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1 Computer Science
Division
University of Memphis
Memphis, Tennessee
38152
(H.C.)
2 Center for Earthquake Research and
Information
University of Memphis
Memphis, Tennessee
38152
(J.-M.C., K.K., S.-C.C.)
3 Department of Earth
Sciences
University of Memphis
Memphis, Tennessee
38152
(J.P.)
4 Institute of Earth
Sciences
Academia Sinica
Nankang
Taipei, Taiwan
(K.K.,
K.-C.C., B.-S.H., Y.-H.Y.)
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Abstract |
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 Top Abstract IntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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Introduction |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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A few relative earthquake location techniques have also been developed in modern times to overcome the discrepancies between the given velocity models and the real Earth. Among them, the joint hypocenter determination (JHD) method has been widely applied to relocate groups of clustered earthquakes using arrival-time information (e.g., Pujol, 1988, 2000). The P- and S-wave station corrections produced during JHD relocation can be used to quantify the lateral variations of velocities in the region (Pujol, 1992). More recently, the double-difference (DD) method has been developed (Waldauser and Ellsworth, 2000) by cross-correlating waveforms to minimize errors in arrival times and by relocating a group of clustered earthquakes using their relative arrival- time differences. Both the JHD and DD methods have successfully provided reliable relative earthquake locations for many regions. However, the JHD and DD methods require clustered earthquakes as an a priori condition. A realistic Earth model is not needed, involved, or resolved in the location process. Thus, neither the traditional earthquake- location programs (e.g., HYPO71, HYPOINVERSE, and HYPOELLIPSE) nor the relative earthquake-location techniques (JHD and DD) are practical and efficient for a single- event location when a 3D Earth model is involved.
Recently, Hauksson (2000) relocated all earthquakes from 1981 to 1998 in southern California using representative 3D VP and VP/VS models derived from a 3D inversion of arrival times from selected local earthquakes (4.3% of total events in the catalog) and from a few surface shots. Lomax et al. (2000) described a probabilistic earthquake- location methodology and introduced an efficient Metropolis–Gibbs, nonlinear, global sampling algorithm to determine local earthquake location over 3D and layered models. Using probabilistic earthquake-location techniques and a 3D VP model with a constant VP/VS ratio, Husen et al. (2003) relocated selected events recorded by the Switzerland seismic network and showed similar results as those obtained from a tomographic inversion. In a routine application for local earthquake location, the travel-time field from a seismic station across the 3D VP model was computed and stored on hard disk (e.g., Husen et al., 2003). In this study, we have developed independently a simple single-earthquake- location algorithm with similar ideas as Hauksson (2000), Lomax et al. (2000), and Husen et al. (2003) to take care of travel time of seismic waves across complicated 3D VP and VS models.
In a traditional 3D tomographic inversion of travel-time data, a selected subset of earthquakes are simultaneously relocated during inversion using the resultant 3D VP and/or VS model. However, it is very common that only a small subset of high-quality earthquakes are selected from the entire earthquake catalog for inversion. Therefore, a large number of smaller events, which are essential for a comprehensive study of spatial and temporal distributions of seismicity, are not selected and must be relocated separately after inversion.
Modern progress in 3D tomographic-inversion techniques and modern advances in seismic networks have offered an opportunity to determine reliable 3D velocity models and earthquake locations (e.g., Shen, 1999; Kim, 2003; H. Chen et al., unpublished manuscript, 2005). Many seismic networks, even the one in southern California, where they could be using Hauksson's 3D models (Hauksson, 2000), are still using a 1D model in their routine earthquake location; and part (but not all) of what is holding them back is how computationally intensive it would be to do all the raytracing during travel-time calculations. In order to reliably and efficiently relocate all earthquakes from a regional earthquake catalog as well as routinely locate every earthquake for any seismic network, it is essential to develop an efficient and stable algorithm to handle travel-time calculation across a 3D Earth model for a single earthquake location, either for previously archived catalogs or for near real- time reports on earthquakes.
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Single-Earthquake Location Using 3D Velocity Models |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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Location Method
Let us assume that an event has been recorded by n stations and that
the arrival time at the ith receiver is
. Let
j, j = 1, 2, 3, and [t with
macron]0 represent the actual unknown hypocentral location and
origin time of the event, and let xj, j = 1,
2, 3, and t0 be the corresponding initial estimates. Under
the assumption that the differences between the actual and initial values are
small, we can write
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| (1) |
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| (2) |
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| (3) |
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| (4) |
ti is the arrival-time
residual
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| (5) |
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| (6) |
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| (7) |
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| (8) |
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| (9) |
t and x are two column vectors; the
superscript T indicates matrix transposition.
In equation (6) the only
unknown is x. After solving for x we use equations
(2) and
(3) to compute new initial
estimates, which in turn gives rise to a new equation
(6). This process is repeated
iteratively until the residual is small enough. To solve equation
(6) we use the least-squares
method. The cost function is
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| (10) |
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| (11) |
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| (12) |
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| (13) |
Introducing the singular value decomposition (SVD) H'
= U
VT (e.g.,
Aki and Richards, 1980), equation
(13) becomes
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| (14) |
Since the initial values used in equation
(1) may be far from the actual
values, we used Levenberg's
(1944) damped least squares,
which has the following cost function:
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| (15) |
2 is a scalar that controls the length of the
solution vector. The damped least-squares solution is given by
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| (16) |
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| (17) |
This formulation is also useful because the condition number of H' may be very large (i.e., H' is ill conditioned), in which case damping decreases it, thus leading to a more stable solution. Damping, however, has the well-known disadvantage that it decreases the resolution of the solution (e.g., Aki and Richards, 1980).
Centering and Scaling
Centering and scaling are commonly used in statistical regression analysis
(e.g., Draper and Smith, 1981)
and has been used by Lee and Lahr
(1975) and Lienert et
al. (1986). Centering of
the data is accomplished by removing the mean values from
t and from each of the columns of H'. Using
the subscript c to indicate centering, we have
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| (18) |
t' is a weighted
vector (see equation 12) and the
weights have been normalized such that their sum is equal to one,
<t'> is a weighted mean. For the element of
H' in the ith row and jth column we have
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| (19) |
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| (20) |
t
can be ignored, which means it is sufficient to solve for the hypocentral
coordinates only.
We define the scaling matrix S as
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| (21) |
:
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| (22) |
, we compute
xcs as follows:
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| (23) |
Calculation of Travel Times
Many different methods have been developed and applied to compute travel
times for seismic waves. In general, the selection of a method for travel-time
calculation depends on the complexity of the velocity model. In this study, the
method of Podvin and Lecomte
(1991) using the finite-
difference technique is selected for travel-time calculation. The method of
Podvin and Lecomte (1991) can be
applied to a 3D velocity model with very significant lateral and vertical
velocity contrasts. The partial hypocenter derivatives can be shown via a
variational argument to satisfy
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| (24) |
Because the finite-difference calculation of travel times through a 3D velocity model is very time-consuming, P- and S-wave travel-time information from every grid point in a 3D grid model covering the study region to every surface station (receiver) can be first calculated and then saved on disk files for later usage. Note that the travel time for a seismic wave from a source to a receiver is the same as that from the receiver to the source. The velocity model and receiver locations will remain the same during earthquake location, and so will the travel-time parameters from all receivers to every grid point. Therefore, travel-time information for any ray path, that is, from a trial hypocenter to a recording station, can be determined efficiently by interpolation from those of the adjacent rays hitting the neighboring grid points, whose travel times have previously been calculated and stored on disk. In this study, a linear interpolation of eight grid points of a cube that contains the trial hypocenter is used for the arrival time and partial derivatives estimation. The simple linear interpolation to compute the arrival time and partial derivatives with regard to the unknowns has been widely used with the finite-difference calculation of travel times (e.g., Benz et al., 1996). When the grid size is carefully chosen, numerical experiments show the inversion is very consistent. By doing so, the computation time required in travel-time calculation for a 3D velocity model during the iteration process of single-earthquake location can be significantly reduced.
To estimate the uncertainty of the hypocenter parameters, we follow this
approach to minimize root mean square (rms) travel-time residual. The
uncertainty in origin time, longitude, latitude, and depth of a hypocenter can
be estimated by perturbing independently each of the hypocenter parameters. When
a parameter is perturbed (e.g., t 
t +
t) a new rms travel-time residual is calculated with respect to
the perturbed hypocenter parameters (e.g., t + t,
x, y, z). The perturbation of a parameter is
determined such that the differences between the new rms travel-time residual
and the final rms travel-time residual reaches ±20%. Assuming the
uncertainty of the origin time and earthquake location follows a normal
distribution, the minimum amount of the positive and negative perturbation is
chosen as the representative uncertainty of the associated parameter.
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Test Examples from the NMSZ in the Central United States |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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The above-mentioned NMSZ earthquakes with more than five reliable P and S picks were selected for a 3D tomographic velocity inversion using the program of Benz et al. (1996), which was revised by P. Shen (personal comm., 1999) to independently invert for VP and VS simultaneously. The selected earthquakes are simultaneously relocated after velocity inversion (H. Chen et al., unpublished manuscript, 2005) and the relocated epicenters are shown in Figure 3. All earthquakes from the original database are then relocated by the newly developed single-event relocation algorithm discussed in this article using the resultant 3D VP and VS models of H. Chen et al., (unpublished manuscript, 2005). The relocated epicenters determined with the new location program are shown in Figure 4. While the distribution patterns of epicenters shown in Figures 2–4 seem to be very similar, there are visible differences in depth views as demonstrated in a few along-strike projections of hypocenters (Figs. 5, 6, and 7). Along the most active central segment of the NMSZ, only randomly selected hypocenters are shown in the along-strike projection (Fig. 6) to make it possible to visualize the hypocenter shift for each individual event. Basically, the relocated hypocenters from the single earthquake location using 3D velocity models are consistent with those obtained from the simultaneous velocity inversion and earthquake relocation. However, the depths of some relocated earthquakes shift visibly from those of the preliminary hypocenters using the horizontally layered PANDA model, particularly in Figures 5 and 7.
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The uncertainties in the earthquake locations for the NMSZ along the x axis (longitude), y axis (latitude), and z axis (depth) are estimated and shown as a function of earthquake number in chronological order in the database (Fig. 8). It is apparent that location uncertainties are, in general, very small, with an average of 0.24 km, 0.30 km, and 0.48 km for ERX, ERY, and ERZ, respectively. Location uncertainties are significantly reduced, mostly below the average value, after the time of earthquake number 500 when the expansion, upgrade, and densification of the modern regional seismic network in the NMSZ was completed. Thus, the shifts in hypocenters shown in Figures 5, 6, and 7 are most likely the consequence of the application of the 3D velocity models in earthquake relocation.
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The 1D PANDA velocity model was built upon the information of well-constrained layer thickness and layer seismic velocity obtained from a few deep and many shallow seismic reflection/refraction lines and from a 1D VP and VS velocity inversion using PANDA data (Chiu et al., 1992). Chiu et al. (1997) showed in terms of statistical results (i.e., rms, vertical error ERZ, and horizontal error ERH) that the PANDA model has provided a very significant improvement in hypocenter determination over other previous models (e.g., Nuttli et al., 1969; Mooney et al., 1983; Andrews et al., 1985; Nicholson et al., 1984). For the first time, images of active faults in the NMSZ could be depicted from the better relocated hypocenters (Chiu et al., 1992, 1997). The homogeneous-layer velocity model beneath the Upper Mississippi Embayment is an appropriate first-order approximation. However, the layer model fails to consider the lateral variations of thickness and seismic velocities inside the crustal layers, especially in the uppermost sedimentary basin. From a JHD analysis of PANDA data in the NMSZ region, Pujol et al. (1997) demonstrated that P- and S-wave station corrections correlate exceptionally well with the lateral variations of the thickness of the thin sediments beneath the stations, revealing that the thin sedimentary basin has a significant impact on earthquake location.
In the Upper Mississippi Embayment, the thickness of the sediments gradually
increases from 0 at the surrounding Paleozoic outcrop boundary to about 1000 m
thickness near Memphis (e.g.,
Chen et al., 1996). Although the thickness of sediments is relatively thin compared to that of all
other crustal layers, the travel times of both P and
S waves inside the sediments cannot be overlooked because of its
extremely low seismic velocity (VP 
1.8 km/sec
and VS 0.6 km/ sec) and the large velocity
contrast with the underlying Paleozoic basement (VP
6.0 km/sec and VS 3.6 km/sec).
Independent of the depth and location of earthquakes, all seismic ray paths will
impinge almost vertically at seismic stations in the NMSZ due to the
extremely low-velocity sediments and the high-velocity contrast across the
bottom of the sediments. For example, a misestimation of 100 m of sediment
thickness in the velocity model will introduce 0.16 sec of travel-time
residuals for the S wave, which will have a significant effect,
particularly on earthquake depth. As an interim solution to accommodate the
lateral variation of sediment thickness beneath the seismic stations, Chiu
et al. (2001) adapted
10 velocity models of different thickness for the uppermost sediment layer to
relocate earthquakes in the NMSZ region.
The thickness of the sediments varies from 200 to 400 m beneath most
seismic stations in the northern Mississippi Embayment. Therefore, the thickness
of the low-velocity sediments was overestimated (h 
250–450 m) in the PANDA model for the northern NMSZ.
The hypocenters beneath the northern NMSZ are expected to be located
shallower than what they should be when the PANDA model is used. An
along-strike cross-sectional view of hypocenters in the northern NMSZ
(Fig. 5) shows that the
relocated hypocenters are deeper in most cases, as expected. The various
horizontal and vertical shifts of the relocated hypocenters between different
events are most probably a reflection of the localized discrepancies between the
PANDA model and the more realistic 3D VP
and VS models (H. Chen et al., unpublished
manuscript, 2005). Therefore, hypocenters, especially their depths, can be
improved either by a simultaneous tomographic inversion and relocation (H. Chen
et al., unpublished manuscript, 2005) or by single earthquake
relocation using 3D velocity models (this study). The thickness of the sediments
is roughly 650 m in the central NMSZ, that is, the sediment is
properly represented in the PANDA model. Thus the relocated
hypocenters as shown in
Figure 6 are very similar
to the original locations using the PANDA model, as expected. On the
contrary, the thickness of the sediments ranges from 700 to 1000 m underneath
those stations in the southern NMSZ, that is, the sediment layer is
underestimated (h 50–250 m) in the PANDA
model. The relocated hypocenters shown in
Figure 7 for a short section of
the southern NMSZ are, in general, shifted toward shallower depths,
again as expected. Therefore, the relocated hypocenters (Figs.
5,
6, and
7) are shifted properly, with
only a few exceptions, from the original PANDA locations to reflect
the deficiencies of the oversimplified homogeneous- layer PANDA
model, which does not represent the sedimentary basin correctly.
The evolution of earthquake location in the NMSZ can be briefly summarized in the following five additional transverse cross-sectional views across the northeast, northwest, north-central, south-central, and southwest segments of the seismicity (Fig. 9). The geometry of the active faults in the NMSZ can be depicted from the better relocated earthquake locations presented in this article. The active faults in the NMSZ consist of the vertical northeast, northwest, and southwest segments and the gently southwest dipping central segment. There are no apparent improvements in the earthquake clusters shown in the AA' and BB' sections even with the 3D models (Fig. 9b). This is probably because of very few local earthquakes along the northeast and northwest segments of the NMSZ and adjacent areas in the northern margin of the NMSZ seismic network. Therefore, there is no sufficient coverage of seismic ray paths to produce high-resolution 3D velocity models beneath the northeast and northwest segments. It is, however, apparent that there are significant lateral variations of fault-zone geometry from north to south in the central segment of the NMSZ, where seismic-network coverage is excellent and seismicity is the highest in the entire region. In order to apply the single-event location program presented in this study for routine earthquake location in the NMSZ seismic network, additional spatial coverage by seismic stations in the northeast and northwest segments of the NMSZ will be needed to improve the resolution of 3D velocity models.
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Test Example from the Hualien Area in Central Eastern Taiwan |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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Since 1991, a modernized island-wide seismic network of 78 three-component stations operated by the Central Weather Bureau (CWB) is responsible for earthquake monitoring in the Taiwan region. In addition to a few CWB stations distributed in the Hualien area, mostly along the north– south-trending tectonic structural belts, a 30-station PANDA II array was deployed in the area (Fig. 10) from 1993 to 1995 (Chen, 1995a). Selected data collected by the island- wide seismic network and the temporary seismic arrays were used to determine high-resolution 3D VP and VS models for the entire Taiwan region (Kim, 2003; Kim et al., 2005).
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All earthquakes in the CWB catalog from 1991 to 2002 were originally located by the island-wide seismic network using a horizontally layered velocity model simplified from a 3D velocity model of Chen (1995b). Chen's model consists of a 3D VP model and a constant VP/VS ratio determined using pre-CWB seismic-network data recorded mostly before 1991. These earthquakes were relocated by the single-event location algorithm presented in this study using the 3D VP and VS models of Kim (2003). The original and relocated CWB catalog data are projected into two transverse cross- sectional views in the Hualien area (Fig. 11). Because of data and methodology differences in the 3D tomographic inversion, the spatial resolution of the 3D VP and VS models of Kim (2003) is superior to that of Chen (1995b) in three major aspects. These aspects include the following: (1) the number of seismic stations covering the same area is more than doubled and all data are three-component in the study of Kim (2003) as compared to that used in Y. Chen (1995), which were mostly single-component; (2) both VP and VS were determined simultaneously and independently in Kim (2003) as compared to VP only in Chen (1995b); and (3) local dense seismic-array data were used in the study of Kim (2003), which significantly increases the spatial resolution of the resultant velocity models, particularly beneath the Hualien region. Because only a few CWB stations are located in the Hualien area, the spatial coverage in the Hualien area is basically poor for local-earthquake location.
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For this test local-earthquake data were selected from the CWB catalog for the period from 1993 to 1995, that is, the same time period when the PANDA II array was deployed. The selected earthquakes were relocated using the single-earthquake-location algorithm presented in this study, the 3D VP and VS models of Kim (2003), and CWB seismic stations. The hypocenters from the original CWB catalog and from the 3D relocation using only CWB stations are projected into two transverse cross-sectional views shown in Figures 11a and 11b, respectively. In spite of the significant improvement of the velocity models and location technique, it is difficult to argue that there is an obvious improvement in hypocenter locations from Figure 11a to Figure 11b when only a few CWB network stations are used. The only differences are that the relocated hypocenters shown in Figure 11b seem to be slightly more clustered and that the clustered seismicity extends to larger depths than in Figure 11a.
While the PANDA II array was deployed from 1993 to 1995 in the
Hualien region, it recorded two to three times more earthquakes than the
island-wide CWB network during the same time period. Most local
earthquakes were very small and either not detected or not locatable by the
CWB network alone. Local earthquakes recorded by the PANDA
II were located by the best available 1D layered model for central Taiwan
(Yeh and Tsai, 1981) and are
shown in Figure 11c along the
two cross-sections used for Figures
11a and
11b. These local earthquakes
have been relocated using the algorithm presented in this study and the 3D
VP and VS models of Kim
(2003), and using CWB
and PANDA II stations. Therefore, these local earthquakes have been
relocated using the best ever seismic-network configuration and the best
possible 3D VP and VS models available
for the region. The westward-dipping planar seismicity from the surface to a
depth of 30 km is slightly more clustered and extends deeper in
Figure 11d than in
Figure 11c, depicting the
geometry of an active fault. This planar seismicity marks the eastern-boundary
fault separating the Central Mountain Range of the Eurasian plate from the
Coastal Range of the Philippine Sea plate
(Kim 2003;
Kim et al., 2005). From
the relocated local seismicity, the width of this boundary fault has further
been reduced to a minimum and its geometry has been better defined
(Fig. 11d) than before (Figs.
11a and
11b). More than half of the
horizontal ground velocity between the converging Eurasian plate and Philippine
Sea plate has been accommodated along this westerly dipping fault
(Kim et al., 2005).
Thus, the seismic-network configuration, the 3D VP and
VS models, and an efficient and stable single-
event-location algorithm constitute three of the most essential components for
efficient and reliable earthquake location.
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Discussion |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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As a rule of thumb in traditional earthquake location, the quality of earthquake location is poor for earthquakes located outside or near the margin of a seismic network. However, Chiu et al. (1997) presented a test case in the NMSZ using PANDA data and concluded that an earthquake outside of a seismic network can still be located reasonably well only if the local and regional crustal velocity models are well determined, even though statistical error estimations for hypocenter location and origin time may indicate otherwise. Thus, the single-earthquake-location technique presented in this article should be applicable not only to local earthquakes but also to regional earthquakes near a seismic network as long as representative 3D VP and VS models for the region are available.
The resolution of a 3D velocity model for a region can be improved, for example, by increasing the spatial coverage of a seismic network, by using a smaller grid size in velocity model, and by improving algorithms for 3D tomographic inversion and for 3D raytracing. Although 3D velocity models are available for many areas (e.g., Hauksson, 2000; Lomax et al., 2000; Husen et al., 2003), the majority of modern seismic networks in the United States and around the world still depend on a horizontally layered 1D velocity model for routine earthquake location. A layered model is always a good first-order approximation. However, modern improvements in seismic instrumentation and an increasing number of seismic stations in most seismic networks have resulted in the collection of earthquake data with high spatial resolution and wide spatial coverage. Such improvements are essential to provide data adequate for the determination of representative regional 3D velocity models. Any future improvement in regional 3D velocity models or raytracing techniques can be easily adapted and applied to the single- earthquake-location algorithm presented in this article. The test example from the Hualien area of central eastern Taiwan demonstrates further that the spatial resolution of 3D VP and VS models and earthquake relocation can be significantly improved by the addition of a dense local seismic array/ network.
In addition, slow computer speed and lack of sufficient disk space have
previously prohibited an effective application of a 3D raytracing technique on
repeated travel-time calculations across 3D velocity models during earthquake
location. Modern advances in computer technology have dramatically improved the
speed of computing travel times across 3D models. Computation times required for
travel- time calculation using 3D raytracing
(Podvin and Lecomte, 1991) and
using the simple method of searching disk files proposed in this study are
estimated for a region of 230 km x 150 km x 17 km. The region was
modeled in terms of 3D models with cubic blocks having sides equal to 0.25, 0.5,
0.75, and 1 km to test the time efficiency and dependence on block size of each
technique. The computations were carried out using a PC computer
running under Linux with a Pentium IV 2.4 GHz CPU and 1 GB
of memory. As summarized in
Table 1 and presented in
Figure 12, the time required
for one travel-time calculation using the 3D raytracing technique of Podvin and
Lecomte (1991) is inversely
proportional to the block size, that is, it varies from 2 sec to 83
sec for block sizes of 1 km and 0.25 km, respectively. However, the time
required for 1000 travel-time calculations using the disk-searching method
proposed in this study is almost a constant, 0.003 sec, in spite of
different block sizes. The computation times required for both methods may vary
significantly when different 3D raytracing methods, different block sizes, or
different computers are used. Nevertheless, the above tests demonstrate that
travel-time calculations using 3D raytracing can be very fast in the modern
computational environment, and that the method proposed here is extremely
efficient, with an increase of about 106 times in computational
speed.
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Recent progress in disk-storage size and faster access time to disk files have made the implementation of the proposed single-earthquake-location algorithm possible. Providing accurate 3D P- and S-wave velocity models for a study region are available, the simple algorithm presented in this article can be efficiently applied for local and regional earthquake location. Most importantly, this single-earthquake-location algorithm using 3D models can be easily adapted by any seismic network for routine earthquake location to produce a high-quality earthquake catalog, which previously seemed to be an impossible goal for any seismic network around the world. With reliable earthquake locations from a routine network operation, it is possible to quickly correlate seismicity with active faults, investigate characteristic features of active faults and their associated seismic hazard, study spatial and temporal variations of seismicity and their implication to precursory studies, and explore regional structural models for tectonics studies.
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Conclusions |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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Acknowledgments |
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TopAbstractIntroductionSingle-Earthquake Location Using...Test Examples from the...Test Example from the...DiscussionConclusionsAcknowledgmentsReferences |
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Manuscript received June 1, 2004
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