a bayesian deconvolution approach

8/10/2019 A Bayesian Deconvolution Approach

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 12, DECEMBER 2010 4151

A Bayesian Deconvolution Approachfor Receiver Function AnalysisSinan Yldrm, A. Taylan Cemgil, Member, IEEE , Mustafa Aktar,

Yaman zakn, and Aysn Ertzn, Member, IEEE

Abstract In this paper, we propose a Bayesian methodology forreceiver function analysis, a key tool in determining thedeepstruc-ture of the Earths crust. We exploit the assumption of sparsity forreceiver functions to develop a Bayesian deconvolution method asan alternative to thewidely used iterativedeconvolution. We modelsamples of a sparse signal as i.i.d. Student-t random variables.Gibbs sampling and variational Bayes techniques are investigatedfor our specic posterior inference problem. We used those tech-niques within the expectation-maximization (EM) algorithm toestimate our unknown model parameters. The superiority of theBayesian deconvolution is demonstrated by the experiments onboth simulated and real earthquake data.

Index Terms Bayesian inference, deconvolution, expectation-maximization (EM), Gibbs sampling, inverse-gamma, MonteCarlo methods, receiver function, sparsity, variational Bayes.

I. INTRODUCTION

A. Receiver Function Analysis

RECEIVER function analysis is an efcient and widelyused method for determining the deep structure of theEarths crust and mantle down to depths of 100 km and evenmore. The approach is based on the conversion of the elasticwaves at structural boundaries, namely P-to-S conversion. Theincoming P-wave from a distant earthquake, considered as thesource signal, generates a converted S-wave whenever it passesthrough a structural boundary. In other words, the S-wave isdened as the signal obtained by the convolution of the sourcesignal with an impulse function, namely the receiver function ,that characterizes the earth structure. Both P- and S-wavescontinue propagating to the surface with different velocitiesand are subsequently recorded at two different components of

Manuscript received August 6, 2009; revised February 26, 2010. Date of publication July 8, 2010; date of current version November 24, 2010. This workwas supported by The Scientic and Technological Research Council of Turkey(TUBITAK) under Grant number 107E050. Part of this work was done whenthe author was at the Signal Processing and Telecommunications Laboratory,University of Cambridge, U.K. During his stay, he was supported in part byProf. W. Fitzgerald.

S. Yldrm is with the Statistical Laboratory, Department of Pure Math-ematics and Mathematical Statistics, University of Cambridge, CB3 0WBCambridge, U.K. (e-mail: [email protected]).

A. T. Cemgil is with the Department of Computer Engineering, BogaziiUniversity, 34342 Istanbul, Turkey (e-mail: [email protected]).

M. Aktar and Y. zakn are with the Department of Geophysics, KandilliObservatory and Earthquake Research Institute, Bogazii University, 34342Istanbul, Turkey (e-mail: [email protected]; [email protected]).

A. Ertzn are with the Department of Electrical and Electronic Engineering,Bogazii University, 34342 Istanbul, Turkey (e-mail: [email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TGRS.2010.2050327

a three-component seismic station. An illustration of a seis-mological process and the resulting receiver function is givenin Fig. 1. The task is then to estimate the receiver functionby deconvolving the composite S-wave from the source signalP-wave which, in practice, reduces down to deconvolving theradial component of the seismogram from the vertical.

The deconvolution approach has been widely used sinceit was proposed 40 years ago [2][5]. The performance of deconvolution heavily depends upon both the quality of therecorded data and the choice of the deconvolution algorithm.Deep and large earthquakes provide signal with sufcient en-ergy and uncontaminated from surface interactions, and arebest suited for such applications. However, they are rare andtherefore require long observation campaigns, leading to atradeoff between the data quality and operational costs. A majoreffort was given to improve the deconvolution procedure, whichinitially used frequency domain methods [6], [7] and later time-domain approaches [6], [8][11]. More effective deconvolutionprocedures provided ways of getting most out of lesser qualitydata sets [12], [13] and reduced the operational costs of longeld campaigns.

Ligorra and Ammon [12] described and applied a time-domain iterative deconvolution approach to receiver-functionestimation and illustrated the reliability and advantages of thetechnique using synthetic simulations and experiments withreal data. The approach is described in [14] as well. zaknand Aktar [15] applied iterative deconvolution for extractingthe crustal structure of Southwestern Anatolia, and reportedthat the iterative deconvolution outperformed the frequency-domain methods like the inverse ltering [4] and water-leveldeconvolution methods [5].

B. Sparsity and Bayesian Deconvolution

Since the convolution model in receiver function analysisarises from the refractions from the P-wave to the S-wave,the receiver function can be assumed as sparse, i.e., there ismostly no activity and the signal level is mostly very closeto zero. Occasional spikes are the indicators of P-wave toS-wave refractions. However, the sparsity assumption is notused in the deconvolution methods stated above. Only in it-erative deconvolution, the sparsity is introduced indirectly bylimiting the number of iterations in an adhoc manner; thecomputational process is not even guaranteed to converge.Moreover, the signals recorded by a seismogram are noisy,which reduces the performance of the deconvolution algorithmsabove as the noise is not taken into account. Finally, iterativedeconvolution needs the complete convolution data to produce

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4152 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 12, DECEMBER 20

Fig. 1. Illustration of the seismological process and the resulting receiverfunction. (b) is taken from [1]. (a) Locations of the source signal and thereceiver. (b) Propagation of P- and S-waves with different velocities and thecorresponding receiver function.

reliable results. However, when this is not the case in seismicrecordings, where the vertical component is as long as the radialcomponent, we have a truncated observation, i.e., missing data.

Bayesian theory allows us to formulate our prior knowledgeabout the data which is ignored by the classical approaches[16]. Since the sparsity is an important information about thecharacteristics of a signal, using Bayesian methods for receiverfunction estimation can improve performance. Moreover, sincethe Bayesian methods are model-based, they can handle ob-servation noise and missing data problems. Sparsity has beenexploited in a Bayesian manner in many application areas.An example, very similar to our application area, is seismicdeconvolution, see [17][23]. Use of sparsity for seismic de-convolution has been justied as well by Larue et al. [24],where they have shown that the sparsity assumption is morerelevant than whiteness assumption. Among other applicationswhere sparsity is exploited in a Bayesian manner are sourceseparation, see [25][28], and image deconvolution, see [29].

The most crucial point of applying Bayesian methods is thechoice of the form of the prior distribution, that is, the statisticalmodeling of sparsity. In literature, one approach is to model thecoefcients of a sparse process by a BernoulliGaussian distri-bution, which is popular particularly in seismic deconvolutionproblems [17][20], [30]. An early example can be seen in apaper of Mendel et al. [17], where the authors used this prior fordeconvolution of reectivity sequences. For possible Bayesianmethods developed for this model for the blind deconvolutionproblem, one can see [19], [20], and [30]. A modication of thisprior-to-model dependencies between successive reectivitysequences is used in [21] and [22], where a BernoulliMarkovrandom eld is introduced for the location of layer disconti-nuities. The BernoulliGaussian model is popular in literaturebecause it is very intuitive and its statistical properties alloweasy implementation of Bayesian methods such as Gibbs sampling [31] and the EM [32]. However, this model suffers frommultimodality and needs very good initializations.

Modeling samples of a sparse signal with a Gaussian random

variable whose variance is an inverse-gamma random variablecould be an alternative approach for the deconvolution of sparsesignals. This prior has already been used in many problemssuch as 1) source separation, [25][28]; and 2) blind imagedeconvolution [29]. Like the BernoulliGaussian prior, thestatistical properties of this model allow easy implementationof Gibbs sampling and the EM algorithm. It also enables usto use variational Bayes techniques for a closed-form approx-imate posterior inference [33]. Moreover, the model has beenreported to have smooth objective functions and to be robust toinitialization [27][29], [34].

C. Contribution and Structure of the Paper In this paper, we introduce a Bayesian deconvolution ap-

proach for the receiver function analysis. Our main motivationis the observation that a receiver function can be modeled asa sparse trace convolved with a narrow smooth pulse, like theGaussian pulse. We are also motivated by the fact that exploit-ing the sparsity in the receiver functions in a Bayesian frame-work has interestingly not been introduced so far, though thereare many such works for very similar problems, such as marineseismic deconvolution. Rather than the standard choice, whichis the Bernoulli prior, we choose the inverse-gamma prior forthe sample variances of the sparse trace. We investigate Gibbssampling and variational Bayes techniques for our specicposterior inference problem. For the estimation of the unknownparameters in our model, we simply use the EM algorithm.

Since our main concern is to have improvement in theperformance of receiver function analysis, we show throughexperiments with both simulated and real data that Bayesian de-convolution can overcome the disadvantages of iterative decon-volution, and produces better results. (A detailed comparisonof the inverse-gamma and BernoulliGaussian priors is alreadyavailable in [34]). For real-life experiments, we use three setsof recordings taken from different seismic stations in Turkeyby Bogazii University Kandilli Observatory and EarthquakeResearch Institute.

The organization of this paper is as follows: In Section II,the deconvolution problem is stated. In Section III, iterativedeconvolution algorithm is summarized. In Section IV, the


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YILDIRIM et al. : BAYESIAN DECONVOLUTION APPROACH FOR RECEIVER FUNCTION ANALYSIS 4153

methodology for Bayesian deconvolution is investigated. InSection V, comparison between the iterative deconvolutionand Bayesian deconvolution are given based on the simulationand real data experiments. In the last section, we discuss theresults with concluding remarks and give possible directions forfuture work.

II. PROBLEM DEFINITION

As stated in Section I, via a three-component seismic station,the vertical and radial components of motion are available. Thevertical components corresponds to the source signal, whereas,the radial component y is the observation signal, which is theconvolution of the source signal and the receiver function rof the earth structure near the receiver. Some additive noise vis present in the observation, and in here, it is assumed to beindependent and identically distributed (i.i.d.) white Gaussian.Therefore, to estimate the receiver function, a deconvolutionoperation must be performed on the noisy observation using

the source.The noisy convolution can be formulated as follows:

yt =T s

i =1

s i r t i +1 + vt , t = 1 , 2, . . . , T y . (1)

For simplicity, we will use the notation

y = sr + v (2)

where denotes the convolution operation, which is the sum-mation part of (1). In (2), the vertical component is representedby a T s 1 vector of coefcients, s = [s1 s2 sT s ] ,where [] is used for the matrix-vector transposition. Like-wise, r = [r 1 r2 rT r ] denotes the vector of receiverfunction coefcients, and v = [v1 v2 vT y ] is the ob-servation noise vector with vk being i.i.d Gaussian with zeromean and variance 2v . Finally, y = [y1 y2 yT y ] isthe vector of samples of the noisy radial components.

Gaussian Filter for the Receiver Function: Rather than aspiky waveform, the receiver function is preferred to be mod-eled as a sequence of smooth and slowly varying pulses. Forthat purpose, the receiver function is assumed to be a sparsetraceh, ltered with a Gaussian pulseg. Therefore, the receiverfunction r can be expressed by

r = gh (3)

where g is the vector with length T g used for thediscrete approximation of the Gaussian pulse and h [h1 h2 hT h ] be the vector of sparse trace coefcientswith T h = T r T g + 1 . In common literature, the Gaussianpulse is dened commonly in the frequency domain as [12]

G() = exp 2

4a2 (4)

where = 2f denotes the angular frequency. Note that thefrequency content is controlled by the Gaussian lter-widthparametera , and the lter is a unit area pulse (i.e., the lter gainat = 0 is unity). It is common in seismology to quantify thelter by the frequency at which G(2f ) has a value of 0.1 [12].

Since the convolution operation is associative, using (3), wecan rewrite (2) as

y = sr + v = s(gh ) + v = ( sg )h + v . (5)

Notice that z sg in (5) is given to us. Hence, wecan perform deconvolution in order to estimate the receiverfunction. Given y and s, we shall follow the steps below fordeconvolution of the receiver function r :

Calculate z = sg . Given y and z , perform deconvolution of h . Calculate the estimated receiver function r = hg .

III. ITERATIVE DECONVOLUTION

Iterative deconvolution is a least squares method used to ndthe minimum least squares solution to y = hz for h given yand z . It was described and used in [12] and [14] for receiver-function estimation and inversion of complex body waves. The

iterative procedure for the discrete version of the algorithm asexpressed in [14] is as follows: Dene the delay operator t as

[ t z ] =0, if tz t , if T z + t > t

. (6)

Start with i = 1 .1) Find t i and m i such that (y m i [ t i z ] )

2 is min-imized. For such a ti , mi = y t i z / z z . (Zero-padeither of the vectors y and t i z if needed.)

2) Update y y m i t i z .3) Update ht i h t i + m i .4) If the algorithm has not converged, seti

i + 1 , go to 1.

The algorithm is easy to implement and very fast. In receiver-function analysis, it is terminated after some number of itera-tions. Therefore, convergence is not assured. Moreover, it doesnot handle sparsity and existence of the observation noise. Also,in case of a truncated convolutiondata, it automatically assumesthe truncated part as zero. We propose a Bayesian approach toovercome these problems.

IV. BAYESIAN DECONVOLUTION

The idea behind the Bayesian deconvolution is to modelh ina statistical way so that it is a sparse process. In other words,a prior distribution is assigned for h . One suitable approach isto model samples of h as i.i.d. Gaussian random variables eachwith zero mean and variance being an inverse-gamma randomvariable. That is

h t N h t ; 0, 2h,t 2h,t IG(2h,t ; , ).Here, the notation N (x; , 2) denotes the very well-knownGaussian distribution for the random variable x with mean and variance 2 [35]

N (x; , 2) = 1 2 2 exp

122

(x )2 .

There also exists the multivariate Gaussian distribution fora random vector x with mean vector m and covariance


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Fig. 2. Sparse trace obtained when = 0 .5 , = 0 .0001 .

matrix [35]

N (x ; m , ) = 1

|2 |exp

12

(x m ) 1(x m ) .Additionally, the inverse-gamma distribution with the shapeparameter and the scale parameter is given by [35]

IG(x; , ) =

()1x

+1

exp x

where () denotes the gamma function [35]. Choosing suitablevalues for and , one can obtain sparse and impulsivetraces (see Fig. 2). Note that in this model, the samples h t areStudent-t random variables [27].

Having introduced our model, the deconvolution problemcan be redened in a Bayesian framework: For the proposedstatistical model, we have

Latent statistical variables: h and 2h 2h, (1: T h ) Deterministic parameters: (2v , , ).Given the observation datay , we wish to calculate the posterior

distribution and thus the mean estimate of the unknown lter h(and 2h ), and nd the maximum likelihood solution for modelparameters .

A. Inference of the Posterior DistributionWe have to calculate the posterior distribution p(h , 2h |y , ) .We can write the logarithm of the unnormalized posterior,

which is the joint density of y , h , and 2h , as in the following:

log p(h , 2h , y |) = log p(2h |) + log p(h |2h )+ log p(y |h , 2h , )

=T h

t =1log IG(2h,t ; , )

+T h

t =1

log

N (h t ; 0, 2h,t )

+ log N (y ; Zh , 2v I T y ) (7)

where Z is the T y T h convolution matrix corresponding to zZ ij =

zij +1 , if 0 i j0, else (8)and I T y is the identity matrix of size T y .

It can be observed from (7) that the posterior distribution p(h , 2h |y , ) does not have a closed-form expression in termsof well-known distributions, hence the expectations of func-tions of (h , 2h ) under this posterior distribution cannot beevaluated easily. In other words, it is intractable. However, thefull conditional densities p(h |2h , y , ) and p(2h |h , y , ) havewell known closed form expressions that can be derived easily.Motivated by this fact, we can use Gibbs sampling [31] andvariational Bayes techniques [33], [36], in order to approximatethe exact posterior of (h , 2h ). The iterations of two methods arevery similar to derive. One uses samples of the latent variables,whereas, the other one denes a xed point iteration betweentheir sufcient statistics.

1) Gibbs Sampling for (h , 2h ): Gibbs sampling is a Markovchain Monte Carlo (MCMC) sampling method [37] which canbe used when the dimension d of the variable to be sampled,say x is at least two. All Monte Carlo techniques are based onthe idea that an analytically inexpressible probability densityfunction (pdf) P (x) is approximated by N number of drawnsamples x(n ) , n = 1 , . . . , N

P (x) = 1N

N

n =1 x x(n ) (9)

where () is the delta-Dirac function. Then, using (9), theexpected value of any function of x under P (x) can be approx-imated as

f (x) P (x ) 1N

N

n =1f x(n ) (10)

where f (x) P (x ) denotes the expectation of f (x) under thedistribution of P (x). This approach is called Monte Carlointegration [38].

The idea of Gibbs sampling is that, even though it is dif-cult to draw samples from P (x)= p(x1 , x2 , . . . , x d ), it maybe easier to sample from the conditional densities p(x i |x1 ,. . . x i1 , x i +1 , . . . , x d ). The general sampling scheme of Gibbssampling forP (x) can be expressed by the following [36], [37]:

x( +1)1 p x1|x( )2 , x

( )3 , . . . , x

( )d

x( +1)2 p x2|x( +1)1 , x

( )3 , . . . , x

( )d

...

x( +1)d p xd |x( +1)1 , x

( +1)2 , . . . , x

( +1)d1 .

It has been shown that, under mild conditions, when ,the samples are guaranteed to come from the exact distributionP (x).

From (7), we can deduce the full conditional densities of h ,and 2h . The posterior of h is given by

p h |2h , y , = N (h ; m h , h ) (11)


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B. Parameter Estimation: EM

To estimate the unknown parameters, we use EM-type algo-rithms [32]. EM contains two iterating steps: 1) Calculating theposterior of the statistical latent variables given the current setof parameters, which is the expectation (E)-step; 2) Updatingof the parameters under the calculated posterior distribution,which is the maximization (M)-step. Performing these stepsiteratively, we eventually hope to nd the posterior of the latent variables for the maximum likely parameter set .

Remind that we have the latent statistical variablesh and 2h ,and the deterministic parameters = ( 2v , , ). Thus, the EMsteps are:E-step: Requires inference of the posterior to compute the EM

quantity

(k ) = log p y , h , 2h | p(h , 2h |y , ( k 1) ) . (19)

M-step: Parameter estimation: Maximize (k ) over

(k ) = argmax

(k ) . (20)

It can be shown that log p(y |(k ) ) monotonically in-creases [32]log p y |(k +1) log p y |(k )

= (k ) (k +1) (k ) ( (k ) )

+ KL p h , 2h |y , (k ) p h , 2h |y , (k +1)

0 (21)since, by (20), (k ) ( (k +1) ) = max (k ) () (k ) ( (k ) ),and the KL distance KL is nonnegative [39]. Also, noticethat we have equality in (21) when (k +1) = (k ) , hence, thealgorithm has converged to a local maximum point.

In (7), we have already found the term in the expectation in(19). Maximizing (19) with respect to 2v gives the followingupdate at iteration k:

2( k )v = 1T y

y y + z C H z 2y H z (22)

where H is constructed in the same manner as Z in (8),C H = H H , and by we mean p(h , 2h |y ; ( k 1) ) . It can beshown that

C H i,j =T h +min( i,j )1

l=max( i,j )

m h ( li +1) m h ( lj +1)

+ h ( li +1 ,l j +1) i, j = 1, . . . , T s .

For (k ) and (k ) , notice from (7) that

(k ) =T h

t =1 log log 2h,t 12h,t log () + C

where C does not depend on (k ) and (k ) , and by ,we mean p(h , 2h |y ; ( k 1) ) . Note that log 2h,t = log h,t ( h ), where (x) = d(log (x)) /dx , is also available for thevariational Bayes method. A closed-form expression for (k )and (k ) that maximize (k ) is not available. However, for agiven estimate of (k ) , we have

(k ) = T h (k )

T ht =1

1 2h,t

. (23)

Using (23), one can obtain (k ) as a function of only.Fortunately, this function is known to be well behaved andmaximization algorithms such as gradient descent [40], orNewtons method [41], work well to determine (k ) .

In order to favor low values of supporting sparsity, may be modeled as an exponential random variable and may be assigned a gamma conjugate prior. Then, the posteriordistributions of and are evaluated rather than the estimationof their maximum likely values, and the hyperparameters of their priors may be estimated using a parameter estimationmethod, most possibly EM. For an example of such a case,see [28].

The general EM procedure applies when we choose thevariational Bayes method to approximate p(h , 2h |y ; (k 1) ),because we have a closed form expression for it, and wecan evaluate the sufcient statistics needed for maximization.This is also called the variational interpretation of EM [33].However, if we choose Gibbs sampling for the state inference,then we do not have closed-form expressions but samples. Inthis case, we perform Monte Carlo-related EM methods asfollows [32].

Start with an initial (0) . for k = 1 , 2, . . . ,

Simulation: Draw N k samples for (h , 2h );(h , 2h )(k, 1) , . . . , (h ,

2h )(k,N k ) from p(h , 2h |y ;

(k1) ). Maximization: Compute ( i ) that maximizes ( i ) ,

where

(k ) = 1N k

N k

n =1log p y , h , 2h

(k,n )

|(k1) .

Note that, the approximation for any sufcient statistics of (h , 2h ), S

(k )f = f (h , 2h ) p(h , 2h |y ; ( k 1) ) is

S (k )f = 1N k

N k

n =1f h , 2h

(k,n ).

Constant or increasingN k are reasonablechoices. Anothervalid approach is to maximize (1 k )(k1) + k (k )for{ k }k 1 > 0, 1 = 1 , and k k +1 1 over [32],which leads to the stochastic approximation

S (k )f = (1 k )S (k 1)f + k 1N kN k

n =1f h , 2h

(k,n ).


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Fig. 3. Illustration of the relaxation effect of the inverse-gamma model: log-prior and log-posterior (superimposed on the log-prior) are shown for both mowith respect to y = 2 h 1 + h 2 + v v . (y = 3 , 20 = 0 .0004 ,

21 = 4 , = 0 .03 , = 0 .5 , = 0 .0001 , and

2v = 0 .05 ). (a) BernoulliGaussian log-prior.

(b) Inverse-gamma prior. (c) BernoulliGaussian log-posterior. (d) Inverse-gamma log-posterior.

C. Relaxing Effect of the Inverse-Gamma Prior

So far, we have not discussed analytically why we have cho-sen the inverse-gamma distribution as our prior for the samplevariances 2h,t yet. The reason for our choice is the inferenceproblems tend to be easier, since inverse-gamma assumptionleads to smoother distributions/objective functions. We can seethis fact better when we compare our model for sparsity withanother well-known one, the BernoulliGaussian model. In thismodel, ht s are assumed as i.i.d. BernoulliGaussian randomvariables

h t N h t ; 0, c t2

, ct {0, 1} Pr( ct = 1) = where Pr( A) denotes the probability of the event A. Therelaxing effect of the inverse-gamma model is illustrated inFig. 3. Fig. 3(a) shows the log-prior distribution of [h1 , h2], if p(h i ) is a BernoulliGaussian random variable with parameters20 = 0 .0004, 21 = 4 , and = 0 .03, whereas, Fig. 3(b) is thelog-prior of [h1 , h2] when they are normal random variableswith their variances coming from the inverse-gamma distri-bution with parameters = 0 .5, and = 0 .0001. Note thatthe two priors produce quantitatively similar traces. Fig. 3(c)and (d) show the log-posteriors of [h1 , h2] given y, for theBernoulliGaussian and inverse-Gamma models, respectively.Here, the system admits y = 2h1 + h2 + v v, with y = 3 and2v = 0 .05. The log-posteriors are illustrated as superimposed

on the log-prior distributions, for the sake of understandability.It can be seen from the bottom gures that, the posterior inthe inverse-gamma model is much smoother than the one inthe BernoulliGaussian model. Roughly speaking, if we startsearching for [h1 , h2] those maximize the posterior from a pointnear to (0, 3), it is much more possible to be stuck in that localmaximum in the BernoulliGaussian model. A more detailedcomparison between those two models is available in [34].

V. EXPERIMENTS AND RESULTS

A. Experiments With Simulated DataWe already stated the disadvantages of the iterative decon-

volution method in Section I, and claimed that a Bayesianapproach might improve the estimations in the cases wherethere is noise and the complete convolution signal is not avail-able. Before dealing with real data, we will now support thosetheoretical arguments with simulation experiments.

Recall that one of the disadvantages is that in iterativedeconvolution, no observation noise is assumed, which maylead to wrong estimates in the noisy convolution case. How-ever, because of the observation noise assumption, we expectbetter results from the Bayesian deconvolution. In our rstexperiment, we aim to compare the performances of iterativedeconvolution and Bayesian deconvolution under noise. To dothis, we evaluated the normalized mean squared error (nMSE)


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Fig. 4. nMSE versus SNR values for iterative deconvolution and Bayesiandeconvolution.

values for the two types of deconvolution using different signal-to-noise (SNR) ratios. The nMSE for an estimate x of x isgiven by

nMSEx (x) = t (x t x t )2

t x2t(24)

By SNR, we mean the ratio obtained by the power of r dividedby the power of the observation noise, which is 2v in our case.In order to make an objective comparison, we generate h usinga different model for sparsity, which is the BernoulliGaussianmodel. We averaged our results over 100 simulations. For eachsimulation:

We generated a sparse process of length T h = 50 withparameters 20 = 0.01, 21 = 1 , and = 0 .05.

The source signal s of length T s = 50 was generated

randomly. For SNR values10, 9, . . . , 50: We generated the observation noise with power corre-sponding to the current SNR, and add it to the convo-lution to obtain the noisy convolution y ,

We perform Bayesian deconvolution and iterative de-convolution separately for the estimation of r ,

For each result, we calculated the nMSE in the estima-tion of r .

Fig. 4 shows the nMSEs versus the SNR values for bothdeconvolution types. It can be observed that, the iterative de-convolution method exhibits larger errors while SNR decreases.Bayesian deconvolution suffers from noise too, however, itis still more reliable than iterative deconvolution. Moreover,Bayesian deconvolution performs even better when there is verysmall, almost no noise. That is due to the fact that in Bayesiandeconvolution, we can introduce prior knowledge about h ,whereas in iterative deconvolution, the algorithm does not takethe statistical properties of h into account.

Another disadvantage of the iterative deconvolution is that itcannot handle missing data correctly. The missing data problemin deconvolution usually occurs when the convolution signalis truncated, that is, length of the convolution less than thesum of the lengths of the convolving components. Indeed, wehave such a problem in our earthquake data, where the sourcesignal, observation signal and the receiver functions are veryclose in length, say, 500, 500, and 450, respectively. (Thereason for these numbers will be apparent in Section V-B.)

Fig. 5. nMSE versus number of truncated samples from the observation signalfor iterative deconvolution and Bayesian deconvolution.

Fig. 6. Mean estimates of r and parameter estimations and the likelihoodfunctions through iterations: d = 2 , SNR = 5 .

This means that almost half of the convolution data is miss-ing. This is a serious problem for iterative deconvolution.However, Bayesian deconvolution method is based on a gen-erative model, and the missing data case can be introducedto the model by suitable choices of the vector lengths s, h ,and y . We can show this fact with a simulation experiment,where we obtain nMSE values versus the number of truncatedsamples in the convolution. The steps and the system pa-rameters of the experiment are the same as the one above, theonly difference is that we truncate the observation signal insteadof adding noise to it. Fig. 5 shows the result of the experiment. Itis obvious that the Bayesian deconvolution keeps estimating hwith an acceptable error even if the length of the observationsignal is equal to that of the source signal, whereas, in theiterative deconvolution method nMSE gets close to one, evengreater than one, while the observation signal is getting shorter.

Fig. 6 shows the results for the mean estimate of thesparse trace, and parameter estimations and likelihood func-tions through iterations both when variational Bayes and Gibbssampling are used in the E-step of the EM algorithm for a singleexperiment. The likelihood functions at iteration k are approxi-mated by Monte Carlo integration as follows: We draw particlesh (n ) from p(h

|(k ) ) for n = 1, . . . , N k for some large N k ,

evaluate the functionf (n ) (y ) = p(y |h (n ) , (k ) ), then calculatelog p(y |(k ) ) log(1/N k ) N kn =1 f (n ) (y ). It can be observed


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Fig. 7. Receiver function estimates of iterative deconvolution and Bayesian deconvolution for the station LOD0. (a) Iterative deconvolution. (b) Bayesdeconvolution.

that around 100 EM iterations are enough for convergence inparameter estimates , , and 2v for both methods. It is moredifcult to sample from an inverse-gamma distribution whenthe shape parameter is smaller, particularly less than one.That is why we observe a slight decrease in the likelihood func-tion for the Gibbs sampling method. Nevertheless, the meanestimates of the sparse trace are too close to be comparable.That is why we only used the variational Bayes technique inthe E-step of Bayesian deconvolution in the experiments above.For the same reason, we used only variational Bayes for ourexperiments on real data.

B. Experiments With Real Earthquake Data

We have tested the method using data from the NationalEarthquake Monitoring Center (UDIM) of Bogazii UniversityKandilli Observatory and Earthquake Research Institute. Wehave chosen three broadband stations: 1) BNN0; 2) LOD0;and 3) KARS, from the west, central and eastern Turkey,respectively. All earthquakes occurred between 20052007, ata distance between 3090 and magnitude greater than 5.5were selected. Horizontal components are rotated to obtainthe radial component according to the back-azimuth angle of each earthquake. Recordings with low SNR were eliminatedbased on the level of polarization of the vertical and radial

components, which is tested using theratio of eigenvalues of thecovariance matrix [15]. This step rejects nearly 30% of the totalnumber of receiver functions and the nal number is reduceddown to about 150.

The receiver functions corresponding to the data were al-ready estimatedby zaknandAktar [15], whoapplied iterativedeconvolution to the records. We employ Bayesian decon-volution to the data and compare our results with the onesobtained in [15]. The frequency at which the Gaussian lterG(2f ) has a value of 0.1 is determined as f = 1 .2 Hz forthis data, which can be realized by selecting a = 2 .5 in (4).Then, the corresponding essential width of the Gaussian lteris approximately 1 s. The original data sampling rate is 50 Hz,but since it is known that no information is contained in thefrequencies higher than 5 Hz, we resample the data to 5 Hzbefore applying Bayesian deconvolution. Therefore, the lengthT g of the Gaussian lter is chosen 5.

For the comparison of the results of iterative deconvolutionand Bayesian deconvolution, the results obtained for each sta-tion are plotted as follows: First, the records are sorted withrespect to their back- azimuth values, which are the anglesof direction from the source of the earthquake to the receiver[42]. Then, all of the estimated receiver functions belongingto that station are placed in a top-to-down order. For the sakeof visuality, the positive and negative parts of each receiver


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Fig. 8. Receiver function estimates of iterative deconvolution and Bayesian deconvolution for the station BNN0. (a) Iterative deconvolution. (b) Baydeconvolution.

function estimate are shaded with different colors. Also, all of the receiver estimates are scaled by a constant to increase thevisual quality of the images.

In fact, there is almost no analytical criteria which can beused to determine and compare the practical quality of our re-sults. In hypothetical situation where all earthquakes occur at axed location on Earth, all receiver functions are expected to beequal and all pulses including both the direct ones and the onesgenerated at boundaries (main pulses as well as reected ones)areexpected to align in a vertical line. However, in real situationwhere earthquakes are scattered around the globe, the verticalalignment can be observed only for a limited part of the back-azimuth range. The deviation from verticality is due to both thenonhorizontality of reecting layers and also to the variationof the incidence angle with different earthquake distances. Thelatter one can be removed by applying a correction term toeach receiver function since the distance of each earthquake areknown with good accuracy. The nonhorizontality of reectivelayers produces different reection arrivals for waves comingfrom different azimuths. This causes the alignment of pulsesin a gently undulating curve instead of a straight line. Thisundulation is often used as the key for the estimation of thedipping angle for the subsurface layers. It is, therefore, of primeimportance to be able to locate with accuracy all coherentpulses in receiver functions. The performance of a deconvolu-

tion technique is therefore directly related to its ability to revealthese coherent pulses.

In this paper, we do not attempt to carry a detailed modelingof the crustal structure as would be the case of a standardapplication of a geophysical survey. Instead, we focus on thecomparison the two deconvolution methods by referring to theirperformance in detecting clear and coherent reection pulses inthe nal receiver functions. In Turkey, for the last ve recentyears, most of the recorded distant earthquakes originated fromtwo given geographical location, the KurileJapan subduction(KJS) zone and the JavaSumatra subduction (JSS) zone, whichcan be assumed to have a xed azimuth and a xed distance.We, therefore, expect that whatever the horizontality of thesubsurface layers, the receiver function from any seismic stationin Turkey should show vertical alignment close to a straightline in the azimuths corresponding to the two subduction zonesabove. This a priori knowledge is used as a checkpoint for thecoherency of the receiver functions.

Figs. 79 show the results of the iterative deconvolutionand the Bayesian deconvolution methods for the three stations.Almost all receiver functions, independently of the recordingstation and the method used, show two azimuth ranges wherevertical lineament of pulses are clearly observed (receiver func-tions 1060 and 90110). This conrms the two subductionzones, KJ and JS, as being the main supplier of the source


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Fig. 9. Receiver function estimates of iterative deconvolution and Bayesian deconvolution for the station KARS. (a) Iterative deconvolution. (b) Bayesdeconvolution.

events and therefore, constitutes a verication for the validity of the receiver functions. Furthermore, the comparison of the twoalternative methods for each station, namely the existence of similar pulses in both cases and their position, shows that the re-sults of the Bayesian deconvolution are consistent with the onesof iterative deconvolution. This is an important test for the pro-posed method since the iterative deconvolution is a well-provenprocedure tested with many types of data from different origins.

A close observation of the vertical patterns in both methodsreveals that the Bayesian deconvolution results have some clearadvantages in terms of detection capability. In all the examples,one can see that Bayesian deconvolution results reveal extravertical lines which are hardly perceptible or even nonexistentin the iterative deconvolution. A clear example of this featureis the receiver functions 3050 in Fig. 7. The image one onthe right hand side, obtained through Bayesian deconvolution,shows a clear vertical alignment of pulses at around 8 s, whichis not seen in the one on the left-hand side, obtained byiterative deconvolution. A similar situation can be observed inthe receiver function image of the KARS station. The Bayesiandeconvolution reveals three primary pulses at 5.5, 12, and 19 sfor the waveforms number 30100. The rst two pulses areknown as the primary Moho conversion (Ps) and reection

(PpPs) and are important in the sense that they constitutethe main basis for the depth estimation of the crustmantleboundary [6]. We notice that only the second one (at 12 s)is clearly seen in the iterative deconvolution case while theother two are blurred in noise. We also note that the verticalline patterns in the images of the Bayesian deconvolution arestraighter than the ones from the iterative deconvolution. Thisdifference can easily be seen in the gures for the stationsLOD0 and KARS. We interpret this as, in low SNR environ-ment, the Bayesian deconvolution is more robust in term of phase estimation of pulses. A corollary of this feature is theimproved pulse separation ability of the Bayesian method. Thiscan be clearly illustrated at station BNN0 (Fig. 8), where therst receivers functions 1120 have a primary conversion (Ps)at 4 s while this conversion jumps to 5.0 s for the latter onesbetween 120180. This can possibly be due to the existenceof different boundary depths when looking from two differentazimuth angles. More interestingly we observe a narrow andintermediate transition zone where both pulses can be seen on asame receiver function (110120). This corresponds to a crustaltransition zone where both boundary coexist. This interestingfeature is completely missed by the iterative deconvolutionwhere the resolution capability is too low to detect such details.


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VI. CONCLUSION

The overall performance of the Bayesian deconvolution sug-gests a potential improvement in the deconvolution processfor receiver-function estimation. It offers a promising tool forthe reevaluation of low SNR data which in turn will not onlyshorten the waiting period for signicant earthquakes, but willalso allow the study of certain azimuth angles where only lowmagnitude events were observed and were therefore discardedtill present. The computational load for the implementationof Bayesian deconvolution is many order of magnitude larger(about 104) than the conventional methods. This however is notexpected to be a serious limitation considering the cost of thealternative which consists of extending the duration of the elddeployment.

We have stated that we expect to see vertical straight lines of pulses when the receiver functions are aligned in an up-downorder with respect to their back azimuth values. Hence, it is easyto come up with the idea that, given a set of receiver functionsordered with respect to their back azimuth values, successivereceiver functions should look alike. It would be interesting toexploit this similarity by using our current receiver function es-timate in the initialization of thenext receiver function estimate.A more principled method to exploit this structure can be basedon a Markov chain between the locations of reections in thesuccessive receiver functions, for a similar approach, see [21]and [22] for reectivity sequence estimation.

The promising results in the Bayesian deconvolution of seismological signals opens the door for further research onreceiver function analysis in a Bayesian framework, such asblind deconvolution of receiver functions when the source is notknown. Though it is known that classical inference methods failin accurate estimation [43] when the source is very long, moresophisticated inference techniques such as sequential Monte-Carlo samplers [44] can be applied for the solution of thischallenging problem.

ACKNOWLEDGMENT

The authors would like to thank UDIM of Bogazii Univer-sity Kandilli Observatory and Earthquake Research Institute forsupplying the earthquake data and to K. Davasloglu for codingsome of the software.

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[36] D. MacKay, Information Theory, Inference and Learning Algorithms .Cambridge, U.K.: Cambridge Univ. Press, 2003.

[37] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain MonteCarlo in Practice . London, U.K.: Chapman & Hall, 1996.

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[42] S. Stein and M. Wysession, An Introduction to Seismology, Earthquakes,and Earth Structure . New York: Wiley-Blackwell, 2003.

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Sinan Yldrm was born in Istanbul, Turkey, on1984. He received the B.S. and M.S. degrees inelectrical and electronics engineering from BogaziiUniversity, Istanbul, Turkey. He is currently workingtoward the Ph.D. degree in the Department of PureMathematics and Mathematical Statistics, Universityof Cambridge, Cambridge, U.K.

His current research interests are in the areasof Bayesian inference, Markov chain Monte Carlomethods, and sequential Monte Carlo methods.

A. Taylan Cemgil (M95) received the B.Sc.and M.Sc. degrees in computer engineering fromBogazii University, Istanbul, Turkey and thePh.D. degree from Radboud University Nijmegen,

Nijmegen, the Netherlands, in 2004.He worked as a Postdoctoral Researcher atthe University of Amsterdam, Amsterdam, theNetherlands on vision-based multi-object trackingand as a research associate at the Signal Process-ing and Communications Laboratory, University of Cambridge, Cambridge, U.K., on Bayesian signal

processing. He is currently an Assistant Professor with Bogazii University,where he cultivates his interests in machine learning methods, stochasticprocesses and statistical signal processing. He also acts as a consultant tothe industry and the professional audio community. His research is focusedtoward developing computational techniques for audio, music, and multimediaprocessing.

Mustafa Aktar was born in 1952 in Istanbul,Turkey. He received the B.Ss. degree in electronics(with honors) from Manchester University Instituteof Science and Technology, Manchester, U.K., theM.Sc. and Ph.D. degrees in electrical engineeringfrom Bogazii University, Istanbul, Turkey, in 1976,1978 and 1984, respectively.

Since 1996, he is at the Kandilli Observatory and

Earthquake Research Institute of Bogazii Univer-sity, where he is currently a Professor and Head withthe Department of Geophysics. He has authored and

coauthored 26 scientic papers in scientic journals and more than 90 papersin proceedings. His current research interests are in the areas of seismology, inparticular source mechanism of earthquakes, earthquake generating processes,analysis of crustal-lithosperic structures, and tectonic processes.

Dr. Aktar is a member of American Geophysical Union (AGU), Seismo-logical Society of America (SSA), and Chamber of Geophysical Engineers of Turkey (JMO).

Yaman zakn was born in Istanbul, Turkey, in1980. He received the B.S. degree in physics and theM.S. degree in geophysics from Bogazii University,Istanbul, Turkey. He is currently working toward thePh.D. degree in geophysics in University of SouthernCalifornia, Los Angeles.

His current research interests are in the areas of seismology and articial neural networks.

Aysn Ertzn (M94) was born in Salihli, Turkeyin 1959. She received the B.S. degree (with hon-ors) from Bogazii University, Istanbul, Turkey, in1981, the M.Eng. degree (with rst class stand-ing) from McMaster University, Hamilton, ON,Canada, in 1984 and the Ph.D. degree from BogaziiUniversity, Istanbul, Turkey, in 1989, all in electricalengineering.

Since 1988, she has been with the Departmentof Electrical and Electronic Engineering, BogaziiUniversity, where she is currently a Professor of

electrical and electronic engineering. She has authored and coauthored morethan 70 scientic papers in journals and conference proceedings. Her currentresearch interests are in the areas of independent component analysis and itsapplications, blind signal processing, Bayesian methods, wavelets and adaptivesystems with applications to communication systems, image processing, andtexture analysis.

Dr. Ertzn is a member of IEEE Signal Processing and CommunicationSocieties, International Association of Pattern Recognition (IAPR), TheInstitute of Electronics, Information and Communication Engineers (IEICE)and Turkish Pattern Recognition and Image Processing Society (TOTIAD).

a bayesian deconvolution approach

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