Developing a Consistent Travel-Time Framework for Comparing Three-Dimensional Velocity Models for Seismic Location Accuracy

Abstract

Location algorithms have historically relied on simple, one-dimensional (1D) velocity models for fast seismic event locations. 1D models are generally used as travel-time lookup tables, one for each seismic phase, with travel-times pre-calculated for event distance and depth. These travel-time lookup tables are extremely fast to use and this fast computational speed makes them the preferred type of velocity model for operational needs. Higher-dimensional (i.e., three-dimensional—3D) seismic velocity models are becoming readily available and provide more accurate event locations over 1D models. The computational requirements of these 3D models tend to make their operational use prohibitive. Additionally, comparing location accuracy for 3D seismic velocity models tends to be problematic, as each model is determined using different ray-tracing algorithms. Attempting to use a different algorithm than the one used to develop a model usually results in poor travel-time prediction. We demonstrate and test a framework to create first-P and first-S 3D travel-time correction surfaces using an open-source framework (PCalc + GeoTess, https://www.sandia.gov/salsa3d/software/geotess) that easily stores 3D travel-time and uncertainty data. This framework produces fast travel-time and uncertainty predictions and overcomes the ray-tracing algorithm hurdle because the lookup tables can be generated using the exact ray-tracing algorithm that is preferred for a model.

1 Introduction

Historically, seismic location algorithms have relied on simple, one-dimensional (1D, with depth) velocity models for calculating event locations rapidly (e.g., iasp91 (Kennett & Engdahl, 1991), ak135 (Kennett et al., 1995)). The speed of using these 1D models made them the preferred type of velocity model for operational needs, mainly due to computational requirements. Higher-dimensional seismic velocity models (e.g., three-dimensional (3D)) are becoming more readily available from the scientific community (e.g., Ballard et al., 2016a; Begnaud et al., 2020b; Hosseini et al., 2019; Li et al., 2008; Myers et al., 2010; Simmons et al., 2012) and can provide significantly more accurate event locations over 1D models (Begnaud et al., 2020a, 2021a, 2021b; Myers et al., 2015). The computational requirements for full ray tracing using these higher-dimensional models tend to make their operational use prohibitive. The benefit of a 1D model is that it is generally used in the form of travel-time lookup tables, one for each seismic phase, with travel-time predictions pre-calculated for event distance and depth. This simple lookup structure makes the travel-time computation extremely fast (i.e., on the order of milliseconds).

The prevalence of 3D seismic velocity models now allows for direct comparison of seismic locations estimated using different 3D models, thereby providing the capability to compare each model’s location accuracy. Comparing location accuracy for 3D seismic velocity models tends to be problematic because each model is usually determined using different inversion parameters and ray-tracing algorithms. More-over, 3D locations are a fundamentally non-linear problem as each updated location estimate requires recalculation of ray paths and travel times. Attempting to use a different ray-tracing algorithm than the one used to develop a model almost always results in poor travel-time prediction compared to those predictions determined using the original algorithm (Rowe et al., 2009). Fitting various 3D models into a common model format has been attempted with significant effort (Reiter et al., 2012), but this format forces each model to use a similar predictor for travel times, regardless of whether that predictor was originally used to create the model.

To remove this predictor inconsistency, we describe the development of a framework to populate 3D station/phase-specific travel-time lookup surfaces (3DTTLS) with the actual predictions from the relevant ray-tracing algorithm associated with an earth model. By using travel-time predictions from the actual algorithms used to generate a 3D model, we remove an inconsistency that can lead to problems comparing higher-dimensional seismic velocity models used for event location. This framework thus enables fair and consistent treatment of each model.

2 3D Travel-Time Lookup Surfaces (3DTTLS)

Travel-time lookup surfaces have been used for correcting travel times from standard 1D models (e.g., iasp91, ak135) to those predicted by 3D models. Source-specific station corrections (SSSCs) have been used at the International Data Centre (IDC) for many years to correct travel times to those predicted by higher-dimensional models for improved location accuracy and precision (Firbas et al., 1998). The IDC is a component of the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO PrepCom). It receives, collects, processes, analyzes, reports on, and archives International Monitoring System (IMS) data and distributes IDC products to the State Signatories of the treaty [public access possible via virtual Data Exploitation Centre (vDEC) https://www.ctbto.org/specials/vdec (Vaidya et al., 2009)].

The IDC uses SSSCs to do fast travel time predictions for regional phases (i.e., Pn, Sn, Pg, Lg) within ~ 18°–20° epicentral distance. The surfaces are developed as a regular latitude/longitude grid of travel-time corrections to the iasp91 model, including estimates of travel-time uncertainty. The IMS is a relatively static network, so development of SSSCs do not require frequent additions or changes. This SSSC framework is how the IDC integrates the Regional Seismic Travel Time (RSTT) model travel-time predictions (Begnaud et al., 2020b, 2021b; Myers et al., 2010). The regular latitude/longitude grid of the SSSCs can present prediction problems at the poles, where the grid spacing becomes infinitely close. The grid spacing also results in corrections that are not equidistant from any given station (Fig. 1). The SSSC format does not allow for variations in corrections with depth, i.e., the corrections are developed for a single depth only and must be extrapolated for all possible event depths during event relocation.

Fig. 1

Grid spacing and structure of the source-specific station correction (SSSC) format. (Left) The regular latitude/longitude grid structure (4° edge lengths) can create prediction problems at the poles due to varying cell sizes, as well as (right) limiting predictions at equidistant positions relative to a station (example is IMS array SPITS)

To produce full 3D travel-time lookup surfaces, we use the PCalc program [https://www.sandia.gov/salsa3d/software/], an open-source program that acts as a prediction calculator for extracting or populating travel times, azimuth, slowness, etc. at specific source-receiver positions and depths from a tessellated model in “GeoTess” format. Users can also extract model values at specified positions within the model, including computing ray path geometries using full 3D ray bending in a GeoTess earth model (Ballard et al., 2009), such as SALSA3D (Ballard et al., 2016a). The GeoTess framework (Ballard et al., 2016b) is also openly available [https://www.sandia.gov/geotess] and is designed with 3D tessellated points that can store any spatially-defined values. GeoTess grids are composed of two-dimensional (2D) triangular tessellations of a unit sphere with 1D radial arrays of nodes associated with each vertex of the 2D tessellations. Variable spatial resolution in both geographic and radial dimensions is supported (Fig. 2). GeoTess can be used to store any earth model (e.g., RSTT, SALSA3D), even those that were not originally developed using GeoTess, with no loss of information. In addition, GeoTess has been used to store station-specific travel times (like those being described here, generated using any ray-tracing algorithm), including any empirical corrections, as well as other model/prediction types such as seismic amplitudes and attenuation at various frequencies (W. Scott Phillips, personal communication). The development of a GeoTess 3DTTLS does not require PCalc, but PCalc does include various embedded ray-tracing algorithms to create and populate a GeoTess surface, including an implementation of the pseudo-bending algorithm by Um and Thurber (1987) identified as “Bender”.

Fig. 2

An example of GeoTess using triangular tessellations of differing resolutions to achieve variable resolution in the radial direction. Points along the radial profiles depict the positions where model parameter values are represented. From Ballard et al. (2016b)

GeoTess travel-time/uncertainty grids do not have the same problems at the poles as regular latitude/longitude grids (i.e., decreasing area per pixel) and can consistently produce values at equidistant positions relative to any station (Fig. 3). PCalc produces travel-time and uncertainty predictions that can be output as simple text files or in a resulting GeoTess grid that can be used by an event location algorithm. These GeoTess travel-time/uncertainty grids are generally produced as corrections to a standard 1D velocity model (e.g., ak135 in Fig. 4) and allow variations with depth as opposed to legacy SSSCs. The 3DTTLS also include estimates of path-dependent uncertainty, which are critical to obtaining more accurate error ellipse sizes and orientations when used with location algorithms (Begnaud et al., 2020a, 2021a) (Fig. 4).

Fig. 3

(Left) Example of 4° triangular tessellation structure demonstrating equal cell area. (Right) Example of a travel-time correction surface for IMS array SPITS out to 30° distance from the array, based on the SALSA3D model (Ballard et al., 2016a). The circle indicates the distance demonstrated in the SSSC surface in Fig. 1. A triangular tessellation does not restrict any distance-related predictions as SSSCs do

Fig. 4

Example of 3DTTLS for station GERES and the first-P phase at 0 km depth. (Top) Correction to ak135 model travel times in seconds, with faster corrections in blue and slower corrections in red. (Bottom) Values for path-dependent travel-time uncertainty in seconds, with lower uncertainties in blue and higher uncertainties in red

3 Building 3DTTLS

3DTTLS are created for each relevant station/phase combination. In general, the standard surfaces created are for the first-P (Pn, P) and first-S (Sn, S) phases that involve rays that turn in the mantle (Pmantle, Smantle, respectively). These phases were selected to target the regional and teleseismic phase predictions commonly used by the IDC and other agencies. For this study, we do not address the common crustal phases Pg and Sg/Lg, as they require additional theory and study for accurately estimating ray paths/travel times. 3DTTLS can be created for any seismic phase (e.g., PKPdf, PcP, pP) that is needed for event location, so long as the ray-tracing algorithm can adequately model and predict that phase. Surfaces could also be combined to include different models and/or phases during relocation/prediction steps, especially if models are specifically designed for certain phases. For example, one set of 3DTTLS could contain the first-P phase using Model “A”, another with the first-S phase using Model “B”, a third with Pg/Lg phases with Model “C”, and another with core phases using Model “D”, depending on the user’s choice of preferred models for various phases. All phases would not have to be defined with the same model, as most models are not developed for all phases.

A benefit of creating 3DTTLS as a correction to a 1D model is that such a surface can be used for other nearby stations, elements of arrays, legacy stations, or even stations from other networks, within a given station-to-station tolerance specified by the user. For example, Fig. 5 shows a set of stations in an area around IMS station ABKT in Turkmenistan. Specifying an example user distance of 10 km, a 3DTTLS for ABKT could be used for array GEYT, temporary array Geyokcha (GY), and single station VAN. Using a single surface for the same phase at various nearby stations/arrays would minimize the number of surfaces to read when using an expanded network or legacy stations for event location.

Fig. 5

Map of stations/arrays within 10 km of IMS array ABKT. 3DTTLS defined as corrections to a standard 1D model can be used for nearby stations that might be outside a specific network, or even for legacy stations (i.e., stations that are no longer active)

Using PCalc and GeoTess, 3DTTLS can be built with various tessellations and structures according to the needs of the user. A tessellation grid can be refined around a station or in any place specified for a denser grid of points to reflect needed travel time resolution. For example, a grid could be varied to be denser at points closer to a station, with less-dense points beyond a certain distance (Fig. 6), or even defined to be denser around a tectonic feature such as a fault or subduction zone. Surfaces can also use a common base grid to save on memory, so a unique grid does not have to be stored for each 3DTTLS.

Fig. 6

Examples of user-defined variable gridding in a GeoTess grid. (Left) Grid with a base tessellation of 2° spacing with a 0.5° spacing for points within 3° of a station. (Right) Grid with a tessellation of 1° within 100° epicentral distance of a station. Beyond 100° distance, the grid overall has a base tessellation of 64° with a transition in tessellation edge length near the 100° distance border

4 Accuracy of 3DTTLS Versus 3D Ray Tracing

3DTTLS are pre-built with travel-time and uncertainty predictions stored at specific lateral points and depths. As such, the use of 3DTTLS within a location algorithm will involve interpolation around the prediction points to extract values at specific event locations and depths during the location procedure. In order to compare the use of 3DTTLS to full 3D ray tracing, we used the SALSA3D model (Ballard et al., 2016a) [https://www.sandia.gov/salsa3d] to build the 3DTTLS as well as for 3D ray tracing (i.e., “Bender”, Ballard et al., 2008; Ballard et al., 2009; Um & Thurber, 1987) using first-P and first-S phases. SALSA3D is a global model for P- and S-velocities (the S-model is unpublished but publicly available), defined on a 1° GeoTess tessellation, with crust, mantle, and core layers. SALSA3D also includes capability to use the full 3D covariance matrix to calculate estimates of full 3D path-dependent uncertainty.

To compare prediction differences between 3DTTLS and 3D ray tracing, 1000 pseudo “events” were randomly selected from − 75° to 75° latitude so as not to concentrate events at the poles (Fig. 7). We randomly select pseudo-event depths for each station-event pair to be within the crust, as crustal events are more relevant for general explosion monitoring. Selected event-station pairs were created using 157 primary and auxiliary IMS seismic stations. Only first-P event-station paths from 0–100° distance were included, resulting in 91,955 distinct paths, and first-S paths from 0–80°, resulting in 65,037 paths.

Fig. 7

Map of the 1000 pseudo “events” (red circles) and IMS primary and auxiliary stations (blue triangles) used to test comparisons of travel times using 3DTTLS and full 3D ray tracing. Events were selected within − 75° to 75° latitude so as not to concentrate events at the poles

3DTTLS were created for several grid sizes and depth spacings to investigate possible preferred optimized grid setups. For first-P, a tessellation was used with 0.5° spacing within 20° of the station, and 1° spacing from 20° to 100° distance. Grids were rotated within the ellipsoidal Earth so a tessellation point was always located at the station position. For first-S, the same grid setup was used, but only to 80° distance. Two constant depth spacings were used: 2.5 km and 5 km. Topography was included in the surface generation to account for the possibility of events above zero elevation.

Given that event depths are not expected to be deep in all areas of the globe, we developed a seismicity-depth model of the Earth (Fig. 8) to use when creating 3DTTLS, with a minimum depth point reflecting topography or bathymetry and a maximum depth point that reflects seismicity patterns related to tectonic features such as subduction zones (using the International Seismological Centre (ISC) catalog (ISC, 2022)). The maximum depth point in a 3DTTLS was always set to be at least 50 km to allow for variable event depths if necessary and was smoothed to allow a wider area for deeper events if such depths were allowed during relocation.

Fig. 8

Seismicity-depth model used to generate 3DTTLS. (top) Minimum depth based on topography and bathymetry, with negative values (in red) representing elevation above sea level and deeper depths are shown in blue. (bottom) Maximum depth based on seismicity patterns from the ISC catalog, smoothed to allow for broader deep areas. The maximum depth was set to be no shallower than 50 km to allow for variations in depth during relocation procedures. Shallower depths are shown in red and deeper depths are shown in blue

Figure 9 and Table 1 show a comparison of travel-time difference (TTD) with epicentral distance between the 3DTTLS and full 3D ray tracing for first-P and first-S phases. For first-P at 5 km depth spacing, the mean TTD is 25.9 ± 91.2 ms, with a median TTD of 17.7 ± 27.3 s. Because of the 0.5° spacing at regional (≤ 20°) distances, the TTD are slightly smaller (mean 19.5 ms, median: 9.0 ms) than the teleseismic values (mean: 26.2 ms, median: 18.2 ms). For the 3DTTLS with 2.5 km depth spacing, overall and distance-specific TTD values are improved with an overall mean of 4.3 ± 78.7 ms and median of 6.5 ± 16.9 ms. All TTD values are positive indicating that the 3DTTLS predictions are consistently faster than those coming from full 3D ray tracing.

Fig. 9

Comparison of travel-time difference (milliseconds) with epicentral distance between 3DTTLS and full 3D ray tracing (i.e., Bender) for first-P (top 2 rows) and first-S phases (bottom 2 rows) with 5 km and 2.5 km depth spacing. Mean is shown as black lines, median as blue. Values for first-P are more consistent with epicentral distance than for first-S

Table 1 Travel-time differences (in milliseconds) between 3DTTLS and 3D Ray Tracing for phases and depth spacing

The TTD values for first-S show more variation than first-P. Using 5 km depth spacing, the first-S mean is 214.8 ± 370.4 ms with median 142.6 ± 117.1 ms. There is a noticeable shift in mean and median at ~ 20° distance, with the 3DTTLS predicting faster travel times. Reducing depth spacing to 2.5 km shows a significant decrease in TTD values with a mean of 134.3 ± 306.3 ms and median 98.6 ± 84.7 ms, along with a more consistent pattern with distance. For first-S, the standard deviation shows a distinct pattern of increasing values to at least 25° distance, with noticeable variations in mean/median around the triplication distance of ~ 18° to 23°. These variations could be the result of small-scale variations in the velocity model near the turning-point depth that aren’t being captured by the spatial and/or depth resolution chosen for the 3DTTLS.

Given the noticeably high standard deviation patterns in first-S along with the variations out to ~ 25°, we decided to create another version of the 3DTTLS with a finer depth spacing and a 0.5° lateral spacing to 25° distance. Having a finer depth spacing at all depths did not seem efficient, given that variability of vertical heterogeneity is greater at lithospheric scales (i.e., < 100–150 km depth) with respect to deeper structures. Therefore, applying finer depth variations at depths < 100 km should improve the accuracy of travel time corrections. We created variable depth spacing surfaces (Fig. 10), with finer spacing at shallower depths and wider spacing at deeper depths while still adhering to the seismicity-depth model in Fig. 8. The seismicity-depth model constrains the depths used from Fig. 10, allowing for one depth above or below the values for minimum and maximum values in the seismicity-depth model, respectively. This variable depth spacing allows for improved shallow depth resolution while limiting unnecessary depth points. Figure 11 shows the TTD with distance using this variable depth spacing and the additional 0.5° resolution out to 25° distance. For first-P, the mean is lower with a lower standard deviation (2.8 ± 68.4 ms) than for the 2.5 km depth spacing example in Fig. 9, also with a lower median (5.4 ± 13.2 ms) value. For first-S, the mean (136.2 ± 307.7 ms) is similar to the 2.5 km depth spacing example, with the median and spread (92.9 ± 78.8 ms) showing some improvement.

Fig. 10

Example of specific depths used for variable depth spacing in 3DTTLS (includes top and bottom edges). Depths are shown as gray lines, with the inset zoomed to show -6–50 km depths. Specific minimum and maximum depth points for a node depend on the seismicity-depth model from Fig. 8

Fig. 11

Comparison of travel-time difference (milliseconds) with epicentral distance between 3DTTLS and full 3D ray tracing (i.e., Bender) for first-P (top) and first-S (bottom) phases with variable depth spacing (see main text). See Fig. 9 caption for figure properties. Y-axis bounds are the same as in Fig. 9

In order to compare location accuracy between full 3D ray tracing and the 3DTTLS, a set of 2279 events was selected from previous validation tests for the SALSA3D model (Begnaud et al., 2020a). These validation events have epicenters known to within 5 km or better [i.e., ground truth, GT5; Bondár & McLaughlin, 2009; Bondár et al., 2004)] (Fig. 12) and were relocated with the SALSA3D model and the 3DTTLS with variable depth spacing using first-P phases (Pn, P). When relocating using full 3D ray tracing, path-dependent uncertainty is not applied and only a distance-dependent uncertainty can be used, due to the high computation times required for calculating the path-dependent uncertainty from the 3D covariance matrix. Therefore, the relocations using the 3DTTLS were run with the same 1D, distance-dependent uncertainty as for full 3D ray tracing. During standard event relocation, the 3DTTLS can utilize the saved path-dependent uncertainty, thus giving more relevant estimates of error ellipse size and orientation.

Fig. 12

Validation events (2279) for comparing location differences between SALSA3D with full 3D ray tracing and the 3DTTLS. All events have epicenters known to within 5 km or better (GT5) (Begnaud et al., 2020a)

For relocation, we used the LocOO3D algorithm developed by Sandia National Laboratories [https://www.sandia.gov/salsa3d/Software.html] (Ballard et al., 2008, 2009) which permits using standard 1D travel-time tables (e.g., iasp91, ak135), ray-bending through 3D GeoTess models (e.g., SALSA3D, RSTT), as well as 3DTTLS (in GeoTess format) for travel-time prediction. Figure 13 shows a one-to-one comparison of mislocation (relative to the GT5 or better event epicenter) for SALSA3D with full 3D ray tracing compared to the 3DTTLS created from SALSA3D, with event color indicating which model resulted in a smaller mislocation. The number of events with a smaller mislocation for a particular run is indicated in the upper left and lower right corners. The relocation run using the full 3D ray tracing is slightly favored (51.9%) over the run using 3DTTLS (48.1%) with an 86-event difference, demonstrating that the 3DTTLS can produce very similar locations to full 3D ray tracing. Results using 3DTTLS will depend on the lateral and depth resolutions chosen when creating the surfaces.

Fig. 13

One-to-one event comparison of mislocation (to ground truth epicenter) using SALSA3D with full 3D ray tracing (blue) and 3DTTLS from SALSA3D (red). Each relocation run used 1D, distance-dependent uncertainty for consistent comparison. (left) One-to-one plot, with event color indicating which model resulted in a smaller mislocation. The number of events with a smaller mislocation for a particular run is indicated in the upper left and lower right corners. Full ray tracing (i.e., bender), is slightly preferred for 51.9% of the events over using 3DTTLS with 48.1%. One event did not locate for the 3DTTLS, resulting in a total of 2278 common events. (right) 2D histogram of the one-to-one plot, with lower counts shown in blue and higher counts shown in red. The histogram shows that most events fall on or close to the one-to-one line

5 Memory and Computation Speed When Using 3DTTLS

For a given seismic velocity model, a 3DTTLS is created for each station-phase combination, allowing for a distance tolerance for similarly located stations. For the primary and auxiliary IMS stations (157) and the variable depth spacing surfaces described above, the total storage requirements for first-P 3DTTLS (travel-time correction and uncertainty) is 1.30 GB; which averages to be ~ 8.5 MB ± 478 K for each surface. For first-S surfaces, the total storage required is 1.09 GB, averaging to ~ 7.12 MB ± 510 K for each surface. For these types of formatted 3DTTLS, these numbers suggest a relatively minor storage requirement for even tens of phase types (20 phase types ≥ ~ 26 GB total). The denser the lateral and depth grid points, the larger the storage requirements will be. Phases that are only defined within specific distances ranges (e.g., a distance annulus) would be created with higher-resolution tessellation nodes only within those ranges, as higher-resolution nodes are not necessary where the phase is undefined.

For many monitoring agencies like the IDC, computational speed of event location is a consideration. To test the speed of the event location process using 3DTTLS, we determined the calculation speed for relocating each validation event with the various models and setups described above (Fig. 14). The relocations were done using one processor (to remove the multi-thread functionality for this test) on a Linux machine (Intel® Xeon® 64-bit CPU E5-2699 v4 @ 2.20 GHz; RedHat7; network-mounted disk drive, timing buffered disk reads: 394 MB/s), and the load time for the model or surfaces was not included in computation times. Granted, timing tests can be inconsistent given that there could be various computational loads occurring at any given time. The 3DTTLS were read once at the onset of the location algorithm, but relevant surfaces could have been read in as needed depending on which surfaces would be required for an event. For the Linux machine used above, each surface took ~ 0.137 s on average to read, noting that each surface also has a unique grid. Sharing a common grid (~ 2.2 MB) for each surface could reduce the RAM footprint by ~ 40% and reduce storage by ~ 23% (to 995 MB total for first-P surfaces).

Fig. 14

Event calculation time during relocation. (top) Results including full 3D ray tracing using SALSA3D (blue), ak135 (black), and 3DTTLS (red), with the y-axis in seconds (log₁₀). For all runs, the load time of the 3D model or 3DTTLS was not included in the computation times. (middle) Average calculation time ratio for 3DTTLS vs 1D ak135. Each bin was only plotted if it contained at least 10 values. (bottom) Results using the 1D model ak135 (black) and the 3DTTLS (using variable depth spacing) based on SALSA3D (red), with the y-axis in milliseconds

Comparing the relative calculation times between a standard 1D model like ak135 and the SALSA3D 3DTTLS, it is clear that the use of 3DTTLS for locations results in very similar times as those for a 1D model, with calculation times for 3DTTLS relocations about 1.42 times greater than for the 1D model alone (3DTTLS predictions also include a 1D prediction). Doing full 3D ray tracing results in computation times ~ 3–4 orders of magnitude greater than those using a 1D model or a 3DTTLS, without the path-dependent uncertainty information included in the 3DTTLS.

We also tested calculation times using 3DTTLS for a much larger set of events to confirm the above results. Using the IDC Late Event Bulletin (LEB) and IMS stations, we selected all LEB events from 2020 and 2021 (14,565) and relocated them using ak135 and the SALSA3D 3DTTLS, with the same locator parameters used for the validation events above. Figure 15 shows that calculation times using 3DTTLS are still very similar to using a 1D model (ak135), with average calculation times about 1.25 times slower. Thus, 3DTTLS can be used for event relocation with overall computation times on the same order as a standard 1D model.

Fig. 15

Event calculation times for 14,565 IDC LEB events (2020–2021) and IMS stations using the ak135 1D model (black) and the 3DTTLS based on SALSA3D (red). Points are plotted for individual events with the solid line being the average (minimum of 10 values) for each count of defining phases. Relocation calculation times using 3DTTLS were about 1.25 times slower than those calculated using the ak135 model

6 Discussion and Considerations

Using 3DTTLS has the benefit of storing actual travel-time predictions for use during event relocation for model comparison. As stated above, Reiter et al. (2012) performed a model comparison for structure and travel times with models reformatted to fit a defined model parameterization, which could possibly lead to loss of information if the original models do not exactly translate to that parameterization. The defined parameterization enabled calculation of travel times using the Podvin-Lecomte ray-tracing algorithm (Podvin & Lecomte, 1991), which was not used in the original development of all of the compared models. This use of a common ray-tracing algorithm, unfortunately, can create an inconsistency when comparing models for location accuracy. Enabling the use of the preferred ray-tracing algorithm for a model to calculate travel times and uncertainty and storing those values in a 3DTTLS allows a fair treatment of each model. Only then can the location algorithm use the 3DTTLS predictions to calculate a hypocenter without any possible inclusion of bias due to model parameterization and/or ray-tracing algorithm.

The development of the 3DTTLS will depend on determining adequate grid resolution, both laterally and with depth. Each model could require the use of specific grids to predict values comparable to those from full 3D ray tracing. The model developer could be required to test various grid sizes to find an optimal one for locating events. Given that each 3DTTLS is distinct, it is not necessary to create a single grid for all surfaces, but rather a developer could create individual grids for any station/phase combination if needed. The LocOO3D algorithm does not require all grids to be defined in the same manner to use them for event location.

If a 3D seismic velocity model is not supported directly in PCalc, surfaces can still be created for the model and used for location. The grid point coordinates may be extracted from an existing GeoTess surface into a simple text format such that the grid points can be used in another ray-tracing algorithm to calculate the travel time and uncertainty for a particular model. Once the new values have been calculated, the GeoTess surface can be updated with these values and saved as a new surface based on the specified model. By using existing grid points from any surface for a station-phase combination, a new surface can be created for a particular model to allow for direct comparison of travel-time and uncertainty predictions. Since the surfaces can be created for the exact same lateral and depth grid points, a model’s specific ray-tracing algorithm does not have to be coded into a location program. Considerations can be made for the resolution needs of any particular velocity model, but the production of the 3DTTLS allows for direct comparison of location results for any velocity model in a fair and consistent manner.

To demonstrate how various models can be used to generate 3DTTLS, we used PCalc to generate surfaces for the RSTT (Myers et al., 2010) and the DETOX-P2 (Hosseini et al., 2019) models (Fig. 16). The RSTT framework is part of the PCalc software, while we had to use the PCalc full ray bending algorithm (i.e., “Bender”) for the DETOX-P2 model since the model did not come with a stand-alone ray tracer. This is not ideal for DETOX-P2 as the PCalc ray bending algorithm was not used to create the model. 3DTTLS were also created for the LLNL-G3D model (Simmons et al., 2012) using the process described above—extracting the 3DTTLS grid points, using the preferred ray-tracing software that was used to develop the G3D model itself to predict travel times at those points, and then creating a new surface with the resulting travel times (Fig. 16).

Fig. 16

Examples of 3DTTLS for example models and the IMS station USRK for the first-P phase (Pn, P). Surfaces are calculated at 0 km depth. a SALSA3D, b RSTT (version pdu202009Du), c DETOX-P2, and d LLNL-G3D. The SALSA3D and RSTT models can be run natively within the PCalc program. DETOX-P2 used the same ray bender as SALSA3D which was not used when creating the model. The LLNL-G3D surface was calculated using the relevant ray-tracing algorithm used to create the model

Many 3D global velocity models are publicly available to use for event location comparisons, but here we point out two critical questions for any user that specifically wants to use these models for improving seismic location accuracy: (1) Did the model developers produce the model for improved seismic location accuracy or mainly for tectonic studies? and (2) If the developers had improving location accuracy as a goal, did they provide a means to calculate travel times and uncertainties for that model? These questions would need to be answered prior to using any 3D model for comparing seismic location accuracy.

7 Conclusions

3DTTLS can be built for any seismic velocity model for a set of stations/phases to be used for event location if relevant ray-tracing algorithms are available. We demonstrate the development and testing of 3DTTLS for the SALSA3D model for first-P and first-S phases using GeoTess grids within the PCalc open-source program. 3DTTLS can be built to specific lateral and depth grid sizes, with dense grid points wherever the model might require them. Comparisons of travel-time predictions using the 3DTTLS show comparable values to full 3D ray tracing, depending on the grid layout. Several tests of various lateral and depth grid sizes were performed, resulting in a preferred setup of a 0.5° tessellation to 25° distance and 1° tessellation to the maximum distance for a phase (100° for first-P, 80° for first-S). Constant (2.5 and 5 km) as well as variable depth spacings were tested, with a variable depth spacing selected as the most efficient for file size and resolution.

Relocation comparisons between 3DTTLS and full 3D ray tracing show that mislocation values are comparable between the methods, with full 3D ray tracing being slightly favored (when using a consistent 1D distance-dependent uncertainty for both 3D ray tracing and 3DTTLS). Storage requirements for the 3DTTLS are minimal for a network like the IMS and would allow for rapid creation and usage of surfaces for many tens of phase types. A location algorithm such as LocOO3D could read in only those surfaces that are required to locate an event or a system could hold all the surfaces in memory as a “service” for a location algorithm. Computation times for 3DTTLS are comparable to using a standard 1D model like iasp91 or ak135 (i.e., < 1 ms per arrival), with full 3D ray tracing being slower by ~ 3–4 orders of magnitude. For a relatively static network like the IMS, 3DTTLS could be used for standard location calculations as quickly as a 1D model but would give the added accuracy and precision of a full 3D model.

The 3DTTLS framework is a straight-forward method to consistently store 3D seismic velocity model travel-time and uncertainty predictions calculated using the exact ray-tracing algorithm used during model development. This framework would thus remove an inconsistency that can be encountered when trying to compare location results using higher-dimensional models. The GeoTess 3DTTLS can be built within the PCalc software or model developers can export simple text versions of surface grids and use their own ray-tracing algorithm to provide travel-time and uncertainty predictions that are valid for a specific model.