Improving Sparse Network Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events

Transcription

1 Bulletin of the Seismological Society of America, 9, 1, pp , February Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events by Stephen C. Myers and Craig A. Schultz Abstract Monitoring the Comprehensive uclear-test-ban Treaty will require improved seismic location capability for small-magnitude events. The International Monitoring System (IMS) is well suited to locate events that are large enough to be recorded at teleseismic distances. However, small events are likely to be recorded on a sparse subset of IMS stations at regional- to upper-mantle distances (less than 3 ), and sparse-network locations can be strongly effected by travel-time errors that result from path-specific velocity model inaccuracies. In an effort to improve sparse network location capability, we outline a procedure that applies empirical corrections to travel times determined with an appropriate velocity model. More specifically, Bayesian kriging and calibration events (constrained with a global network) are used to estimate epicenter-specific travel-time corrections. For a test (sparse) network of stations, we calculate travel-time residuals for the calibration events relative to the ak135 velocity model. Travel-time residuals are assigned to the respective calibration epicenter, forming a set of spatially varying travel-time correction points. The spatial set of correction points is declustered to reduce the dimension of the observations with minimal reduction in accuracy of the travel-time corrections. We then use the declustered set of calibration points and Bayesian kriging to form continuous traveltime correction surfaces for each station of the test network. The effectiveness of travel-time correction surfaces is evaluated by locating, with and without corrections, a subset of the 1991 Racha earthquake sequence (Caucasus Mountains), for which we have accurate locations that were independently determined with a dense local network. When no travel-time correction is applied, the mean horizontal distance between the local and test network locations is km, and there is a distinct bias in sparse-network locations toward the north-northwest. The mean difference between local and sparse network locations is cut to 13 km when corrections are applied, and the bias in location is significantly reduced. When calibration events in the Racha vicinity are not used to make the correction surfaces, there is still a significant improvement in location, with mean mislocations of 15 km. When corrections are not applied, only one of the locally determined locations lies within the associated 9% coverage ellipse determined with the test (sparse) network. However, by using traveltime corrections and estimates of model uncertainty determined using kriging, representative error ellipses are obtained. This study demonstrates that kriging correction surfaces based on global-network-constrained calibration events can improve the ability to accurately locate lower magnitude events while providing representative coverage ellipses. Introduction Determining accurate seismic locations with representative uncertainty estimates is a fundamental step in seismic investigation. Location is one of the most basic attributes in seismology, frequently providing a starting point for subsequent investigation. In addition to being the first step in many seismic studies, seismic-event location is a topic unto itself, with applications ranging from seismotectonics to seismic risk. Accurate seismic location will be particularly important for monitoring the Comprehensive uclear-test- Ban Treaty (commonly referred to as the CTBT). Seismic monitoring will constitute one of the first lines of detection and location for underground nuclear explosions, and seis- 199

2 S. C. Myers and C. A. Schultz mic locations may be called upon to guide ensuing CTBT investigations such as on-site inspections. One goal of seismic monitoring is to locate events of all magnitudes as accurately as possible. The International Monitoring System (IMS) is well suited to locate events that are large enough to be recorded globally; however, as magnitude decreases, signals are likely to be recorded on a subset of the IMS at regional- to near-teleseismic distances (less than 3 ). At these distances, sparse-network locations are subject to significant error resulting from strong regional variations in crustal and upper-mantle velocity structure. If seismology is to effectively guide CTBT investigations for small events, then accurate sparse-network locations must be achieved in regions with significant three-dimensional heterogeneity. A number of techniques can be used to improve seismic location. For instance, improvement can be achieved by more accurately predicting the travel times of seismic phases that are used in location. Velocity models are fundamental in predicting travel times, and Smith and Ekström (1996) demonstrate that locations can be improved by using a more general three-dimensional model to predict travel times as opposed to a one-dimensional model. However, even threedimensional velocity models are designed to minimize a network average of residuals, leaving room for improved traveltime prediction along a specific ray path. In addition to improved velocity models, other studies demonstrate that corrections to travel-time predictions based on nearby calibration events can lead to significant improvement in location (e.g., Herrin and Taggart, 1966; Hales et al., 1968; Lienert, 1997). In another technique the travel-time-correction method can be refined by determining the optimum set of station correction for a group of events. This is the basic concept behind relative location techniques such as joint hypocenter determination (JHD) (i.e., Douglas, 1967; Dewey, 197; Jordan and Sverdrup, 1981, Pavlis, 1986; Pujol, 1995). Relative location techniques are appropriate for events that have highly correlated station corrections. Two scenarios in which relative location techniques can be highly effective are (1) when the events are in close proximity to one another, resulting in similar ray paths to network stations (and therefore similar travel-time residuals) (e.g., Dewey, 197); and () when travel-time-prediction errors are dominantly the result of a near-surface-velocity anomaly that affects all arrivals at a given station (e.g., Pujol, 1995). Both travel-time correction and relative location techniques are appropriate if travel-time corrections for the calibration events and the event of interest are highly correlated. In CTBT monitoring, however, an event of interest could occur anywhere, and the closest calibration event may be hundreds of kilometers away. In this case, travel-time residuals for the calibration events and the event of interest will only be partially correlated. It is in this case that statistical interpolation techniques are invaluable (Schultz et al., 1998) because they can accurately account for a more general covariance between the travel-time residuals of spatially separated events. In this study, we explore the use of epicenter-specific travel-time corrections to account for travel-time bias. The most fundamental correction is a static (bulk) correction that is applied to all arrival times at a station. Station corrections can be refined by defining region-specific corrections that account for broadscale-velocity-model inaccuracies on the source side of the ray path (e.g., Engdahl et al., 1998). The resolution of the source-side correction can be further increased such that the correction becomes epicenter specific, in which case a correction surface is defined. There are many methods that can be used to formulate correction surfaces, but Schultz et al., (1998) argue the merit of kriging to interpolate a set of calibration events to form a continuous correction surface. Kriging provides an optimal estimate of the travel-time correction, thus improving the travel-time prediction accuracy. Unlike many other methods for determining correction surfaces, kriging provides an estimate of the travel-time correction uncertainty, thus characterizing the post-correction error distribution. The kriging method has been used to correct other seismological measurements, such as amplitude. Rodgers et al., (1999) and Philips (1999) both compare kriging to other methods of amplitude correction such as distance corrections, path-specific crustal waveguide parameters, and cap averaging (running-mean smoothing), and in both studies kriging was found to be the single best predictor of amplitude residuals. We test the improvement in sparse-network location achieved using kriging travel-time-correction surfaces by measuring the mislocation vectors relative to benchmark events. We also test whether the error ellipses for the test data set are representative of the true uncertainty. The test data set is the 1991 Racha earthquake sequence, which occurred in the Caucasus Mountains between the Black and Caspian Seas. The M s 7. mainshock was followed by a vigorous aftershock sequence that prompted the deployment of a dense local network (Fuenzalida et al., 1997). Using the dense local network, many of the Racha events are precisely located, with error bounds of less than km. These high precision locations provide an opportunity to benchmark the accuracy of event location under normal monitoring conditions. Under such conditions small magnitude Racha events would typically be recorded on a sparse network, consisting of stations at regional to near teleseismic distances. We use a sparse test network of six stations (Fig. 1) to locate a subset of the Racha events with and without kriging correction surfaces. We then use the local network epicenters (Fuenzalida et al., 1997) as benchmarks to test the accuracy of the testnetwork locations. The correction surfaces are based on calibration events throughout the Middle East that are located with a global seismic network (we refer to these events as teleseismically constrained ). Although the highest quality calibration events are always desirable, in many regions of the world teleseismically constrained events will be the most precise source of calibration available. Because there is a degree of uncertainty in these calibration events themselves, we develop methods to propagate uncertainty from calibration events through to the correction surfaces and ultimately

3 Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events Stations used to relocate Racha events Racha vicinity KAS KVT Turkey Iran ARU SVE Former Soviet Union GAR KHO 3 3 E E 5 E 6 E 7 E Figure 1. Map showing the test, sparse-network stations, and the area covered by the 1991 Racha earthquake sequence. The test network is meant to be representative of an IMS station configuration for small-magnitude events. to the sparse-network locations. We show that sparse-network locations can be improved by using correction surfaces based on kriging and teleseismically constrained calibration events. We also find that characterization of picking and travel-time-prediction error, which is part of the kriging method, results in representative error ellipses. These findings validate kriging with teleseismically constrained calibration events as a method to improve IMS locations for small events. Data The 1991 Racha earthquake sequence provides a test data set to evaluate sparse network locations that make use of kriging correction surfaces. There are three sets of locations used in this study: (1) sparse-network locations that are used to test the improvement achieved using kriging corrections; () teleseismically constrained calibration locations that are used to construct the correction surfaces at the sparse-network stations; and (3) local-network locations, which we use as benchmarks to test the sparse-network locations. The Sparse etwork Seismic stations ARU, GAR, KAS, KVT, and SVE comprise the test (sparse) network used to relocate the Racha events (Fig. 1). This network forms three station pairs at distinct distances and azimuths from the Racha vicinity. The primary reasons for choosing this configuration are (1) an extensive database of calibration events is available for each station, which enables the construction of correction surfaces; () it allows a test of the consistency between correction surfaces for nearby stations; (3) the absence of an arrival pick at one station does not significantly change the geometry of the test network, thus increasing the number of usable events. The average distances from the Racha events to stations ARU, GAR, KAS, KHO, KVT, and SVE are 1.7, 5., 11.,.8, 1.8, and 18.1, respectively. Stations KAS and KVT are at far-regional distance, with the travel times strongly affected by crustal and shallow mantle properties along a path that crosses the southern margin of the Black Sea. At stations ARU, GAR, KHO, and SVE, arrivals from the Racha area are strongly affected by upper-mantle-velocity discontinuities (upper mantle distance). Observed arrival-time picks come from two sources. Lawrence Livermore ational Laboratory analyst picks are used at stations where we were able to acquire data: all of the test events at station ARU and 3 events at station GAR. In all other instances International Seismic Center (ISC) picks are used (Engdahl et al., 1998). Calibration Data High-quality-calibration events are the corner stone on which the kriging correction surfaces are built. Ideally, a dataset of spatially distributed events with perfectly known locations would be used for calibration. Unfortunately, this type of global dataset is not currently available, and the effort to develop such a dataset would be prohibitive. Therefore it is important to evaluate the effectiveness of calibration events with locations that are not exactly known. Candidate calibration sources range from events recorded on a dense local network, for which epicenter uncertainty may be as low as 1 km, to events located with a global network, for which epicenter uncertainty is on the order of 1 to km. Future studies will likely combine calibration events with varying degrees of precision. However, the least precise calibration events (constrained with a global network) will have to suffice in some instances. It is therefore important to test the performance of corrections based on these events alone. In this study we use high-quality teleseismically constrained hypocenters as calibration events (Engdahl et al., 1998). These events were located using reassociated ISC picks, making use of multiple arrivals including depth phases. Billings et al. (199) found that location estimates based on global datasets with multiple-phase arrivals are far less susceptible to bias than locations based on only one or two phases, suggesting that the data set of Engdahl et al. (1998) is minimally biased and suitable for use in calibration. Engdahl et al. (1998) report that differences between their relocations and a worldwide set of known locations is about 1 km on average, and this degree of precision is confirmed in the Racha area by comparing locations for individual events that have both teleismic and local network locations. Depending on the station, the correction surfaces in this study make use of 5 to 15 teleseismically constrained calibration events that span the Middle East. For

4 S. C. Myers and C. A. Schultz most of the stations, there were about 1 P-wave arrivals, which provided a rich set of calibration sources throughout the region. Benchmark Data Set Sparse-network locations are evaluated relative to epicenters determined with a dense local network, for which epicenter uncertainty is generally less than km (Fuenzalida et al., 1997). The dense seismic network was in place within two weeks of the mainshock, providing precise locations for many low-magnitude events. Fuenzalida et al. (1997) report that Racha events occurring prior to the installation of the temporary network (including the mainshock) were relocated using permanent stations and travel-time corrections based on events that were recorded on both the dense temporary network and the permanent stations. The permanent network included three stations in Georgia that are local to the Racha earthquake sequence. Thirteen of the events had arrivals at enough of the test-network stations to afford a location (Table 1). Test-event magnitudes range from 7. to., as determined using the coda envelope method of Mayeda and Walter (1996). Event depths range from.3 to 13.3 km, with a mean of 7. km. Four of the calibration events were located with both the local and teleseismic networks (Table 1). However, it is important to note that the global (calibration) and local network (benchmark) locations were determined independently. either the calibration events nor the sparse network locations make use of the local network for either arrival times or travel-time corrections. Methods We relocate a subset of the 1991 Racha aftershocks, with and without the aid of spatial correction surfaces. The sparse-network locations are then compared to epicenters determined with a local network. Three tests cases are evaluated as follows. (1) Correction surfaces are not used; this is the standard approach where travel times and distancedependent uncertainties are determined using the ak135 velocity model (Kennett et al., 1995). () Travel-timecorrection surfaces are used, and the surfaces make use of all the calibration events, including events in the Racharegion. This is the optimal situation with calibration events in close proximity to the test events. (3) Lastly, correction surfaces are used, but the surfaces are made without events in the Racha region (excluding events inside the box defined by latitudes between and 3 and longitudes between 3 and.3 ). This is a more typical monitoring situation in which calibration events are some distance from the test events, and appreciable interpolation is required. We relocate events using a modified version of the EvLoc program provided by Sandia ational Laboratory. See the article by Bratt and Bache (1988) for details. This routine provides the flexibility to determine error ellipses using a Bayesian approach with end members defined by (1) travel-time-residual misfit (F statistic) and () an a priori distribution for picking and travel-time prediction error (chisquared statistic). For sparse data sets, which we are interested in here, Evernden (1969) argues that error ellipses based on the F statistic can be unrepresentative, because this approach unrealistically inflates the error ellipse as the num- Table 1 Local etwork Locations for Racha Aftershocks with Distance and Azimuth to Sparse-etwork Relocations* Local etwork Location o Correction Event Date and Time Lat. Lon. Dist(km) Az( ) Correction Area of o Racha Cal. Ellipse (km ) Dist(km) Az( ) Correction Area of All Cal. Events Ellipse (km ) Dist(km) Az( ) Area of Ellipse (km ) # Max Arrivals Az Gap m b /9/91 9:1: /9/91 15:8: /9/91 17:1: /9/91 18:3: /9/91 :1: /3/91 16:7: /1/91 1:: /3/91 6:8: /3/91 :19: /1/91 9:36: /15/91 1:8: /3/91 8:1: //91 7:59: *The size of the confidence ellipse is given for each of the test cases. It is important to remember that the ellipses for the case with no correction are unrepresentatively small. The number of arrivals used to locate each event, the maximum azimuthal gap for the sparse-network geometry, and the bodywave magnitude are also listed. Correction surfaces made without events in the Racha area. Correction surfaces made with all available teleseismically constrained data, including events in the Racha area. Indicates events with both local network (benchmark) and teleseismic (calibration) locations.

5 Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events 3 ber of observations becomes small. The chi-squared approach, which is preferred by Evernden (1969), is better suited to determine a coverage ellipse for sparse data sets if a priori error distributions for picking and travel-time prediction can be adequately defined. In this study we test the ability to locate with a sparse network, so we adopt the chisquared statistic for determining error ellipses. When kriging correction are not used, we characterize the travel-timeprediction error by using a distance-dependent-error estimate based on the misfit of a global set of travel-time residuals relative to the ak135 predictions. When kriging corrections are used, a conservative uncertainty estimate for the kriging correction (see following section) is used to characterize travel-time-prediction error. When locating the Racha aftershocks we solve for latitude, longitude, and origin time. Depth is fixed at 15 km due to poor vertical resolution. We found that fixing the depth of test events at various depths does not significantly affect our analysis of event epicenters. Bayesian Kriging We use the Bayesian kriging method of Schultz et al. (1998) to produce continuous travel-time-correction surfaces (first arrival P waves) from empirical observations. There are a number of techniques that can be used to develop the travel-time-correction surfaces (see Schultz et al., 1998). However, kriging has the advantage of being a minimum variance solution that provides an accurate estimate of the prediction uncertainty, which can be passed to the location algorithm as model error. Schultz et al. (1998) modified simple kriging (e.g., Srivastava and Isaaks, 1989) for applications to seismological data sets. Standard kriging works well when interpolating a dense set of data points. However, estimation accuracy, especially estimation of prediction uncertainty, decreases where data are sparse and in zones of extrapolation (when all of the observations are off to one side of the predicted point). Because seismic epicenters are commonly clustered along tectonic boundaries, creation of an empirical seismic correction surface necessitates predictions in areas of extrapolation. Schultz et al. (1998) incorporate an a priori statistical model (Bayesian prior) to constrain the kriging prediction in areas of extrapolation. For this study we set the a priori model to the overall mean and variance of the traveltime residuals throughout the entire region. Therefore, rather than use an extrapolated prediction value and an overly optimistic estimate of the uncertainty, the predicted value is a smooth transition between the kriging solution in areas of good data coverage and the background statistics in areas of extrapolation. Although this often has the effect of increasing the prediction uncertainty in areas of extrapolation, this method provides a more accurate statistical representation of the correlation surface. An additional modification to kriging is the incorporation of sample-specific uncertainty (Schultz et al., 1998). Most kriging techniques allow a uniform level of sampling uncertainty, which is referred to as the nugget is geostatistical literature. For seismological data sets, however, sample variance can vary considerably from one observation to the next. For instance, the arrival-time uncertainty for a high signal-to-noise (S/) event is likely to be lower than for a low S/ event. A detailed presentation of the kriging technique can be found in the article by Schultz et al. (1998), and we give a brief overview here. The Bayesian kriging equation (equation [] of Schultz et al., 1998) is: Z*(x ) x {B(x,x )[Z(x ) E(x )]} (1) p ip p i i i i 1 where Z is a spatial random variable (e.g., travel-time-residual surface), which is a function of the location vector x; Z* is the prediction of Z; E is the random sampling error (e.g., picking error), which is sample specific; x is the weight assigned to each observation in the kriging prediction; and B is the blending function that smoothly varies between the simple kriging prediction and the Bayesian prior. The variance of the error for [1], (Z* Z), can be written as: Var[Z*(x ) Z(x )] {x xb(x,x,x ) p p i j p i j i 1 j 1 [C(x i,x j) r(e i)r(e j)]} () {x B(x,x )C(x,x )} C(x,x ) j p j p j p p j 1 where C(x i,x j ) is the covariance of Z at points x i and x j ; B(x p,x i,x j ) is the product of the blending functions B(x p,x i ) and B(x p,x j ); r(e i ) is the standard deviation of the random (uncorrelated) error associated with the ith observation; and is the number of observations bearing on the prediction of Z(x p ). Minimizing () with respect to the weights yields the kriging system of equations, where the ith equation is of the form: j 1 C(x i,x j)b(x p,x i,x j) d ij[c(x i,x j)(1 B(x p,x i,x j ))] [x i] d B(x,x,x )r(e )r(e ) ij p i j i j C(x i,x p)b(xpxx) p i d ip[c(x i,x p)(1 B(xpxx))] p i (3) where d ij is the Kronecker delta (1 if i j and otherwise). See Schultz et al. (1998) for further details and derivation. Solving [3] for the weights allows Z* and var(z* Z) to be predicted at point x p via [1] and [3], respectively. Variogram Modeling. The spatial statistics needed for kriging (e.g., C(x i,x j )) are determined by variogram modeling (see Srivastava and Isaaks, 1989). The variogram is a statistical parameter based on the covariance of values (e.g., travel-time residuals plotted at the epicenter) as a function of the distance between them. The variogram is defined as:

6 S. C. Myers and C. A. Schultz 1 i j i 1 c(d) (Z(x ) Z(x )) () where c is the variogram as a function of distance (d) between points x i and x j, is the number of observations, and Z is the spatial variable (travel-time residual). We follow the common practice of binning variogram values in predefined distance ranges and averaging the variogram values within each bin, providing a robust estimation of the variogram. An iterative least-squares technique is then used to fit an analytical curve (model variogram) to the data variogram. The model variogram is used to calculate the covariance between observations and points of prediction. Assuming Gaussian statistics, the expected value of the variogram can be expanded to give: c(d) ri rj q(d)rr i j (5) where r i and r j are the standard deviations of the random variable Z at the locations x i and x j. If the random process is stationary (statistics do not change across the region) than r i r j, which is the background standard deviation of Z over the entire region. For stationary surfaces, the variogram can be simplified to: c(d) rbg q(d)rbg or c(d) rbg C(d) (6) where C and q are the covariance and correlation structures as a functions of distance (d), respectively. Equation [6] shows that a stationary variogram is just the background variance minus the covariance of points a distance d apart. If the correlation between points is perfect (q(d) 1), then the variogram is. As points become less correlated (q approaches zero), the variogram approaches the variance of the overall population ( r bg ). Four fundamental parameters are determined in variogram modeling: (1) the variance of the background (r bg ), which is the variance of the surface as a whole; () the distance at which the correlation between points becomes zero; (3) the covariance of colocated data, which is the variogram value at zero lag (i.e., an estimate of the random sampling error at a given point); and () the shape of the correlation function as a function of distance. In geostatistical literature, the first three parameters are often referred to as the sill, range, and nugget, respectively. Using (6) we estimate the background variance over the entire region and determine the model covariance as a function of distance. These parameters are used to formulate the kriging covariance matrix (Equation 3). Figure is an example of the data and model variograms for P-wave residuals at the stations used in this study. ote that the variograms do not approach zero for points that are colocated (i.e., nearly coincident data are not perfectly correlated) due to errors associated with determining travel-time residuals. However, it is apparent that the variograms reach minima (correlation is a maximum) for points that are close together, and the variograms increase (correlation decreases) as points become separated by greater distance. Declustering. Variograms at each of the test-network stations (Fig. ) demonstrate that uncorrelated noise is present for nearly colocated points (the zero lag of the variogram). Although reduction of random error by averaging can be achieved in kriging, it is far more efficient to average these essentially redundant calibration points prior to kriging. We term the process of averaging nearby points prior to kriging as declustering. Declustering can be accomplished by spatially binning data, averaging data in the bin, and determining the confidence of the average. By averaging nearly colocated observations, random error is reduced, and a representative mean of many observations can be input to kriging. If the correlation of travel-time residuals within a local bin is high compared to the correlation in the region as a whole, then variations within the bin are primarily associated with random (uncorrelated) processes, and standard statistical formulas can be used to calculate the declustered value (mean) and the confidence of the declustered value (standard error). However, the correlation of travel-time residuals will not be a static value if the bin size is too large, and in this case the correlation of the travel-time residuals cannot be ignored when computing the standard error. We wish to generalize declustering so that the decluster bin size can be arbitrarily large, and therefore cover distances over which spatial correlation cannot be ignored. By allowing for an arbitrarily large decluster bin, we add considerable flexibility to the methodology, while maintaining proper propagating of errors from calibration events to the travel-time corrections and ultimately to the event locations. By treating the travel-time residuals as random variable, the variance of the declustered mean can be formulated as i i i 1 1 s var t(x ) e(x ), (7) where s is the variance of the mean, is the number of data, t is the travel-time residual, and e is the random error associated with picking the seismic phase. Assuming no correlation between t i and e i, equation [7] can be expanded to give: i j i j i 1 j 1 1 s C(t(x ), t(x )) C(e(x ), e(x )) (8) Further, assuming that random picking errors (e) are uncorrelated: 1 s [q ij(d)rr i j drr] ij ei ei (9) i 1 j 1

7 Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events 5 Variogram Variogram 6.5 KAS Distance (degrees) 5.5 KVT Distance (degrees) Variogram Distance (degrees) Stations used to relocate Racha events Racha vicinity Racha KAS Vicinity KVT Turkey ARU Iran ARU SVE Former Soviet Union GAR KHO 3 E E 5 E 6 E 7 E Variogram Variogram 7.5 SVE Distance (degrees) GAR Distance (degrees) KHO 3.5 Variogram 3.5 Figure. Variograms of travel-time residuals for stations used in this study. Residuals are relative to the ak135 velocity model. Crosses are the data variogram values determined in one-degree bins (see text). Solid lines are the model variograms determined by curve fitting. These statistical models were used to construct the kriging surfaces in Figure Distance (degrees) where q is the spatial correlation between data points, r is the standard deviation of the correlated travel-time residuals, r e is the standard deviation of the uncorrelated picking error. Equation (9) shows that the standard error of the declustered value can be decomposed into two end-member components: (1) uncorrelated errors for which s 1/, and () correlated errors for which s r, regardless of. Fortunately, we have estimates for both the correlated and uncorrelated error components from the variogram, with C(x i, y i ) (q(d)r i r j ) c(d) r bg and re nugget. Figure 3 is an example of the raw and declustered calibration epicenters for station ARU. For ARU the number of epicenters is reduced by about 55% during declustering, resulting in more than an order of magnitude increase in computational efficiency when determining the kriged surface. Because the teleseismically constrained calibration events have a bias term in the location uncertainty and averaging does not reduce this bias, we add a bias term to the standard error of the declustered value. We assumed a maximum bias of 1 km (see section on calibration data). For the ak135 velocity model a 1-km-epicentral mislocation can result in a travel-time uncertainty of about 1.5 seconds for the event-station distances applicable to this study. We believe that 1 km bias is representative of the bias distribution at the 95% confidence level. Therefore, we added a 1.5 sec variance to the standard error of the declustered points. Travel-Time-Correction Surfaces Figure shows the kriged P-wave travel-time-correction surfaces for the stations in this study. The amplitude

8 6 S. C. Myers and C. A. Schultz 5 5 a) 95 Events b) 3 Events E E 5 E 6 E 3 E E 5 E 6 E Figure 3. Teleseismically located (calibration) epicenters for station ARU (a) before and (b) after declustering. The number of calibration epicenters before and after declustering are 95 are 3, respectively. ote that the coverage in parts (a) and (b) look similar, even though the number of events is reduced by more than half. This process eliminates redundant data, and in the case of station ARU decreases the kriging computation time by about an order of magnitude. (peak-to-peak) of travel-time-residual surfaces in the Middle East, as estimated at the stations in this study, can be as high as 1 seconds. In the Racha region (a subregion of the kriged surfaces) the maximum correction is 3.8 seconds and most corrections are under 1. second. We emphasize that this study does not sample the correction surfaces at their extrema (maxima or minima). Below we show that the mislocation in the Racha area is km, suggesting that the mislocation associated with the extreme values of the correction surfaces could be in excess of 1 km. Test stations that are close to one another have correction surfaces with similar patterns and amplitudes (Fig. ). This observation holds regardless of whether the stations are at upper mantle or regional distance from the Middle East, suggesting slowly varying deviations from predicted travel times. Although the overall patterns and amplitudes of the correction surfaces are similar for stations that are close to one another, the corrections in the Racha region can differ by 1 second for these stations. For the Racha area, kriging predicts early P-wave arrivals ( and..1) for stations ARU and SVE and late arrivals for stations GAR and KHO (.13.9 to.3.8 seconds). Both of these station pairs are at upper mantle distance. For the two stations at regional distance, kriging predicts early arrivals at KVT (.8. seconds) and late arrivals at station KAS (.5.3 seconds). Although the sign of the Racha-region correction is different at the two regional stations, the magnitude of the difference between the corrections at these two stations is similar to the variation in corrections for nearby upper-mantle-distance stations, and Figure shows that the Racha region is one of the few areas where the kriging predictions are different in sign for stations KAS and KVT. We test the predictive power of the kriging technique by cross validating each of the calibration data sets. Cross validation is accomplished by leaving out one data point and using the remaining points to predict the value at the leftout point. This procedure is repeated for each point in the data set, and a statistical distribution of the misfit between observed and predicted values is constructed. Cross validation results for each of the stations is shown in Figure 5. The average prekriging variance of travel-time residuals is reduced from. sec to 3.17 sec after kriging (see Figure 5 for individual station statistics). The reduction in variance of the travel-time distributions demonstrates that kriging provides good predictions even in the presence of considerable uncorrelated noise. It is important to note that kriging does not attempt to reduce the uncorrelated component of the travel-time-residual distribution. However, the postkriging error distribution, which is primarily uncorrelated noise, is accurately characterized (see demonstration in Schultz et al., 1998), enabling proper propagation of error to the location algorithm. Variogram modeling in Figure shows that the uncorrelated noise ranges from 1.5 seconds to 3 seconds, which is in rough agreement with the postkriging distributions. Results Figure 6 and Table 1 demonstrate the improvement in location that can be achieved by using travel-time-correction surfaces. Uncorrected locations are biased toward the northnorthwest with a mean mislocation of km. When the kriging corrections are applied, the mislocation is reduced to 13 km on average, which constitutes an improvement of about 7%. A more rigorous test of the kriging corrections is conducted by removing all calibration events in the Racha aftershock region before determining the correction surface

9 Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events 7 Figure. Correction surfaces determined using the Bayesian kriging method of Schultz et al. (1998). Travel-time residuals from teleseismically located events were used as calibration. ote that correction surfaces for nearby stations are similar, suggesting that coherent variations between the one-dimensional velocity model and the true velocity structure cause the travel-time-residual pattern. Kriging-determined traveltime corrections and model uncertainties were used when locating the Racha aftershocks. (Fig. 6b). This results in a mean mislocation of 15 km, which constitutes an improvement of 65%. This improvement is encouraging, considering that the calibration events are accurate to within 1 km. Figures 7 and 8 demonstrate that coverage ellipses are representative of the true location uncertainty when the uncertainty of the kriging travel-time correction is propagated through the location algorithm as travel-time-prediction error. For the case with no correction, we use a distance-dependent travel-time prediction uncertainty applicable to the ak135 velocity model. This distance-dependant estimate of model uncertainty is derived from the fit of the predicted travel-time curve to a worldwide set of arrival-time picks. Figure 7 shows that the 95% coverage ellipses obtained us- ing this method are too small, with only a few of the coverage ellipses encompassing the associated benchmark epicenter. However, when kriging corrections are used as model error, the 95% coverage ellipses encompass the known location for all of the events, regardless of whether the correction surface is created with or without events in the Racha region. In a more rigorous test, we compute the coverage ellipses at a level of 1%, %... up to 9%, and count the number of times the local network location lies within the test network ellipse (Fig. 8). If the coverage ellipses are representative of the true error, then the confidence histogram should increase linearly from zero ( confidence) to the total number of events considered (confidence of 1). In this test case, coverage ellipses based on the ak135 error

10 8 S. C. Myers and C. A. Schultz umber of Occurances Before Kriging After Kriging Station = ARU Var=6.s Var=3.58s Before Kriging After Kriging Station = GAR Var=3.93s Var=.8s 6 6 umber of Occurances umber of Occurances Before Kriging After Kriging Station = KAS Before Kriging After Kriging Var=6.18s Var=.86s Station = KVT Var=.13s Var=3.6s Travel-Time Residual (s) Before Kriging After Kriging Station = KHO Before Kriging After Kriging Var=3.5s Var=.6s Station = SVE Var=.6s Var=3.37s Travel-Time Residual (s) Figure 5. Distributions of uncorrected and corrected travel-time residuals. Corrected residuals are determined using cross validation, in which a kriging prediction is calculated at a point that is excluded from the data set. Repeating this procedure for each point and taking the difference between observed and predicted values develops an error distribution that is used as a noncircular test of travel-time prediction improvement. The variance of travel-time residuals is reduced from an average prekriging value of.73 s to 3.7 s. distribution are not representative of the true error. When the kriging correction and uncertainty are propagated through the location algorithm, there is good agreement between observed and predicted occurrences of events lying within the coverage ellipse. Discussion and Conclusions In this study we demonstrate the improvement in sparsenetwork location that can be achieved by using travel-timecorrection surfaces determined using Bayesian kriging and teleseismically constrained calibration events. The traveltime residuals and uncertainties for calibration events are propagated through declustering and kriging algorithms to produce travel-time-correction surfaces that include a measure of the correction uncertainty. The travel-time corrections are applied during sparse-network location (Fig. 1) to remove path-specific, travel-time prediction error that results from inaccuracies in the one-dimensional (ak135) velocity model. Uncertainty in the kriging correction is then used as an estimate of travel-time-prediction error, which propagates uncertainty in calibration event locations and uncertainty associated with spatial interpolation of travel-time corrections through to the sparse-network location. It is important to note that the kriging correction surfaces can be based on travel-time residuals for any velocity model, including a three-dimensional model. Therefore, in areas where sufficient data are available to construct a region-specific velocity model, it is desirable to remove trends in the travel-time residuals with improved velocity models, then use kriging to further improve travel-time prediction along specific ray paths. The magnitude of mislocation vectors are significantly reduced when kriging is used to correct for travel-timeprediction inaccuracies. For the test data set, the average difference between the sparse and local network locations is reduced from to 13 km when correction surfaces are used (Fig. 6; Table 1). This technique is robust to the exclusion of nearby calibration events, with the mislocation only increasing to 15 km when calibration events in the Racha area are not used to make the correction surfaces. When no correction is applied, there is a distinct bias in the locations toward the north-northwest. With the application of corrections, the bias is significantly reduced, but it is not completely removed. Relocation bias is probably due to bias in the calibration events themselves, which is known to approach 1 km. Determining representative error ellipses is a crucial requirement for seismic monitoring. If an error ellipse does not adequately describe the random chance that the true location lies within its bounds, then unsound decisions could be made based on the seismic location. On the other hand,

11 Improving Sparse etwork Seismic Location with Bayesian Kriging and Teleseismically Constrained Calibration Events 9 3 Locations from local seismic network Locations from sparse test network o Correction Average mislocation = km Corrections Made Without Events in Racha Area Average mislocation = 15 km Corrections Made With All Events Average mislocation = 13 km Latitude 5 Bias Bias is Significantly Reduced when corrections are applied 3 a) b) c) 3 E 3 3 E E 3 3 E E Longitude Figure 6. Sparse network locations are significantly closer to the benchmark (local network) epicenters when kriging correction surfaces based on teleseismically constrained calibration events are applied. (a) Sparse-network locations for the 13 Racha events determined without correction surfaces are an average of km from the associated benchmark location and there is a strong mislocation bias towards the northnorthwest. (b) When events in the Racha area (see text) are omitted during construction of the correction surfaces, test events relocated using correction surfaces are mislocated by 15 km on average. This demonstrates the effectiveness of kriging correction surfaces in regions of interpolation between calibration events. (c) When all of the teleseismically constrained calibration events are used, including events in the Racha area, the event mislocation is reduced to 13 km on average E E o Correction Corrections Made Without Events in Racha Area Corrections Made With All Events 3 Latitude a) b) c) 3 E E 3 E E Longitude Figure 7. When kriging corrections are not used during event location, 95% confidence ellipses are unrepresentative of location error (assessed with benchmark locations). When corrections and estimates of model error from kriging are incorporated into the location, the location coverage ellipses are representative of the true error (see Figure 8 for quantification). 3 E E

12 1 S. C. Myers and C. A. Schultz # Events Within Confidence Ellipse a) o Correction Expected Result b) Correction Without Racha-Area Events c) Correction With All Events Error Ellipse Confidence Interval Figure 8. The use of Bayesian kriging corrections during location results in coverage ellipses that are representative of the true error. Conversely, coverage ellipses are not representative when kriging corrections are not applied. The histograms show the number of benchmark epicenters lying inside the 1%, %,...upto9%coverage ellipse determined with the test sparse network. (a) When model error is estimated using a distance-dependent variance of global travel-time observations relative to ak135 predictions, all but two of the events lie outside of the 9% coverage ellipse. (b) Using the uncertainty of the kriging correction as model error, the coverage ellipses are far more representative, even when calibration events in the Racha area were not used to construct the correction surfaces. (c) By including calibration events in the Racha area, the coverage ellipses are even more representative. This result suggests that the use of kriging correction uncertainty as model error during location can bridge the gap between formal and actual confidence ellipses. if error ellipses are representative of the true uncertainty, then uncertainty can be propagated to studies that make use of the location. We use a chi-squared statistic to compute error ellipses for sparse network locations (see Methods section), and this approach assumes that travel-time predictions are unbiased and that picking and travel-time prediction error distributions are accurately characterized. When distance-dependent travel-time uncertainty for the ak135 velocity model is used, only a few of the benchmark events lie within the associated 95% coverage ellipse, suggesting that the travel-time-error distribution is not adequately characterized for the test case. Conversely, characterization of travel-time-prediction uncertainty via variogram modeling and propagation of error through the travel-time-correction procedure allows us to obtain representative error ellipses (Fig. 7 and 8). The kriging method minimizes travel-timeprediction bias and characterizes the residual error distribution, therefore providing the proper front end for algorithms that make use of a chi-squared statistic to calculate error ellipses for sparse network locations (e.g., Evernden, 1969; Jordan and Sverdrup, 1981; Bratt and Bache, 1988). The use of kriged travel time corrections, as presented above, is distinctly different than relative location techniques. Relative location techniques (e.g., Douglas, 1967; Dewey, 197; Jordan and Sverdrup, 1981, Pujol, 1995) minimize the effect of velocity model inaccuracies by finding optimal station corrections that best fit travel-time residuals for a set of calibration events or a single master event. This methodology assumes that station corrections are highly correlated for all events in the data set. The kriging method stands apart from relative location techniques, because kriging allows for a more general (spatially varying) travel-timeresidual covariance structure. By making use of the more general travel-time-residual covariance structure, kriging can take advantage of calibration events that are hundreds of kilometers away from an event of interest to determine an optimal travel-time correction. Therefore, the kriging method is better suited to monitoring situations where traveltime corrections from spatially distributed calibration events are used to find the optimum location for an arbitrary event in a large region. An important conclusion of this study is that teleseismically constrained calibration events can be used to improve sparse-network location. Validating the usefulness of globalnetwork locations for travel-time corrections vastly increases the number and coverage of calibration events, allowing improved sparse-network performance throughout the world s seismogenic regions. It is important to note that the use of regional-network locations and calibration explosions, both of which improve upon the precision of teleseismically constrained locations, can further improve sparse-network locations; higher precision calibration events afford higher precision travel-time corrections. Additionally, globally recorded events could be more accurately located by constructing correction surfaces based on regional-network locations and calibration explosions. Just at sparse network locations are seen to approach the accuracy of global-network locations, global-network locations could approach the accuracy of regional-network locations and calibration events. Improvement in sparse-network location is directly applicable to CTBT monitoring. Because the CTBT aims to monitor the world for all nuclear explosions, regardless of magnitude, locations of small events that are recorded on a