estimated pulse sequene we produe estimates of the delay time struture of ripple-red events. Formulation of the Problem We model the k th seismi trae

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1 Bayesian Deonvolution of Seismi Array Data Using the Gibbs Sampler Eri A. Suess Robert Shumway, University of California, Davis Rong Chen, Texas A&M University Contat: Eri A. Suess, Division of Statistis, One Shields Avenue, Davis, CA 9566 ( Key Words: Bayesian analysis, Gibbs Sampling, time series, losions, ripple-ring, nulear test monitoring. Abstrat The problem of monitoring for low magnitude nulear losions, using seismi array data, under a Comprehensive Test Ban Treaty (CTBT requires a apability for distinguishing nulear losions from other seismi events. Industrial mining losions are one type of seismi event that needs to be ruled out when trying to detet nulear tests. We onsider a Bayesian approah to the problem of deteting ripple-red mining losions. Seismi array data are ressed as multidimensional time domain onvolutions of an unknown pulse funtion, representing the ripple-red delay pattern, with unknown signalpath eets sequenes on eah hannel, whih are assumed to follow independent AR proesses. Using the Gibbs Sampler a proposal of Cheng, Chen Li [] for blind deonvolution, we develop an approah to estimating the delay parameters the unknown signal-path eets sequene at eah sensor. Results for a ripple-red mining losion reorded at the Arti Experimental Seismi Station (ARCESS will be presented. Finally, the impliations of our method for monitoring a nulear test ban treaty are onsidered. Introdution The problem of monitoring seismi events for possible nulear losions is an important one that has been studied extensively is still open to further development. Muh work has been done on this problem due to the many treaties that have been signed in the past between the U.S., the former U.S.S.R., other nulear powers related to testing issues. The fous in the past has been on distinguishing possible nulear losions from earthquakes. Currently, beause the above treaties have put limitations on the permissible sizes of the nulear losions for testing other smaller seismi events suh as mining losions have beome of interest in the disrimination problem. The work we are proeeding with here is related to distinguishing low level nulear losions from ripple-red mining losions that are on the same seismi level. This tehnique of quarry mining involves the use of multiple rows of losives that are detonated with approximately equal time delays between the rows. The objetive of this paper is to develop an estimation proedure that estimates delay times in seismi events. The delay struture of a single seismi event is haraterized by the size (or amplitude of the delayed losions the delay times between the single losions. If a delay struture an be estimated, this will be useful in disriminating ripple- red mining losions from other, possibly nulear, seismi events. In addressing this problem, we have developed a model for seismi reordings olleted by an array of sensors for ripple-red events a deonvolution method based on work by Rong Chen, in the univariate ase, work by Cheng, Chen, Li [] that is used to estimate parameters in our model related to the delay time struture. Our method works by separating, or deonvolving a pulse sequene from the underlying signal-path effets sequenes. The pulse sequene is used to model the detonation design of a mining losion. The underlying signal-path eets sequenes are used to model the ommon underlying signal sent by eah detonation in a ripple-red event the eets of the path on the signal as it traes to the sensors. We approah the problem from a Bayesian perspetive. The Gibbs sampler is implemented to produe posterior estimates of the parameters in our model, using the data available prior information. From the

2 estimated pulse sequene we produe estimates of the delay time struture of ripple-red events. Formulation of the Problem We model the k th seismi trae y k (t, t ::: n,in an array of q-sensors, that is suspeted to have been produed by a ripple-red mining losion, using the following multivariate onvolution model: y k (t m j0 a j s k (t ; j+" k (t: ( We refer to the vetor of parameters a [a ::: a m ] 0 as the relative pulse sequene. This set of parameters is assumed to reet the delay pattern used in the design of the ripple-red event. We dene the k th underlying signal-path eets sequene, s k (t, t ::: n, as the ombination of the \signal" produed by eah sub-losion in a ripple-red event along with the \path eets" that are due to the rom imperfetions in the material the signal travels through on the k th hannel to the k th seismometer. We assume that the pulse sequene a is independent of the underlying signal-path eets sequene s k (t, t ::: n, on eah hannel k, k :: q. Assuming that the rst losion in the ripple-red event is used as the referene signal, for whih wex a 0 to be, that the remaining a j are alulated relative toa 0, the model an be written in a more insightful form as y k (t s k (t+ m j a j s k (t ; j+" k (t: ( This model assumes that eah delayed losion in a ripple-red event sends a signal having the same form as the rst. The only dierene is that its amplitudes a j may vary. Further modeling assumptions on the pulse sequene a, the underlying signal-path eets sequenes s k (t, k :: q, are needed to insure that the parameters in the onvolution model are identi- able. To model the pulse sequene we useavari- ant of the Bernoulli-Gaussian model ommonly used to model reetion seismology data. A disussion of the model an be found in Mendel [4]. For the pulse sequene a, wexa 0 model the relative amplitudes as follows: We dene j, j ::: m,tobe a sequene of independent Bernoulli rom variables with p( j ; p( j 0. The nonzero values of j index the possible detonation times in the ripple-red event of duration m, measured in points per seond. The parameter is the probability of observing a zero value in the pulse sequene. The parameter m is the length of the vetor a, m is assumed to be known. Next, we dene the distribution of eah a j, onditional on j, j ::: m. In ripple-red events the amplitudes will be positive beause mining losions are usually performed at shallow depths. We dene a j, j ::: m, onditional on j, j ::: m, as p(a j j j 0ifa j 0 0 otherwise. And p(a j j j ( p (3 where ( Z 0 ; (a j ; p ; (x ; I(a j > 0 dx: I(a j > 0 0 if a j > 0 0 otherwise. The trunated normal distribution is used beause the a j, the relative amplitude of the delayed detonation at time j, is assumed to be positive rom. Trunating at zero restrits the values of a j to positive values. It is reasonable to use the trunated normal distribution to model the variation in the size of a single detonation sine the distribution of the size of a single losion should be entered at a mean value with a variane. Alternatively, we write the density of the independent a j, j ::: m,as p(a j j I(a j 0+(;p(a j j j I(a j > 0 (4 whih is a mixture of a Bernoulli distribution a trunated Gaussian distribution. The mean is used to model the average size of the delayed losions in ripple-red events in the monitoring region the variane is used to model the variane of suh losions. So the a j are independent identially distributed with a density dened in (4. Note that for the Bernoulli sequene j, j ::: m, will have n 0values m ; n unit values, whih implies that the pulse sequene has m ; n nonzero values or pulses. Note that n is rom. Dene J to be the index set that inludes the values of j for nonzero values of j or thereby the nonzero values of a j. Hene, we dene the density ofa [a ::: a m ] 0 as p(aj n ( ; m;n ;n ( ( ; (a j ; J n p : (5 This modeling assumption is based on the Bernoulli- Gaussian model used by Cheng, Chen, Li [].

3 To model the underlying signal-path eets sequene s k (t, in the k th seismi trae y k (t, t ::: n, we use a low-order autoregressive, AR(p, model 0 s k (t+ s k (t ; + ::: + p s k (t ; p e k (t (6 where wex 0 let e k (t be independent identially distributed, over t k, normal with mean 0 variane. The use of this model is justied by Dargahi-Noubary [] Tjstheim [8] who gave theoretial empirial arguments for modeling seismi waveforms using low-order autoregressive models.the preision of s k (t is dened as, whih isused later in our appliation of the Gibbs sampler. We use the same set of autoregressive oeients, [ ::: p ] 0, to model the k th underlying signal-path eets sequene s k (t, sine the same event is assumed to generate the trae y k (t reorded at the k th seismometer. Hene, we believe the same autoregressive model will be appliable for eah underlying signal-path eet sequene s k (t, k ::: q, t ::: n, although the funtions generated will be dierent. Finally, we assume that the additive observation noise term " k (t in ( is distributed independent identially, over t k, normal with mean 0 variane, where >0 is xed. The parameter an be thought ofastheinverse of the signalto-noise ratio, dened as SNR e ", where e is the variane of the underlying signal-path eets sequene on eah hannel " is the variane of the noise term. Reall that e ", whih implies that SNR. The use of the xed parameter alleviates a saling problem that exists if two separate variane terms are inluded in our model. 3 The Bayesian Approah In this paper the deonvolution of seismi array data for ommon delay patterns is ahieved using Bayesian methods. Speially in the model under onsideration in (, the parameter set is f a Sg: (7 The set ontains +m+p++nq elements, where we dene a [a a ::: a m ] 0, [ ::: p ] 0, S [s s ::: s q ] 0, suh that eah row of S is s k [s k ( s k ( ::: s k (n] 0. The given data onsists of the olletion of q seismi traes reorded for a duration of n points. Here we dene Y [y y ::: y q ] 0, suh that eah row of Y is y k [y k ( y k ( ::: y k (n] 0. Hene in this problem only qn data points are available to estimate all of the parameters in. To develop a Bayesian deonvolution method we follow the Bayesian approahto statistial data analysis. We develop a model likelihood p(yj. Prior distributions for the parameters are hosen. The overall prior is p(. Lastly, the posterior distributions of the parameters in, given the data Y, is alulated. By using Bayes' Rule the posterior distributions an be alulated as p(jy / p(yjp(. Then, to draw inferenes about the unknown parameters in our model, we alulate the marginal posteriors. For point estimates we alulate the posterior means. In our deonvolution problem, is a vetor of many parameters, due to our modeling assumptions the marginal posterior distributions are diult to alulate. However, the onditional marginal distributions are available. From the latter we an implement the Gibbs sampler, a Markov Chain Monte Carlo tehnique that simulates \rom samples" from the onditional marginal distributions, whih an be used to alulate posterior estimates of the parameters in our model. Point estimates are alulated by taking the means of the Markov hains samples after a suient \burn in" period. See Gelf Smith [3] Tanner [7] for desriptions of the Gibbs sampler. 3. Prior Distributions To perform our Bayesian analysis we hoose prior distributions for eah unknown parameter in the parameter set. We speify priors on,, sine they are unknown. We have modeling assumptions that determine the distributions of the rom parameters a S. The probability of seeing a zero at any point in the pulse sequene a j, j ::: m, is assumed to have a beta prior, with predetermined hyperparameters > 0 > 0. For the AR(p oeients [ ::: p ] 0, we assume a p-variate normal prior distribution where 0 0 are predetermined hyperparameters. To speify the prior for the ommon preision of the underlying signal-path eets, for eah hannel k, we assume that has a gamma prior where > 0 > 0 are predetermined hyperparameters.

4 3. Overall Prior, Likelihood, Conditional Posterior Distributions The overall prior distribution on the parameter set is dened using independene assumptions is p( p(p(p( my j p(a jj qy k p(s k j :(8 We speify the onditional likelihood use it as an approximation of the full likelihood. Due to the dependene of y k (t on the past m values of s k (t, whih an be seen in (, the dependene of s k (t on its past p values, whih an be seen in (6, we dene the likelihood onditional on y k ( ::: y k (l, k ::: q, where l m + p. We now redene the data matrix as Y [y y ::: y q ], where eah y k, k ::: q, starts at l +(m + p +, so y k [y k (l + ::: y k (n] 0. Using the model assumption that the noise, " k (t, is independent identially distributed normal with mean 0 variane, the onditional likelihood is p(yj ( ;q(n;l ( q(n;l ; " k(t P m where " k (t y k (t ; j0 a js k (t ; j. The joint density of the unknown parameters the data Y is p(y p(yjp(, where the prior is given in (8 the likelihood is given in (9. The joint posterior of the unknown parameters, given the data, an be ressed using Bayes' Rule as proportional to the likelihood times the prior, here p(jy / p(yjp(. For our deonvolution, alulations of p(jy, p(ajy, p(jy, p( jy, p(sjy are not easily alulated analytially. So we now proeed to develop the onditional marginal posterior distributions of the parameters in our model, whih are used in the Gibbs sampler. It is through the Gibbs sampler that we produe estimates of the parameters in our model. We proeed to develop the onditional marginal posterior distributions of the sets of parameters in our model. The following onditional marginal posteriors distributions are alulated: p(jy rest, p(ajy rest, p(jy rest, p( jy rest, p(sjy rest, where rest refers to the remaining parameters in for a spei parameter or set of parameters of interest.. The onditional posterior of is (9 jy rest beta( (0 with parameters n + m ; n +.. The onditional posterior of a j, j ::: m is p(a jjy rest / ji(aj 0 +( ; j (aj a j p aj ; where j [ +(; d j ], d j (aj a j ( " aj a j + (a j ; aj a j I(a j > 0 ( aj aj a j ; " k(t[;j]s k (t ; j P where " k (t[;j] y m k(t ; i0 i6j a is k (t ; i, a j " + s k(t ; j : 3. The onditional posterior of is jy rest N p ( ( where ~s k (t [s k (t ; ::: s k (t ; p] 0, "; s k (t~s k (t ~s k (t~s 0 k(t + 0 : 4. The onditional posterior of is jy rest gamma( (3 where q(n ; l ; p+ " k(t+ k tp+ e k(t+ : 5. For k :: q i (p + ::: n, s k (i has a Normal posterior distribution with s k (ijy rest N( sk (i s k (i (4

5 with sk (i s k 4 min((i+m n (i " 0 k(t[;i]a t;i tmax(i (l+ min((i+p n ; e 0 k(t[;i] t;i where " 0 k(t[;i] y k (t ; e 0 k(t[;i] s k (i ti m j0 j6t;i p j0 j6t;i 4 min((i+m n tmax(i (l+ + a j s k (t ; j j s k (t ; j a t;i min((i+p n ti t;i 6. For k :: q the onditional posterior of ~s k (p +[s k (p ::: s k (] 0 : We dene t [ t ::: p 0 ::: 0] 0 for t ::: p also dene k (t s k (p + t + s k (p + t ; + ::: + ts k (p + for t f ::: pg. Therefore, we have ~s k (p + N p ( ~sk (p+ ~s 0 k (p+ (5 where " p ~sk (p+ ~sk (p+ ; k (t t " ~s 0 k (p+ p t t t 0 t + 4 Results for Real Data We present in this setion the results produed by applying our Bayesian deonvolution tehnique to real data. The array data we analyze is Event 054, a mine blast reorded at the Arti Experimental Seismi Station (ARCESS in northern Norway, previously analyzed by Shumway, Baumgart, Der [5]. The seismi soure of this data is known to be a ripple-red mining losion that was reorded at a regional distane with a sampling rate of 40 points per seond. : : 4. Hyperparameters Fixed Model Parameters In this setion we rst present our hoie of hyperparameters for the prior distributions on the parameters in our hoie of the xed model parameters. For, thehyperparameters we use in our beta prior are For the autoregressive oeients themeanvalue we use for our multivariate normal prior on is 0 [;0:9 0:7 ;0:] 0 the ovariane matrix is 0 4 (0:45 ;(0:5 (0:5 ;(0:5 (0:30 ;(0:5 (0:5 ;(0:5 (0:5 3 5 : (6 The hyperparameters we hoose for the gamma prior distribution of the preision of the underlying signal-path eets sequene s k on eah hannel k are 5 4. The hyperparameters that determine the normal distribution used to model the independent nonzero elements of the pulse sequene a j are 0:7 0:5. The value of the parameter is hosen to reet the signal-to-noise ratio dened as SNR e ". We x 0:0. And we xm Results for Event 054 Here we present the results produed by applying our Bayesian deonvolution program to the P n -phase of Event 054, see Figure for plots of the q 5 reordings of the P n -phase. Estimates of the model parameters are alulated from a typial run of our Gibbs sampling program. For the results presented here the program was run 40,000 iterations a \burn in" of 35,000 iterations was used. The estimates were omputed by alulating the means of the Markov hains after \burn in" for eah parameter. The peak values of a estimated from the P n -phase our at the times 4, 8,, 4, 8, 5, 8, 35, whih an be seen in Figure. The estimated delay struture ontains multiple delays, indiating that the P n -phase of Event 054was generated by ripple- ring. 5 Conlusions To summarize our work, we have developed a Bayesisan deonvolution method for seismi array data that detets a ommon delay pattern. This work was based on the univariate work of Rong Chen Cheng, Chen, Li []. The priors we hose were derived from the previous work

6 Pn054 Channel One - Observed Trae Pn054 Channel Two - Observed Trae Pn054 Channel There - Observed Trae the statistial analysis ert opinion related to the parameters in our model or researh related to the seismi ativity of the region being monitored. Seondly, we see the possibility of automating the analysis so a quik rst analysis an be run to hek seismi events for ommon ripple-red mining losions. 6 Aknowledgments This work was supported by the Center for Statistis in Siene Tehnology, U.C. Davis, under grant DMS Pn054 Channel Four - Observed Trae Pn054 Channel Five - Observed Trae Figure : P n -phase Event 054: Saled data, y k, k ::: q, whereq 5 n 0. aest Pn054 - Delay Amplitude Estimates Figure : P n -phase Event 054: Posterior estimates of the pulse sequene, ^a j, j ::: m, where m 40. of Tjstheim [8], Smith [6], Shumway, Baumgart, Der [5]. We tested our program extensively on simulated data we used the lessons learned from the simulated data analysis to implement our program on the data from Event 054 reorded at ARCESS. We foresee two main impliations of our method for monitoring seismi ativity for low level nulear losions. First, our method an inorporate into j Referenes [] Cheng, Q., Chen, R., IL, T.-H. (996 Simultaneous Wavelet Estimation Deonvolution of Reetion Seismi Signals IEEE Transations On Geosiene Remote Sensing 34, [] Dargah-Noubary, G.R. (995 Stohasti modeling identiation of seismi reords based on established deterministi formulations. J. Series Analysis 6, 0-9. [3] Gelf, A.E. Smith, A.F.M. (990 Sampling-Based Approahes to Calulating Marginal Densities Journal of the Amerian Statistial Assoiation 85, [4] Mendel, J.M. (990 Maximum-Likelihood Deonvolution: A Journey into Model-Based Signal Proessing. New York: Springer-Verlag. [5] Shumway, R.H., Baumgart, D.R. Der, Z.A. (998 A Cepstral F Statisti for Deteting Delay-Fired Seismi Signals. Tehnometris 40, [6] Smith, A.T. (993 Disrimination of Explosions from Simultaneous Mining Blasts Bulletin of the Seismologial Soiety of Ameria 83, [7] Tanner, M.A. (996 Tools for Statistial Inferene, Methods for the Exploration of Posterior Distributions Likelihood Funtions, 3rd edition. New York: Springer-Verlag. [8] Tjstheim, D. (975 Autoregressive Representation of Seismi P-wave Signals with an Appliation to the Problem of Short-Period Disriminants Geophys. J. R. Astr. So. 43, 69-9.