Seismic TomographyA Window Into Deep Earth

5/19/2018 Seismic TomographyA Window Into Deep Earth

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Physics of the Earth and Planetary Interiors178 (2010) 101135

Contents lists available atScienceDirect

Physics of the Earth and Planetary Interiors

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p e p i

Review

Seismic tomography: A window into deep Earth

N. Rawlinson a,, S. Pozgay a, S. Fishwick b

a Research School of Earth Sciences, Australian National University, Mills Rd., Canberra, ACT 0200, Australiab Department of Geology, University of Leicester, Leicester LE1 7RH, UK

a r t i c l e i n f o

Article history:

Received 2 May 2009Received in revised form

10 September 2009Accepted 5 October 2009

Edited by: G. Helffrich.

Keywords:

Seismic tomographyInversionBody waveSurface waveEarth structureRay tracing

a b s t r a c t

The goal of this paper is to provide an overview of the current state of the art in seismic tomographyand trace its origins from pioneering work in the early 1970s to its present status as the pre-eminen

tool for imaging the Earths interior at a variety of scales. Due to length limitations, we cannot hope tocover every aspect of this diverse topic or include mathematicalderivationsof theunderlying principlesrather, we will provide a largely descriptive coverage of the methodology that is targeted at readernot intimately familiar with the topic. The relative merits of local versus global parameterization, raytracing versus wavefront tracking, backprojection versus gradient based inversion and synthetic testinversus model covariance are explored. A variety of key application areas are also discussed, includingbody wave traveltime tomography, surface wave tomography, attenuation tomography and ambiennoise tomography. Established and emerging trends, many of which are driven by the ongoing rapidincreases in available computing power, will also be examined, including finite frequency tomographyfullwaveformtomographyandjointtomographyusingmultipledatasets.Severalpracticalapplicationsoseismic tomography, including body wave traveltime, attenuation and surface waveform, are presentedin order to reinforce prior discussion of theory.

2009 Elsevier B.V. All rights reserved

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1. What is seismic tomography? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2. Pioneering work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3. The last three decades: a brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

1.3.1. Local studies of the crust and upper mantle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031.3.2. Regional and global tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

1.4. Recent trends: ambient noise and finite frequency tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5. Seismic tomography and computing power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Representation of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1. Common regular parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2. Irregular parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. The data prediction problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1. Ray-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1. Shooting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2. Bending methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3. Grid-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1. Eikonal solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2. Shortest path ray tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.4. Multi-arrival schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.5. Finite frequency considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4. Solving the inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1. Backprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2. Gradient methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Corresponding author. Tel.: +61 2 6125 5512; fax: +61 2 6257 2737.E-mail address:[email protected](N. Rawlinson).

0031-9201/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.pepi.2009.10.002
http://www.sciencedirect.com/science/journal/00319201http://www.elsevier.com/locate/pepimailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_7/dx.doi.org/10.1016/j.pepi.2009.10.002http://localhost/var/www/apps/conversion/tmp/scratch_7/dx.doi.org/10.1016/j.pepi.2009.10.002mailto:[email protected]://www.elsevier.com/locate/pepihttp://www.sciencedirect.com/science/journal/00319201


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102 N. Rawlinson et al. / Physics of the Earth and Planetary Interiors 178 (2010) 101135

4.2.1. Solution strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2.2. Frchet matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.3. Fully non-linear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4. Analysis of solution robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.1. Joint inversion of teleseismic and wide-angle traveltimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2. Attenuation tomography in a subduction zone setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3. Regional surface wave tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6. Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

1. Introduction

1.1. What is seismic tomography?

Seismic tomography is a data inference technique that exploitsinformation contained in seismic records to constrain 2D or 3Dmodels of the Earths interior. It generally requires the solution of alargeinverseproblemtoobtainaheterogeneousseismicmodelthatis consistent with observations. More formally, provided that wecan establish an approximate relationship d = g(m) between seis-

mic datad and seismic structure m so that for a given model mwe can predict d then the seismic tomography problem amountsto finding msuch that d explains the data observations dobs. Inmost casesdandmare discrete vectors of high dimension, whichmeans that many data records are used to constrain a detailedmodel. Implicitly, this detail must apply to both vertical and lat-eral structure. As such, the radial Earth model produced byBackusand Gilbert (1969),based on the theory in their seminal paper oftheprecedingyear(Backus and Gilbert, 1968), is notusuallyviewedas an early example of seismic tomography despite the similarityin methodology.

A simple example of seismic traveltime tomography, whichserves to illustrate several features typical to most applications, isshown inFig. 1. In this artificial test, a synthetic model in spherical

shell coordinates is generated (Fig. 1a) which consists of 780 gridpoints evenly spaced in latitude and longitude with cubic B-splinefunctions used to describe a smooth velocity field. For a given set ofsources and receivers, first-arriving geometric ray traveltimes arethen computed (Fig. 1b) through the model. These traveltimes con-stitute the synthetic data set that is equivalent to the informationone may obtain from seismograms recorded in the field. The rela-tionship d = g(m), where d represents the traveltime dataset andmthe velocity model, is non-linear in this case because the pathtaken by the seismic energy is a function of velocity. Almost with-out exception, only the first-arrivals of any phase are exploited intraveltime tomography, because laterarrivals due to multi-pathing(wavefront folding) are difficult to pick. One property of firstarrivals is that they tend to avoid low velocity anomalies, and pref-

erentially sample high velocity anomalies, as can be see inFig. 1b.Due to the non-linearity of the inverse problem, the traveltime

misfit surface (some measure of the difference between observa-tion and model prediction) may not be a simple smooth functionwith a well defined minimum. While a fully non-linear solutiontechnique may therefore seem appropriate, the size of the prob-lem usually makes this computationally prohibitive. Instead, somegradient-based technique is often used, which relies on having aninitial model close to the solution model. Fig. 1cshows the ini-tial model used in this case, which has a uniform velocity,resultingin great circle paths. The node spacing is identical to that of thesynthetic model (Fig. 1a), which will favourably bias the recoveryof structure. Repeated application of forward ray tracing and lin-earized inversion eventually produces the solution model shown

inFig. 1d, which remains unchanged with further iterations, and

satisfies the synthetic dataset. This basic approach, which relieson accurate a priori information in the form of an initial model,some class of forward solver, and a local inversion technique, isubiquitous to most forms of tomography, be it traveltime, surfacewaveform, anisotropy or attenuation.

Comparison ofFigs. 1aand d reveals a number of interestingsimilarities and differences between the synthetic and recoveredmodels. Clearly, regions near the edge of the model that have nopath coverage do notdeviate from the initial model. In cases whererays exist but have a similar azimuth, recovered anomalies have

a tendency to be severely smeared out in the dominant ray pathdirection (e.g. the two high velocity anomalies in the southwestand northeast corners of the model). Within the bounds of thereceiver array, where path coverage is dense, the recovery of struc-ture appears to be accurate, with the exception of the four distinctlow velocity anomalies, whose amplitudes are severely underesti-mated.Thisproblemarisesfromthefactthatfirst-arrivalsavoidlowvelocity regions, as shown clearly inFig. 1b, and therefore poorlyconstrainthem.Mostformsof seismic tomography, even those thatdo not directly exploit traveltimes (e.g. attenuation tomography),are affected in some way by this phenomenon, because they usu-ally rely on the paths provided by first-arrival tomography to solvethe data prediction problem. Other issues, including solution non-uniqueness (where more than one solution satisfies the data to

the same extent) and the validity of geometric ray theory, will bediscussed in the following sections.

1.2. Pioneering work

Thenamemost commonly associated with theorigins of seismictomography is that of Keiiti Aki, who published a seminal paper in1976 on 3D velocity determination beneath California from localearthquakes (Aki and Lee, 1976).In this paper, traveltime data col-lected at 60 stations from 32 local earthquakes are inverted for3D crustal structure, described by a total of 264 constant slowness(inverse of velocity) blocks, and hypocenter corrections. The inver-sion is linear, because ray paths are assumed to be straight, and adampedleast squares approach is used to find a solution. Estimates

of model covariance andresolution arealso made to assess solutionrobustness A year later, this publication was followed by an equallyinfluential paper which employs teleseismic tomography to imagethe 3D velocity structure beneath the Norwegian Seismic Array(Norsar) in southeast Norway (Aki et al., 1977).Traveltime resid-ual information from distant (teleseismic) earthquakes is used toconstrain structure, which is confined to a local region beneath thearray.Constantslownessblocksareagainusedtodescribethelitho-sphere, although this time, the initial model is defined by constantvelocity layers, so ray paths are permitted to bend. However, theinversion is still linear as path geometry is not updated to accountfor the recovered heterogeneity.

Theearlywork of Akiundoubtedlycatalyzed thenumerousseis-mic tomography studies of the crust and lithosphere that soon

followed, but a number of otherinfluential developments, arguably


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N. Rawlinson et al. / Physics of the Earth and Planetary Interiors 178 (2010) 101135 10

Fig. 1. Synthetic reconstruction test illustrating several typical characteristics of seismic traveltime tomography. (a) Synthetic test model with sources (grey stars) andreceivers (blue triangles) superimposed; (b) same model as in (a) but with all first arrival paths plotted; (c) starting model and path coverage for the iterative non-lineainversion; (d) recovered model, which can be compared with (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web versioof the article.)

of similar importance, occurred at around the same time. In globaltomography, Adam Dziewonski publisheda paper in 1977 that usesnearly 700,000 P wave travel time residuals from the bulletins ofthe International Seismological Centre (ISC) to image the velocitystructure of the Earths mantle, described using a spherical har-monic parameterization (Dziewonski et al., 1977).Despite the sizeof the traveltime dataset, the number of unknowns in the inverseproblem is restricted to only 150, presumably due to limitations incomputing power.

Although not as frequently cited as Aki and Dziewonskis sem-inal works, an earlier paper by Bois et al. (1972) implements ascheme that clearly conforms to the above definition of seismictomography. In this study, the authors use cross-hole (or well towell) active source seismic imaging to examine part of the Lacq

oil field of southwest France. Small charges were inserted downonehole,and their detonation recorded bygeophones placeddownanother hole. Traveltimes picked from the resultant seismogramsare then inverted for the 2D velocity structure of the cross-sectionseparating the two boreholes. Rather than use a constant blockparameterization, a regular grid of nodes is specified together withan interpolant that ensures continuity of the velocity field andits first derivative at every point. The traveltime prediction prob-lem is solved using a shooting method of ray tracing that fullyaccounts for isotropic heterogeneity, and an iterative non-linearapproach, similar to that demonstrated inFig. 1,is used to recon-cile observed and model traveltimes (Bois et al., 1971).Althoughthe number of unknowns that are solved for is 110, and the max-imum number of ray paths used is 90, the proposed technique is

sophisticated, particularly considering the minimal development

that had occurred in the field prior to this application. One mighargue that seismic tomography implies 3D imaging, but in terms othe underlying theory, there is no real difference, except for thesize of the inverse problem, and the complexity of the forwardsolver.

1.3. The last three decades: a brief history

1.3.1. Local studies of the crust and upper mantle

Following the pioneering efforts in seismic tomographydescribed above, a veritable cascade of new applications anddevelopments soon followed. In cross-hole tomography, varioutechniques for ray tracing and inversion were trialled (McMechan1983, 1987; Bregman et al., 1989), but essentially, the underly

ing method ofBois et al. (1971) was not significantly advancedupon.Backprojection inversion techniques, inherited from medicaimaging, were generally more popular than gradient-based methods, perhaps due to similarities in acquisition geometry. Greateinnovation came in the form of diffraction and wave equationtomography (Pratt andWorthington, 1988;Prattand Goulty, 1991Song et al., 1995; Pratt and Shipp, 1999),which attempt to exploimore of the recorded waveform. Other classes of seismic tomography that have their origins in exploration include reflectiontomography and wide-angle (refraction and wide-angle reflection) tomography, which use artificial sources such as explosionsairguns and vibroseisto generate seismic energy. Reflection tomography is a natural compliment to migration imaging, because ioffers a means to constrain velocity and interface depth using

traveltimes and, less commonly, geometric spreading amplitude


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and reflection/transmission coefficients. One of the first studiesto implement reflection tomography was that ofBishop et al.(1985),which combines ray shooting and a gradient based inver-sion technique to constrain a 2D model described by constantvelocity gradient blocks and cubic spline interfaces. Similar stud-ies have also been carried out by Farra and Madariaga (1988)and Williamson (1990). In general, coincident reflection travel-time data alone appears to be insufficient to satisfactorily resolvethe trade-off between interface depth and layer velocity. Con-sequently, more recent efforts have been directed towards jointinversion of traveltime and amplitudes (Wang and Pratt, 1997;Wangetal.,2000),joint inversion of reflection and wide-angle trav-eltimes (Wang and Braile, 1996; McCaughey and Singh, 1997)andfull waveform tomography (Hicks and Pratt, 2001; de Hoop et al.,2006).

Wide-angletomography is similar to reflectiontomography, butthe sourcereceiver offset tends to be much greater in order todetect refracted rays from significant depths (e.g. Pn waves fromthe Moho). Both 2D and 3D experiments are common, and over thelast few decades have played a major role in unravelling the crustalarchitectureof continents andmarginsin various parts of theworldincluding Canada (e.g.Hole, 1992; Kanasewich et al., 1994; Cloweset al., 1995; Zelt and White, 1995; Morozov et al., 1998; Zelt etal., 2001, 2006)and Europe (e.g.Riahi and Juhlin, 1994; Staples etal., 1997; Darbyshire et al., 1998; Louden and Fan, 1998; Mjelde etal., 1998; Korenaga et al., 2000; Morgan et al., 2000; Bleibinhausand Gebrande, 2006).Early efforts in this field tended to treat thewide-angle reflection and refraction data separately, but it wassoon recognised (e.g.Kanasewich and Chiu, 1985)that their jointinversion dramatically increased the likelihood of resolving bothinterface structure and velocity variation. In recent years, wide-angle tomography has been the subject of much interest in theemerging field of full waveform tomography, where theprospectoffar greater resolution has motivated a number of different studies(Pratt et al., 1996; Sirgue andPratt, 2004; Brenders andPratt, 2007;Jaiswal et al., 2008).

Followingthe early work ofAkiand Lee(1976), localearthquake

tomography (orLET) hasbecome a popular tool forimaging subsur-face structure in seismogenic regions. One distinguishing featureof the technique is the need to relocate hypocenters in tandemwith recovering seismic structure. Although the conceptual basisof LET has not really changed since Akis original paper, severaladvances have been made, including full 3D ray tracing and itera-tivenon-linearinversion(Eberhart-Phillips,1990); direct inversionforVP/VSor QP/QS ratio (e.g.Walck, 1988);development of meth-ods for constraining 3D anisotropic velocity variations (Hirahara,1988; Eberhart-Phillips and Henderson, 2004) and attenuationstructure (Sanders, 1993; Tsumura et al., 2000);and double differ-ence tomography (Zhang and Thurber, 2003; Monteiller and Got,2005),which aims to significantly improve hypocenter relocation.In subduction zone settings, recent advances include tomographic

inversionofshearwavesplittingmeasurementsforanisotropicfab-ric (e.g. Abt and Fischer, 2008), and of velocity and attenuationanomalies for water content, temperature and composition (Shitoet al., 2006).

Teleseismic tomography has been used extensively to mapthe structure of the crust and lithosphere in 3D (e.g. Oncescuet al., 1984; Humphreys and Clayton, 1990; Benz et al., 1992;Glahn and Granet, 1993; Achauer, 1994; Saltzer and Humphreys,1997; Graeber et al., 2002; Rawlinson et al., 2006b; Rawlinson andKennett, 2008).Compared to the original technique ofAki et al.(1977), mostteleseismic tomography now usesiterativenon-linearinversion coupled with 3D ray racing or wavefront tracking (e.g.VanDecar and Snieder, 1994; Steck et al., 1998; Rawlinson et al.,2006b). In most cases, teleseismic tomography is still based on

the recovery of isotropic velocity models from arrival time residu-

als, although attempts have been made to recover anisotropy (e.g.Plomerov et al., 2008).

Detailed local studies of the upper mantle have also been con-ducted using data from surface waves. For regions with closespacing of broadband seismometers, interstation measurementsor array techniques (Friederich and Wielandt, 1995; Forsyth andLi, 2005; Pedersen et al., 2003)can be used to estimate the localdispersion characteristics within the zone of interest. These meth-ods have been applied in a number of locations to produce detailedtomographic images of the lithospheric mantle (e.g. Weeraratne etal., 2003; Bruneton et al., 2004; Li and Burke, 2006; Darbyshire andLebedev, 2009).

1.3.2. Regional and global tomography

The different classes of seismic tomography discussed abovetend to use temporary deployments of recorders to target a lim-ited geographical region; hence they can be described as localmethods.By contrast,regionaland global tomography studies morecommonly utilize information from permanent networks that spanlarge continental regions or much of the globe, such as the GSN(Global Seismic Network), in addition to any available data fromtemporary arrays. Targets include the upper mantle, whole mantleor the entire Earth. Since the pioneering work ofDziewonski et al.(1977),which used the traveltimes of P-waves, efforts have beenfocused on improving resolution by exploiting an ever increasingvolumeofrecordeddata.CurrentglobalP-wavemantlemodelsthatexploit traveltime data from the ISC commonly constrain structureat a scale length of a few 100 km or less using millions of paths(Zhao, 2004; Burdick et al., 2008).

In addition to direct P-waves, other phases such as PcP andPKP are now commonly used to improve coverage, particularly inthe core (Vasco and Johnson, 1998; Boschi and Dziewonski, 2000;Karason and van der Hilst, 2001).While spherical harmonics arestill preferred in some cases, most body wave studies now opt forlocal parameterizations, such as blocks or grids, which are bettersuited for recovering detailed structures such as mantle plumes orsubducting slabs (van der Hilst et al., 1997; Bijwaard et al., 1998;

Karason and van der Hilst, 2001; Zhao, 2004).The highly unevendata coverage that typifies regional and global body wave studies due largely to irregular distribution of earthquakes and recordingstations has stimulated the idea of using irregular parameter-izations, where blocks or nodes are placed only where they arerequired by the data. Bijwaard et al. (1998), Bijwaard and Spakman(2000) and Spakman and Bijwaard (2001) use a spatially vari-able cell size parameterization based on ray sampling, in whichan underlying regular grid is used to construct a mosaic of non-overlapping irregular cells.Sambridge and Gudmundsson (1998)propose a more sophisticated scheme based on Delaunay andVoronoi cells,whichis subsequentlyappliedto whole Earth tomog-raphy (Sambridge and Faletic, 2003; Sambridge and Rawlinson,2005).

Body wave tomography using S-waves is also common inregional and global studies (e.g. Grand et al., 1997; Vasco andJohnson, 1998; Widiyantoro et al., 2002), and can either be done inisolation or simultaneously with P-waves to obtain VP/VSratio as inLET. An alternative is to jointly resolve bulk sound and shear veloc-ity (Su and Dziewonski, 1997; Kennett, 1998),quantities that canbe more readily linked to experimental laboratory measurementsof the physical properties of mantle minerals.

Surface waves and normal modes can also be used to constructtomographic images of the Earths interior. Compared to bodywaves, surface waves have the advantage that they can sample theupper mantlebeneath ocean basins at sufficient density to producewell constrained models of oceanic lithosphere; on the other hand,theycannotprobeintothedeepmantleathighresolution,andhave

difficultlyresolvingcrustalstructure. A variety of differentmethod-


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ologies have been applied to obtain information from the surfacewavetrain. Some globalstudies use long paths andattempt to mea-sure phase velocity directly for the fundamental mode for eachpath (e.g.Ekstrm et al., 1997; Laske and Masters, 1996).Groupvelocities can be extracted using filter analysis, and have beenused to produce maps at both regional (e.g. Ritzwoller and Levshin,1998; Danesi and Morelli, 2000; Pasyanos and Nyblade, 2007), andglobal scales (Shapiro and Ritzwoller, 2002).Additional informa-tion from surface wave overtones provides better resolution withdepth;withthisinmind,van Heijstand Woodhouse(1997)developa new method for measuring overtone phase velocities. Combiningthese data with information from body waves, Ritsema et al.(2004)produce a shear wavespeed model of the mantle, with particularemphasis on the upper mantle transition zone. Alternatively, aninversion procedure can be used to fit the surface waveform (andin some cases long period S-waves). This style of approach has alsobeen used to produce shear wavespeed models at both regional(e.g.van der Lee and Nolet, 1997; Simons et al., 1999; Friederich,2003; Heintzet al., 2005; Fishwick et al., 2005; Priestleyet al., 2008)and global scales (Debayle et al., 2005; Lebedev and van der Hilst,2008).

Normal modes or free oscillations of the Earth, which can beviewed as very long period standing surface waves, also offer ameans to constrain seismic structure. Individual peaks of the dis-crete spectrum are often split due to Earth rotation, ellipticity andlateral heterogeneity. Isolating the latter effect enables both man-tle structure (Li et al., 1991; Resovsky and Ritzwoller, 1999),andcore structure (Ishii andTromp, 2004) to be imaged. The advantageof this approach is that data coverage is relatively uniform, but dueto the very low frequencies of detectable normal modes, the scalelength of recovered heterogeneity tends to be extremely broad.

Another area of active research in global seismology is atten-uation tomography, in which lateral variations in the anelasticparameter Qare retrieved. A key challenge with this techniqueis to successfully extract the anelastic signal from the recordedwaveform, which is dominated by elastic effects. Studies to datetend to use surface waves and hence focus on the upper man-

tle (Romanowicz, 1995; Billien and Lvque, 2000; Selby andWoodhouse, 2002; Gung and Romanowicz, 2004; Dalton andEkstrm, 2006; Dalton et al., 2008),although body wave studieshave also been carried out (Bhattacharyya et al., 1996; Reid et al.,2001; Warrenand Shearer, 2002). Oneof theattractions of attenua-tion tomography is its strong sensitivity to temperature variations,and therefore its potential to image hot spots, mantle plumes andsubduction zones.

Anisotropy is a potentially complex issue in all tomographicstudies from localto globalscales,as itpervadesmanyregionsof theEarth includingthe crust, uppermantle,coremantle boundary andinner core. The main barrier to itsaccurate recovery in tomographyistheunder-determinednatureoftheinverseproblem;itisdifficultenoughto resolve isotropicvelocityvariations, letaloneall 21 inde-

pendent elastic constantsrequired to describe arbitraryanisotropicanomalies. As a result, studies that attempt to include anisotropydo so with a limited subset of the elastic moduli. One of the firststudies to resolve upper mantle transverse isotropy with a verticalaxisofsymmetryotherwiseknownasradialanisotropy(requiringfive independent parameters) was that ofNataf et al. (1984), whoinverted both Love and Rayleigh wave data for velocity structure,described by degree 6 spherical harmonics, to a depth of approxi-mately 450 km. By assuming this class of anisotropy, downwellingand upwelling features associated with slab subduction were suc-cessfully imaged. Radial anisotropy is now frequently incorporatedinto global shear velocity studies (e.g.Panning and Romanowicz,2006; Kustowski et al., 2008).A form of anisotropy that is morecommonly assumed in surface wave tomography studies is that

of azimuthal anisotropy (e.g. transverse isotropy with a horizon-

tal axis of symmetry), which allows velocity to vary as a functionof horizontal direction, and is therefore more well tuned to uppemantle dynamics associated with contemporary plate tectonicsEarly work in this area was carried out byTanimoto and Anderson(1984, 1985), whofoundvariations of anisotropy in theuppermantle to be as large as 1.5%, albeit with low order spherical harmonicsMontagner and Nataf (1986)andMontagner and Tanimoto (1991develop a scheme which they describe as vectorial tomographywhichallows radial and azimuthalanisotropy to be simultaneouslconstrained by inversion of surface waveforms and regionalizatioof phase or group dispersion curves. The incorporation of seismicanisotropy in one form or another in surface wave tomography hanowbecomealmostroutine(e.g. Debayle,1999;Simonsetal.,2002Debayle et al., 2005; Sebai et al., 2006),but issues still remain as tothe appropriate choice of elastic parameters, and how they maytrade-off in an intrinsically under-determined inverse problem.

Shear wave splitting provides insight into the strength andorientation of anisotropy by measuring the differential arrivatime between orthogonal components of an arriving shear waveHowever, due to the path integral nature of the measurementsit provides limited information on the spatial distribution oanisotropy. In the last few years, this limitation has been addressedin the form of shear wave splitting tomography (e.g.Zhang et al2007;AbtandFischer,2008), whichattemptstomaptheanisotropyinferred from the splitting measurements into a volumetric modeIn related developments, splitting intensity measurements fromSKS waves (Favier and Chevrot, 2003)have also been used to perform anisotropy tomography (Chevrot, 2006; Long et al., 2008).

1.4. Recent trends: ambient noise and finite frequency

tomography

Recordings of identifiable wavetrains from sources such aearthquakes or explosions form the basis of traditional methods oseismic tomography as describedabove. However, since theturn othe millennium, virtual-source seismology has gradually emergedto become an important field in modern seismology, thanks to the

work of a number of researchers who have both theoretically andexperimentally demonstrated a remarkable property of fully diffuse or randomwavefields: information they accumulate about thmedium through which they propagate can be extracted by thelong-term cross-correlation of waveforms recorded at two separate locations (e.g.Lobkis and Weaver, 2001; Campillo and Paul2003; Shapiro and Campillo, 2004; Snieder, 2004; Wapenaar et al2005; Sabra et al., 2005; Wapenaar and Fokkema, 2006).It turnout that the cross-correlation produces an estimate of the Greenfunction between two points; that is, the signal that would arriveat one point if the source waveform were a delta function (or poinimpulse)locatedattheotherpoint.Thisisaparticularlyusefulpieceof information, because the travel time and shape of the waveleare purely a function of the properties of the intervening medium

In the seismic case, the cross-correlation of ambient seismic noiserecorded at two stations (Shapiro and Campillo, 2004),or the seismic coda associated with distant earthquakes (Campillo and Paul2003),can be used to extract empirical Greens functions. For theseismic coda, multiple scattering from small-scale heterogeneityin the lithosphere appears to generate a sufficiently diffuse wavefield. Oceanic and atmospheric disturbances, further randomizedbyscatteringcausedbysolidEarthheterogeneity,isoneofthemaienergy sources for ambient noise tomography.

Ambient noise tomography has now become an establishedtechnique for imaging Earth structure at a variety of scales, buits development continues at a rapid pace. The most commonapproach is to extract Rayleigh wave group traveltimes from thecross-correlated waveforms and invert for group velocity at differ

ent periods (e.g.Shapiro et al., 2005; Sabra et al., 2005; Kang and


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Shin, 2006; Yao et al., 2006; Yang et al., 2007; Zheng et al., 2008).This is often done under the assumption of straight/great circle raypaths, but several studies have used bent rays via solution of theeikonal equation, and have therefore addressed the non-linearityof the problem (Rawlinson et al., 2008; Saygin and Kennett, 2009).In the seminal study ofShapiro et al. (2005),only one month ofdata from the US Array stations was required to produce high res-olution images of the California crust, which clearly discriminatesbetween regions of thick sedimentary cover and crystalline base-ment. More recent efforts have been directed towards recoveringphase velocity in addition to group velocity (Benson et al., 2008),and attempting to resolve 3D shear wave velocity structure fromthe inversion of Rayleigh and Love wave dispersion maps (Bensonet al., 2009).

Geometric ray theory forms the basis of the forward predictionproblem in most forms of seismic tomography, but its validity islimited tocaseswherethe seismic wavelength is much smaller thanthe scale length of heterogeneity that characterizes the mediumthrough which it passes. In fact, unless the seismic energy hasinfinitelyhigh frequency(whichof course is unphysical), theactualground motion recorded by a seismometer will have a partialdependence on the medium in the neighbourhood of the geomet-ric ray. Unless properly accounted for, this finite frequency effectwill essentially blur the final image. Recognition of this fact hasbeen longstanding in the seismic imaging community, but untilrecently, a workable solution was impeded by limits in both com-puting power and theoretical development. One of the first surfacewave studies that attempted to account for finite frequency effectswas that ofSnieder (1988a,b),who used so-called first-order per-turbation theory (or Born theory) to account for scattering. Thenew technique was used in the inversion of waveform phase andamplitude to construct phase velocity maps of Europe and theMediterranean.

In the context of body wave tomography, sensitivity kernels fortraveltimes or waveforms have been formulated by a variety ofresearchers (e.g. Luo and Shuster, 1991; Yomogida, 1992; Vascoand Mayer, 1993; Li and Romanowicz, 1995; Friederich, 1999;

Marquering et al., 1999; Dahlen et al., 2000; Zhao et al., 2000).Theintriguing result that body wave traveltimes are insensitive to het-erogeneity exactly along the geometric ray path led Marqueringto use the terminology banana doughnut kernel. Using such sen-sitivity kernels, finite frequency body wave tomography has beenapplied to a number of different datasets with often interestingresults (e.g.Montelli et al., 2004; Yang et al., 2009),not least ofwhich are the well defined mantle plumes revealed in the study ofMontelli et al. (2004).

It is briefly worth noting that the beginnings of finite frequencytomographywereaccompaniedbysomediscussionastoitsvalidityin general heterogeneous media and the degree of improvement itbrought to conventional ray-based tomography (de Hoop and vander Hilst, 2005a,b; Dahlen and Nolet, 2005; Montelli et al., 2006;

Trampert and Spetzler, 2006).However, with increasing use of thetechnique, and validation against wave equation solvers (Tromp etal., 2005), these discussions have become less relevant. Besides thestudy ofMontelli et al. (2004), others to have used finite frequencytomography include Hung et al. (2004), who report increased reso-lutionintheuppermantletransitionzonebeneathIceland; Chevrotand Zhao (2007),who use finite frequency Rayleigh wave tomog-raphy to image the Kaapval craton; andSigloch et al. (2008),whoexploit teleseismic P-waves to elucidate the structure of subductedplates beneath western North America.

Compared to seismic traveltime tomography based on geo-metric ray theory, the advantage of finite frequency traveltimetomography is that a larger range of phase information is usedto constrain structure. For a single sourcereceiver arrival, filter-

ing over a large range of frequencies will produce a set of delay

Fig. 2. Increasein thenumber of transistorsas a function of time fora range of Intelmicroprocessors. [Source: 60 years of the Transistor: 19472007, Intel website.]

times (e.g. extracted using cross-correlation with a synthetic pulse- seeNolet, 2008)that can be inverted for structure. The advan-

tage of phase information is that it behaves more linearly thanthe waveform, and is hence more amenable to inversion by lin-earized techniques.Anotherbenefit of finitefrequencytomographyis that it is feasible to invert amplitude information (e.g.Sigloch etal., 2008)due to the phenomenon of wavefront healing. Geomet-ric ray amplitudes behave in a much more non-linear fashion, andare therefore difficult to incorporate in tomography. Provided thatbroadband observables are available, finite frequency tomographyhas the potential to improve seismic imaging on many fronts.

1.5. Seismic tomography and computing power

The rise of seismic tomography is inextricably linked with therapid advances in digital computing and microprocessor technol-

ogy that began in the 1960s. This branch of seismology wouldsimply not be feasible without the ability to make millions to tril-lions of calculations persecond. An often used proxy forthe growthin computing power is Moores law, which stems from his semi-nal paper (Moore, 1965)in which he predicted that the number ofcomponentson an integrated circuit would increase exponentially,approximately doubling every two years up until at least 1975.Today, this rule of thumb is applied to the number of transistors ona microprocessor. Fig. 2 shows the Moores law plot for Intel pro-cessors between 1971 and 2007. Linear regression demonstratesthat an exponential increase appears to be a valid approximation,at least until recently. With recent emphasis on cluster comput-ing and multiple core processors, the rapid increases in computingpower appear set to continue.

Early applications of seismic tomography were challenged bywhat we would probably today regard as breathtakingly limitedhardware. For example, at about the time that Keiiti Akis pio-neering work on seismic tomography was published (Aki and Lee,1976; Aki et al., 1977),the cutting edge in computing power wasthe Cray I, the first commercially successful vector computer. Itwas capable of a peak performance of 250 million FLOPS (float-ing point operations per second)though usually ran at about80 million FLOPS, had about 8 megabytes of main memory, andweighed nearly 2.5 tons (Schefter, 1979).Although impressive forits time, the stunning advances in computing power over the lastfew decades means that a standard desktop computer is now manytimes faster. For example, computers using a single Intel Core i7processor can expect performance of around 60+ billion FLOPS

(source:www.hardcoreware.net), some 750 times faster than the
http://www.hardcoreware.net/http://www.hardcoreware.net/


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CRAY 1 in normal operating mode. While this increase in poweris considerable, it should be considered in the context of the vastquantities of high quality digital seismic data that are now beingrecorded and archived, and the need to represent the Earth bymany parameters to properly accommodate such large quantitiesof information in tomographic studies. Forexample, recentregionalandglobal body wave tomography studies useover 107 traveltimesto constrain models with 105106 unknowns (e.g.Burdick et al.,2008).

The remainder of this review paper will describe methodsused in seismic tomography for representing structure, solvingthe forward and inverse problems, and assessing solution non-uniqueness. Several case studies of local and regional tomographyarethen presentedto provide thereader with a broad cross-sectionof the different types of studies that are commonly carried out, andthe particular issues associated with them. The final section of thepaper will discuss future directions in seismic tomography. Withinthe confines of a relatively short review paper, it is not possibleto cover every aspect of this large and diverse field. In addition tothe many references that are provided in specific subject areas, wecanrecommend several other reviewarticlesand books.The editedvolumes ofNolet (1987)andIyer and Hirahara (1993)are notablefor being two of the earliest books to be published on the subjectand contain a wealth of useful information. However, they do notprovide a gentle introduction to the subject. The recent book byNolet (2008)provides an authoritative, coherent and wide rangingdissertation on seismic tomography and is highly recommended.The review article ofRawlinson and Sambridge (2003b)providesgood coverage of crustal and lithospheric traveltime tomography,and the review paper ofRomanowicz (2003) is a good introductionto global mantle tomography.

2. Representation of structure

Ideally, one would like to extract structural information fromseismic data without first imposing limitations on the nature of its

spatial variation. In the synthetic example shown inFig. 1,cubicB-splines on a regular grid were used to represent velocity struc-ture, which limited the minimum scale length of heterogeneityto the chosen grid spacing, and only allowed smooth variationsin wavespeed. In reality, the Earth may contain both continuousand discontinuous (e.g. Moho, faults) variations in wavespeed, andexhibit structural heterogeneity at multiple scale-lengths. Hence,our choice of parameterization immediately restricts the field ofpermissible models, and can be viewed as a form of ad hoc regular-ization. The use of splines on a regular grid to represent structure,asin Fig.1, is an example of a regular staticparameterization, whichis by far the most common approach used in seismic tomogra-phy. Other options include irregular parameterizations, where theminimum scale lengthof structureis variable,and adaptive param-

eterizations, where the inversion process plays a role in adjustingthe number and/or location of parameters to suit the resolvingpower of the data. While several studies have used static irregularparameterizations, they are generally applied within an adaptiveframework. Apart from limiting the range of structure that can berecovered, the choice of parameterization is important because itimpacts on the solution technique chosen for both the forward andinverse problems.

2.1. Common regular parameterizations

Regular parameterizations are attractive because they areconceptually simple, easy to formulate, and generally do not com-plicate theforward andinverse solvers. Cells or blocks(Fig.3a) with

uniform seismic properties (e.g. velocity or slowness) are the most

basic form of parameterization, and make initial value ray tracingsimple because path segments in each block are straight lines. Onthe other hand, the artificial discontinuities between each blockare unrealistic, and can lead to unwarranted ray shadow zonesand triplications, which may make the two-point ray tracing problem more non-linear. Using a large number of blocks with someform of smoothing regularization can mitigate these problems, buit will be at the expense of increased computing time. Constanslowness/velocity blocks have been widely used in most forms otomography, including teleseismic (Aki et al., 1977; Oncescu eal., 1984; Humphreys and Clayton, 1988, 1990; Benz et al., 1992Achauer, 1994; Saltzer and Humphreys, 1997), local earthquak(Aki and Lee, 1976; Nakanishi, 1985),wide-angle (Zhu and Ebel1994; Hildebrand et al., 1989; Williamson, 1990; Blundell, 1993and global (Grand et al., 1997; Vasco and Johnson, 1998; van derHilst et al., 1997; Boschi and Dziewonski, 1999).A slightly morsophisticated approach is to use triangular cells (2D) or tetrahedra (3D) with a constant velocity gradient, which like constanvelocity blocks, facilitates analytic ray tracing (e.g. Chapman andDrummond, 1982; White, 1989).

An alternative to block parameterizations is to define seismic properties at the vertices of a regular grid ( Fig. 3b) togethewith some interpolation function. One of the first implementations of this approach was byThurber (1983),who used trilineainterpolation between a rectangular grid of nodes to define a continuously varying velocity field for local earthquake tomographyThis scheme is now commonly used in earthquake tomography(Eberhart-Phillips, 1986, 1990; Zhao et al., 1992; Eberhart-Phillipand Michael, 1993; Scott et al., 1994; Graeber and Asch, 1999)and can be found in other forms of tomography, including teleseismic tomography (Zhao et al., 1994; Steck et al., 1998). Thuse of higher order interpolation results in a smoother continuum,but requires a larger basis. For example,trilinearinterpolatiomeans that any point within a cell is defined by the 8 points thadescribe the cell, but produces C0 continuity (i.e. continuous, bunot differentiable everywhere). On the other hand, the use of cubiB-splines means that any point within a cell is a function of 64

surrounding points, but results in C2 continuity (i.e. continuousecond derivatives). Thus, there is generally a trade-off betweensmoothness, the width of the local basis, and consequently, computing time. Exceptions include natural cubic splines, which arecubic polynomials that interpolateeach grid point andhavea globabasis (i.e. any point defined by the spline is a function of all gridpoints). Cubic spline functions with a local basis are used widely intomography:Thomson and Gubbins (1982)andSambridge (1990use Cardinal splines in teleseismic and local earthquake tomography respectively;Farra and Madariaga (1988)andMcCaughey andSingh (1997)use cubic B-splines in wide-angle tomography; andRawlinson et al. (2006b)use cubic B-splines in teleseismic tomography. Splines under tension (Smith and Wessel, 1990)is a flexibleform of parameterization that essentially allows variation between

quasi-trilinear interpolation and cubic spline interpolation. Theideal tension factorresults in a smoothmodel that minimizesunrealistic oscillations yet maximizes local control. Neele et al. (1993)VanDecaretal.(1995) and Ritsemaetal.(1998) allusethisapproacin teleseismic tomography.

In regional and global tomography, regular blocks or grids inspherical coordinates are faced with the additional challenge of anartificialincreaseinspatialresolutiontowardsthepolesandcentralaxis. To address this problem Wang andDahlen (1995)and Wangeal. (1998) developspherical surface splines whichessentially correspond toa cubic B-spline basis on a triangular grid of approximatelyequally spaced knot points. In global waveform tomography, thso-called cubed-sphere (Ronchi et al., 1996), which is an analytic mapping from the cube to the sphere, has become popular

particularly in conjunction with the spectral element method fo


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Fig. 3. 2D velocity field defined using (a) constant velocity blocks; (b) cubic B-spline patches.

numerical solution of the elastic wave equation (Komatitsch et al.,2002).

A common alternative to the discretization of seismic proper-tiesinthespatialdomainistoinsteadusethewavenumberdomain.Spectral parameterizations thatuse someform of truncated Fourierseries fall into this category (e.g.Wang and Houseman, 1997);theunknown parameters in the inversion problem then become theamplitude coefficients of the harmonic terms, rather than the val-ues at grid nodes or within blocks that is generally the case when

the spatial domain is parameterized. At the local scale, spectralparameterizations have been used in wide-angle traveltime inver-sion by Hildebrand et al. (1989), Hammer et al. (1994) and Wigginset al. (1996). In global tomography, spherical harmonics are fre-quently used for structural representation (Dziewonskiet al., 1977;Dziewonski and Woodhouse, 1987; Li et al., 1991; Trampert andWoodhouse, 1995; Reid et al., 2001; Romanowicz and Gung, 2002)due to their natural affinity with the shape of the Earth, their rela-tivesimplicity in controlling thewavelengthof recovered structure,and their common usage in other global geophysical studies (e.g.gravity, magnetism), which helps facilitate direct comparison. Thedrawback of infinitely differentiable functions of this type is thatthey have a global basis (i.e. adjustment of any single harmonicterm will have a global influence on the model), so poorly resolved

portions of a model may detrimentally influence (or leak) intoother regions. Furthermore, compute time can become significantfor models described by a large number of harmonic terms, sincethey all contribute to the value of the function at any given point.Amirbekyan et al. (2008)attempt to address these shortcomingsby developinga harmonicspherical spline parameterization, whichcombines spherical harmonics with the spatial localization prop-erties of spline functions.

Representingthe Earth by a function which assumes continuousvariation of seismic properties is valid in many circumstances, butthere arecaseswhereexplicitinclusion ofinterfaces is required.Forexample, in wide-angle tomography, refracted and reflected wavesare the primary observables, and cannot be synthesized withoutthe presence of discontinuities (one could argue that sharp veloc-

ity gradients will give rise to similar phenomena, but the data willgenerally not be able to resolve such features, so explicit interfacesare a valid approximation). There are two basic styles of interface

parameterization that are used in seismic tomography. The mostcommon represents the subsurface as one or more sub-horizontallayers overlying a half-space (Fig. 4a); each layer laterally traversesthe entire model, but may pinch together in one or more places(Rawlinson and Sambridge, 2003b). This is often used in coincidentreflection and wide-angle tomography, where ubiquitous inter-facessuchastheMohoarewellsuitedtothisformofrepresentation(e.g. Chiu et al., 1986; Farra and Madariaga, 1988; Williamson,1990; Sambridge, 1990; Wang and Houseman, 1994; Zelt, 1999;

Rawlinson et al., 2001a; Rawlinson and Urvoy, 2006).The velocity(or other seismic property) within each layer can be representedusing any of the techniques described above, and need not nec-essarily be linked to the interface geometry or adjacent layers. Therelativesimplicityofthisrepresentationmakesitamenabletorapiddata prediction, yet allows many different classes of later arrivingphases to be computed.

In some instances, a priori information is sufficiently detailedthat more sophisticated parameterizations that mix continuousand discontinuous variations in seismic properties are warranted.For example, in exploration seismology, data coverage is usuallydense, and near surface complexities (such as faults) often needto be accurately represented. Furthermore, there is often detailedinformation from field mapping and other geophysical techniques

that is available. A parameterization that may be more suitablein these circumstances involves dividing the model region upinto an aggregate of irregularly shaped volume elements (Fig. 4b),within which seismic properties vary smoothly, but is discontinu-ous across element boundaries (e.g.Pereyra, 1996; Bulant, 1999).This allows most geological features such as faults, folds, lenses,overthrusts, intrusions etc. to be faithfully represented, but makesboth the forward prediction and inverse problems more challeng-ing to solve.

The mathematical functions used to describe interfaces arelargely analogouswith those used to describe seismic continua.Forexample, piecewise linear segments are somewhat equivalent toconstant velocity cells, and produce artificial ray shadow zones onaccount of the gradient discontinuities between each line segment

(Williamson, 1990; Zelt and Smith, 1992).The logical extension ofthis to 3D is to represent surfaces using piecewise triangular areaelements (Sambridge, 1990; Guiziou et al., 1996),as illustrated in

Fig. 4. Twoschemesfor representing mediawhich contain bothcontinuousand discontinuousvariations in seismic property (a) laterally continuous interfaces within which

seismic structurewi(x,z) varies smoothly; (b) flexible framework based on an aggregate of irregular blocks within which seismic structurewi(x,z) varies smoothly.


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Fig. 5. Multi-valued surface constructed using (a) a mesh of triangular area elements; (b) a mosaic of cubic B-spline surface patches.

Fig.5a. Bothoftheseparameterizationsmakemulti-valuedsurfacesstraightforward to represent, but cause two-point ray tracing andhencethe dataprediction problem to become lessrobust. Interfacesmay also be defined on a grid of depth nodes, with some inter-polation function used to describe the complete surface (Fig. 5b).Cubic splines are widely used in 2D and 3D wide-angle tomog-raphy, where sub-horizontal interfaces are commonly included

(e.g.Farra and Madariaga, 1988; White, 1989; Lutter and Nowack,1990; Pereyra, 1996; McCaughey and Singh, 1997; Rawlinson andHouseman, 1998; Rawlinson et al., 2001a; Rawlinson and Urvoy,2006).

2.2. Irregular parameterizations

In regional and global tomography, it has long been recognisedthat the limited geographical distribution of sources and receiversoften leads to highly irregular sampling of the subsurface by therecorded seismic energy. This problem alsoexists for moretargetedexperiments such as localearthquake and teleseismic tomography,although station distribution tends to be more uniform. Studieswhich control the location of sources, such as vibroseis, explo-

sions, air-guns and ambient noise experiments, are less liable toexperience uneven data coverage, but it still remains an issue. Theuse of uniform basis functions, as described above, to representstructural information extracted from such data is therefore incon-sistent, because it does not recognise its spatially varying resolvingpower. An alternative approach is to use a parameterization whichcan itself adapt to the spatially varying constraints supplied by thedata.

Pioneering work in this area goes back several decades, with thestudies ofChou and Booker (1979)andTarantola and Nercessian(1984), who propose block-less parameterizations for seismictomography. These allow local smoothing scale lengths to varyspatially, and are in principle similar to the more recent and com-monly used variable mesh schemes. Continuous regionalization,

as developed byMontagner and Nataf (1986),is one manifesta-tion of the block-less approach to structural representation thatis commonly used in surface wave tomography (e.g. Debayle, 1999;Debayle and Kennett, 2003).It produces a smooth model of vari-able scale length by using a Gaussian prior covariance function toenforce correlation between adjacent points. This takes the form ofa prior variance and horizontal correlation length, which constrainthe allowable amplitude and lateral length scale of anomalies. Thechoice of correlation length can be based on ray path coverage,which helps address the problem of uneven data sampling. One ofthe main drawbacks of the scheme is computational cost, whichscales with M2, where Mis the number of data. Montagner andTanimoto (1990) introduce several approximations to the orig-inal scheme to improve efficiency, and Debayle and Sambridge

(2004) implement sophisticated geometrical algorithmsto exclude

regions that contribute little to the prior covariance function. Thihas the dual benefit of further improving efficiency andmaking thalgorithm highly suited to parallelization. Consequently, the newscheme can be applied to much larger problems (of the order o50,000 paths for example).

In an alternative approach, Fukao et al. (1992) use nonuniformly sized rectangular 3D blocks to account for uneven ray

sampling, andAbers and Roecker (1991)introduce a scheme inwhich fine scale uniform 3D blocks are joined to form larger irregular cells (a bottom-up approach).Sambridge et al. (1995)andSambridge and Gudmundsson (1998)were the first to propose thuse of Delaunay tetrahedra andVoronoi polyhedra, which arecompletely unstructured meshes, in seismic tomography (seeFig. 6foan example of Delaunay triangulationa continuum can be readily described for any arbitrary distribution of nodes). The mainchallenges in using such schemes include: (1) increased computetime to solve the forward problem; (2) developing an appropriate technique for fitting the mesh to the data constraints; (3interpreting the results, which will exhibit structure at multiplescalelengths. Static schemes use a fixed parameterization throughout the inversion, while adaptive schemes dynamically adjust the

parameterization during the inversion.One of the first studies to use an adaptive scheme was that o

Michelini (1995),who adjusts the velocity and position of cubicB-spline control vertices in 2D cross-hole tomography. While thetopology of the control mesh in this case is regular, the use oparametric splines allows some irregularity in the position o

Fig. 6. Irregularparameterization using optimal Delaunay triangulation to describ

a continuum based on a discrete set of control nodes.


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Fig. 7. Principle of the (a) shooting method; (b) bending method of ray tracing. In both cases, iterative refinement of some initial path is required to locate the correct twopoint path.

nodes.Curtis and Snieder (1997)also consider the 2D cross-holeproblem, but use Delaunay triangulation to represent structure. Agenetic algorithm is used to find the position of the node whichminimizes the condition number of the tomographic system ofequations. In 3D reflection tomography,Vesnaver et al. (2000)andBhm et al. (2000)develop an adaptive scheme which uses Delau-nay triangles and Voronoi polyhedra.Zhang and Thurber (2005)also devise an adaptive scheme based on tetrahedral and Voronoidiagrams to match the data distribution, andapply it to local earth-quake and shot data to image the 3D structure beneath Parkfield,California.

Bijwaard et al. (1998), Bijwaard and Spakman (2000) andSpakman and Bijwaard (2001)perform global P-wave traveltimetomography usingan approachsimilar to Abers and Roecker (1991)

in which the 3D mesh is matched to the ray path density prior toinversion (i.e. a static approach). One of the first studies to carryout adaptive whole Earth tomography was that ofSambridge andFaletic (2003),who parameterize the Earth in terms of Delaunaytetrahedra. A top-down approach to mesh adaptation is used, inwhich new nodes are added to the edge of tetrahedra where thelocal velocity gradient is highest. Four updates are performed, witha linear tomographic system based on rays in a laterally homoge-neous Earth solved after each update. This approach to adaptationis simple to implement, but regions of good data constraints arenot always characterized by significant velocity gradients. Otherstudies to use Delaunay tetrahedra in global body wave tomogra-phy include those ofMontelli et al. (2004)andNolet and Montelli(2005).In fact, most global body wave imaging studies now use

irregular meshes of one sort or another (e.g.Burdick et al., 2008).

A review of this topic can be found inSambridge and Rawlinson(2005).

As noted earlier, spectral parameterizations such as sphericalharmonics are not well suited to problems that exhibit signif-icant variations in data coverage. An alternative approach thatshows great promise in addressing the multi-scale nature of seis-mic tomography is the use of wavelet decomposition. Chiao andKuo (2001)investigate the use of Harr wavelets on a sphere forrepresenting lateral shear wave speed variations in D, as con-strained by S-SKStraveltimes. They conclude fromtheir results thatwavelets provide a natural regularizationscheme basedon ray pathsampling, with recovered detail varying according to the resolvingpower of the data.Tikhotsky and Achauer (2008)invert both con-trolled sourceseismic andgravity data for3D velocity andinterface

structure also represented using Haar wavelets.Loris et al. (2007)use more sophisticated wavelets that allow for smoother repre-sentations of structure than the discontinuous Haar wavelets. Theyalso minimize an objective function that, in addition to the usualL2 data misfit term, contains an L1-norm measure of the waveletcoefficients, the aim being to promote a parsimonious descriptionof structure that only has detail where required by the data.

A statistical method known as partition modelling, which is anensemble inference approach used within a Bayesian framework,has recently been introduced to seismic tomography (Bodin andSambridge, 2009). It uses a dynamic parameterization which is ableto adaptto theuneven spatial distributionof information that char-acterizesmost datasets,and doesnot require explicit regularization(damping and smoothing terms can be discarded). In the paper of

Bodin and Sambridge (2009),a Markov chain Monte Carlo method


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is used to invert traveltime data (assuming straight rays) with amodel comprising a small number of constant velocity Voronoicells.Remarkably, even thougheachmodelin thepoolof best fittingsolutions has a very blocky appearance, their average is a smoothmodel that is superior to that obtained using a conventionalregulargrid approach.

3. The data prediction problem

Seismic tomography may exploit one or more observables froma seismic record, including traveltime, amplitude, frequency con-tent orthe full waveform.The need foraccurate,efficient androbustpredictions of these quantities has driven the development of awide range of techniques, most of which are based on the high-frequency assumption of geometric optics. The descriptions anddiscussion below will focus mainly on ray and grid-based tech-niques for solving the two-point problem of finding the path takenby seismic energy between source and receiver.

3.1. Ray-based methods

The full elastic wave equation can be greatly simplified in cases

where the high frequency assumption is valid. It can be shown forboth P and S waves in an isotropic medium (e.g.Rawlinson et al.,2007)that the elastic wave equation will reduce to:

|T| = s, (1)2A T+A2T= 0 (2)where Tis traveltime, s is slowness andA is amplitude. Eq. (1) istheeikonal equation, which governs the propagation of seismic wavesthrough isotropic media. Eq.(2) is the transport equation, whichcan be used to compute the amplitude of the propagating wave. Infully anisotropic media ( Cerveny, 2001),the eikonal and transportequationshave a slightly more complex form dueto thepresenceofthe elastic tensor c. Instead of directly solving theeikonal equation,it is possible to only consider its characteristics, which are trajec-

tories orthogonal to the wavefront (in isotropic media). These aredescribed by the kinematic ray equation:

ddl

s

drdl

= s. (3)

where l is path length and r is a position vector of a point along theray. In anisotropic media, the Hamiltonian formalism of classicalmechanics ( Cerveny, 2001; Chapman, 2004)is a more convenientform of representation. The behaviour of rays in the presence ofinterfaces is simply described by Snells law, which can be gener-alized for anisotropic media (e.g.Slawinski et al., 2000).Dynamicray tracing can be applied to yield amplitudes, andthis can be donemost easily by using the paraxial ray approximation (Cerveny andPsencik, 1983;Cerveny and Firbas, 1984;Cerveny, 1987; Farra and

Madariaga, 1988; Virieux and Farra, 1991;Cerveny, 2001;Cervenyet al., 2007; Tian et al., 2007a), which essentially involves usingfirst-order perturbation theory to deduce characteristics of thewavefield in the neighbourhood of a reference ray.

3.1.1. Shooting methods

Shooting methods of ray tracing formulate Eq. (3)as an initialvalue problem, which allows a complete raypath to be traced (withapplication of Snells law at interfaces if necessary) given some ini-tial trajectory. The two-pointproblem of locating a sourcereceiverpath is more difficult to solve, because it is essentially a (potentiallyhighly) non-linear inverse problem, with the initial raydirection asthe unknown, and some measure of the distance between receiverand ray end point as the function to be minimized. In media

describedbyconstantvelocity(orslowness)blocks,theinitialvalue

problem is simple to solve (via repeated application of Snells law)but the two-point problem is not (Williamson, 1990).Analytic raytracing can also be used in media with a constant velocity gradient (e.g.White, 1989; Rawlinson et al., 2001a),constant gradienof ln v, and constant gradient of the nth power of slowness 1/v(Cerveny, 2001).Other than these few cases, numerical solution oEq. (1) is required.In the presence of interfaces, one potentially difficult problem is to efficiently locate the ray-interface intersectionpoint, particularly when sophisticated interface parameterizationare used. However, a number of practical methods are available(Sambridge,1990;VirieuxandFarra,1991;Rawlinsonetal.,2001a )

The boundary value problem is most commonly solved usingan iterative non-linear approach, in which the source trajectory osome initial guess ray path is perturbed until it hits the desiredend point (seeFig. 7a). Julian and Gubbins (1977) propose twodifferent iterative non-linear techniques for solving the two-poinproblem: one is based on Newtons method, and the other on themethod of false position. Both techniques have been widely used(e.g.Sambridge, 1990; Rawlinson et al., 2001a).Fig. 8shows twopoint paths computed using the shooting method ofRawlinson eal. (2001a). A variety of methods have been proposed for locating asuitably accurate initial guess ray, including shooting a broad fanorays towards the receiver and then iteratively refining the ray fan(Virieux and Farra, 1991),and using the correct two point path foa laterally averaged version of the model (Thurber and Ellsworth1980; Sambridge, 1990). As thenon-linearity of theboundaryvaluproblem increases, iterative non-linear solvers require more accurateinitialguessrays(seeFig.10ofRawlinsonetal.,2007, foracleaillustration). Although not frequently acknowledged in the literature, practical applications of shooting, particularly in regions osignificant heterogeneity, often settle for some acceptable tradeoff between the percentage of two-point paths located, and totacompute time.

Fully non-linear shooting methods, based on sampling algorithms like simulated annealing, have been devised and tested (e.gVelis and Ulrych, 1996, 2001),but they have not proved popularPerhaps this is because ray tracing is at its most useful when veloc

ity heterogeneity is not too severe, so that local sampling of the rayfield is still a valid approach for the detection of two-point pathsWhen this is no longer the case, global techniques like grid basedeikonal solvers (see below) will be much more efficient. Shootingmethods of ray tracing are widely used in seismic tomography, duto their conceptual simplicity, and potential for high accuracy andefficiency (Cassell, 1982; Benz andSmith, 1984; Langanet al., 1985Farraand Madariaga, 1988; Sambridge, 1990; Zelt and Smith, 1992VanDecar et al., 1995; McCaughey and Singh, 1997; Rawlinson eal., 2001b).

3.2. Bending methods

Bending methods of ray tracing iteratively adjust the geome

try of some arbitrary two point path until it becomes a true raypath (seeFig. 7b) i.e. it satisfies Fermats principle of stationarytime. A common approach to implementing the bending methodis to derive a boundary value formulation of Eq. (3), which canthen be solved iteratively (Julian and Gubbins, 1977).Pereyra eal. (1980)devise a bending method similar to that of Julian andGubbins (1977),but extend it to allow for the presence of interfaces. In complex mediaPereyra (1996)use ray shooting to helplocate an initial guess ray.

Pseudo-bending methods use the same principle of adjustingray geometry to locate a true ray, but avoid direct solution of therayequations. One of the first pseudo-bending schemes was developed byUm and Thurber (1987),who describe a ray path by aset of linearly interpolated points. For some initial arbitrary path

described by a small number of points, the scheme proceeds by


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Fig. 8. Two point paths through a 3D medium computed using the shooing method ofRawlinson et al. (2001a). (a) Reflected rays; (b) refracted rays.

adjusting the location of each point by directly exploiting Fermats

principle of stationary time. Once some convergence criterion issatisfied, new points are interpolated between pre-existing points,and the iterative procedure continues until sufficient accuracy isachieved. Despite the relatively crude approximations made inpseudo-bending,Um and Thurber (1987)find it to be much morecomputationally efficient than conventional bending schemes.Consequently, it has become popular for problems that requirelarge travel-time datasets to be predicted, such as in 3D localearthquaketomography (Eberhart-Phillips, 1990; Scottet al., 1994;Eberhart-Phillipsand Reyners, 1997; Graeber andAsch, 1999). Zhaoetal.(1992)modifythepseudo-bendingschemeofUmandThurber(1987)to allow for the presence of interfaces, and Koketsu andSekine (1998)devise a similar scheme in 3D spherical coordinates.

Like ray shooting, fully non-linear bending methods have also

been devised; for example,Sadeghi et al. (1999)develop a methodwhich uses genetic algorithms to globally search for the minimumtime path between two fixed points. Again, like fully non-linearshooting, one could argue that the exhaustive interrogation of theray field for each sourcereceiver pair would make other classesof techniques that guarantee to find the global minimum (likeeikonal solvers) more practical. Apart from shooting and bendingmethods, the boundary value problem can also be solved usingtechniques based on structural perturbation ( Cerveny, 2001). Inthis class of scheme, a known two-point path exists in a referencemedium, and the aim is to locate the equivalent two point path in aslightly modified medium. Solution of this class of problem can beobtainedusingrayperturbationtheory(Farraand Madariaga, 1987;Snieder and Sambridge, 1992; Snieder, 1993; Pulliam and Snieder,1996).Although relevant to iterative non-linear tomography, thisapproach is not widely used.

3.3. Grid-based methods

Instead of tracing rays between source and receiver, an alterna-tive strategy is to compute the global traveltime field as defined bya grid of points. This will implicitly contain the wavefront geome-try as a function of time (i.e. contours ofT(x)), and all possible raytrajectories (specified byT). Compared to conventional shootingand bending methods of ray tracing, grid-based traveltime schemeshave several clear advantages: (1) they compute traveltimes to allpoints in the medium, including (in most cases) diffractions in rayshadow zones; (2) they exhibit high stability in strongly hetero-

geneous media; (3) they efficiently compute traveltime and path

information, particularly when the ratio of sources to receivers

(or vice versa) is high; (4) they consistently yield first-arrivals.The advantages of grid-based schemes are offset somewhat bythe following drawbacks: (1) their accuracy is a function of grid-spacingin 3D,halving thegrid spacing will increase compute timeby at least a factor of eight; (2) in most cases, they only producefirst-arrivals; (3) they have difficulty computing quantities otherthan traveltime (e.g. amplitude); (4) anisotropy, easily dealt withby ray methods, is more of a challenge. Two grid-based schemes finite difference solution of the eikonal equation and shortest pathmethods have emerged as popular alternatives to ray tracing.

3.3.1. Eikonal solversThe use of eikonal solvers in seismology was largely pioneered

by Vidale (1988, 1990), who developeda techniquefor finite differ-

ence solution of the eikonal equation on a grid. Relatively simplecentred difference stencils are formulated for approximating thegradient terms in Eq.(1),so that traveltimes can be computed atnew points using known values at adjacent points. An expand-ing square is adopted for the computational front, which sweepsthrough themedium from thesource point until thecompletetrav-eltime field is computed. Ray paths can be found retrospectively bysimply following the traveltime gradient from each receiver backto the source. The resulting scheme is fast and sufficiently accuratefor most seismic applications, with CPU time being approximatelyproportional to the number of points defining the grid. The use ofan expanding square formalism to define the shape of the com-putational front cannot always respect the direction of flow ofinformation through the medium. For example, it is possible that

a first arrival will need to sample a region outside the expandingsquarebefore returning back into it. Consequently, first arrivals arenot always guaranteed, which can lead to instability. Nonetheless,the basic scheme proposed byVidale (1988, 1990)remains popu-lar, andits stabilityhas been improved thanks to new features suchas special headwave operators (Hole and Zelt, 1995; Afnimar andKoketsu, 2000),and post-sweeping to correct for the non-causalnature of the expanding square (Hole and Zelt, 1995). van Trier andSymes (1991)use entropy-satisfying first-order upwind operatorsto improve computational efficiency and deal with wavefront dis-continuities. Comparable improvements are made byPodvin andLecomte (1991),who employ Huygens principle in the finite dif-ference approximation.

The above techniques, which have largely been independently

developed in seismology, bears some similarity with essentially


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Fig. 9. Reflected wavefront and traveltime field computed using the FMM schemeofRawlinson and Sambridge (2004a).

non-oscillatory (ENO) finite difference schemes developed in thefield of computational mathematics (Shu and Osher, 1988, 1989).The attraction of ENO schemes is that they can be readily expandedto very high orders of accuracy, yet remain stable. Kim and Cook(1999) devise a scheme which they describe as ENODNOPSto efficiently compute first-arrival traveltime fields. The DNO (ordown n out) refers to an expanding box computational front, andPS refers to post-sweeping. Therefore, apart from the finite differ-ence stencil, itis very similar tothe scheme ofHole andZelt (1995).WENO (weighted ENO) schemes, which offer improved computa-tion time andstability compared with ENO schemes, have also beendeveloped (Liu et al., 1994; Jiang and Shu, 1996; Jiang and Peng,2000).Qian and Symes (2002)use a WENO scheme with adaptivegridding to compute traveltimes, andBuske and Kstner (2004)implement a WENO scheme in polar coordinates to compute trav-eltimesthataresufficientlyaccuratetosolvethetransportequationfor amplitudes.

In order to overcome the limitations of the expanding squareformalism,Qin et al. (1992) and Cao and Greenhalgh (1994) usethe first-arriving wave-front as the computational front, which islocally evolved by always choosing the point with minimum trav-eltime along the edge of the computed field to update adjacentpoints. This ensures that the shape of the computational front con-forms to the first-arrival wavefront, and will not result in causalitybreaches.The drawbackof thisapproachis the additional computa-tional expense involved in locating the global minimum traveltimepoint along the computational front. For example, if heap sorting isused, then computing time will be proportional toNlog N, whereNis the total number of grid points used to describe the velocityfield.

Another eikonal solver that was developed in the field of com-

putational mathematics is the so-called Fast Marching Method orFMM (Sethian, 1996; Sethian and Popovici, 1999; Popovici andSethian, 2002).It uses upwind entropy-satisfying finite differencestencils to solve the eikonal equation, and a computational front(narrow band) that encapsulates the first-arriving wavefront. Thefinite difference stencils account for the fact that the first-arrivaltraveltime field is not always differentiable (i.e. the Tterm inEq.(1)is not necessarily defined everywhere), and result in a veryrobust method. Rawlinson and Sambridge (2004a,b) extend thescheme to improve accuracy in the source neighbourhood wherewavefront curvature is high (and therefore poorly described bya regular grid), and compute phases comprising any number ofrefractedandreflectedbranchesinmediacontaininginterfaces(seeFig. 9).Eikonal solvers are now widely used in tomography, partic-

ularly 3D wide-angle and teleseismic studies (Hole, 1992; Zelt et

al., 1996, 2001; Riahi et al., 1997; Zelt and Barton, 1998; Zelt, 1999Day et al., 2001; Rawlinson et al., 2006a,b; Rawlinson and Urvoy2006; Rawlinson and Kennett, 2008).

3.3.2. Shortest path ray tracing

Another class of grid-based scheme that has been used in seismic tomography to compute traveltimes to all points of a griddedvelocity field is shortest path ray tracing or SPR (Nakanishi andYamaguchi, 1986; Moser, 1991; Fischer and Lees, 1993; Cheng andHouse,1996; Zhao etal.,2004;Zhouand Greenhalgh,2005). Insteadof solving a differential equation, a network or graph is formedby connecting neighbouring nodes with traveltime path segmentsDijkstra-like algorithms can then be used to find the shortest timepath between a source and receiver, which, according to Fermatsprinciple of stationary time, will correspond to a valid ray pathShortest path networks are commonly defined in terms of eithea cell or grid centred framework. In the latter case, the connection stencil is often referred to as the forward star ( Klimes andKvasnicka, 1994).In 2D a forward star with 8 connections will joinany node with all of its immediate neighbours, but will not allowvariations in ray path trajectory less than 45 (for a square grid)By allowing direct connections between the central node and thneighbours of the 8 nodes, a forward star with 16 connections canbedefined,whichwillpermitgreaterflexibilityoftheraygeometry.

The difference between a scheme like FMM and SPR is actuallynotallthatgreat;theybothusetheshapeofthefirst-arrivingwavefront as the computational front and use the same approach fochoosing a new node for a local update of the traveltime field. Theonly obvious change is in theupdate stencil that is used to computnew traveltimes. As such, many of the extensions that have beenapplied to FMM, such as grid refinement and the location of laterarriving phases consisting of reflected and refracted branches, areequally applicable to SPR (Moser, 1991).Although less frequentlyused than eikonal solvers, SPR has been implemented in a numbeof tomographic studies to solve the forward problem of traveltimprediction (Nakanishi and Yamaguchi, 1986; Toomey et al., 1994Zhang and Toksz, 1998; Bai, 2005).

3.4. Multi-arrival schemes

All of the schemes described above are really only suitable fotracking a single (usually the first) or a limited number of arrivalsbetween two points. However, even relatively small variations inseismic wavespeed can produce a phenomenon known as multipathing, which is simply defined as when more than one ray pathconnects two points in the medium. In order for this to occur, thepropagating seismic wavefront must distort to such an extent thait self-intersects (or folds over on itself). In such situations, eikonaand shortest path methods will yield the first arrival only, whilmost shooting and bending methods will only produce a singlearrival (with no real guarantee as to whether it is a first or late

arrival). With sufficient probing of the medium, multiple arrivalcan be produced with ray tracing, but usually not in a robust oefficient enough manner for applications such as tomography. ThheterogeneityoftheEarth,particularlynearthesurface,meansthatmulti-pathingcommonlycontributes to thecomplexity of recordedwaveforms. Therefore, any method that can accurately predict alarrivals of significant amplitude has important implications foEarth imaging.

To date, a number of methods have been developed to solve thmulti-arrival problem. These include both grid-based (Benamou1999; Steinhoff et al., 2000; Engquist et al., 2002; Fomel andSethian, 2002; Osher et al., 2002; Symes and Qian, 2003), ray based(Vinje et al., 1993; Lambar et al., 1996; Vinje et al., 1996, 1999Hauser et al., 2008)and hybrid (Benamou, 1996)schemes. Due to

therelativeinfancy of grid-based schemes, which have been largely


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Seismic TomographyA Window Into Deep Earth

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