eikonal tomography: surface wave tomography by phase-front ... · cwp-661 eikonal tomography:...

CWP-661

Eikonal tomography: Surface wave tomography by

phase-front tracking across a regional broad-band

seismic array

Fan-Chi Lin1, Michael H. Ritzwoller1 & Roel Snieder2

1Center for Imaging the Earth’s Interior, Dept. of Physics, University of Colorado at Boulder2Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401

ABSTRACT

We present a new method of surface wave tomography based on applying theEikonal equation to observed phase travel time surfaces computed from seismicambient noise. The source-receiver reciprocity in the ambient noise method im-plies that each station can be considered to be an effective source and the phasetravel time between that source and all other stations is used to track the phasefront and construct the phase travel time surface. Assuming that the amplitudeof the waveform varies smoothly, the Eikonal equation states that the gradient ofthe phase travel time surface can be used to estimate both the local phase speedand the direction of wave propagation. For each location, we statistically sum-marize the distribution of azimuthally dependent phase speed measurementsbased on the phase travel time surfaces centered on different effective source lo-cations to estimate both the isotropic and azimuthally anisotropic phase speedsand their uncertainties. Examples are presented for the 12 and 24 sec Rayleighwaves for the EarthScope/USArray Transportable Array stations in the westernUS. We show that: (i) the major resulting tomographic features are consistentwith traditional inversion methods; (ii) reliable uncertainties can be estimatedfor both the isotropic and anisotropic phase speeds; (iii) “resolution” can beapproximated by the coherence length of the phase speed measurements and isabout equal to the station spacing; (iv) no explicit regularization is required inthe inversion process; and (v) azimuthally dependent phase speed anisotropycan be observed directly without assuming its functional form.

Key words: seismic interferometry, anisotropy, crustal structure

1 INTRODUCTION

The seismic surface wave tomography inverse problemis normally approached in one of two ways that can bethought of as either “single-station” or “array-based”methods. Both methods have proven effective at reveal-ing the spatial variability of surface wave speeds fromglobal to regional scales.

The first (single-station) approach to surface wavetomography is based on travel time measurements be-tween a set of seismic sources (typically earthquakes)and a set of receivers one receiver at a time. The traveltimes are then interpreted in terms of wave speeds in themedium of propagation using ray theory with straight or

potentially bent rays (e.g., Trampert and Woodhouse,1996; Ekstrom et al., 1997; Ritzwoller and Levshin,1998; Yoshizawa and Kennett, 2002) or finite frequencykernels (e.g. Dahlen et al., 2000; Ritzwoller et al. 2002;Levshin et al., 2005). This method results in a set offrequency-dependent dispersion maps of either Rayleighor Love wave group or phase speed. This approach alsohas been applied to ambient noise data (e.g. Sabra etal., 2005; Shapiro et al., 2005; Yao et al. 2006; Moschettiet al., 2007; Lin et al. 2007; Yang et al., 2007; Bensenet al., 2008), which provides wave travel times betweenpairs of receivers. In this case, one station can be con-sidered to be an “effective” source, but it is equivalent

278 F-C. Lin, M. H. Ritzwoller & R. Snieder

to the earthquake tomography problem in which thesources excite the wavefield. A variant of this methodinvolves waveform fitting which in some cases bypassesthe dispersion maps to construct the 3-D variation ofshear wave speed directly in earths interior (e.g. Wood-house & Dziewonski 1984; Nolet, 1990; van der Lee andFredriksen, 2005).

The second approach to surface wave tomographydeals with stations as components of an array and in-terprets the phase difference observed between wavesrecorded across the array in terms of the dispersioncharacteristics of the medium. In doing so, this methodeither applies geometrical constraints on the stations,typically that they lie nearly along a great circle withthe earthquake (e.g. Brisbourne & Stuart 1998; Prindle& Tanimoto 2006), or inverts for the characteristics ofthe incoming wave-front along with the surface wave dis-persion characteristics of the medium lying within thearray (e.g. Alsina et al. 1993; Friederich 1998; Yang &Forsyth 2006).

In both approaches, the surface wave dispersionmaps result from a regularized inverse problem that istypically solved by matrix inversion. Regularization inmost cases is ad-hoc, and includes spatial smoothingas well as matrix damping. As in many geophysical in-verse problems, a trade-off between the amplitude ofthe heterogeneity and the resolution emerges that af-fects confidence in the smaller structural scales whenhigh resolution is desired. This trade-off is most severefor azimuthal anisotropy, as has been well documentedby previous studies (e.g. Laske & Master 1998; Levshinet al. 2001; Trampert and Woodhouse 2003; Smith et al.2004; Deschamps et al. 2008), in which the amplitudeof anisotropy is particularly poorly determined. Theseproblems are exacerbated by the fact that uncertaintyinformation that emerges for the maps tends to be un-reliable. Theoretical approximations made in the inver-sion, such as the assumption of straight (great-circle)rays or approximate sensitivity kernels, also affect thequality of the resulting maps. This particularly calls intoquestion the robustness of information about azimuthalanisotropy because the magnitude of the travel time ef-fects of azimuthal anisotropy and ray bending, for ex-ample, is similar.

The purpose of this paper is to present a newmethod of surface wave tomography that complementsthe traditional methods. The method is based on track-ing surface wavefronts across an array of seismometers(Pollitz 2008) and should, therefore, be seen to lie withinthe tradition of array-based methods, although as willbe seen in the discussion below the method degener-ates to phase measurements obtained at single stations.The method is applicable, in principle, to surface wavesgenerated both by earthquakes and ambient noise, butapplications in this paper will concentrate on ambientnoise recordings across the Transportable Array (TA)component of EarthScope/USArray (Fig. 1). Because it

235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

Figure 1. The 499 stations used in this study are identifiedby black triangles. Waveforms are taken continuously fromOctober, 2004 until November, 2007. Most stations are fromthe EarthScope/USArray Transportable Array (TA), but afew exceptions exist, such as NARS Array stations in Mex-ico. The four red symbols identify locations used later in thepaper.

is an array-based method, however, an array is needed.The TA provides an ideal setting, but large PASSCALexperiments are suitable for the method and the emer-gence of large-scale arrays in Europe and China thatmimic the station spacing of the TA also provide nearlyoptimal targets.

The method described in this paper is performedin three steps. We discuss the method here in the con-text of ambient noise tomography such that each stationcan be considered to be an effective source as well as areceiver. The relevance of the method to earthquake to-mography is discussed later in the paper. In the firststep, a phase delay (or travel time) surface is computedacross the array centered on each station. We refer tothis step as wavefront or phase-front tracking. In thesecond step, the gradient of each travel time surface iscomputed at each spatial node. Invoking the Eikonalequation, the magnitude of the gradient approximateslocal phase slowness and the direction of the gradientis the direction of propagation of the geometrical ray.Steps 1 and 2 are performed with every station in thearray as the effective source for the travel time surface.

Eikonal tomography 279

Finally, in step 3, for each spatial node the local phasespeeds and wave path directions are compiled and aver-aged from the travel time surfaces centered on each in-dividual station in the array. Because step 2 invokes theEikonal equation, we refer to the method as “Eikonaltomography”.

Eikonal tomography complements traditional sur-face wave tomography in several ways. First, there isno explicit regularization and, hence, the method islargely free from ad-hoc choices. The method as we im-plement it does, however, involve smoothing in track-ing the phase-fronts. Second, the method accounts forbent rays, but ray tracing is not needed. The gradientof the phase front provides information about the lo-cal direction of travel of the wave. The use of bent raysin traditional tomography would necessitate iterationwith ray tracing performed on each iteration. Third,the method naturally generates error estimates for theresulting phase speed maps. In our opinion, this is moreuseful than relying on global misfit obtained by tradi-tional inversion methods. Fourth, in the context of es-timating azimuthal anisotropy, Eikonal tomography di-rectly measures azimuth dependent phase velocities ateach node. Unlike the traditional tomographic method,no ad-hoc assumption about the functional dependenceof the phase velocity with azimuth is made. Finally, inthe construction of phase speed maps, the ray tracingand matrix construction and inversion of the traditionalmethods have been replaced by surface fitting, computa-tion of gradients, and averaging. The method, therefore,is computationally very fast and parallelizes trivially.

Although we have applied Eikonal tomography suc-cessfully from 8 sec to 40 sec period across the westernUS, we present results here only for the 12 sec and 24sec Rayleigh waves. In principle, the same method canbe applied to Love waves as well. The results shown inthis study are presented to illustrate the method. In-terpretation of the results will be the subject of futurecontributions.

2 THEORETICAL PRELIMINARIES

The traditional approach to seismic tomography beginswith a statement of the forward problem that links un-known earth functionals (such as seismic wave speeds,surface wave phase or group speeds, etc.) with obser-vations. In surface wave tomography, when mode cou-pling and the directionality of scattering are neglected,this involves the computation of travel times from the2-D distribution of (frequency dependent) surface wavephase speeds, c(r), that can be written in integral formas

t(rs, rr) =

Z

A(r, rs, rr)dxm

c(r)(1)

where rs and rr are the source and receiver lo-cations, r is an arbitrary point in the medium, and

m = 1 or 2 denotes line and area integrals, respec-tively. For “ray theories”, m = 1 and the integral ker-nel, A(r, rs, rr), vanishes except along the path, whichis typically either a great-circle (straight ray) or a pathdetermined by the spatial distribution of phase speed(geometrical ray theory) which is known only approx-imately. Ray theories are fully accurate at infinite fre-quency and approximate at any finite frequency. Form = 2, the integral is over area, and the integral kernelrepresents the finite frequency spatial extent of struc-tural sensitivity. The sensitivity kernel may be ad-hoc(e.g., Gaussian beam) or determined from a scatter-ing theory (e.g, Born/Rytov) given a particular 1D orhigher dimensional input model. Spatially extended ker-nels are referred to as finite frequency kernels, to con-trast them with ray theories. Much of recent theoreti-cal work in surface wave seismology has been devotedto developing increasingly sophisticated, and presum-ably accurate, representations of the integral kernel inequation 1 (e.g. Zhou et al. 2004; Tromp et al. 2005),although debate continues about whether approximatefinite frequency kernels are preferable practically to raytheories based on bent rays with ad-hoc cross-sections(e.g., Yoshizawa and Kennett, 2002; van der Hilst & deHoop 2005; Montelli et al. 2006; Trampert and Spetzler,2006).

Equation 1 defines travel time as a “global” con-straint on structure; that is, it is a variable that de-pends on the unknown structure over an extended re-gion of model space and is defined to be contrasted with“local” constraints. The traditional primacy of the for-ward problem in defining the inverse problem necessi-tates that the inverse problem is similarly global in char-acter. Travel time observations constrain phase speedsnon-locally, that is over an extended region of modelspace.

In contrast, Eikonal tomography places the inverseproblem in the primary role once the phase travel timesurfaces, τ (ri, r), for positions r relative to an effectivesource located at ri are known. The Eikonal equation(e.g., Wielandt, 1993; Shearer, 1999) is based on thefollowing

1

ci(r)2= | ▽ τ (ri, r)|

2 −▽2Ai(r)

Ai(r)ω2(2)

which is derived directly from the Helmholtz equa-tion. When the second term on the right is small, then

k̂i

ci(r)∼= ▽τ (ri, r) (3)

Here, ci is the phase speed for travel time surface i

at position r, ω is frequency, and A is the amplitude ofan elastic wave at position r. The gradient is computedrelative to the field vector r and k̂i is the unit wavenumber vector for travel time surface i at position r.The Eikonal equation, equation 3, derives by ignoringthe second term on the right hand side in equation 2.In this case, the magnitude of the gradient of the phase


travel time is simply related to the local phase slownessat r and the direction of the gradient provides the localdirection of propagation of the wave. Thus, the Eikonalequation places local constraints on the surface wavespeed.

Dropping the second term on the right hand sideof equation 2 is justified either at high frequencies orif the spatial variation of the amplitude field is smallcompared with the gradient of the travel time surface.The latter is the less restrictive constraint and willhold if lateral phase speed variations are sufficientlysmooth to produce a relatively smooth amplitude field.Moreover, when repeated measurements are performedwith phase travel time surfaces from different effectivesources, the errors caused by dropping the amplitudeterm are likely to interfere destructively, but will con-tribute to the estimated uncertainty especially when thewavelength is shorter than the length-scale of velocitystructure (Bodin & Maupin, 2008). We take this inter-pretation as the basis for the use of the Eikonal equationand use synthetic tests, presented in section 5.1, to con-firm that the effect of dropping the amplitude term isnot a significant source of error in this study. In addi-tion, in ambient noise tomography, absolute amplitudeinformation is typically lost due to time- and frequency-domain normalization prior to cross-correlation (Bensenet al., 2007). In this circumstance, the computation ofthe second term on the right hand side of equation 2 isimpossible.

The question may arise whether Eikonal tomogra-phy should be considered to be a geometrical ray the-ory or a finite frequency theory. The question is moti-vated by considering globally constrained inverse prob-lems and is somewhat inapt for a locally constrainedinversion. We believe, however, that the answer is thatEikonal tomography has elements of both. Certainly,the Eikonal equation presents information about the lo-cal direction of propagation of a wave and is, there-fore, not a straight ray method but is “geometrical”in character. But, the phase travel time surfaces thatare taken as data in the inversion possess spatially ex-tended sensitivity (finite frequency information) and Linand Ritzwoller (On the determination of empirical sur-face wave sensitivity kernels, manuscript in preparation,2009) shows how approximate empirical finite frequencykernels can be determined from them. Thus, ignoringthe second term on the right hand side of equation 2does not equate with rejecting finite frequency infor-mation. However, the resulting interpretation of the lo-cal gradient of the phase travel time surface in termsof a wave propagating with a single well-defined direc-tion, k̂, is consistent with a single forward scatteringapproximation. If there were more than one scatterer,i.e., multipathing, then the equation could not be inter-preted as defining an unambiguous direction of travelat each point. Thus, we do not see Eikonal tomographyas a ray method, but summarize it as an approximate

finite frequency, geometrical (i.e., bent ray), single for-ward scattering method.

3 PHASE-FRONT TRACKING

Eikonal tomography for ambient noise begins by con-structing cross-correlations between each station-pair.The ambient noise cross-correlation method to estimatethe Rayleigh and Love wave empirical Greens functions(EGFs) is described by Bensen et al. (2007) and Linet al. (2008). We use the method to produce Rayleighwave EGFs and phase velocity curves between 8 and40 sec period and have processed all available verticalcomponent records from the USArray/TA observed be-tween October 2004 and November 2007. These stationsare shown in Figure 1. The symmetric component cross-correlation (average of positive and negative lag wave-forms) between each station pair is used to constructthe EGFs.

Each phase travel time surface is defined relative toa given station location, ri, which is coincident with theeffective source location of the wave field. If r denotes anarbitrary location, then the travel time surfaces relativeto effective sources i is given by τ (ri, r)for1 ≤ i ≤ n,where n is the number of stations. The construction ofthe phase travel time surfaces across the array startsby mapping the phase travel times in space centeredon the effective source locations. Figure 2a presents ex-ample great-circle ray paths for an effective source atTA station R06C and Figure 2b shows the EGFs to allother TA stations plotted as a record section band-passfiltered from 15 to 30 sec period. The coherence of theinformation contained in this record section can be seenin wavefield snap-shots such as those in Figure 3, inwhich the amplitude of the normalized envelope func-tion for each EGF is color coded. Plots such as theseillustrate that the entire Rayleigh wavefield can be seento propagate away from the effective source. The plotalso illustrates how the amplitude of the EGF varieswith azimuth, with the largest amplitudes pointing di-rectly toward or away from the coast relative to thecentral station. Nevertheless, reliable phase times aremeasurable at nearly all azimuths, which is essential inorder to map the phase travel time surface.

Phase travel times to all stations from an effec-tive source are measured using the method of Lin etal. (2008) on each EGF between 8 and 40 sec period.For a fixed frequency, the measured phase travel timeis assigned to each station whose EGF has a signal-to-noise ratio (SNR) exceeding 15, where SNR is definedby Bensen et al. (2007). To construct a phase travel timesurface, these phase travel times must be interpolatedonto a finer, regular grid. To do this, we fit a minimumcurvature surface onto a 0.2◦×0.2◦ grid across the west-ern US. The result for central station R06A for the 24sec Rayleigh wave is shown in Figure 4a. Variations inthe method of interpolation have minimal effect on the


Dis

tan

ce (

km

)

0

500

1000

1500

Time (sec)0 100 200 300 400 500 600

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

(a) (b)

Figure 2. (a) Great circle paths linking station R06C (southeast of Lake Tahoe, identified by the white star) with all TAstations where cross-correlations were obtained. (b) Symmetric component record section for 15-30 sec period band-passedvertical-vertical cross-correlations with station R06C in common. More than 450 cross-correlations are shown. Clear move-outnear 3km/s is observed.

resulting surface, averaging less than 0.2 sec except nearthe central station and on the maps periphery. An ex-ample is shown in Figure 4b in which a second interpola-tion scheme invokes an extra tension term in the surfacefitting (Smith and Wesson, 1990). The difference nearthe center is expected because the real travel time sur-face will have singular curvature at the effective source.Accurate modeling of the phase time surface near thesource, therefore, would require a different method of in-terpolation than that used here. In addition, travel timemeasurements obtained between stations separated byless than 1-2 wavelengths are less reliable than thosefrom longer paths. Thus, from each travel time surfacewe remove the region within two wavelengths of the cen-tral station and also any region in which the phase traveltime difference between the two interpolation methodsis greater than 1.0 sec. Finally, as an added quality con-trol measure, for each location we include measurementsfrom this location only when at least three of the fourquadrants of the East-West and North-South axes areoccupied by at least one station within 150 km. Theresulting truncated phase travel time map centered onstation R06A for the 24 sec Rayleigh wave is shown inFigure 5a. Several other examples with either a differ-ent central station or a different period are also shown inFigure 5. This method of phase front tracking is not per-fect, as several irregularities in the contours of constant

travel time in Figure 5c testify. Statistical averaging isneeded to reduce the effects of these irregularities, asdiscussed later in section 4.

The phase-front tracking process introduced hereis essentially the only place in the Eikonal tomographymethod where the inverter has the freedom to make ad-hoc choices. The choice of using a minimum curvaturesurface fitting method as our interpolation scheme min-imizes the variation of the gradient and hence gives thesmoothest resulting velocity variation. With this inter-polation scheme, however, the phase travel time sur-face within an area bounded by the three to four closeststations will always have similar gradients. This spatialcoherence of the variation of the gradient, as we willdiscuss later on in section 4.2 and 5.1, limits our abil-ity to resolve velocity anomalies much smaller than thestation spacing. If higher resolution is desired, a moresophisticated interpolation scheme will be required.

4 EIKONAL TOMOGRAPHY

For the Eikonal equation, equation 3, the magnitude ofthe gradient of the phase travel time is simply relatedto the local phase slowness at position r and the direc-tion of the gradient provides a measure of the directionof propagation of the wave. Taking the gradient on the


235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

0 15 25 35 50 70 100

235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

100s 200s

normalized amplitude

(a) (b)

Figure 3. Snapshots of the normalized amplitude of the ambient noise cross-correlation wavefield with TA station R06C (star)in common at the center. Each of the 15-30 sec band-passed cross-correlations is first normalized by the rms of the trailing noise(Lin et al. 2008) and fit with an envelope function in the time domain. The resulting normalized envelope functions amplitudes arethen interpolated spatially. Two instants in time are shown, illustrating clear move-out and the unequal azimuthal distributionof amplitude.

phase travel time surface gives the local phase speed asa function of the direction of propagation of the wave.Hence, there is no need for a tomographic inversion. Ifthe Eikonal equation is looked at as an inverse problem,the gradient is seen as the inverse operator that mapstravel time observations into model values (phase slow-nesses) and is applied without the need first to constructthe forward operator.

4.1 Isotropic wave speeds

Figure 6 shows the result of applying the Eikonal equa-tion to the phase travel time surface for the 24 secRayleigh wave shown in Figure 5a centered on stationR06A. For each individual central station i, the resultingphase speed map is noisy (Figure 6a) due to imperfec-tions in the phase travel time map. This is caused byerrors in the input phase travel times which, in a similarmeasurement, Lin et al. (2008) estimated to be about1 sec, on average. This is a significant error when spac-

ing between stations is small. But, there are n stations,which in the present study for the TA is about 490.This allows the statistics of the phase speed estimatesto be determined. For example, Figure 7a shows the 455Rayleigh wave phase speed measurements at a period of24 sec as a function of the propagation direction for thepoint in Nevada identified by the star in Figure 1. Todetermine the isotropic phase speed and its uncertaintyfor each point, we first calculate the mean slowness, s0,and the standard deviation of the mean slowness, σs0

,from the distribution of slowness measurements, si:

s0 =1

n

nX

i=1

si (4)

σs0

2 =1

n(n − 1)

nX

i=1

(si − s0)2 (5)

where n is the number of effective sources. This in-termediate step properly accounts for error propagation.


235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0 100 200 300 400 500

Phase travel time (sec) Phase travel time difference (sec)

(a) (b)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

-2.0 -1.0 -0.5 -0.1 0.1 0.5 1.0 2.0

Figure 4. (a) The phase travel time surface for the 24 sec Rayleigh wave centered on TA station R06C (star). Contours areseparated by 24 sec intervals. (b) The difference in phase speed travel time using two different phase-front interpolation schemes.The 48 sec contour is identified with a grey circle centered on station R06C.

The isotropic phase speed, c0, and its uncertainty, σc0 ,are then determined by

c0 =1

s0

(6)

σc0 =1

s02σs0

(7)

The local phase speed uncertainty, σc0 , is mappedfor the 24 sec Rayleigh wave in Figure 8a where only theregion in which the number of measurements is greaterthan half the total number of the effective sources isshown. The average uncertainty across the map is about7 m/sec or about 0.2% of the phase speed. Note thatthis uncertainty estimate only accounts random errorswithin travel time measurements. Systematic errors in-troduced by the tomography method itself will be dis-cussed in Section 5.1.

Example phase speed measurements and the uncer-tainty map for the 12 sec period Rayleigh wave are dis-played in Figures 7b and 8b, respectively. Uncertaintyat this period is largest along the western and northernedges of the region which is most likely due to smallscale wave-front distortion resulting from large velocitycontrasts. The average uncertainty is about 8 m/sec,

which is slightly larger than at 24 sec. This is not un-expected because the validity of the Eikonal equationrelies on smoothly varying velocity structures and thisis a less robust assumption for surface waves at shorterperiods.

The isotropic phase speed maps at periods of 24sec and 12 sec are plotted in Figures 9a and 10a, re-spectively. For comparison, the phase speed maps de-termined from the phase speed measurements using atraditional tomographic method based on the straightray approximation (Barmin et al., 2001) are shown inFigures 9b and 10b. Differences between the methodsare illustrated in Figures 9c and 10c.

Agreement between the isotropic maps producedwith Eikonal tomography and the traditional straightray tomography is generally favorable, but there are re-gions of significant disagreement. At 24 sec period, thedifferences are greatest near the western boundary ofthe map where Eikonal tomography seems to recovercrisper, more highly resolved features that correlate bet-ter with known geological structures. For the 24 secRayleigh wave, the phase velocity contrast between thefast and slow anomalies is generally too gentle to makeray paths deviate significantly from great circle paths.


235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0 100 200 300 400 500Phase travel time (sec)

(a) (b)

(c) (d)

Figure 5. Rayleigh wave phase speed travel time surfaces at periods of (a,b) 24 sec and (c,d) 12 sec centered on two “effectivesources”: stations R06C (eastern California) and F10A (northeastern Oregon). Travel time level lines are presented in incrementsof the wave period. The maps are truncated within 2 wavelengths of the central station and where the three out of four quadrantselection criterion is not satisfied. These two criteria usually take effect only near the periphery of the station coverage.


Phase velocity (km/s)

(a) (b)

235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

3.00 3.25 3.35 3.45 3.55 3.65 3.75 3.85 4.10

Angle difference (deg)

0 1 3 6 12 18

235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

Figure 6. (a) The phase speed inferred from the Eikonal equation for the 24 sec Rayleigh wave travel time surface shown inFig. 5a centered on station R06A. (b) The propagation direction determined from the gradient of the phase travel time surfaceat each point is shown with arrows. The difference between the observed propagation direction and the straight ray prediction(radially away from stations R06A) is shown as the background color.

This is also indicated in Figure 6b where the average de-viation of propagation direction from great circle pathis only about 3◦. It is not likely, therefore, that the dif-ferences observed between Eikonal and traditional to-mography at this period are purely because Eikonal to-mography accounts for bent rays. Differences more likelyresult from the regularization applied in the straight rayinversion, which tends to distort the velocity anomaliesnear the edges of the map. At 12 sec period, however,velocity contrasts are more significant and the off-great-circle effect is more pronounced. The effect of modelingbent rays in Eikonal tomography can be seen in at leasttwo features of the 12 sec map. First, a lineated anomalyassociated with the Cascade Range is better observedwith Eikonal tomography. Second, Eikonal tomographyalso produces wave speeds that are systematically slowerthan the straight ray inversion (Figure 10c) in most ofthe region. The bent rays travel faster than the straightrays (Roth et al. 1993) and to fit the data equally wellwith bent rays requires depression of wave speeds, onaverage. This can be seen clearly in the histograms ofdifferences presented in Figure 11, where the mean dif-ference between the two 12 sec maps is about 10 m/sec

(about 0.3% of the phase speed), whereas the 24 secmaps differ, on average, only by ∼5 m/sec.

4.2 Coherence length of the measurements

Traditional estimates of resolution typically are basedon applying the inverse operator (relating observationsto model variables) to the forward operator (relatingmodel variables to observations) in an inverse problem.With Eikonal tomography, neither an inverse nor a for-ward operator are constructed explicitly, so resolutionis not straightforward to determine. Checkerboard testsare possible, but numerical simulations would need toaccurately calculate the phase travel time between eachstation pair.

We take a different approach and attempt to es-timate the resolution based on the coherence length ofthe measurements. To do so, we first estimate the statis-tical correlation, ρ, of slowness measurements betweenlocations j and k by

ρjk =

ˆPn

i=1(sji − sj0)(ski − sk0)

˜

2

Pn

i=1(sji − sj0)2

Pn

i=1(ski − sk0)2

(8)


3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

0 50 100 150 200 250 300 350

(a)

(b)

c0 = 3.556 km/s

σ = 0.006 km/s

Azimuth (deg)

Vel

oci

ty (

km

/s)

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

0 50 100 150 200 250 300 350

c0 = 3.191 km/s

σ = 0.004 km/s

Azimuth (deg)

Vel

oci

ty (

km

/s)

0c

0c

24 sec

12 sec

C. Nevada

C. Nevada

Figure 7. (a) Example of the azimuthal distribution of the Rayleigh wave phase velocity measurements at 24 sec period forthe point in central Nevada indicated by the star in Figure 1. (b) Same as (a), but for the 12 sec Rayleigh wave phase speed atthe same location. The mean and standard deviation of the mean are identified at upper left in each panel.


Phase speed uncertainty (m/s)

(a)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

(b)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0 3 6 9 12 15

Figure 8. (a) The 24 sec period isotropic Rayleigh wave phase speed uncertainty map, determined from the distribution ofphase speed measurements based on applying the Eikonal equation to each of the phase travel time maps at each point. (b)The 12 sec isotropic Rayleigh wave phase speed uncertainty map.

3.30 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.80

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

Phase speed (km/s)

(a) (b)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

Phase speed difference (km/s)

-0.20 -0.10 -0.03 0.00 0.03 0.10 0.20

(c)

Figure 9. (a) The 24 sec Rayleigh wave isotropic phase speed map derived from Eikonal tomography. The isotropic phasespeed at each point is calculated from the distribution of local phase speeds determined from each of the phase travel timemaps. (b) Same as (a), but the straight ray inversion of Barmin et al. (2001) is used. The black line is the 100 km resolutioncontour. (c) The difference between Eikonal and straight ray tomography is shown where positive values indicate that theEikonal tomography gives a higher local phase speed.


Phase speed (km/s)

(a) (b)

Phase speed difference (km/s)

-0.20 -0.10 -0.03 0.00 0.03 0.10 0.20

(c)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

2.90 3.00 3.10 3.15 3.20 3.25 3.30 3.40 3.50

235˚ 240˚ 245˚ 250˚ 255˚

30˚

35˚

40˚

45˚

50˚

Figure 10. The same as Figure 9, but for the 12 sec Rayleigh wave. The result of Eikonal tomography is slightly slower(yellow-red shades), on average, than the straight ray tomography because it models off-great-circle propagation.

0

2

4

6

8

10

12

14

16

18

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

24s 12smean = -0.005 km/s

σ = 0.03 km/smean = -0.01 km/s

σ = 0.04 km/s

per

cen

tag

e (%

)

per

cen

tag

e (%

)

Phase speed difference (km/s) Phase speed difference (km/s)

0

2

4

6

8

10

12

14

16

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Figure 11. Normalized histograms of the Rayleigh wave phase speed difference across the studied region between Eikonaltomography and straight ray tomography at 12 and 24 sec period. The mean differences result because Eikonal tomographymodels off-great-circle propagation, which is more significant at 12 sec than 24 sec period.

where i is the index of the effective sources andsj0 and sk0 are the mean slowness at locations j andk, respectively. The statistical correlation, ρ, varies be-tween 0 and 1 and represents the degree of coherence orindependence between the measurements made at thetwo locations. Using the point in central Nevada (Fig-ure 1) as an example again, the statistical correlationbetween the phase speed observations at that point andthe neighboring points is summarized as a correlationsurface shown in Figure 12a. We follow Barmin et al.(2001) and estimate the coherence length of the mea-surements by fitting the correlation surface with a cone,where the base radius of the cone is taken as the coher-ence length estimate R.

Although this is different from the traditional def-

inition of resolution, it does provide information aboutthe length scale of features that can be resolved in a re-gion. The coherence length estimated in this way for the24 sec Rayleigh wave is shown in Figure 12b. In mostregions, coherence length is somewhat smaller than theaverage inter-station spacing of 70 km across the west-ern US. Although this result is comparable to the reso-lution estimated by the straight ray tomography (Lin etal. 2008), there are fundamental differences between thetwo. When the observed phase travel times are affectedby a velocity structure much smaller than the inter-station distance, without a more sophisticated interpo-lation scheme, the minimum curvature fitting methodwe use will smear the travel time anomalies to an areaconfined by the few closest nearby stations. This smear-


(a)

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

40 50 60 70 80 90 100R (km)

(b)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

242

243

24439.8

40.841.8

Corr

elat

ion

Longitude Latitiude

Corr

elat

ion

Figure 12. (a) An example of the spatial coherence of the measurements for the 24 sec Rayleigh wave at the point in centralNevada indicated by the star in Figure 1. (b) The radius (R) of the cone fit to the coherence surface at each location, whichbears a similarity to resolution.

ing effect is further evidenced in out synthetic tests insection 5.1. Thus, the station spacing constrains the co-herence length as well as the smallest scale of structurethat can be confidently resolved. Increasing the numberof effective sources will tend to reduce the estimated un-certainty, but most likely will have little impact on thecoherence length.

4.3 Azimuthal anisotropy

Eikonal tomography also provides an estimate of az-imuthal anisotropy. In traditional surface wave inver-sions, it is commonly assumed that the Rayleigh wavephase speed exhibits the following functional depen-dence on azimuth, which is derived based on theoreticalstudies of weakly anisotropic media (Smith & Dahlen,1973),

c(Ψ) = c0 + Acos[2(Ψ − ϕ)] + Bcos[4(Ψ − α)] (9)

where Ψ is the azimuthal angle measured positiveclockwise from north, A and B are the amplitude ofanisotropy, and ϕ and α define the orientation of theanisotropic fast axes for the 2Ψ and 4Ψ componentsof anisotropy. Although the estimated 2Ψ fast direc-tions may be robust in the traditional inversion, theamplitude of the anisotropy almost inevitably dependson the regularization parameters chosen (e.g., Smith etal., 2004). In Eikonal tomography, the velocity as a func-

tion of azimuth of the wave is measured directly and itis then determined if the relationship reflects a simplefunction of azimuth.

As with the measurement of isotropic phase veloc-ity, the estimation of anisotropy begins with the setof phase speeds estimated at a single spatial locationfrom the set of phase speed travel time maps segre-gated by azimuth, as in the example shown in Figure7a for the 24 sec Rayleigh wave for a point in centralNevada. Due to phase travel time errors in the maps,the measured phase speeds are significantly scatteredand any azimuthally dependent trend is obscured. Scat-ter is reduced substantially by stacking and binning intwo stages. First, we combine the azimuthally depen-dent phase speed measurements obtained at the tar-get point with measurements at the eight surroundingspatial points (3 × 3 grid with the target point at thecenter). We use a 0.6◦ grid separation approximatelyequal to the coherence length estimate described in thelast section, which effectively guarantees that measure-ments are statistically independent from one another.To reduce mapping the lateral variation of isotropicphase speed into azimuthal anisotropy, we remove theisotropic speed difference between each point and thecenter point of the 3 × 3 grid for all of the measure-ments. This stacking process increases the number ofmeasurements for the center point, but does so at theexpense of reducing spatial resolution. Second, we com-


bine all of the azimuthally dependent phase speed mea-surements in each 20◦ azimuthal bin into a mean speedand its standard deviation of the mean for that bin.Here, again, the mean slowness and the standard devi-ation of the mean slowness are first calculated and thenconverted to the mean speed and its uncertainty.

Figure 13 shows examples for four different geo-graphical locations of the stacked azimuthally depen-dent phase speed measurements with their uncertain-ties for the 24 sec Rayleigh wave. For the examplesin Utah and Nevada, Figures 13a and b, where goodazimuthal data coverage exists, a clear 2Ψ variation isobserved for the entire 360◦ of azimuth. On the otherhand, Figures 13c and d show two examples near thewestern boundary of the map where azimuthal cover-age is limited. Nevertheless, the 2Ψ velocity signal isstill observed robustly because measurements cover atleast 180◦. Based on these observations, for each pe-riod and location, we adopt the assumption of a weaklyanisotropic medium, fit the results with the 2Ψ part ofthe cosinusoid, and use it to estimate the amplitude andfast direction of anisotropy with associated uncertain-ties. Here, robust statistics are used. Measurements thatcannot be fit within 2 standard deviations are removedto minimize the effect of significant outliers, but thedifference between the robust statistics and non-robuststatistics is small overall. Adding the 4Ψ term does notimprove the data fit appreciably which indicates thatthe 4Ψ variation of Rayleigh waves is weaker and ourdataset is not sufficient to constrain it. The observed 2Ψazimuthal anisotropy exhibits different amplitudes andfast directions in different locations. This minimizes con-cern about systematic errors in the input phase traveltimes due to azimuthally inhomogeneous ambient noisesources which could result in a uniform fast directionfor the entire region.

Azimuthal anisotropy for the 24 sec Rayleigh waveis summarized in Figure 14a. The peak-to-peak ampli-tude of anisotropy is presented in Figure 14b. Figure 15apresents the variance reduction after introducing the 2Ψanisotropy term. Significant improvements (> 80%) areobserved over extended regions, which not only indi-cates the robustness of the measurements but also sug-gests that azimuthal anisotropy is a general feature ofRayleigh waves in the western US. We note that theregions with poor variance reduction (< 40%) are gen-erally accompanied by weak anisotropy (< 0.5%), whichmay be a real feature or may be due to a spatially rapidand unresolvable change in fast direction. The estimateduncertainty of the observed azimuthal anisotropy fastdirections and amplitudes are summarized in Figure 15band 15c, respectively. As in traditional anisotropy to-mography, the fast directions are generally robust fea-tures. We estimate the uncertainties of the fast direc-tions to be less than 6◦ in most of regions. Again, re-gions with larger uncertainties in the fast direction gen-erally result from weak anisotropy. Uncertainties in the

amplitude of anisotropy are generally smaller (< 3m/sor 0.1% of the isotropic phase speed) in regions withnearly complete azimuthal data coverage than near theperiphery of the studied region where only part of entireazimuthal range has measurements.

For comparison, the 2Ψ 24 sec Rayleigh wave phasespeed anisotropy determined by traditional straight rayinversion (e.g., Barmin et al., 2001) with two differentsmoothing strengths is summarized in Figure 16a and16d with amplitudes plotted in Figure 16b and 16e.The difference in fast directions compared to Eikonaltomography is also summarized as histograms in Figure16c and 16f, where only regions with anisotropy am-plitude larger than 0.5% in the Eikonal tomography areincluded. Overall, the observed anisotropy fast directionpatterns are consistent between the two traditional in-versions and the Eikonal tomography inversion. This isnot unexpected because the off-great-circle effect is rel-atively weak at 24 sec period. The anisotropy amplitudeis significantly smaller in the second case of the straightray inversion, which indicates that the smoothing regu-larization was too strong. Most places with a significantdifference in fast directions (> 30◦) occur near a transi-tion in the fast direction of anisotropy where the resultsof neither model are robust.

With the traditional inversion method, it is trickyto select the right regularization parameters and meth-ods to do so are typically ad-hoc. Many studies usetrade-off curves between misfit and model roughnessor the number of degrees of freedom to select thepreferred regularization parameters (e.g. Boschi 2006;Zhou et al. 2005). This is, however, difficult for az-imuthal anisotropy because by including 2Ψ azimuthalanisotropy, for example, the number of degrees of free-dom at each node increases to 3 from 1 for an isotropicwave speed inversion despite the fact that the improve-ment in misfit is usually modest. For traditional to-mography applied to the 24 sec Rayleigh wave phasespeed data, the standard deviation of travel time mis-fit drops from around 3 sec for a homogeneous refer-ence model to 1.57 sec after the straight ray isotropicspeed inversion (Figure 9b). However, it then only de-creases slightly to 1.53 sec and 1.54 sec for the two 2Ψazimuthal anisotropy inversions (Figure 16a and 16d).With Eikonal tomography, through the stacking andbinning process, we effectively separate the velocity vari-ation due to measurement error from anisotropy andare able to inspect the observed azimuthally dependentphase speed measurements visually. In this way, the ob-served variance reduction is statistically meaningful andcan be used to indicate the confidence level of the result.

The 12 sec Rayleigh wave 2Ψ azimuthal anisotropyresults based on Eikonal tomography are presented inFigure 17. Overall, the anisotropy is robustly mea-sured despite the fact that the amplitudes of anisotropyare generally weaker and the fast direction pattern isslightly different than the 24 sec results. Figure 18a


3.4

3.45

3.5

3.55

0 50 100 150 200 250 300 350

3.65

3.7

3.75

3.8

0 50 100 150 200 250 300 350

3.45

3.5

3.55

3.6

3.65

0 50 100 150 200 250 300 350

3.35

3.4

3.45

3.5

3.55

0 50 100 150 200 250 300 350

(a) (b)

(c) (d)

Azimuth (deg)

Ph

ase

Vel

oci

ty (

km

/s)

Azimuth (deg)

Ph

ase

Vel

oci

ty (

km

/s)

Azimuth (deg)

Ph

ase

Vel

oci

ty (

km

/s)

Azimuth (deg)

Ph

ase

Vel

oci

ty (

km

/s) A = 0.031 ± 0.003 km/s

ϕ = 66o ± 4o

A = 0.034 ± 0.004 km/s

ϕ = 167o ± 5o

A = 0.073 ± 0.006 km/s

ϕ = 125o ± 4o

A = 0.057 ± 0.005 km/s

ϕ = 76o ± 3o

Utah

N. CA

Nevada

C. CA

Figure 13. Examples of the azimuthal dependence of phase velocity measurements for the 24 sec Rayleigh wave at four pointsin the western US where large amplitude 2Ψ azimuthal variation can be observed: (a) Utah, (b) Nevada, (c) northern California,and (d) central California. The locations are indicated by the circle, star, square, and diamond in Figure 1, respectively. Errorbars are estimated based on the distribution of phase velocity measurements in each 20◦ azimuthal bin for the given locationand its 8 nearest neighboring grid points. For each case, the solid line is the best fit of the 2Ψ azimuthal variation.

shows an example of the 12 sec 2Ψ azimuthal anisotropydetermined by our traditional straight ray inversionwith anisotropy amplitude plotted in Figure 18b. Thedifference in fast directions compared to the Eikonaltomography is summarized in the histogram in Figure18c. Compared to 24 sec period, more significant dif-ferences in both the fast directions and the amplitudepatterns are observed, particularly near regions wherethere are discrepancies between the two isotropic wavespeed maps (Figure 10). We believe that the off-great-circle effect, which is more important for 12 sec Rayleighwaves, is responsible for most of the observed differencesbetween the methods at this period.

5 DISCUSSION

5.1 Numerical simulations to test for

systematic errors

To assess possible systematic errors due to approxima-tions in the Eikonal tomography method, which include

both dropping the amplitude term in equation 2and us-ing a minimum curvature surface fitting method to in-terpolate the phase travel time surface, we perform aseries of 2D finite difference simulations to solve theHelmholtz equation numerically and obtain a synthetictravel time database. We invert this database basedon Eikonal tomography and evaluate the difference be-tween the tomography result and the input phase speedmodel to constrain the systematic errors.

Two cases, 12 and 36 second periods, are studiedhere which represent periods at the short and long pe-riod ends of our study. The isotropic wave speed mapsderived from the USArray dataset and Eikonal tomog-raphy are used here as the input models (Figure 19aand 20a). In each simulation, a periodic source cen-tered at one station location is used to generate a singlefrequency out-going wave which propagates in the 2Dmedium of the input wave speed model. The resultingwaveforms observed at all other station locations areused to measure the phase travel times between thosestations and the effective source, where the measure-


235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

2%

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

Anisotropy amplitude (%)

(a) (b)

0.00 0.25 0.50 1.00 1.50 2.00 3.00

Figure 14. (a) The 24 sec period Rayleigh wave azimuthal anisotropy fast axis directions and peak-to-peak amplitudes, 2A/c0,which are proportional to the length of the bars. (b) Peak-to-peak amplitude of anisotropy presented in percent.

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0 20 40 60 80 90 100

Variance reduction (%) Fast direction uncertainty (deg) Anisotropy amplitude uncertainty (m/s)

(a) (b) (c)

0 3 6 12 18 30 0 2 3 4 6 8

Figure 15. (a) Variance reduction of the 24 sec Rayleigh wave 2Ψ azimuthal anisotropy relative to the isotropic speed at eachpoint. (b) The uncertainty in the angle of the fast direction, ϕ. (c) The uncertainty of the amplitude of anisotropy.


2%

2%

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0.00 0.25 0.50 1.00 1.50 2.00 3.00

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚



Per

centa

ge

(%)


Per

centa

ge

(%)

-80 -60 -40 -20 0 20 40 60 80

-80 -60 -40 -20 0 20 40 60 80

(a) (b) (c)

(d) (e) (f)

2%

2%

0

5

10

15

20

25

30

0

5

10

15

20

25

30

σ = 26˚

σ = 24˚

Figure 16. (a)-(b) Same as Figure 14a-b, but here the 24 sec Rayleigh wave azimuthal anisotropy result is determined withthe traditional straight ray method of Barmin et al. (2001) with a regularization chosen to approximate the amplitudes in Fig.14b. The black line is the 100 km resolution contour. (c) The normalized histogram of the difference in fast directions betweenthe Eikonal tomography result (Fig. 14a) and the straight ray tomography result. (d)-(f) same as (a)-(c) but with strongersmoothing regularization. Patterns of anisotropy remain largely unchanged, but amplitudes diminish with greater the damping.

ments are made when the waveform stabilizes after sev-eral cycles. Although synthetic travel times are avail-able between all station pairs, to be comparable withthe inversion with real data, only those measurementsincluded in the original datasets are included. We fol-low the same procedure described in Section 3 and 4 toinvert these synthetic datasets based on Eikonal tomog-raphy and both the isotropic and anisotropy results areshown in Figures 19b, 19c, 20b, and 20c.

Unsurprisingly, the resulting isotropic speed maps,for both 12 and 36 sec, closely replicate the largescale features of the input models, although small scaleanomalies in the input models tend to be smoothed out.This smoothing effect is expected, as discussed in sec-tion 4.2. To assess other systematic errors, we smooththe input models with a spatial Gaussian filters with astandard deviation of 35 km and summarized the dif-ferences between the isotropic inversion results and thesmoothed input models in Figure 19d and 20d. Devi-

ations are most significant near the periphery of ourstation coverage, particularly near regions with large ve-locity contrasts such as regions near the Central Valleyof California and the Sierra Nevada for the 12 secondcase and the Southern Sierra Nevada for the 36 secondcase where delamination is inferred by previous stud-ies (e.g. Yang & Forsyth 2006). Similar anisotropic de-viations are also observed for both the 12 and 36 seccases (Figure 19c and 20c), where the amplitude of theanisotropy tends to correlate with the observed isotropicwave speed deviations. This suggests that rapid veloc-ity contrasts near the periphery of the maps tend todistort the wavefront dramatically and the method be-comes less robust. The observed isotropic (Figures 19dand 20d) and anisotropic (Figures 19c and 20c) devia-tions are also summarized as histograms in Figures 19e,20e, 19f, and 20f, respectively.

We test whether we can reduce these deviationsby including amplitude measurements in our synthetic


235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚


(a) (b)

0.00 0.25 0.50 1.00 1.50 2.00 3.00

2%

Figure 17. Same as Figure 14, but for the 12 sec Rayleigh wave.

2%

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0.00 0.25 0.50 1.00 1.50 2.00 3.00Anisotropy amplitude (%)


Per

cen

tage

(%)

-80 -60 -40 -20 0 20 40 60 80

(a) (b) (c)

2%

0

5

10

15

20

σ = 34˚

Figure 18. Same as Figure 16, but for the 12 sec Rayleigh wave. Agreement between the Eikonal and straight ray tomographyis worse at 12 sec than 24 sec because of the larger effect of off-great-circle propagation.

datasets. Again, minimum curvature surface fitting isused to first interpolate the synthetic amplitudes mea-sured at each station to construct amplitude surfacesbefore calculating the second term in equation 2 . Theeffect of including the amplitude term is in general un-noticeable, which is partly because the surface interpo-

lation schemes we use here provides relatively smoothamplitude surfaces which tend to minimize the Lapla-cian term in the equation 2 . This is inevitable unless adenser station network is available.

The observed isotropic and anisotropic amplitudedeviations, with standard deviations approximately


235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

2.90 3.00 3.10 3.15 3.20 3.25 3.30 3.40 3.50

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

-50 -30 -20 -10 10 20 30 50

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

2%

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

Phase speed (km/s)

Phase speed difference (m/s)

per

centa

ge(

%)

per

centa

ge(

%)

Phase speed difference (m/s) Anisotropy amplitude (%)

(a) (b) (c)

(d)

(e) (f)

σ = 11m/s σ = 0.3%

0

5

10

15

20

25

30

-40 -20 0 20 40

Figure 19. (a) The input wave speed model for the 12 sec simulations. The model is derived based on the isotropic resultof Eikonal tomography with real data (Figure 10a) where the model gradually smears into a homogeneous model near theboundary of the station coverage. (b)-(c) The isotropic and anisotropic inversion results from Eikonal tomography with the12 sec synthetic dataset. (d) The difference between the synthetic inversion and smoothed input model where positive valuesindicate that the synthetic inversion gives a higher local phase speed. (e) Normalized histogram of the speed difference acrossthe studied region between the synthetic inversion and the smoothed input model. (f) Normalized histogram of the anisotropicpeak-to-peak amplitude of the synthetic inversion across the studied region.

equal to 10m/s and 0.3% peak-to-peak (or 6m/s assum-ing 4km/s isotropic speed) (Figures 19e, 19f, 20e, and20f), respectively, are generally small relative to the ob-served isotropic velocity variations (Figure 9a and 10a)and anisotropy amplitudes (Figures 14b and 17b). Theyare, however, approximately on the same scale as the es-timated uncertainties derived from our statistical anal-ysis (Figures 8a, 8b, and 15c). This suggests that the es-timated uncertainties described in Section 4, which onlyaccounts the random measurement errors, may under-estimate the difference between the tomography resultsand the real medium properties. When numerical solu-

tions are available, such as here, systematic errors due tothe tomography method can be numerically estimatedand a better estimation of the uncertainty can be madeby summing the effects of the systematic and randommeasurement errors. However, this may prove impracti-cal due to the heavy computation required. Consideringthe positive correlation between random (Figures 8a, 8b,and 15c) and systematic errors (Figures 19c, 19d, 20c,20d), here we propose 1.5 as a rule of thumb scalingfactor to multiply the random error uncertainty estima-tions to provide a more realistic uncertainty estimate.

We would like to emphasis here that the systematic


30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚ 235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

235˚ 240˚ 245˚ 250˚ 255˚30˚

35˚

40˚

45˚

50˚

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

Phase speed (km/s)

Phase speed difference (m/s)

per

centa

ge(

%)

per

centa

ge(

%)

Phase speed difference (m/s) Anisotropy amplitude (%)

(a) (b) (c)

(d)

(e) (f)

σ = 9m/s σ = 0.3%

3.50 3.60 3.65 3.70 3.75 3.80 3.85 3.90 4.00

2%

0

5

10

15

20

25

30

-40 -20 0 20 40

-50 -30 -20 -10 10 20 30 50

Figure 20. Same as Figure 19, but for the 36 sec simulations.

errors discussed here are solely due to the imperfectionin the tomography method and do not account for sys-tematic errors in travel time measurements. Systematictravel time measurement errors can arise, for example,due to timing errors or inhomogeneous noise source dis-tributions for noise cross-correlation measurements. Webelieve that the effect of inhomogeneous noise sourcedistribution in our results is small, however. Travel timeerrors due to inhomogeneous source distribution arelikely similar between nearby stations. When the gra-dient is calculated in Eikonal tomography, these errorswill cancel.

5.2 Advantages and limitations of Eikonal

tomography

There are several significant advantages of Eikonal to-mography over traditional surface wave tomographymethods.

First, the implementation of the inverse operatorfor Eikonal tomography depends on operations to thedata without explicitly solving the forward problem. Fora wave propagating in an inhomogeneous medium, theobserved wave properties such as phase travel time areonly linearly related to the local velocity structure whenstructural perturbations are small. In other words, anylinearized forward operator, such as the ray or finitefrequency sensitivity integrals, and the inverse operatorderived from it can only be considered approximate. Er-rors caused by this linearization are often overlooked orare unknown, and moving beyond them requires itera-


tive simulations which are computationally expensive.Eikonal tomography extracts the information about lo-cal velocity structure directly from the data without ex-plicitly constructing the forward operator. It, therefore,finesses the nonlinear nature of the problem and shouldresult in a better estimate of both the local isotropicand anisotropic phase speeds, especially where off-great-circle propagation is important.

Second, uncertainties in local phase speeds can beestimated with Eikonal tomography. Instead of minimiz-ing a penalty functional that usually includes some com-bination of global misfit and model norm or roughnessconstraints, Eikonal tomography directly estimates lo-cal phase speed from independent measurements basedon different phase travel time surfaces. Therefore, theuncertainties of the resulting local phase speeds can bedetermined statistically in a straightforward way. Theuncertainties are important for later 3D inversion andquantitative comparisons between different models.

Third, Eikonal tomography is free from explicitmodel regularization. The method, therefore, eliminatesthe need to make ad-hoc choices of the damping andregularization parameters which are sometimes contro-versial and may result in dubious models. This particu-larly is a problem for studies of surface wave azimuthalanisotropy because the increased number of degrees offreedom is often not offset by a comparable improve-ment in misfit. Eikonal tomography with the additionalsmoothing intrinsically embedded in the phase fronttracking process has no explicit regularization and thesubjectivity of the inverter to affect the tomographicresult is restricted.

Fourth, the azimuthal dependence of phase speedscan be measured directly without assuming its para-metric form. Unlike classic studies of Pn azimuthalanisotropy (e.g., Morris et al. 1969) where the wavespeed variation with the direction of propagation isobserved directly, traditional surface wave tomographytypically posits the relationship between phase speedand the direction of wave propagation based on theo-retical studies of weakly anisotropic media (e.g. Smith& Dahlen 1973). The ability to measure and observe theazimuthal dependence of phase speeds directly leads togreater confidence in the information about anisotropy.

There are several limitations on Eikonal tomogra-phy worthy of note. First, unlike traditional inversionmethods where the resolution is controlled by path orkernel densities, Eikonal tomography estimates the co-herence length of the measurements which is controlledby station spacing. Without applying a more sophis-ticated travel time surface interpolation method, thisprohibits the use of this technique to resolve structuressmaller than the inter-station spacing.

Second, when long period or more complicated sur-face waves are considered, the second term in equation2 can have values more similar to the magnitude ofthe phase speed anomalies that we seek to resolve. Al-

though our simulations shows that the amplitude termis relatively unimportant for our dataset, other theo-retical and numerical studies, such as Wielandt (1993)and Friederich et al. (2000), suggest that when eitherthe velocity anomaly is smaller than a wavelength orthe incoming wave is complicated by multipathing, ne-glect of the amplitude term by the Eikonal equationcan blur the velocity anomaly and cause systematic er-rors in the phase speed measurements. It is possible tosolve this problem by inverting both phase and ampli-tude together which amounts to recasting the problemin terms of the Helmholtz equation. Amplitude mea-surements are, however, less accurate than phase mea-surements and the second spatial derivative of the am-plitude variation tends to be unstable and is underesti-mated, particularly when the station spacing is sparse.The situation is even worse for measurements based onambient noise cross-correlations where amplitudes havebeen separately normalized for different stations so thatmeaningful amplitude information has been lost. Am-plitude anomalies then mainly reflect the distributionof ambient noise sources not structural gradients.

Third, travel time interpolation schemes usually areunreliable near the periphery of the station coveragewhich results in increasing both random and system-atic errors. Hence, the area that can be imaged by theEikonal tomography method is generally smaller thanwhen a traditional tomography method is applied (Fig-ure 9 and 10). It requires a large scale array, such asthe TA, to really take the advantages of the Eikonal to-mography method where both applicable area can beextended and measurement uncertainties can be signif-icantly reduced when ambient noise method is applied.

5.3 Applicability to earthquake tomography

To construct the phase travel time surfaces in this studywe use measurements of ambient noise. In principle,however, Eikonal tomography can be applied to phasetravel time measurements based on earthquake wave-forms. There are a few differences, however, consideringthe nature of earthquake measurements.

First, surface waves emitted by a distant sourceusually develop a certain amount of multipathing thatcan potentially invalidate the assumption of smoothlyvarying amplitudes. In fact, this is the fundamental con-cept of the two plane wave inversion method (e.g., Yang& Forsyth 2006). Friederich et al. (2000) showed numer-ically how wave complexity can contribute to uncertain-ties in the local phase speeds inferred from the Eikonalequation. This problem is relatively minor for measure-ments based on ambient noise cross-correlations in thewestern US because the effective sources (i.e., the sta-tions in the ambient noise method) usually are relativelyclose, with average distances near 700 km. Other thanat the short period end of our study and near regionswith sharp velocity contrasts, this is usually too short


for multipathing to be well developed. Second, surfacewave studies based on teleseismic events usually focuson longer periods (> 25 sec) due to the strong scatter-ing and attenuation of shorter period signals. At longerperiods, when a wavelength is larger than the size ofa velocity anomaly, the second term in equation 2 canblur and distort the velocity anomaly which we wish toresolve (Friederich et al. 2000).

Considering these factors, the amplitude term mayplay a bigger role in Eikonal tomography based onearthquake measurements and the second term in equa-tion 2 should probably be properly taken into account.Unlike ambient noise cross-correlation measurementswhere only the phase information is retained, the am-plitude of the surface wave emitted by an earthquakecan be used in the inversion as well. By including am-plitude information, the Helmholtz equation can be ap-plied instead of the Eikonal equation, and may resolvethe local phase velocity structure with greater certainty(Wielandt 1993; Friederich et al. 2000; Pollitz 2008).

6 CONCLUSIONS

We present a new method of surface wave tomographycalled Eikonal tomography and argue that this methodpresents an improvement over traditional methods ofambient noise tomography, particularly as the methodis applied to data from the Transportable Array com-ponent of EarthScope/USArray. The method initiatesby tracking phase fronts across the array to producephase travel time maps centered on each station, con-sidered as an “effective source”. The method culminatesby interpreting the local gradients of the phase time sur-faces in terms of local phase speed and the direction ofpropagation of the wave. The most significant advan-tages of Eikonal tomography compared with traditionalstraight-ray tomography is its more accurate represen-tation of wave propagation, its ability to produce mean-ingful uncertainty information about the inferred phasespeed maps, and its production of more reliable informa-tion about azimuthal anisotropy. Improvements in theisotropic dispersion maps result predominantly from themethods ability to track the direction of propagation ofwaves, which is tantamount to use of off-great-circle ge-ometrical rays but without the need for iteration. Im-provements in information about azimuthal anisotropyderive from the methods freedom from ad-hoc choices inregularization. This provides more reliable informationabout the amplitude of anisotropy, in particular. In ad-dition, the method provides a local visualization of howphase speeds vary with azimuth, which we believe addsconsiderably to confidence in the results. Eikonal tomog-raphy is an approximate method. It accurately tracksthe direction of wave propagation but only approxi-mately incorporates what may be traditionally thoughtof as finite-frequency effects and assumes a single wave

propagating at each point in space. To improve the abil-ity to resolve small scale feature and reduce systematicerrors, future work will focus on finding more sophisti-cated interpolation schemes as well as incorporating theamplitude term of equation 2 .

REFERENCES

Alsina, D, Snieder, R, & Maupin, V. 1993. A test of the greatcircle approximation in the analysis of surface waves.Geophys. Res. Lett., 20(10), 915–918.

Barmin, M. P., Ritzwoller, M. H., & Levshin, A. L. 2001.A fast and reliable method for surface wave tomography.Pure Appl. Geophys., 158(8), 1351–1375.

Bensen, G. D., Ritzwoller, M. H., Barmin, M. P., Levshin,A. L., Lin, F., Moschetti, M. P., Shapiro, N. M., & Yang,Y. 2007. Processing seismic ambient noise data to ob-tain reliable broad-band surface wave dispersion mea-surements. Geophys. J. Int., 169(3), 1239–1260.

Bensen, G. D., Ritzwoller, M. H., & Shapiro, N. M.2008. Broadband ambient noise surface wave tomographyacross the United States. J. Geophys. Res., 113(B5).

Bodin, Thomas, & Maupin, Valerie. 2008. Resolution poten-tial of surface wave phase velocity measurements at smallarrays. Geophys. J. Int., 172(2), 698–706.

Boschi, L. 2006. Global multiresolution models of surfacewave propagation: comparing equivalently regularizedBorn and ray theoretical solutions. Geophys. J. Int.,167(1), 238–252.

Brisbourne, A. M., & Stuart, G. W. 1998. Shear-wave veloc-ity structure beneath North Island, New Zealand, fromRayleigh-wave interstation phase velocities. Geophys. J.

Int., 133(1), 175–184.Dahlen, FA, Hung, SH, & Nolet, G. 2000. Frechet kernels

for finite-frequency traveltimes - I. Theory. Geophys. J.

Int., 141(1), 157–174.Deschamps, F., Lebedev, S., Meier, T., & Trampert, J. 2008.

Azimuthal anisotropy of Rayleigh-wave phase velocitiesin the east-central United States. Geophys. J. Int.,173(3), 827–843.

Ekstrom, G., Tromp, J., & Larson, E. W. F. 1997. Measure-ments and global models of surface wave propagation. J.

Geophys. Res., 102(B4), 8137–8157.Friederich, W. 1998. Wave-theoretical inversion of teleseis-

mic surface waves in a regional network: phase-velocitymaps and a three-dimensional upper-mantle shear-wave-velocity model for southern Germany. Geophys. J. Int.,132(1), 203–225.

Friederich, W., Hunzinger, S., & Wielandt, E. 2000. A noteon the interpretation of seismic surface waves over three-dimensional structures. Geophys. J. Int., 143(2), 335–339.

Laske, G., & Masters, G. 1998. Surface-wave polarizationdata and global anisotropic structure. Geophys. J. Int.,132(3), 508–520.

Levshin, A. L., Barmin, M. P., Ritzwoller, M. H., & Tram-pert, J. 2005. Minor-arc and major-arc global surfacewave diffraction tomography. Phys. Earth Planet. Int.,149(3-4), 205–223.

Levshin, AL, Ritzwoller, MH, Barmin, MP, Villasenor, A, &Padgett, CA. 2001. New constraints on the arctic crust


and uppermost mantle: surface wave group velocities, P-

n, and S-n. Phys. Earth Planet. Inter., 123(2-4, Sp. Iss.SI), 185–204.

Lin, F. C., Moschetti, M. P., & Ritzwoller, M. H. 2008. Sur-face wave tomography of the western United States fromambient seismic noise: Rayleigh and Love wave phase ve-locity maps. Geophys. J. Int., 173(1), 281–298.

Montelli, R., Nolet, G., & Dahlen, F. A. 2006. Comment on’Banana-doughnut kernels and mantle tomography’ byvan der Hilst and de Hoop. Geophys. J. Int., 167(3),1204–1210.

Morris, G. B., Raitt, R. W., & Shor, G. G. 1969. Velocityanisotropy and delay-time maps of mantle near Hawaii.J. Geophys. Res., 74(17), 4300–&.

Moschetti, M. P., Ritzwoller, M. H., & Shapiro, N. M. 2007.Surface wave tomography of the western United Statesfrom ambient seismic noise: Rayleigh wave group velocitymaps. Geochem. Geophys. Geosys., 8(AUG 21), Q08010.

Nolet, G. 1990. Paritioned wave-form inversion and 2-dimensional structure in the Network of AutonomouslyRecording Seismographs. J. Geophys. Res., 95(B6),8499–8512.

Pollitz, Fred F. 2008. Observations and interpretationof fundamental mode Rayleigh wavefields recorded bythe Transportable Array (USArray). J. Geophys. Res.,113(B10), B10311.

Prindle, K., & Tanimoto, T. 2006. Teleseismic surfacewave study for S-wave velocity structure under an array:Southern California. Geophys. J. Int., 166(2), 601–621.

Ritzwoller, MH, & Levshin, AL. 1998. Eurasian surfacewave tomography: Group velocities. J. Geophys. Res.,103(B3), 4839–4878.

Roth, M, Muller, G, & Snieder, R. 1993. VELOCITY SHIFTIN RANDOM-MEDIA. Geophys. J. Int., 115(2), 552–563.

Sabra, K. G., Gerstoft, P., Roux, P., Kuperman, W. A., &Fehler, M. C. 2005b. Surface wave tomography from mi-croseisms in Southern California. Geophys. Res. Lett.,32(14), L14311.

Shapiro, N. M., Campillo, M., Stehly, L., & Ritzwoller, M. H.2005. High-resolution surface-wave tomography from am-bient seismic noise. Science, 307(5715), 1615–1618.

Shearer, P. 1999. Introduction to seismology. Cambridge:Cambridge University Press.

Sherburn, Steven, White, Robert S., & Chadwick, Mark.2006. Three-dimensional tomographic imaging of theTaranaki volcanoes, New Zealand. Geophys. J. Int.,

166(2), 957–969.Smith, M. L., & Dahlen, F. A. 1973. Azimuthal Dependence

of Love and Rayleigh-Wave Propagation in a SlightlyAnisotropic Medium. J. Geophys. Res., 78(17), 3321–3333.

Trampert, J, & Spetzler, J. 2006. Surface wave tomography:finite-frequency effects lost in the null space. Geophys.

J. Int., 164(2), 394–400.Trampert, J, & Woodhouse, JH. 1996. High resolution global

phase velocity distributions. Geophys. Res. Lett., 23(1),21–24.

Trampert, J, & Woodhouse, JH. 2003. Global anisotropicphase velocity maps for fundamental mode surface wavesbetween 40 and 150 s. Geophys. J. Int., 154(1), 154–165.

Tromp, J., Tape, C., & Liu, Q. Y. 2005. Seismic tomography,adjoint methods, time reversal and banana-doughnut ker-

nels. Geophys. J. Int., 160(1), 195–216.

van der Hilst, R. D., & de Hoop, M. V. 2005. Banana-doughnut kernels and mantle tomography. Geophys. J.

Int., 163(3), 956–961.Van Der Lee, S, & Frederiksen, A. 2005. Surface wave to-

mography applied to the North American upper mantle.In: Levander, A., & Nolet, G (eds), Seismic Earth: Ar-

ray Analysis of Broadband Seismograms. American Geo-physical Union, Washington, D.C.: American Geophysi-cal Union Monograph.

Wielandt, E. 1993. Propagation and Structural Interpreta-tion of Nonplane Waves. Geophys. J. Int., 113(1), 45–53.

Woodhouse, J. H., & Dziewonski, A. M. 1984. Mapping theUpper Mantle - 3-Dimensional Modeling of Earth Struc-ture by Inversion of Seismic Waveforms. J. Geophys.

Res., 89(NB7), 5953–&.Yang, Y., & Forsyth, D. W. 2006. Rayleigh wave phase ve-

locities, small-scale convection, and azimuthal anisotropybeneath southern California. J. Geophys. Res., 111(B7),B07306.

Yang, Y. J., Ritzwoller, M. H., Levshin, A. L., & Shapiro,N. M. 2007. Ambient noise rayleigh wave tomographyacross Europe. Geophys. J. Int., 168(1), 259–274.

Yang, Yingjie, Ritzwoller, Michael H., Lin, F. C, Moschetti,M. P., & Shapiro, Nikolai M. 2008. Structure of thecrust and uppermost mantle beneath the western UnitedStates revealed by ambient noise and earthquake tomog-raphy. J. Geophys. Res., 113(B12), B12310.

Yao, H. J., van der Hilst, R. D., & de Hoop, M. V. 2006.Surface-wave array tomography in SE Tibet from am-bient seismic noise and two-station analysis - I. Phasevelocity maps. Geophys. J. Int., 166(2), 732–744.

Yoshizawa, K, & Kennett, BLN. 2002. Determination of theinfluence zone for surface wave paths. Geophys. J. Int.,149(2), 440–453.

Zhou, Y., Dahlen, F. A., & Nolet, G. 2004. Three-dimensional sensitivity kernels for surface wave observ-ables. Geophys. J. Int., 158(1), 142–168.

Zhou, Y., Dahlen, F. A., Nolet, G., & Laske, G. 2005.Finite-frequency effects in global surface-wave tomogra-phy. Geophys. J. Int., 163(3), 1087–1111.

eikonal tomography: surface wave tomography by phase-front ... · cwp-661 eikonal tomography:...

Documents