ANALYSIS OF HIGH DIMENSIONAL TIME SERIES: OCEAN BOTTOM SEISMOGRAPH DATA Genshiro Kitagawa () and Tetsuo Takanami (2) () The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 06-8569 Japan, kitagawa@ism.ac.jp (2) Hokkaido University, Research Center for Seismology and Volcanology Kita-ku, Sapporo 060-080, Japan, ttaka@eos.hokudai.ac.jp Abstract. To explor underground velocity structure based on OBS (Ocean Bottom Seismograph), it is necessary to extract reection or refractions waves from the data contaminated with relatively large direct waves. In this paper, we consider the time series decomposition, spacial decomposition and time-space decomposition of the data. In spatial decompsition and time-space decomposition, the dierence of the travel time (delay) corresponding to underground layer structure is considered. Key words and phrases: Explosion seismology, underground structure, Bayesian modeling, general state space model, Monte Carlo lter, self-organization. Introduction In a cooperative research project with University of Bergen, Research Center for Seismology and Volcanology, Hokkaido University, performed a series of experiments to observed articial seismic signals by ocean bottom seismogram (OBS) near Norway (Berg et al. (200)). The objective was to explore the underground velocity structure. In one experiment, for example, 39 OBSs were set on the bottom of the sea (500{ 2000 m depth) with distances 0{30 km and observed the signals generated by air-gun equipped on board. The ship moved with a constant speed and generated signal 982 times at each 200m (70 seconds). At each OBS, four-channel time series (2 horizontal and 2 vertical (high-gain and low-gain) components were observed with sampling interval of /256 second. As a result, 982 4-channel time series with 5366 observations were obtained at each OBS. In this article, we consider methods of extracting information about underground velocity structure from multi-channel time series obtained from array of OBS. In particular an important problem is the estimation of the reection waves from the observed seismographs. For that purpose, we rst applied a time series decomposition model. We then develop a spatial model that takes into account of delay of the propagation of the direct and reection waves. We then extend the model to a space-time model to take into account of both the time series structure and spatial delay structure.
Sea Surface Bottom OBS Fig.. Exploring underground structure by OBS data 2. State Space Decomposition of Time Series 2. Separation of Direct Waves and Reection/Refraction Waves The problem of extracting reection waves and refraction waves from relatively large direct waves by the state space modeling is considered here. Note that in this section, we consider the modeling and smoothing of single channel and thus the subscript identifying the channel is omitted. For the extraction of the reection (or refraction) waves from direct waves, we consider the model (2.) y n = r n + s n + " n where r n, s n and " n represent the direct wave, the reection wave and the observation noise, respectively. To separate these three components and to extract information about the underground structure from complicated observed time series, it is assumed that both r n and s n are expressed by the autoregressive models (2.2) r n = mx i= a i r n;i + u n s n = `X i= b i s n;i + v n where the AR orders m and ` and the AR coecients a i and b i are unknown and u n, v n and " n are white noise sequences with u n N(0 2 ), v n N(0 2 2 ) and " n N(0 2 ), respectively (Kitagawa and Takanami (985)). The models in (2.) and (2.2) can be combined in the state space model form (2.3) x n = Fx n; + Gw n y n = Hx n + " n where x n is (m+`)-dimensional state vector dened by x n =(r n ::: r n;m+ s n ::: s n;`+ ) T, and w n = (u n v n ) T is a two dimensional system noise. F, G and H are respectively 2
(m + `) (m + `), (m + `) 2and (m + `) matrices dened by (2.4) F = 2 64 a a 2 ::: a m... 0 0 b b 2 ::: b`... H = [ 0 0 j 0 0 ] 3 75 G = 2 64 0 0 0.. 0 0 0 0 0.. 0 0 3 75 and the variance of w n is given by (2.5) Q n = " # 2 0 0 2n 2 : 2.2 Separation of the Time Series and Estimation of the Parameters If all of the parameters m, ` and = (a j b i 2 2 2 2 ) T are given, the state vector x n can be estimated by the Kalman lter and the xed interval smoother (Anderson and Moore 979). In actual estimation, however, the parameters of the model is unknown. autoregressive coecients of the model, a i and b i, were estimated by tting the AR model with observation noise (2.6) y n = r n + " n r n = mx i= a i r n;i + u n to the data where apparently only the direct wave or reection wave exists. The state space representation for this model can be obtained by considering the special case when ` = 0 or m = 0 in (2.2). The log-likelihood of this AR(m) (or AR(`)) plus noise model is obtained by (2.7) `( m )=; N 2 log 2 ; 2 NX n= log r n ; 2 where " n = y n ; Hx njn; and r n = HV njn; H T + 2 with x njn; and V njn; being the mean and the variance covariance matrix of the one-step-ahead predictor of the state obtained by the Kalman lter (Jones (980)). 2.3 Estimation of the Time Varying Variance The variances of the autoregressive model for the direct and reection waves, 2 and 2 2, are related to the amplitude of the waves and are actually time varying. Namely, the variance is almost zero before the direct or reection wave arrives, becomes large NX n= " 2 n r n The 3
depending on the amplitude of the wave and then goes back to zero as the signal dies out. These variance parameters play the role of signal to noise ratios, and the estimation of this parameters is the key problem for the extraction of the reection waves. A self-organizing state space model was successfully applied for the estimation of the time-varying variance (Kitagawa 998). In this method, the original state vector x n is augmented with the time-varying parameter n as (2.8) z n =[x n n ] T where the parameter n, for the present problem, is dened by (2.9) n = [log 0 2 n log 0 2 2n] T : The logarithm of the variance is used to assure the positivity of 2 n and 2 2n. We further assume that this parameter n changes according to the random walk model (2.0) log 0 2 j n = log 0 2 j n; + j n j = 2 where j n is the Gaussian white noise with j n N(0 2 j ). The state space model for this augmented state is easily obtained from the original state space model for x n and (2.0). It can be expressed in nonlinear state space model form: (2.) z n = F n (z n; v n ) y n = H n (z n w n ): Then by applying the Monte Carlo lter/smoother (Gordon et al. 993, Kitagawa 996), we can estimate the state z n from the observations. Since the augmented state z n contains x n and n, this means that the marginal posterior distributins of x n and n can be obtained simultaneously and that the variances of the direct and reection waves are obtained automatically. 3. Statial Filtering/Smoothing 3. Time-lag Structure of the Data and Spatial Smoothing The actual time series observed at OBS contains signals of direct waves, reection waves, refracton waves and observation noise. Just beneath the air-gun, the direct wave (compression wave with velocity about.48km/sec.) that travels through the water arrives rst and dominates in the time series. However, since the velocity of the waves in the ground are faster than that in the water (2-8 km/sec.), a refection (or refraction) wave comes rst for the epicentral distance larger than approximately 5 km. As an example, assume the following three-parallel-lay structure: the depth and the velocity of the water layer: h 0 km and v 0 km/sec., the width and the velocities of three layers: h, h 2 h 3 km and v, v 2 v 3 km/sec., respectively. 4
Table. Wave types and arrival times Wave type Arrival time Wave(0 2k; ) v 0 ; q (2k ; ) 2 h 2 0 + D2 Wave(0 2k; ) (2k ; )v 0 ; q h 2 0 + d2 0 + v; (D ; (2k ; )d 0 Wave(0 2k; 2) (2k ; )v 0 ; q q h 2 0 + d2 02 +2v; h 2 + d2 2 + v; 2 d 2 Wave(0232) v 0 ; q q q h 2 0 + d2 03 +2v; h 2 + d2 3 +2v; 2 h 2 + d2 23 + v; 3 d 3 where d ij = v i h i = q v 2 j ; v 2 i, d 2 = D ; (2k ; )d02 ; 2d2, d3 = D ; d03 ; 2d3 ; 2d23. Wave 0 Wave 000 Wave 02 Fig. 2. Left: Examples of wave types: Wave(0), Wave(000) and Wave(02) Right: Arrival times of various waves. Horizontal axis: epicentral distance D (km), vertical axis: arrival time t ; D=6(sec.). From bottom up in vertical axis, Wave (0), (0), (02), (000), (000), (0232), (0002), (00000), (000232). The wave path is identied by the notation Wave(i i k ), (i j = 0 2 3), where Wave(0) denote the direct wave (compression wave) that travels directly from the air-gun to the OBS, Wave(0) denotes the wave that travels on the surface of the bottom of the sea, Wave(0002) denotes the wave that reected at the bottom and the surface of the sea and travels through the surface between the rst and the second layers (Figure 2). Table shows the travel times of various waves. At each OBS these waves arrive succesively (Telford et al. (990)). Figure 2 shows the plot of the arrival times, t, versus the epicentral distances, D, for various wave path. The parameters of the 3-layer structure are assumed to be h0 = 2:km, h = 2km, h2 = 3km, h3 = 5km, v0 = :5km/sec, v = 2:5km/sec, v2 = 3:5km/sec, v3 = 7:0km/sec. It can be seen that the order of the 5
Table 2. Wave types and delay of arrival times for various epicentral distance Wave type Epicentral Distance (km) 0 5 0 20 50 Wave(0).2 3.5 33.4 33.9 34. Wave(0 3 ) 0.4 2.2 28.9 32.6 33.9 Wave(0 5 ) 0.2 4.7 23.5 30.2 33.4 Wave(0) 20.5 20.5 20.5 20.5 Wave(02) 4.6 4.6 4.6 4.6 Wave(0232) 7.3 7.3 7.3 arrival times changes in a complex way with the horizontal distance D, even for such simplest parallel ray structure. 3.2 Dierence of the Arrival Time and Spatial Smoothing At each OBS, 982 time series were observed, with the location of the explosion shifted by 200 m. Therefore the consecutive two series are cross-correlated and by using this it is expected that we can detect the information that was dicult to obtain from a single time series. Table 2 shows the dierence of the arrival times between two consecutive time series, computed for each wave type and for some epicentral distance, D. The dierences of the waves that travel on the surface between two layers, such aswave(0) (02), (0232), are the constants independent on the epicentral distance D. The delay time becomes small for deeper layer or faster wave. On the other hand, for the direct waves that path through water, such as Wave(0), (000), (00000), gradually increases with the increase of the epicentric distance D, and converges to approximately 34 for distance D > 0km. This indicates that for D>0, the arrival time is approximately a linear function of the distance D. Taking into account of this fact, we consider the following model. In this modeling, time series structure is ignored and we use (3.) s n j =2s n;k j; ; s n;2k j;2 + v n j y n j = s n j + w n j where k is the dierence of the arrival times between channels j ; and j. By dening the state vector by x n j =[s n j s n;k j;] T we obtain the state space representation, x n j = Fx n;k j; + Gv n j y n j = Hx n j + w n j : Therefore, if the delay time k is given, we can easily obtain estiamtes of the \signal" by the Kalman lter and the smoother. If one value of k dominates in one region, we can estimate it by the maximum likelihood method. However, in actual data, several dierent waves may appear in the same time and the same channel. 6
To cope with this situation, we consider a mixture-lag model dened by (3.2) KX s n j = n j k^s n j k k= y n j = s n j + w n j where ^s n j k is the one step ahead predictor of the s n j dened by ^s n j k = 2s n;k j; ; s n;2k j;2 and n j k is the mixture weight at time n and channel j. In the recursive ltering, this mixture weight can be updated by (3.3) n j k / n;k j; k exp n ;(y n j ; ^s n j k ) 2 =2r njn; o 4. Space-Time Filtering/Smoothing In general, the sequential computational method for ltering and smoothing cannot be extended to space-time ltering/smoothing problems. However, for our special situation where a signal propagates in one direction, a reasonable approximate algorithm can be developed. We consider a multi-variate version of the decomosition model in (2.) (4.) y n j = r n j + s n j + " n j where r n j, s n j and " n j denote the direct wave, reection wave and the observation noise component of channel j. The direct wave and the reection wave components are assumed to follow AR models (4.2) mx r n j = i= `X a i j r n;i j + v r n j s n j = i= b i j s n;i j + v s n j : Considering the delay structure, we also use the model (4.3) r n j = r n;kr j; + u r n j s n j = s n;ks j; + u s n j : An approximate estimation algorithm can be developed by combining the ltering and smoothing algorithm in time and in space (channel). However, as mentioned in section 3, we need to consider mixture of verious wave types (or various time lag components). 5. Summary Several methods for the analysis of OBS data for exploring underground structure are shown. In time sereis decomposition, a smoothness prior approach is taken. In spatial and spatial-temporal ltering/smoothing the delay structure of various types of waves are considered. In the presentation, we shall show the numerical results of the actual OBS data. 7
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