a). This random location is proposed from the discreteprobability *|q kz z( , )z

s s( ) ( ) (Eq. (16b)). The birth increases the di-mensionality of the parameter space by two, the difference beingaccounted for one discrete variable, the new interface position

*zi , and one continuous variable, the new velocity *ci (i.e.,Θ Θ− =*dim( ) dim( ) 2k k s( ) , where Θk s( ) denotes the space of

current state and Θ *k is the candidate parameter space). In orderfor the Markov chain to jump between subspaces Θk s( ) and Θ *k ,θ s( ) is augmented with a two dimensional auxiliary variable udrawn from a proposal distribution ψ u( ). The new state ofthe chain θ* is found from the transformation such thatθ θ* = * u( , )k k

s,

( )s( ) . *k k,s( ) is a deterministic mapping, the so-

called dimension matching transformation. The proposed layer

3. The RJMCMC transitions: Birth, Death, Move and Perturb. By random combination ofion.

velocities *c from the transformation * uc( , )k ks

,( )

s( ) is given by

ς

ς

=

=⋮=

= −

= +⋮=

*

*

*

−*−

*

+*

*

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

u

c c

c c

c c u

c c u

c c

c( , )

(14)

k ks

s

i is

i is

c

i is

c

k ks

,( )

1 1( )

1 1( )

( )

1( )

( )

s

s

( )

( )

This implies that the velocity of the chosen layer is perturbedfrom a Gaussian proposal to attain the velocity of the twoemerged layers. ςc is a variance measure, defining size of thesearch step. Notice that the hyperparameters remain un-changed in the Birth (also in Death) move. The candidate stateis accepted with probability

*

* * * * * * *

* * *

*

θ θ

θ

θ

θ θ

θ θ

ψ

ψ=

|

|

′| |

| |

×∂

∂ ∂

*

* *

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

r

p k

p k

p k

p k

q u q k

q u q k

u

u

d

d

c z z

c z z

c

c

( , )

min 1,( , )

( , )

( , , )

( , , )

( ) ( , )

( ) ( , )

( , )

(15)

k s k Ps

P

P

Ps s

obs P Hs

obs Ps

Hs s

k k s k k s z s

k s k k s ks z s s

k s ks

s

( ),( )

( ) ( )

prior ratio

( )

( ) ( ) ( )

likelihood ratio

, ( ) , ( )( )

( ), ( ),( ) ( ) ( )

proposal ratio

( ),( )

( )

Jacobian

where the prior ratio is

θθ

σ λ

σ λ

*=

*| * *| * *|

| | |

*p k

p k

p v k p k p k

p v k p k p k

c z

c z

( , )

( , )( , , ) ( ) ( )

( , , ) ( ) ( )

P

Ps s

s s s

s s s s s s s s( ) ( )

2( ) ( ) ( )

( ) 2( ) ( ) ( ) ( ) ( ) ( ) ( )

and

| * * ∝* −

q kk

z z( , )1

1 (16a)zs( )

*| ∝− −

q kT k

z z( , )1

( 1) (16b)z

s ss

( ) ( )( )

these four types of transitions, MCMC searches the parameter space for the

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–6862

ψ ′| * ∝* u c( ) 1 (16c)k k, s( )

ψς

| ∼ |*⎛⎝⎜

⎞⎠⎟u uc( )

10, 1

(16d)k ks

c,

( )s( )

= =* ⎪

⎪⎧⎨⎩

qk1/2, 1

1/4 otherwise (16e)k k

s

,

( )

s( )

= =* ⎪

⎪⎧⎨⎩

qk K1/3,

1/4 otherwise (16f)k k

s

,

( )

s( )

ς∂

∂ ∂=* u

u

c

c

( , )2

(16g)

k ks

sc

,( )

( )

s( )

θ θ|p kd( , , )obs P H is the likelihood function, which is constructedaccording to Eq. (3).Notice that the parameter space for the interface positions isdiscrete (whereas it is continuous for the velocity space), and adiscrete uniform random variable was drawn to propose theposition of the new interface (16b). Denison et al. [9] showthat the Jacobian term is always unity for discrete transforma-tions. Therefore only where continuous model spaces changein dimensions, determining the Jacobian is required, and thedimension matching transformation maps the continuous

subspace: ⇌ * *= =

−

⎛⎝⎜

⎞⎠⎟c u c({ } , ) { }s

ik

ik( )

1 1s( )

1.

Death: * = −k k 1s( ) .

�

With probability = *p qk

Dk k

( ),s s( ) ( ) , a Death move is proposed. A

current interface i is randomly chosen from the probability*|q kz z( , )z

s s( ) ( ) and removed (Fig. 3b). The proposed velocityprofile *c is determined from the deterministic Death trans-formation (which is the reverse Birth transformation)

=

=⋮=

= +

=⋮=

*

*

*

−*−

*+

+*+

*

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎛⎝⎜

⎞⎠⎟u

c c

c c

c c c

c c

c c

c( , )12

(17)

k ks

s

i is

i is

is

i is

k ks

,( )

1 1( )

1 1( )

( )1

( )

1 2( )

( )

s

s

( )

( )

The acceptance probability is the same as Eq. (15), with thefollowing modifications:

| * * ∝− −

q kT k

z z( , )1

( 1) (18a)z

ss

( )( )

*| ∝* −

q kk

z z( , )1

1 (18b)zs s( ) ( )

ψς

′| * ∼ |* u uc( )1

(0, 1 )(18c)k k

c, s( )

ψ | ∝* u c( ) 1 (18d)k ks

,( )

s( )

= =* ⎪

⎪⎧⎨⎩

qk1/3, 1

1/4 otherwise (18e)k k

s

,

( )

s( )

= =* ⎪

⎪⎧⎨⎩

qk K1/2,

1/4 otherwise (18f)k k

s

,

( )

s( )

ς

∂

∂ ∂=* u

u

c

c

( , ) 12 (18g)

k ks

s c

,( )

( )

s( )

Move: * =k k s( ).

� With probability = *p q

kM

k k( )

,s s( ) ( ) , a “Move” move is proposed. Alayer interface is randomly chosen from a uniform probability,and moved to an available knot location (Fig. 3c). A new set ofhyper parameters θ *

H is drawn from the probability θ θ|*q ( )H Hs( ) .

Log-normal proposals are used to update all the hyperpara-meters.In a Move step, as the number of material layers is fixed, thealgorithm reduces to the regular Metropolis–Hastings MCMCwith the acceptance probability of the following form: (Noticethat the hyperparameters of the model are also updated inMove and Perturb.)

θ θ

θθ

θ θθ θ

θ θθ θ

*

=* |

||

|

*

* * *

*

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

r

p

p

p k

p k

q

q

d

d

( , )

min 1,( )( )

( , , )

( , , )

( )

( )

(19)

k ks

s

obs P Hs

obs Ps

Hs s

Hs

H

H Hs

,( )

( )

prior ratio

( )

( ) ( ) ( )

likelihood ratio

( )

( )

proposal ratio

s( )

Perturb: * =k k s( ).

� With probability = *p q

kP

k k( )

,s s( ) ( ) , a Perturb move is proposed. Alayer is randomly picked from a uniform density, and its ma-terial property is perturbed with a Gaussian proposal (Fig. 3d).It is also attempted to update the model hyperparametersfrom log-normal proposal densities (similar to the M move).The probability of accepting the candidate state is found fromEq. (19). Notice that the uniform and Gaussian proposals toupdate θP do not appear in this ratio (also in the M step), forreasons of symmetry.

8. Application to a synthetic case

The inversion scheme outlined in the preceding sections isapplied to a synthetic data set to deduce the subsurface elasticproperties of a soil model. We consider the horizontally stratifiedsemi-infinite soil medium depicted in Fig. 4. The medium ismodeled as a one-dimensional PML-truncated domain, with theregular domain extending to z¼100 m, and the PML buffer zonethickness being 10 m. Fig. 4 illustrates the target wave velocityprofile, which reflects sharp transitions between different mate-rials in depth. The medium is probed with a Gaussian pulse-typeexcitation p t( ) applied at the soil surface (Fig. 5a). The maximumfrequency of the excitation is 40 Hz and the peak amplitude is10 kPa. Fig. 5b depicts the frequency spectrum of the excitation.

Fig. 6a shows the displacement time history response of thesoil model depicted in Fig. 4 to the prescribed load. This is ob-tained by solving the forward problem (4) and (5) using a mixedfinite element method. 220 elements of length 0.5 m are used inthe analysis. The readings are recorded every 0.001 s for a totalduration of 2 s. Displacement response, as a measurable char-acteristic of the wave field, will serve as the input to our inversionscheme. We generate the synthetic data by perturbing the

200 300 400 500

0

20

40

60

80

100

c (m/s2)

z(m

)

Fig. 4. Benchmark soil velocity profile.

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 63

displacement response of the soil model computed at the groundlevel v t(0, ) with 20% Gaussian noise. Fig. 6b illustrates this dataset. The attenuation effect is disregarded in this study, and the soildensity is assumed to be known a priori (ρ = 2000 kg/m3).

9. Results

In this section, we illustrate the applicability of the Bayesianvarying dimensional inversion, and model determination using themethodology introduced in the preceding sections. The inversionis allowed for a maximum of 40 soil layers (up to the truncationinterface), which indicates maximum number of 83 model un-knowns. This maximum resolution is attributed to the frequencyof the exerted load (maximum frequency 40 Hz). The simplestearth model is k¼1, which corresponds to the state of a homo-geneous medium. No additional assumption is made concerningthe regularization of the deduced velocity profile.

We started the inversion with homogeneous initial guess( = =k c1, 200 m/s). The RJMCMC sampler was run, and a total of100K iterations were stored as the generated samples. The first20K samples were discarded as burn-in iterations. Every fifthvisited sample was kept in the chain as high dependency is ex-pected, especially between successive values of k, since the

0 0.5 1 1.5 20

2

4

6

8

10

12

p(t)

(kP

a)

t (s)

Fig. 5. (a) Time history of the applied stress p t( ). (b

difference between the current and the proposed k values could beat most one. Fig. 7 illustrates the first 300 RJMCMC sampling se-quence for the model index (number of layers), starting from k¼1.This figure shows that k increases rapidly up to k¼10 and in about200 iterations, then it settles down to the five layer target model.This figure also implies that even though our sampling strategydoes not force the model to undergo dimension changes at everyiteration (we are pointing to M and P moves) the waiting time at asingle model is not long. Hence the sampler promptly explores thespace of plausible models until it converges to the target modelk¼5. The rest of the simulation effort is committed to arriving atthe stationary condition in sampling the soil parameters of the fewfavored models. This observation confirms the efficiency of thealgorithm design and of the proposal density formulations.

Fig. 8a depicts the full sampling history for the same parameter,to further emphasize the stability of the RJMCMC chain. Themarginal posterior probability mass function of k is shown inFig. 8b, which quantifies the level of certainty in accepting eachhypothesis. According to this figure, 6 layer profile is also a likelymodel to describe the observations with much less probability.The figure manifests the Bayesian inversion capability to deducethe true nature of the underlying process without imposing anyregularization constraint to penalize overly complex models.

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Four

ier

tran

sfor

m o

f th

e G

auss

ian

puls

e (k

Pa)

f (Hz)

) Frequency spectrum of the applied stress p t( ).

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t(s)

v (k

g/m

2 )

0 0.5 1 1.5 20.5

0

0.5

1

1.5

2

2.5

t(s)

v (k

g/m

2 )

Fig. 6. (a) Computed displacement response at the surface (b) Synthetic data: computed displacement response at z¼0 perturbed with 20% Gaussian noise.

0 50 100 150 200 250 3001

2

3

4

5

6

7

8

9

10

11

Iteration

k

Fig. 7. Number of basis functions (layer interfaces) variation in first 300 RJ-MCMCiterations.

0 2 4 6 8 10

x 104

1

2

3

4

5

6

7

8

9

10

Iteration

k

Fig. 8. Number of layers k( ) in the partition model. (a) Po

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–6864

Figs. 9 and 10 illustrate the marginal posterior densities of themodel specific parameters, given the true model k¼5. Fig. 9 showsthe posterior estimates for the layer thicknesses | =p kz d( , 5)obs ,and their associated uncertainties. The target values are also su-perimposed on each histogram (dashed lines as reference). Thefigure indicates the ability of the inversion scheme to deduce thetarget parameters (notice that the deviation of the posterior meanfrom the target values are about one to two element dimensionsand that the scales of the figures are not the same). The thicknessof the fifth layer is not included here, as it is considered to besemi-infinite.

Theoretically, the PML is assumed to be located at a depthbeyond which homogeneity is ascertained. Fig. 10 shows the in-verted acoustic soil velocities of the true model | =p kc d( , 5)obs ,together with the target values (notice the variation of scales onthe figures as well).

Fig. 11 shows inference for model hyperparameters. Althoughthese parameters might not be incorporated directly in modelpredictions, they are highly influential in attaining reliable para-meter estimates. In Fig. 11a the standard deviation of the ob-servational error term s2 is displayed, which is relatively centeredaround the target added Gaussian noise (signal to noise ratio,SNR¼5).

1 2 3 4 5 6 7 8 9 10k

sterior samples of k. (b) Posterior mass function of k.

Fig. 9. Marginal posterior density of the layer thicknesses given k¼5 and the corresponding target values (dashed line).

Fig. 10. Marginal posterior density of the layer velocities given k¼5 and the cor-responding target values (dashed line).

Fig. 11. Marginal posterior density for model hyper-parameters. (a) Noise standarddeviation (s). (b) Precision parameter (v). (c) Rate parameter in poisson prior (λ).

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 65

Fig. 11b depicts the dispersion parameter v. This parameter is ofcrucial significance in our model determination framework, sincefixing v to small values (choice of relatively sharp priors on c)limits the flexibility of each basis function coefficient, thereforemany partitions (layers) are required to adequately model thetarget process ( |k d( )obs grows). The definition of the basis func-tions in a Bayesian partition model is given in Eq. (7). By contrast,large v (relatively diffuse prior on c) results in a more flexibleregression function posterior mean c z( ) (see Eq. (6)), which can

0

10

20

30

40

50

60

70

80

90

100100 200 300 400 500 600 700 800

z(m)

c(z)(m/s)

0

10

20

30

40

50

60

70

80

90

100100 200 300 400 500 600 700 800

z(m)

c(z)(m/s)

Fig. 12. Prior and posterior model predictions. (a) Superimposed e5 2 likely prior soil models. (b) Superimposed e5 3 likely posterior soil models.

Fig. 13. Posterior mean estimate together with 95% credible intervals for the meanposterior and the mean prior.

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–6866

accommodate wilder oscillations in its behavior. Hence, fewerbasis functions are needed to reflect the true underlying process( |k d( )obs becomes increasingly small), as each basis function hasmore degrees of freedom. Notice that here we did not choose toset up a fixed value for v, rather this parameter is considered as arandom variable (Eq. (13c)), and its value is deduced from the datasuch that the marginal likelihood is maximized. The marginalposterior density of the rate parameter λ in the Poisson prior (Eq.(13e)) is provided in Fig. 11c. We can see that the Bayesian pointestimate for λ is closely approximated by λ≊5. This parameter isthe mean of the Poisson prior equation (13e), which reflects thenumbers of layers k accommodated in the model c z( ). Fig. 12 de-monstrates the essence of Bayesian updating and uncertainty re-duction as a result of introducing the experimental observations.Fig. 12a presents 5�102 superimposed likely prior soil models (Eq.(6)), with the coefficients of each curve drawn directly from thedefinition of the priors (Eq. (13)). These curves show the state ofminimum knowledge about the subsurface structure. No stratifi-cation and velocity measure is discernible at this initial state.Fig. 12b depicts 5�103 posterior soil model realizations, whichmimic accurately the general trend of the target process.

Fig. 13 quantifies the observations of the previous figure. Theposterior mean soil profile c z( ) (black solid curve) is illustratedtogether with 95% credible intervals for the posterior predictions(dark shaded area). The prior credible region is also included in thefigure (light shaded area), which occupies the entire space (andextends to infinity). The mean posterior prediction of the dis-placement time history response of the media =v z t( 0, ) is pic-tured in Fig. 14. The figure also provides a measure of uncertaintyaround the posterior mean estimate v t(0, ). This plot accentuatesthe high fidelity of the posterior estimates to the experimentalobservations.

Fig. 14. Posterior displacement prediction together with 95% credible intervalsaround the mean.

10. Conclusions

This paper introduces a probabilistic calibration approach via aBayesian formulation for the solution of inverse problems, definedby the random field characterization of heterogeneous media, foran acoustic one-dimensional velocity field with horizontallylayered structure. A self-regularized varying structure model isformulated based on the notion of Bayesian partition models inorder to parameterize the acoustic wave velocity random field. Themethod offers a reduced dimensional inversion technique by di-viding the velocity random field into an unknown number of soil

Fig. 15. A Schematic presentation of Bayesian Occam's razor.

S. Esmailzadeh et al. / Probabilistic Engineering Mechanics 39 (2015) 56–68 67

layers within a certain depth interval. Number of layers, their ve-locities and thicknesses are inferred, conditioned on the observa-tions. The reward of the approach is that the explicit regularizationof the inverted profile by global damping procedures or eventhrough imposition of priors, which carry smoothness constraints(and might introduce subjectivity to the inference process), is notrequired. The reversible jump MCMC algorithm was implementedto carry out the simulation of the resulting varying dimensionalposterior density. The provided synthetic case indicates significantfunctionality of the inversion scheme to retrieve the benchmarksubsurface profile.

Acknowledgments

The authors would like to thank Simula School of Research andInnovation (Computational Geoscience) and Statoil for sponsoringthe development of this work. Also, the constructive comments ofour reviewers are gratefully acknowledged.

Appendix A. Bayesian Occam's Razor

A simple explanation of why the Bayesian model selectionadheres to the concept of parsimony is presented in Fig. 15. Thisfigure illustrates the Bayesian embodiment to the latter concept[21,30,8]. The horizontal axis presents the data space, and thevertical axis shows the measure of the marginal likelihood. 1

and 2 are two competing explanations of a same process. Model1 is the simpler theory, and 2 is the more complex one. The

simple model is only capable of reaching a limited subdomain inthe data space 1, whereas the more complex model is able toembrace a wider space due to its flexibility. As both |p d( )1 and

|p d( )2 are integrated to one over the data space, if the observeddata lies in 1 which is accessible by both the models, then 1 isfavored over 2, as it assumes higher probability in this region.

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