# Accounting for lateral variations of the upper mantle gradient in *P*_{n} tomography studies

## Abstract

[1] The effect of an upper mantle velocity gradient on regional arrival times has been approximated by a cubic distance term, which can be extended to two dimensions for use in tomographic studies. To demonstrate this, we add a laterally varying upper mantle gradient to the standard *P*_{n} time-term tomography technique, and apply to a data set from Asia compiled using ground truth, event location criteria. We observe strong lateral variations in the gradient, ranging from −0.001 to 0.003 s^{−1}, with high gradients associated with the Tethys convergence zone. The gradient patterns may reflect lateral variations in the thermal gradient of the mantle lid. Variance reduction is 63% with respect to *P*_{n} tomography without gradients. Adding gradients allows the use of longer path lengths, improving velocity image definition in high-gradient regions with sparse station distribution, such as Tibet.

## 1. Introduction

[2] The dramatic increase in seismic velocity at the base of the earth's crust causes waves to refract through the underlying mantle, and ray paths for a large range of travel distances concentrate within a narrow range of mantle depths. This allows seismologists to resolve high-precision, two-dimensional images of upper mantle velocity using arrival-time data collected over regional distances (roughly 2.5° to 18°).

[3] Regional *P* waves are not actually critically refracted along the top of the mantle as vertical velocity gradients and earth sphericity combine to cause longer rays to sample deeper regions. To accommodate this behavior, *Hearn et al.* [2004] split Chinese regional network arrivals into short (1.8° to 9.5°) and long (9.5° to 15°) path length sets, demonstrating a better overall fit to data, and tectonically significant differences between velocity images. Rather than splitting data by path length to obtain images for different depth ranges, we present a characterization of the upper mantle in terms of lateral variations in both velocity and vertical velocity gradient. This will lead to seamless inversion of, and hence, prediction of travel times over the entire regional distance range.

[4] *Zhao* [1993] and *Zhao and Xie* [1993] approximated the effect of an upper mantle gradient on refracted or near-refracted waves, assuming small, laterally uniform gradients and shallow ray bottoming depths. The approximation was used to show that accounting for a uniform gradient and laterally varying *P*_{n} velocity eliminated the need for upper mantle anisotropy in the Basin and Range; however, that assertion proved controversial [*Beghoul and Barazangi*, 1995; *Zhao*, 1995].

[5] Here, we extend the upper mantle gradient technique of *Zhao* [1993] and *Zhao and Xie* [1993] to two dimensions. The method will be tested using an Asian data set based on seismic events, primarily earthquakes, with high quality locations. This is a well studied region, and previous velocity tomography work [*Zhao and Xie*, 1993; *McNamara et al.*, 1997; *Hearn et al.*, 2004; *Liang et al.*, 2004; *Sun et al.*, 2004; *Phillips et al.*, 2005; *Liang and Song*, 2006; *Sun and Toksöz*, 2006; *Pei et al.*, 2007] will allow evaluation of our results. We will show that lateral variations of the gradient are easily obtained along with upper mantle velocity, that the fit to data improves significantly, and that velocities are better recovered in high gradient regions. We anticipate that this improvement in the *P*_{n} tomography technique will aid in understanding the tectonics of the region, and will lead to efficient travel-time prediction for inclusion in location procedures.

## 2. Method

*Zhao*[1993] and

*Zhao and Xie*[1993] the time-term equation for

*P*

_{n}tomography can be written,

*i*and

*j*are event and station indices, respectively, and

*k*is a velocity model discretization index. The

*α*and

*β*terms represent travel times along crustal legs at the source and receiver ends of raypaths. Travel time for the mantle leg is obtained by summing products of distance increments associated with each model node at Moho depth,

*δ*

*x*

_{ijk}, with mantle lid slownesses (inverse velocity) of each node,

*s*

_{k}. The gradient term for a particular path (dropping

*i*and

*j*indices) is given by,

*s*

_{0}is an average slowness,

*X*

_{m}is the length of the mantle leg, and

*c*is related to the vertical gradient,

*g*, and includes a sphericity correction [

*Helmberger*, 1973],

*Zhao*[1993], and

*Zhao and Xie*[1993], and involves Taylor expansions of an analytic solution based on small

*ch*, where

*h*is the bottoming depth of the ray below the Moho. Calculations showed that approximate and analytic traveltimes match to within 1.0 s for a high gradient mantle (0.003 s

^{−1}) at 18°. Further, traveltimes ray-traced through a laterally varying gradient model similarly matched approximate traveltimes based on path averaged gradients.

[7] The *c*^{2} term can be allowed to vary in two dimensions by discretizing in the same manner as for the upper mantle velocity and taking a path average, for inclusion in the tomographic inversion. The gradient term is weakly non-linear because slowness and gradient multiply each other; however, we set *s*_{0} constant and adjust until the regional average slowness is matched. Alternatively, the *s*_{0} term could be taken as a raypath average, requiring iterative solution, but we use the simpler regional average technique in this initial study.

[8] To integrate through slowness and gradient grids, we first obtain equally spaced points, representing centers of ray segments, along great circle paths at 50 km depth for a WGS84 ellipsoid [*Vincenty*, 1975]. Mantle pierce points are set using 50° incidence angles. Model values at each ray segment center point are obtained using bilinear interpolation from the surrounding four nodes [e.g., *Thurber*, 1983]. For each arrival datum, we obtain partial derivatives with respect to grid-interpolated model parameters as well as source and site terms, establish a linear system of equations, and solve using conjugate gradient methods (LSQR) [*Paige and Saunders*, 1982] after applying first-difference constraints to adjacent grid parameters [e.g., *Shaw and Orcutt*, 1985; *Phillips and Fehler*, 1991]. Two difference-constraint weights, one for slowness and one for the gradient, are chosen to maximize model variation while avoiding artifacts that would be introduced by noise.

## 3. Data

[9] To test the gradient tomography method, we considered events that pass the *Bondar et al.* [2004] criteria for ground truth (GT, absolute epicenter error) less than or equal to 25 km (GT25). We started the culling process by analyzing data from individual bulletins to determine events that passed GT criteria. Those events that did not immediately pass were relocated, holding depths fixed, using the IASP91 earth model [*Kennett and Engdahl*, 1991], after merging first-P arrivals from all available global and regional bulletins, and subsetting the arrivals into *Bondar et al.* [2004] GT-constraint distance ranges (0°–2.5°, 2.5°–10°, 2.5°–20°, 28°–91°). Relocated events were then reconsidered under the various GT criteria. Roughly 95% of our selected events required relocation before they passed, many to eliminate non-first-P arrivals that the GT criteria rejects or to eliminate first-P arrivals outside the acceptable GT distance ranges.

[10] Following application of the GT criteria, we selected events within a latitude 0°–60°, longitude 30°–135° box covering a large portion of Asia, limiting to depths less than 40 km. Of these event origins, 80% were supplied by the EHB bulletin [*Engdahl et al.*, 1998], the remainder were obtained from regional network solutions, special earthquake studies, and high precision nuclear and chemical explosion lists.

[11] We then selected arrival data from the GT25 or better events, limiting event-station distances to between 2.5° and 18°. In this way, we obtained 10,396 events, and 186,822 arrivals recorded at 1287 stations [*Begnaud et al.*, 2006] (Figure 1). The vast majority of these arrival times originated from International Seismological Centre (ISC), Annual Bulletin of Chinese Earthquakes (ABCE) [*Lee et al.*, 2002], and Earthquake Data Report (EDR) bulletins, in that order. While considerable attention was paid to high-quality event locations, we only performed a cursory 10 s residual trim (with respect to the IASP91 model), to eliminate extreme arrival time outliers at this stage. Further trimming will be performed as part of our inversion procedure. Note the sparse station density in and around the Tibetan Plateau in Figure 1.

## 4. Results

[12] We performed the inversion in two steps. First, we enforced smooth models by applying heavy regularization, after which residuals greater than 4 s, were trimmed (just over 2*σ*). These data were then re-inverted with lighter regularization. For comparison, we also inverted the final data set without considering gradients, using the same regularization factors.

[13] We obtained residual RMS of 2.57 s and 1.57 s for the non-gradient and gradient calculations, respectively. Setting degrees of freedom equal to the number of event plus station terms, we obtained a variance reduction of 63%; thus, inclusion of the gradient provided a significant improvement in data fit. The numbers of effective velocity and gradient terms were estimated to be 592 and 127, respectively, by summing diagonal elements of the resolution matrix, and were insignificant relative to the number of crustal time terms for purposes of calculating residual variance.

[14] Maps of upper mantle *P*-wave velocity and vertical gradient variations are shown in Figures 2 and 3, respectively. Resolution, expressed as a linear measure across a fully resolved feature (square root of node area divided by diagonal resolution element), remains below 2° for velocity, and below 4° for the gradient, over wide areas of high ray density. Velocities range from 7.6 to 8.2 km/s and follow patterns similar to those of previous studies in this region (see earlier list). High velocities appear in stable regions such as the Indian, Arabian, and Kazakh cratons, micro continents such as the Sichuan, Tarim, Junggar, Ordos, and Qaidam basins, and trapped oceanic crust under the E. Mediterranean, Black and S. Caspian seas, whereas low velocities appear in north-central Tibet, the Songpan-Ganzi terrane, the Panxi, Shanxi and Baikal rifting areas, the North China Basin, the western Tian Shan, the Hindu Kush, the Anatolian and Iranian plateaus, the Red Sea and adjacent Arabian Shield, and oceanic convergence zones. Gradients range from −0.001 to 0.003 s^{−1}. High gradients are associated with the Tethys convergence zone, including the Anatolian and Iranian plateaus, the Hindu Kush, the Tibetan Plateau, and extend through the Sichuan Basin. High gradients are also observed in oceanic convergence zones. Low gradients are observed surrounding the Tethys high gradient zone, including southeastern and northeastern China, southern India, the Red Sea-Arabian Shield, and the Tajik Basin. Station terms reflect variations in crustal thickness, similar to earlier studies in this region [e.g., *Hearn et al.*, 2004], and event terms are a strong function of hypocentral depth.

## 5. Discussion

[15] We obtain a significant improvement in fit to data by solving for two-dimensional upper mantle velocity and gradient in the *P*_{n} tomography. However, our initial and final residual standard deviations are 2.57 s and 1.57 s, which are fairly high. In contrast, previous studies in this area [*McNamara et al.*, 1997; *Hearn et al.*, 2004; *Liang et al.*, 2004; *Sun et al.*, 2004; *Pei et al.*, 2007] report residual standard deviations of 0.55, 1.3, 1.33, 0.65 and 1.9 s, respectively, while a differential arrival *P*_{n} tomography study [*Phillips et al.*, 2005] reports residual levels of 0.96 s. These values should be compared to our non-gradient residual level of 2.57 s. This is caused by the longer distance ranges and the liberal quality control applied to arrival time data employed in this study. The evaluation of arrival-time error, without having access to original waveforms, must be developed, and a technique based on arrival differences has shown promise [*Rowe et al.*, 2003; *Phillips et al.*, 2005].

[16] We observe dramatic differences in the magnitude and extent of low velocity anomalies in high gradient areas by comparing velocity images obtained with and without gradients. For example, the two, adjacent low velocity zones in north-central Tibet and in the Songpan-Ganzi terrane are much better defined in the gradient case. We also see increased definition in south-east areas of the low velocity Iranian Plateau. Using the longer, thus deeper and faster, paths in these areas degrades our ability to image low velocity zones when gradients are not accommodated. For example, imaging Tibet and surrounding areas relies on long raypaths because of the sparse station density. Incorporating upper mantle gradients improves the imaging of low velocity zones, which are of great tectonic interest, and, in the case of north-central Tibet, may represent areas of upwelling mantle related to the underthrusting of Indian lithosphere to the south [*Zhao and Xie*, 1993; *Hearn et al.*, 2004]. We note that our Tibet and Songpan-Ganzi images are remarkably similar to those of *Liang and Song* [2006], which were obtained by adding data from temporary deployments, dramatically improving coverage in this area.

[17] We observe vertical gradients of upper mantle *P*-wave velocity from −0.001 to 0.003 s^{−1} in Asia, with an average level across Tibet of 0.0025 s^{−1}. We can compare these values with the value of 0.0031 s^{−1} found for Tibet by *Zhao and Xie* [1993]. Because we employ regularization, peak values of the gradient will be suppressed somewhat by the tomography procedure, holding our results slightly below those of a more focused study. Thus, our gradient results are in accordance with a previous measurement in this area. The continental high gradient pattern is of broader scale than patterns of upper mantle velocity. The central portion of the gradient high encompasses the low velocities in north-central Tibet and in the Songpan-Ganzi terrane, as well as the high velocities in the Sichuan Basin. The gradient image appears to correlate best with crustal thickness, and highs are expected to indicate low thermal gradients in the mantle lid. For example, high gradients determined by *P*_{nl} modeling of southern Africa cratons have been related to the effects of deep, cool, lithospheric roots [*Clouser and Langston*, 1990].

## 6. Conclusions

[18] We have extended standard, time-term methods used to invert regional *P*_{n} arrivals to image upper mantle velocities, by adding laterally varying gradient terms. The gradient terms are implemented using an approximation to an analytic solution for small *hc* (*h*, ray bottoming depth below the Moho; *c*, normalized gradient), which was used to account for uniform gradient in *P*_{n} tomography studies by *Zhao* [1993] and *Zhao and Xie* [1993].

[19] We applied the two-dimensional gradient technique to an Asian data set based on high quality event locations, specifically ground truth levels (epicentral location error) of 25 km and under. Residual variance was reduced by 63% by adding gradients. Residual standard deviations were 1.57 s and 2.57 s, with, and without gradients, respectively, which are high compared to previous studies. Residual levels depend on arrival-time quality control efforts, which were minimal in this study. We believe that significant improvement can be made if data error can be estimated without access to the original waveforms; the use of differential arrival techniques hold promise.

[20] Gradients ranged from −0.001 to 0.003 s^{−1}, with highest gradients along the Tethys convergence zone. We also observed high gradients along oceanic convergence zones. In the absence of compositional effects, velocity gradients likely vary inversely with the thermal gradient of the mantle lid, with high velocity gradient associated with low thermal gradient, and low velocity gradient with high thermal gradient. Upper mantle velocities follow familiar patterns with respect to previous work in this area; however, low velocity anomalies in high gradient areas, such as Tibet, were much better resolved when gradients were used. The use of long ray paths in Tibet, an area of low station density, caused degradation of low velocities in calculations carried out without gradients. Using gradients improves coverage and resolution as a wide range of raypath lengths can be used without fear of image degradation. The use of gradients may help to resolve anisotropic effects in a similar manner.

## Acknowledgments

[21] We thank Tom Hearn for directing us to the upper mantle gradient studies by L.S. Zhao. In addition, Richard Stead and Hans Hartse placed catalog information into CSS database tables and performed background quality control. We also thank two anonymous reviewers, who provided valuable comments. GMT software [*Wessel and Smith*, 1998] was used for all figures. This research was supported by the US DOE under Contract DE-AC52-06NA25396.