array analysis of two-dimensional variations in surface...

Array Analysis of Two-Dimensional Variations in Surface Wave Phase Velocityand Azimuthal Anisotropy in the Presence of Multipathing Interference

Donald W. Forsyth

Department of Geological Sciences, Brown University, Providence, Rhode Island

Aibing Li

Department of Geosciences, University of Houston, Houston, Texas

Multipath propagation of surface waves introduces distortions in waveformsthat can bias array measurements of phase velocities. We present a method forarray analysis of laterally and azimuthally varying phase velocities that representsthe incoming wavefield from each earthquake as the sum of two interfering planewaves. This simple approximation successfully represents the amplitude and phasevariations for most earthquakes recorded in the MELT Experiment on the EastPacific Rise in the period range from 16 to 67 s. The inversion for velocitiesautomatically reduces the importance of data from earthquakes or periods thatare not described well by this approximation. Each iteration in the inversioninvolves two stages: a simulated annealing inversion for the best wave parameterdescription of each event, and a linearized inversion for velocities and changesin the wave parameters. At 29 s period, the two-plane-wave solutions indicatethat nearly every signal is significantly affected by multipathing. The larger ofthe two plane waves typically has an apparent azimuth of propagation that iswithin a few degrees of the great circle path. The smaller wave is more scattered,differingin apparentazimuthfrom the largerwaveby an averageof about 13°at 29 s. Both lateral and azimuthal variations in Rayleigh wave phase velocity inthe study area are significant, although it is possible to trade off azimuthal anisot-ropy with rapid and probably unrealistic lateral variations in velocity. Apparentazimuthal anisotropy reaches 5 to 6%, with the fast direction approximatelyperpendicular to the ridge.

I. INTRODUCTION apparent direction of propagation assuming the incomingwave is planar. Surface waves, however, encounter manyheterogeneities between the source and receivers that distortthe waves, causing deviations from great-circle azimuths[Evernden, 1953; Alsina, et at., 1993; Laske, 1995], waveinterference effects sometimes described as multipathing[Capon, 1970], and scattering [Snieder and Nolet, 1987].Non-planar energy can dominate the signals for Rayleighwaves at periods less than 50 s [Friederich et at., 1994].Neglecting wave interferenceby selecting only those recordsfrom an event that have a "clean" appearance can lead toa systematic bias in the apparent phase velocity [Wielandt,

The measurement of phase velocities of surface wavescrossing an array of seismometers is a powerful way todetect variations in lithospheric and asthenosphericstructure.Traditional methods of array-analysis solve for slowness and

Seismic Earth: Array Analysis of Broadband SeismogramsGeophysical Monograph Series 157Copyright 2005 by the American Geophysical Union1O.1029/156GM06

81

82 ARRAY ANALYSIS WITH ANISOTROPY AND MULTIPATHING

1993] if the clean appearance stems from constructive inter-ference of waves that have traveled different paths.

To overcome the inaccuracies in phase velocity determina-tions caused by non-planar waves, Friederich and Wielandt[1995] have developed a method that simultaneously solvesfor phase velocity variations within an array and for thestructure of the incoming wavefield from each event. Thewavefields are represented by a set of basis functions inthe form of Hermite-Gaussian functions that describe the

perturbations from a plane wave. This elegant approach hasbeen successfully applied in studies of regional structure inGermany [Friederich, 1998] and California [Pollitz, 1999].There are, however, some practical difficulties that arise.Given a finite number of stations, the wavefield solution is

non-unique [Wielandt, 1993] and must be constrained by anadditional condition, such as that the model wavefield have

the same total energy as is estimated from the observationsat the existing stations [Friederich et ai., 1994]. Even withthis condition, the inversions tend to put the maximum pre-dicted amplitude variations in gaps between stations or out-side the array (see examples in Figure II of Friederichet ai., 1994); a typical problem with any orthogonal basisfunction expansion with incomplete coverage.

Another problem is that many parameters may be neededto represent relatively simple interference patterns of largeamplitude but unknown wavelength. With Hermite-Gaussianexpansion up to order twenty, 44 parameters are required todescribe the wavefield [Friederich et ai., 1994; Friederich,

1998] at any given frequency. With this large number ofwavefield parameters for each source event, it is not surpris-ing that it is difficult to resolve the relatively small changesin phase of the waves associated with variations in phasevelocity within the study area. The spatial pattern of phasevelocity variations seems to be robust in the joint inversionsfor structure and wavefields, but the amplitude ofthe velocityvariations is highly dependent on the relative damping ofthe wavefield and the velocity parameters [Friederich, 1998;Pollitz, 1999].

Motivated by the relatively simple interference patternsrevealed by wavefield modeling with orthogonal basis func-tions and by the need to find a more stable inversion withfewer parameters for arrays with gaps between stations, wehave developed a procedure for joint inversion of velocityand wavefield structure in which each incoming wavefield

is represented by the interference between two plane waves.This approach reduces the number of wavefield parametersfor each event to six. In most cases, this representation

provides an adequate description of the amplitude variationsacross an array. The procedure automatically reduces theinfluence of events with more complex wavefields that arenot well-represented by this model of two-plane-wave inter-

ference. This approach is illustrated with an example ofRayleigh waves propagating across the array of ocean-bot-tom seismometers deployed in the MELT Experiment[MELT Seismic Team, 1998; Forsyth et ai., 1998]. Shearwave splitting [Wolfeand Solomon, 1998],seismic refractionexperiments [Raitt et ai., 1969], ophiolite samples [Chris-tensen and Salisbury, 1979], and propagation of Rayleighand Love waves [Forsyth, 1975]all indicate that the oceaniclithosphere should be anisotropic in the MELT study area.Consequently, the two-dimensional structure in the inversionincludes terms for azimuthal anisotropy of phase velocity.Other papers [Forsyth et al., 1998; MELT Seismic Team,1998] describe more details of the experiment and discussthe tectonic significance of the results. This paper focuseson the two-plane-wave method of array analysis, which hasalso been successfully applied in continental settings [Li etai., 2002, 2003; Weeraratne et ai., 2003].

2. DATA AND DATA ANALYSIS

To illustrate the method, we use Rayleigh wave data from21 earthquakes recorded at ocean-bottom stations of theMELT Experiment on the East Pacific Rise between Novem-ber 1995 and May 1996. Sensors were either three-compo-nent, I-Hz seismometers (model L4C, Mark Products) orCox-Webb [Cox et al., 1984] differential pressure gauges(DPGs). The instruments came from four different instru-ment pools with different recording and filtering operations,so before analysis, all responses for seismometers and pres-sure gauges were equalized to duplicate the response ofthe seismometer type that produced the most abundant data[Forsyth et ai., 1998]. Despite the short natural period ofthe seismometers, in the stable thermal environment of theseafloor, they yield reliable responses out to periods onthe order of 100 s. The DPGs were designed initially foroceanographic purposes and have flat response to accelera-tion out to periods of thousands of seconds [Cox et ai.,1984], although at periods greater than 35 s, pressure varia-tions from infra-gravity waves [Webb, 1998] in the 2700 to3500 m water depths of the MELT Experiment increasinglyinterfere with the recording of earthquake signals. The phaseresponses of all the DPGs were well matched, but the abso-lute gain varied somewhat from instrument to instrument,so in the inversions described in later sections, terms wereadded to find the best gain factor for each station. Stationswere deployed primarily in two lines across the axis (Platela), an arrangement intended to provide optimum resolutionfor small features beneath the ridge for body wave tomogra-phy, rather than optimum coverage for surface waves. Anaverage of 30 stations recorded each event with good signal-

Plate 1. (a) Location of MELT Experi-ment seismic array on the East PacificRise. Bathymetry is from Smithand Sand-well [1997]. The spreading center ismarked by the narrow, axial high regionshallower than 2850 m, trending aboutNI2°E. Open circles mark locations ofdifferential pressure gauges, open trian-gles are three-component seismometers.Small diamonds indicate grid of nodalpoints used in the velocity models. Large,open, squares indicate comers definingedges of the region of interest; travel times(text equation 5) are calculated from thepoint where a wavefront, assumed to beperpendicular to the great circle path forthis purpose, first encounters one of thesecomers. Double red lines mark thesewavefront intersections with the comersfor an earthquake along the Kurile trench(Plate Ib and Figure 5a and b) ap-proaching from the northwest and anearthquake off the coast of Oaxaca, Mex-ico (Figure 5c and d) approaching fromthe north-northeast. (b) Multipath interfer-ence effects for the Kurile earthquake (Ms7.1 at 2136 on 7 Feb. 1996) at 0.035 Hz.At selected stations, chosen so that thedisplayed seismograms do not overlap,500 s records are plotted along a line inthe direction of great circle propagation.Time increases away from the stationsymbol. These records are generated bynarrow-passband filtering the original,observed seismograms with comer fre-

quencies at 0.030 and 0.040 Hz. Colors indicate predicted amplitudes for two-plane wave representation found in inversion for velocitymodel 4, Table I. Amplitudes are normalized to the maximum, observed amplitude for each event, hence the predicted amplitude canexceed 1.0. To illustrate the two plane waves found in the solution, two of the peaks of the model, sinusoidal wavefields for the two planewaves are shown as green, dashed and red, dotted lines, representing a spatial snapshot at the reference time for the event. One entirewavelength is shown for each wave, although in the idealized, Fourier representation of a single frequency, the sinusoidal oscillationsextend across the entire area. Constructive interference occurs wherever the waves are in phase, as in lower left, and destructive interferencewherever the waves are out of phase, as along violet band striking diagonally across center of map. The strike of the interference patternis along the bisector of the two wavefronts, which in this case is nearly, but not exactly, along the great-circle direction.

-120' a, -118-10'

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FORSYTH AND LI 83

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-120' -118' -116" -114' -112' -110' -108' -106' -104'

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depth mb,

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0.0 0.2 0.4 0.6 0.8

Normalized Amplitude

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to-noise ratio, ranging from a minimum of 20 to a maximumof 38 stations.

The amplitudes and phases of Rayleigh waves at selectedfrequencies were determinedby Fourier analysis after instru-ment correction, filtering and windowing the records. Fre-quencies analyzed ranged from 15 to 60 mHz. A narrowbandpass filter (4thorder Butterworth, zero-phase shift) cen-tered at the frequency of interest with comer frequenciesseparatedby 10mHz was applied to each record. The filteredrecords were then windowed (boxcar with 50-s cosine taperson each end) to isolate the fundamental mode Rayleighwave from other phases. The width of the window varieddepending on the dispersion of the incoming wave aroundthat frequency, but the same frequency-dependent windowlength was applied to all stations of the array. Amplitudeswere corrected for geometrical spreading on a sphericalearthand differential attenuation. These corrections are relativelyminor, but significant across an array spanning 800 km. Toavoid an emphasis on earthquake size in the inversion, themeasured spectral amplitudes were normalized to unit RMSamplitude for each event.

Earthquakes from around the Pacific rim were used assources (Figure 1). To facilitate separation of fundamentalmode Rayleigh waves from S phases, a minimum epicentraldistance of 3000 km was employed. The earthquakes werechosen to be as well distributed in azimuth as possible, butthere is a substantial azimuthal gap, because no earthquakesof sufficient size occurred to the south of the array duringthe recording period. Nevertheless, the distribution ofsources and receivers yielded an excellent set of crossingpaths within and to the north of the array (Figure 2) thatprovide the basis for tomographic imaging of the velocityvariations. Paths that traveled across ocean-continent bound-aries or along island arcs were avoided as much as possibleto reduce the severity of multi-pathingor scattering [Frieder-ich et ai., 1994]. The period range used for each eventdepended on the magnitude of the earthquake and the natureof the path. At short periods for more complex paths or pathstraveling across the deep water of the northwest Pacific,Rayleigh waves were highly scattered and incoherent fromstation to station; those periods for particular events werediscarded. Scattering in the northwest Pacific has previouslybeen described by Lerner-Lam and Park [1989]. Despite thecareful selection of events and period ranges, significantinterference and wavefield complexity remains that cannotbe ignored in phase velocity analysis. The interference canbe recognized in individual records in the form of nodes orbeats in thedispersed waveforms, or by systematic amplitudevariations across the array (Platelb). Some of the nodes oramplitude minima may be associated with the excitationfunction at the source; the effects of multipathing can be

Figure 1. Epicenters of earthquakes used as Rayleighwave sourcesin this study. Projection is azimuthalequidistant centered on middleof array, with distance in degrees noted by concentric circles.Geometry of the array indicated by two short, parallel lines. Thinlines mark plate boundaries. The two earthquakes far to the westof the array in the Solomon and Banda Seas were useful only atperiods greater than 30s, at which multi-pathinginterference effectsare less severe.

extreme when the waves leave the source in a near nodal

direction where significant phase shifts in the interferingwaves can be introduced by small changes in azimuth.

3. FORWARD PROBLEM

3.1 Coordinate Systems

A local cartesian coordinate system is set up for eachearthquake source based on the epicenter and the locationof a reference station (Figure 3). The reference station isestablished as the origin of the coordinate system, with the+x direction in the direction of propagation along a greatcircle path from the epicenter and the +y direction 90° coun-terclockwise from the x direction. The x-coordinate is equalto the difference between the epicentral distance at eachpoint and the epicentral distance to the reference station,thus removing the curvature of the expected wavefront ona uniform, spherical earth. The y-coordinate is defined asthe distance along a small-circle about the epicenter fromthe point to the x-axis (the great-circle path through thereference station). This earth-flattening coordinate system

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FORSYTH AND LI 85

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Figure 2. Great circle paths in the vicinity of the array used in the analysis at 0.035 Hz. Double line marks locationof plate boundary, circles and triangles the locations of instruments. Some tomographic control is possible whereverthere are crossing paths.

represents a compromise between (1) a mercator projectioncentered about an axis about 90° from the study area, and(2) a mercator projection centered on the epicenter. Option(]) preserves the relative geometry of points within the studyarea, but leaves the wavefront curved. Option (2) flattensthe wavefront and makes all great circle paths lie along thex-direction, but distorts distances between points in the y-direction. The coordinate system used in this paper flattensthe wavefront, but preserves distances between points betterthan option (2), which is important because the waves maynot be approaching the array along a great circle path andthe interpolation scheme we use for phase velocity is basedon distances to fixed grid points.

Rather than describe variations in phase velocity in termsof a set of orthogonal basis functions or a set of cells ofconstant velocity, we use a continuous function that is aweighted average of velocities at neighboring grid points.The characteristic velocity at thejth grid point is azimuthallyanisotropic and given by

;'0 = Bo,j + Bl,j cos 2 ;(Jj + B1,j sin 2 ;(Jj (1)

where ;Bj is the backazimuth from the jth grid point tothe ith event in the original geographic coordinate system.Higher order azimuthal terms are neglected as they are ex-pected to be small for Rayleigh waves [Smith and Dahlen,

]973]. We use a 2-D Gaussian weighting function that hasa characteristic scale Lw.The phase slowness at every point(x,y) in the medium, including at the grid points themselves,is given by

N W. /N

.S = "V...! !... ~.w-I £.J £.ill

j=1 ;'0 j=l(2)

where N is the number of grid points, the weights aregiven by

and CXj,iYj)is the location of the jth gridpoint in the coordi-nate system of the ith event. The weighting function acts asan averaging or smoothing function, with Lwcontrolling thelength scale for variations in velocity of the medium. Withthis approach, velocities are described everywhere, evenoutside the actual grid of points employed, and will returnto the average velocity of the grid at infinite distance.

There is no absolute criterion for selecting Lw.As is usualfor this type of inverse problem, there is a tradeoff betweenmodel resolution and model variance or between data misfitand model length. Decreasing Lw increases the resolution,


(X,y) or (r, 1jI)

- -(0,0) (O,X)

Figure 3. A different coordinate system is employed for eachevent. Upper diagram shows two great-circle paths from the epicen-ter to stations in the array. x-coordinate is distance along greatcircle path relative to distance to reference station (triangle). They-coordinateof any other point (circle) is distance along the perpen-dicular, small-circle passing through that point to the great circlepath through the reference station. These coordinates are used asCartesian system for local analysis of phase velocities, convertedto localpolarcoordinates(r,ifJ)forconveniencein describinglocal

. direction of propagation e of an incoming plane wave (lower di-agram).

allowing more details of velocity variations to be resolvedor represented, usually resulting in smaller data residuals,but at the expense of greater amplitude, shorter wavelength,perhaps unrealistic, oscillations in the velocity model thathave greater uncertainty. It is a choice between resolvingmore with greater uncertainty and less with greater certaintythat must be made in the light of the particular tectonicquestion that is being addressed by the tomographic study.

The actualgrid of points employed in this study is showninPlate la. There is no requirement that the points be regularly

-

spaced on a rectangular grid. Here we base our grid on thenatural coordinate system provided by the ridge, as it isexpected that the dominant variations in velocity will occuras a function of the age of the seafloor. Points are regularlyspaced in latitude, but at each latitude, the center of the gridis tied to the ridge axis, which makes it easy during theinversion process to impose requirements such as that veloc-ity vary only as a function of distance from the ridge. It isimportant that the grid of points extend well-outside the areaof interest that will have well-resolved velocities constrainedby crossing paths; in the tomographic inversion, these outerpoints "absorb" travel-time variations that represent addi-tional deviations from the idealized two-plane wave repre-sentation of the incoming wavefield. Describing the resolu-tion of the tomographic inversion is easier if the grid pointsare regularly spaced in the region of interest, but to reducethe total number of variables in the inversion, we employ asparser grid in the outer, surrounding region. The total num-ber of grid points in this model is 315.

3.2 Two-Plane-Wave Representation

At each frequency w for each event, the incoming waveis regarded as the sum of two horizontally propagating planewaves so that the vertical displacement is of the form

Uz(w) = Al (w) exp [-i(kl .x - wt)]+ A2 (w) exp [-i(k2 .x - wt)] (3)

where k is the horizontal wavenumber of each wave and x

is the position vector. In practice, we describe the positionwithin the array relative to the reference station and use acommon reference origin time for all records of a singleevent, so that the predicted displacement at the kth stationfor the ith event at each frequency is simply

where

and

?cPl and? cP2are the phases of the first and second planewaves at the reference station, 7T and? T are the travel times

along the great circle path from the edge of the study areato the kth and reference stations, i{tl and i{t2are the angular

deviations from the great circle path of the first and second

waves, 7S is the average slowness, and the (x,y) coordinatesof the stations are described also in terms of polar coordinates(r,1/1)centered on the reference station and oriented in terms

of the great circle path. The incoming wavefield at each

frequency is thus described by six parameters; the amplitude,reference phase and direction of each of two waves. The

directions and relative phases of the two waves are illustrated

in Plate Ib for one of the earthquakes of this study by plotting

peaks of the two sinusoidal oscillations for one frequencyat one instant, the reference time. Wherever the waves are

in phase, they will add constructively, and where they are

out of phase, they will interfere destructively, producing a

characteristic interference pattern that varies slowly in space.The travel times are found by integrating along the great

circle path (x-direction) from the edge of the study area to

each station. The edge is defined by the planar wavefrontperpendicular to the great circle path, i.e. constant value

jXedgein the coordinate system for each event, at the pointwhere the wavefront intersects one of the specified comersof the study area (Plate la). Thus,

In the correction for deviations from a great circle path(second terms in the expressions for phase in equation 4),the average slowness is defined as

Both the average slowness and the travel times for each pathare functions of the velocity coefficients at all grid pointsas defined by the Gaussian averaging functions, althoughof course the points nearest the great circle ray path aremost heavily weighted. The expressions for phase are exactin Cartesian coordinates if the waves lie along the greatcircle path or if the velocity in the medium is uniform, but ifthe waves deviate from a great circle path in a heterogeneousmedium, the second term in these expressions for phaseprovides only a first order correction. This approximationis justified because, as will be shown later, deviations froma great circle path are small and typical velocity variationsare only a few percent.

FORSYTH AND LI 87

4. INVERSE PROBLEM

Solution of the inverse problem for the wave parametersand velocity parameters is accomplished through two setsof iterations with two stages of inversion in each iteration.Rather than trying to match the observations in terms ofnormalized amplitude and phase, we choose to minimizemisfit to the real and imaginary components. This approachhas the advantage that misfits in phase have little importanceat those stations where destructive interference causes lowamplitudes and rapid phase fluctuations; a satisfactory fit isfound if the predicted real and imaginary components areboth small, regardless of phase. To begin the inversion, thestation with the largest amplitude is chosen as the referencestation, because the two plane waves are most likely tobe in phase at that station and the amplitude should berepresentative of the sum of the amplitude of the two waves.Observed amplitudes at other stations are normalized by thereference station amplitude and phase, and the phases of thetwo plane waves in the starting model are set to zero at thispoint (but are not fixed during the inversion). With 6 waveparameters required to describe the incoming wavefield fore~chevent and 2 pieces of information per station (amplitudeand phase or real and imaginary component), a minimumof four stations per event is required to have any informationabout the velocity structure. In practice, a local array of 5or 6 stations yields reasonable results with a measure ofuncertainty, but clearly better results are obtained with morestations.

In the first stage of each iteration, the velocity model isfixed either with starting values or with the current valuesfound in the previous iteration. In this stage, we find thebest fitting wave parameters in a least squares sense usinga downhill simplex method of simulated annealing [po444,Press et at., 1992].We found that this stage was necessary,because the combination of the periodic non-linearity of theproblem and the ambiguity in the solution if the two planewaves have similar azimuth makes it very difficult to achieveglobal minimization of the objective function with standardlinearized inversion techniques. With the velocityfixed, eachevent is independent, so we perform a separate simulatedannealing search for the 6 wave parameters for each event.To insure (almost always) that a global minimum is found,we start the simulated annealing in each iteration for eachevent with several different wave-parameter models, one ofwhich is the best model from the previous iteration.

In the second stage of each iteration, we use a standardlinearized inversion technique [Tarantolaand Valette, 1982]to solve simultaneouslyfor corrections to the current velocitymodel (values of velocity and azimuthal anisotropy at the

kix

77 = f iSdx (5)

i x edge


grid points) and wave parameters for each event. The solu-tion to the general non-linear least-squares problem is

where m is the current model, mo is the original startingmodel, dm is the change to the model, dd is the differencebetween the observed and predicted data for the currentmodel, G is the partial derivative or sensitivity matrix relat-ing predictedchangesin d to perturbationsin m, and Cnn

and Cmmare the apriori data and model covariance matrices,respectively. The observed data are the real and imaginarycomponents at a single frequency (we invert each frequencyindependently) for each of the filtered and windowedseismograms. In the examples discussed later at 0.035 Hz,we have 629 seismograms from 21 earthquakes, or 1258observations (real and imaginary from each record). Thepredicted data are functions of the wave parameters andthe velocity parameters at the nodes or grid points, whichtranslate to integrals of the weighting functions along thegreat circle paths. The velocity and wave parameters togethermake up m. With Bo, BJ and B2 at each of 315 nodesand 126 wavefield parameters there are potentially 1071parameters, leading to non-unique or underdeterminedmodel solutions. In addition, there may be other parameterssuch as the DPG gains or an attenuation coefficient. In theterminology of Menke [1984], it is a mixed determinedproblem. It is overdetermined in the sense that there aremore observations than model parameters, but it is underde-termined in the sense that not all the model parameters arewell resolved.

Regularization or damping of underdetermined solutionsis achieved by introducing diagonal and off-diagonal termsinto C-r:rn.We use either a minimum length criterion em-ploying diagonal terms only [Marquardt, 1970], or bothminimum length and a smoothing criterion using off-diago-nal terms also [Constable et al., 1987]. For the minimumlength criterion, we employ uniform variance for points inthe interior of the grid. We increase the variance for pointson the edge of the grid by a factor of 10 so that theseessentially undamped values can "absorb" travel-time devi-ations from the two-plane-wave model without introducingunnecessary velocity variations into the interior area of inter-est. Only very light damping is applied to the wave parameterterms so that analysis of the uncertainty or resolution of thevelocity model will include full trade-offs with the waveparameter terms. Table I indicates what we mean by lightdamping; the difference between the total rank and the veloc-ity rank is the number of independent combinations of wave-field parameters that are resolved, which range from 94 to97% of the total of 126 wavefield parameters. Starting mod-

els for the wave parameters are from the simulated annealinginversions in the previous stage and the m -moterms (equa-tion 7) penalizing changes from the original starting modelare dropped for the wave parameters.

When the smoothing criterion is applied, the inversionminimizes the 2D second derivative for velocities [Maureret aI., 1998] in the interior of the grid and the first derivativeat the edges. For simplicity, we use the same variance (orweight) for the diagonal component of Cmm in the smoothingcriterion as for the minimum length criterion, althoughclearly one could choose to vary the relative importanceassigned to these regularization criteria. No smoothing isused for the wave parameter terms.

In the first set of iterations, all the observations are as-signed equal variance. The data covariance matrix is as-sumed to be diagonal. Experience shows that a typical misfitto the normalized real and imaginary terms is on the orderof 0.1, which we chose as the initial, a priori estimate ofstandard deviation in the first set. Typically, after 4 or 5iterations, no further significant changes in velocity occur,although minor fluctuations in the model continue with fur-ther iterations due to the small random variations in wave

parameters introduced in the simulated annealing steps. Weusually terminate the inversion after 10 iterations. At thatpoint, we estimate the a posteriori standard deviation ofthe data from each individual earthquake. Because someearthquakes have better signal-to-noise ratios than othersand the wavefield from some earthquakes at some periodsmay not be well-described by the two-plane-wave model,there is a wide variation in how well the observations fromdifferent earthquakes are fit. The dominant source of vari-ance in the original data is the deviation from the ideal planewave character. If this deviation is not well-described byour simple representation of the incoming wavefield, we donot want the poor wavefield model to bias the velocitymodel. Therefore, in the second set of iterations, we assigna priori standard deviations to the observations based onthe a posteriori standard deviations found for each earth-quake after the first set of iterations. Our experiments indi-cate that with this scheme, eliminating one or two eventswith large residuals has little effect on the velocity model.

Our inversions are based on least squares minimizationwhich implicitly assumes a Gaussian distribution of errors.We find no a posterior evidence in the form of an excessnumber of outliers in the residuals to individual events thatwould indicate a violation of this assumption, provided thatthe data have been carefully cleaned to avoid processingerrors, glitches in the data, and noisy records. After theinitial inversions, we routinely reexamine the data for anyevent with large residuals. If we identify clear problems, weeither correct the error or eliminate that record if we had

inadvertently accepted data with poor signal-to-noise ratio.If there is no obvious cause for the large residual, we leavethe poorly fitting data in the analysis.

The renormalization of standard deviations changes theweight assigned to each earthquake based on the result ofthe first inversion. As a result, there is no one measureavailable to straightforwardly compare the quality of fit ofdifferent velocity models; every inversion of the same datawith different model parameters or smoothing length ordamping parameters will have somewhat different weightsassigned to the data. Because the weights or standard devia-tions are different, we cannot directly point to the reductionin variance to describe how much better one model is than

another in fitting the data. Because renormalization reassignsstandard deviations based on how well each model fits thedata, each model will tend to fit equally well in terms ofminimizing the final objective function, i.e., the square ofthe ratios of the misfits to the standard deviations renormal-ized after the first set of iterations. This is a small price topay for an inversion method that is insensitive to bad dataor poor fits to one or two earthquakes. We can provideapproximate descriptions of the quality of fit by removingthe renormalization in a variety of ways at the end of theprocess.

We employ three approximate descriptions of quality offit to the data (Table I): the root-mean-square (rms) misfitin terms of real and imaginary components, unnormalizedby multiplying by the standard deviations for each event asused in the second set of iterations; the rms phase misfitconverted to equivalent seconds, based on all stations andevents; and the median rms misfit in phase for an individualearthquake source. The first of these measures is closest torepresenting our actual minimization that is performed onreal and imaginary components rather than phase and ampli-tude. The second represents the misfits that are most directly

related to travel times and the velocity structure. Both ofthese measures, however, are sensitive to one or two poorlyfit events that are given little weight in the renormalized,final set of iterations. The median rms phase misfit is morerepresentative of the effect of renormalization, as it charac-terizes the "typical" event that is given intermediate weightin the inversion.

5. TWO-PLANE-WAVE SOLUTIONS

The solutions for relative amplitudes of the two planewaves for each earthquake demonstrate that wavefield com-plexity or interference is the dominant source of variancein travel times or phase from the simplest model of a planewave propagating across a region of uniform velocity. If theratio of the amplitude of the second, smaller plane wave tothe first, larger wave is Rw, then the maximum possiblephaseshift 'Y introduced by the second wave is sin') Rw.Ifthe phase shift throughout the study area is approximatelyrandom, then the expected root-mean-square (rms) time de-viation is approximately T'Y/(23/21f),where T is the periodof the wave. Figure 4 shows the distribution of Rwfor twofrequencies. These ratios are taken from solutions in whichthe phase velocity is 2-D and anisotropic, model 4 in Tablel. At 0.035 Hz or 28.6 s, roughly in the middle of our usefulfrequency range, seven of 21 events have Rw > 0.5. Themedian ratio is 0.28, which corresponds to an expected rmsscatter of 0.91 s in travel-time at 0.035 Hz. In contrast, inthe simultaneous solution for wave parameters and velocityvariations, the median rms misfit in phase at this frequencycorresponds to 0.42 s. If the velocity is assumed to be uni-form, the median misfit is 0.76 s after solving for the bestvelocity and wavefield parameters (Table 1). Thus, wave-field complexity is expected to be a greater contributor tovariance than either velocity variations or noise (given ourselection of high-quality signals).

FORSYTH AND LI 89

Table 1. Velocitv Models Used in Inversions at 0.035 Hz.

RMS MedianVelocity Averaging Total Velocity RMS Misfit Phase RMS

Model Variations Len th Dam in Rank Rank Real&lma Misfit Misfituniform 65 km 123.6 1.0 .119 .95 s .76 sisotropic

2 uniform 65 km 123.8 3.0 .114 .90 s .74 sanisotropic

3 I-D 65 km min length 138.4 16.8 .087 .65 s .57 sisotropic smoothing

4 2-D & 65 km min length 219.2 101.0 .075 .51 s .42 sI-D aniso smoothing

5 2-D 50 km min length 255.5 133.2 .075 .51 s .42 sisotroDic


-

oo 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

AmplitudeRatio AmplitudeRatioFigure 4. Amplitude ratios of the smaller of the two plane waves to the larger. Solutions from model 4, Table 1. (a)0.035 Hz and (b) 0.025 Hz. Note that typical ratio decreases with decreasing frequency, indicating decreasing effectof multipathing.

At 0.035 Hz, virtually every event deviates significantlyfrom a plane wave, despite the simplicity of the paths thatare confined almost entirely to the Pacific basin. Figure 5illustrates the details of fits for two events at this frequency.The Kurile earthquake has the largest amplitude variations,roughly an order of magnitude difference between thelargest and smallest records (Plate lb). The great-circlepaths from this event traverse the deep water of thenorthwest Pacific and pass close to the Hawaiian hotspot.Despite the large variations, the interference pattern isrelatively simple and the two-plane-wave model fits theobserved amplitudes well with an Rw of 0.87 (Plate Iband 5a). This event also has the largest rms deviations(1.09 s) between observed and predicted phase of any ofthe sources at this period. The phase misfits, however,stem largely from stations where destructive interferenceis occurring and the amplitudes are small (Figure 5b). Atthese stations, slight deviations from the model cause largephase shifts. It is important to keep these stations in theinversion, however, because their amplitudes constrain thewavefield parameters. No damage is done to the velocitymodel by inclusion of these stations, because there is nodirect penalty in the least squares minimization for phasemisfit; we minimize the real and imaginary componentmisfits, which are well satisfied by the modeled lowamplitudes. The quality of overall fit for this event isaverage as measured by the rms misfit of real and imaginarycomponents.

A more typical source at 0.035 Hz is the Oaxaca earth-

quake. It shows amplitude variations of about a factor of 2(Figure 5c) and is characterized by an Rwof 0.27, comparedto the median value of 0.28. These amplitude variations arewell-resolved and the phases are extremely well-fit at allstations (Figure 5d). The limit of resolution of Rwfor thisdata set is about 0.1, corresponding to amplitude variationsof +/- 10% and rms phase variations of about 0.3 s. Atshorter periods, scattering becomes more severe,particularlyfor events from the western Pacific traversing regions wherethe short periods are sensitive to the thick water layer and arehighly dispersed. The spectrum of Rwfor events employedinthe inversion, however, is not much different than at 0.035Hz because we discard highly scattered arrivals. At 0.06 Hz,the median value of Rwis 0.31, but 6 of the 21 events wererejected. At longer periods, wavefield complexity is reduced[Friederich et af., 1994]. At 40 s period, the median valueof Rwis 0.19, which means that significant improvement infit is obtained by using two plane waves for about half ofthe events. Thus, as expected, the degree of scattering ormultipathing decreases with increasing period.

It is probably not realistic to interpret the two plane wavesolutions literally as they are intended to be approximationsto a more realistic and more complicated wavefield. Never-theless, it is interesting to examine the character of theinterference pattern implied by the solutions. The intensityof the interference is controlled by the amplitude ratio. Theinterference pattern consists of alternating maxima and min-ima aligned along a direction that is the bisector of thenormals to the two wavefronts, typically nearly aligned in the

a. b.7 6

.035Hz I I I 1.025Hz6 5

5 4

C 4 C::J 30

()()3

22

a.1.0

0.8Q)

't)

.a'E.0.6E«'t)Q)

:§ 0.4't)Q)...a..

0.2

oo 0.2 0.4 0.6 0.8

Observed Amplitude

1.0

b.1.0

en 0.8Q)(3:>.(,)

0.6Q)enttS.ca..'t) 0.4Q)t5'6Q)Q: 0.2

FORSYTH AND LI 91

c.1.0

0.8Q)

't)

.a'E.0.6E«

't)Q)

:§ 0.4't)~a..

0.2

oo 0.2 0.4 0.6 0.8

Observed Amplitude

1.0

d.1.0

~ 0.8(3:>.(,)

0.6Q)enttS.ca..'t) 0.4Q)t5'6Q)

Q: 0.2

o . I I I I . 0o 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

Observed Phase cycles Observed Phase cyclesFigure 5. Comparison of observed and predicted (a) amplitudes and (b) phases for the Kurile event at 0.035 Hz andthe observed and predicted (c) amplitudes and (d) phases for the Oaxaca earthquake (Ms 7.0 at 0308 on 25 Feb. 1996).Scatter appears to be least at intermediate amplitudes, because many of the intermediate values happen to come fromthe very closely spaced stations near the center of the southern line of the array (Plate 1b). In (b), phases from stationswith observed amplitudes less than 0.2 are shown with open symbols. Note decreased accuracy of predicted phasesfor these stations with maximum, destructive interference.

'IKurileEO .

.....

....

. ..,...4 ....

. ..

."KurileEO . ..

0. 0.

0 ...o.o.

..ce.

OaxacaEO .:..... .., ..,. ... .

.OaxacaEO ......

I...,

,.....-.


great circle propagation direction (Plate Ib). The wavelengthbetween maxima is controlled by the angle separating thetwo normals, 13,and is given by

AI = cT cos f3/2sin /3

(8)

where c is the phase velocity of the plane waves (assumedto be equal for both). In the example given above for theKurile earthquake at 0.035 Hz, the direction of propagationof the larger plane wave is 0.45° counter-clockwise (-) fromthe great circle path and the direction of the smaller waveis 3.98° clockwise (+). For the Oaxaca event, the primarywave is +3.85° and the smaller is -1.64°. All but one of the21 primary waves at 0.035 Hz lie within 5° of the greatcircle path (Figure 6a). The secondary waves are more scat-tered and the scatter is greater the smaller the secondarywave (Figure 6b). All of the secondary waves for eventswith Rw> 0.5 lie within 12° of the great circle path. Of thesix secondary waves that deviate by more than )20, fourhave Rw < 0.25. In most cases, the interference pattern isaligned within a few degrees of the great circle path, similarto the trends found in Germany using the more complicatedbasis function expansions of wavefields [Friederich et al.,1994; Friederich, 1998]. The mean difference in angle be-tween the two waves is 12.8°, the median is 7.6°, and themaximum is 35.4°, corresponding to AIbetween interferencemaxima of about 490, 820, and 180 km, respectively. Thus,

a. PrimaryWave .035Hz7

o-10 o s

degrees

.s

Deviation from Great Circle

although there are substantial variations, the typical scaleof interference we detect is on the order of the maximum

dimension of the array. This result is not surprising, becauseinterference on much smaller wavelength scale would leadto rejection of the data as being incoherent from station tostation and interference at much larger wavelengths wouldbe difficult to resolve.

One of the common questions we have received fromothers who have used this method for their data sets is "How

do you know when the two-plane-wave method will help?"We also have received skeptical comments such as "It worksin the Pacific because the paths are so simple, but it won'twork in my study area" and "What about other effects onamplitude, such as attenuation, radiation pattern and focus-ing/defocusing due to 3D structure?". Using the two-plane-wave method as opposed to the usual single plane wavemethod never hurts; if a single plane wave is an adequaterepresentation, then the model finds a solution that is equiva-lent. The two-plane-wave method is most effective in thetransition period range between a simple, planar wavefrontand incoherent scattering. If, in examining the filtered, nar-row-bandpass records from the array the amplitude is highlyvariable between adjacent stations or there are more thantwo beats in the bandpassed seismogram, then two planeswaves will probably not provide a good representation ofthe scattering. The orthogonal polynomial approach of Fried-erich and Wielandt [1995] is effectively a multi-plane-waverepresentation of the incoming wavefield [Pollitz, 1999],

10 -30 .20 .10 o 10 20 30 40

Deviation from Great Circle degrees

Figure 6. Distribution of differences between azimuths of the model plane waves and the great circle direction at0.035 Hz. (a) Azimuths of the waves with the larger amplitudes of the pair for each event typically vary only a fewdegrees from the great circle direction. (b) Azimuths of the smaller waves are more scattered than the larger wave andscatter increases with decreasing amplitude ratio of the smaller to the larger. Note change in scale from lefthand plot.

6- - - -

sr-

T

4E:::>0t)

3-- -

2

b. Secondary Wave .035 Hz6

I I I I

' . "---'t+ 11 1---1

Rallo>.6 L ----+----+

4r-rL t--LL- I I I I

'rfllll'T2 iT- __ -h+

but, unless there is a very high density of stations in a 2-Darray, we find there is a rapid transition with decreasingperiod between the case where two-plane waves is an ade-quate description and highly incoherent scattering.

This approach has proved to be useful for arrays in avariety of settings, including Tanzania, New England, andColorado. The complexity of paths to the arrayjust changesthe relative amplitudes of the two plane waves and theperiodof the transition to incoherent scattering. The radiation pat-tern is relatively unimportant as long as the dimensions ofthe array are much less than the distance to the source, sothat range of azimuths from source to receiver is small. Inthis study, the range of azimuths is typically about 3 degrees.Attenuation is a minor effect across a small array, but wehave performed inversions in which we solve for attenuationcoefficients. As we will report in another paper, typical Qfor Rayleigh waves in the MELT study area is about 80.Focusing and defocusing inside the array can have a biggereffect. We are incorporating into the method amplitude andphase response kernels for each of the plane waves to repre-sent the internal scattering effects. Of course, the wholepoint of using two plane waves is to approximately representthe scattering effects of unknown 3-D structure outsidethe array.

6. PHASE VELOCITY SOLUTIONS

To explore the information contained in the data aboutthe variations in phase velocity, we have performed a largeset of inversions with different degrees of complexity al-lowed in the velocity model. Models range from constantvelocity throughout the study area to 1-D variations that area function only of distance from the ridge to 2-D variationsin both isotropic velocity and azimuthal anisotropy. Withinthe models allowing lateral variations, resolution can becontrolled by varying the averaging length Lwor by adjustingthe a priori model covariance. For all the models summa-rized in Table I, we employed an a priori model standarddeviation of velocities at the grid points of 0.2 kmls for bothBoand the anisotropy terms B) and B2.This choice stabilizesthe inversion, but is not very restrictive as the standarddeviation is equal to or larger than the deviations from themean velocity at each period within the well-resolved partof the grid inside the array. Increasing the standard deviationbeyond this level makes little difference in the amplitude ofthe velocity variations for our particular problem and choicesof Lw, but decreasing it dampens the velocity variations.With this choice, resolution is determined primarily by Lw.

Resolving power can be briefly summarized by the rankof the problem, equivalent to the number of pieces of infor-mation extracted from the data. The rank can be estimated

FORSYTH AND LI 93

by summing the diagonal elements of the resolution matrixR, defined by

The contribution to the rank from the velocity terms can besummed separately from the wave parameters; both the totalrank and the rank from the velocity terms are listed in Table1. The rank from the wave parameters in this linearizedinversion is nearly independent of the rank from velocityterms and typically about 97% of 6N, where N is the numberof earthquakes. In addition, the wave parameters are uncon-strained in the simulated annealing stage. Thus, the lightdamping of these parameters has essentially no effect on thevelocity solution and the linearizeddescriptions of resolutionand variance of the velocity solutions include nearly thecomplete tradeoff with wave parameters.

The inversions in Table 1 do not include the gains of theDPGs as variables. These inversions are for a single period,but the gain is best estimated from all periods. The approachwe use is to solve for the gain factors in an initial inversionequivalent to model I with uniform velocity and wave pa-rameters. We then find the weighted average of the gainfactors at all periods for each station and apply them asstation corrections before redoing all the inversions.

The simplest models are those with constant velocitythroughout the area. If no azimuthal anisotropy is allowed,the average velocity Bo at 0.035 Hz is 3.758 ::t:.004 kmls(modell, Table I). Adding 28 terms reduces all three mea-sures of misfit (model 2, Table 1), yielding estimates Bo =3.736::t: .006, BI = -0.067 ::t:.007, and B2 =-0.021 ::t:.006kmls. The peak-to-peak variation in velocity with azimuthis nearly 4% with the fast direction orthogonal to the ridgeaxis. The direction is in agreement with that inferred foryoung sea floor on the East Pacific Rise in previous studiesof Rayleigh wave anisotropy using long paths to land stations[Forsyth, 1975; Nishimura and Forsyth, 1988; Montagnerand Tanimoto, 1990; Laske and Masters, 1998; Larson etaI., 1998], but the amplitude is about twice the estimatefrom those previous studies. It also agrees well with thedirection found for shear wave splitting in the MELT Experi-ment [Wolfe and Solomon, 1998]. In tomographic studies,it is often assumed that neglecting azimuthal anisotropy willcause no significant bias in the isotropic model, because thepaths are sufficiently distributed in azimuth to average outthese effects. Here we find that the change in Bo caused byintroducing azimuthal terms is small, but significant at the99% confidence level.

The uniform velocity model is used as a starting modelfor an inversion in which azimuthal anisotropy is neglectedand Bo is allowed to vary only as a function of distance


from the spreading center. Given the design of the grid, wefind the velocities by simplycollecting the partial derivativesfrom all points at common distances from the ridge axis andreducing the number of velocity variables to the 21 sets ofthese points (Plate la), but it could also be done by introduc-ing off-diagonal terms into Cmmthat force perfect correlationbetween grid points sharing a common distance. The im-provement in fit achieved by introducing these lateral varia-tions (model 3, Table I) is much greater than that achievedby the introducing the uniform anisotropic terms. In thissetting, we expected that the variation in velocity wouldbe dominated by the effects of the cooling of the oceaniclithosphere as a function of the age of the seafloor. Thisexpectation is largely confirmed by the large velocity varia-tions (Plate 2a) and large reduction in data variance foundfor this model compared to model 2 (Table I). The minimumin velocity is offset somewhat from the ridge axis and thegradients are asymmetric, leading to higher velocities to theeast, on the Nazca plate side of the East Pacific Rise. Theseasymmetries are discussed more thoroughly in Forsyth etal. [1998]. The pattern is similar to that found at longerperiods and greater distance scales by Ekstrom et al. [1997].

Maps of phase velocities are constructed using the sameGaussian weighting function of neighboring grid points(Equation 2) as employed in the inversion. Thus each pointon the map represents a tapered average of phase velocitiesin the surrounding region, but this value is an accurate repre-sentation of velocities actually assumed in the inversionrather than a smoothed or filtered version of velocities in

cells. Uncertainties in the maps are described by the varianceof the weighted averages found by linear error propagation[Clifford, 1975] and given by

(10)

where q is the vector of weights for all the grid points fora particular position and CMMis the complete, a posteriori,model covariance matrix

(11)

where we have extracted the elements dependent on velocityonly. It is important to use the complete covariance matrixin the sense of including off-diagonal terms, because theweighted averages are much better known than would beinferred from the diagonal variance terms alone. Typically,there is a large, negative covariance between adjacent gridpoints; their individual values may not be well-constrained,but their sum is. When we combine grid points, as in model3, we still use the Gaussian weighting functions to generatethe maps and equation 10 to estimate the uncertainties, ex-

cept in this case there is a perfect positive correlation be-tween combined points represented in CMM.

The isotropic solution with velocities dependent on dis-tance from the axis is employed as the starting model formodels with 2-D variations. Model 4 (Plate 2c and TableI) allows 2-D variations in Bo,but the azimuthal anisotropyterms are allowed to vary only with distance from the axis.The characteristic averaging length Lw is 65 km and bothminimum length and smoothing conditions are applied.As iswell known, minimum length solutions tend to be oscillatory.The smoothing criterion alone tends to project or continuelinear variations in well-constrained parts of the map intoless well-constrained parts. Applying a combination of bothcriteria seems to minimize the appearanceofthese undesired,unresolved artifacts in the inversion. The map of variancein Bo (Plate 2d) indicates that some of the along-axis varia-tions in velocity are significant (exceed two standard devia-tions), but the dominant effect is still variations with distancefrom the axis. Note that there is some resolution outside thearray, particularly to the north where there are many crossingpaths (Figure 2). As expected, the best resolution is insidethe array where there is a high density of stations and cross-ing paths. The fit to the data with this model is remarkablygood, with typical misfits in phase on the order of O.4sorabout 1/70th of a cycle.

The average azimuthal anisotropy in model 4 is similarto that found in the uniform velocity model, but there is aminimum at orjust to the east of the axis that is well resolved.Maximum anisotropy is about 5 to 6%. We can constructmodels in which anisotropy is allowed to vary along-axisas well, but we find that the uncertainty in the inversion istoo large to be useful unless Lw is increased to the pointwhere only one value is resolved across the along-axis di-mension of the array.

A more critical issue than resolving 2-D variations inanisotropy is resolving whether azimuthal anisotropy is re-quired at all. As has been pointed out many times previously,travel times or phase shifts caused by anisotropy can alwaysbe matched by a model of lateral heterogeneities if the varia-tions are sufficiently strong and small-enough in scale. Inthis case, data from the MELT array forces a reduction inscale of the needed heterogeneity by at least an order ofmagnitude from previous regional or global studies, yet thefast direction is consistent with those studiesand the apparentdegree of azimuthal anisotropy is greater. In addition, thefast direction inferred from the Rayleigh waves agrees withshear wave splitting measurements at the same stations[Wolfe and Solomon, 1998] and the direction expected forsimple geodynamic models of the spreading process [Black-man et al., 1996]. Nevertheless, a laterally heterogeneous,isotropic model can satisfy the observations just as well as

3.603.683.703.723.743.763.783.803.823.843.863.88 3.90 3.924.00

phase velocity km/s-116' -114' -112' -110'

-14'

-18'

28.6 s

-16'

o DPGA VERT

-16'

-14'

b

-18'o DPGA VERT

-116' -114' -112' -110'

0.00 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1.002 x standard error km/s

3.60 3.68 3.70 3.72 3.74 3.76 3.78 3.80 3.82 3.84 3.86 3.88 3.90 3.92 4.00

phase velocity km/s-116' -114' -112' -110'

-14'28.6 s

e

-16'

-18'ODPGA VERT

-14'

-16'

-18'ODPG

A VERT

-116' -112' -110'

0.00 0.03 0.04 0.05 0.06 0.07 0.06

2 x standard error km/s0.09

FORSYTH AND LI 95

3.60 3.68 3.70 3.72 3.74 3.76 3.78 3.80 3.82 3.84 3.86 3.88 3.90 3.92 4.00

phase velocity km/s.116' -114' -112' -110'

.14'

-16'

-18'

4%

ODPGA VERT

-14'

-110'

d...

__.J

-16'

ODPGA VERT

-18' -.116' .112'

0.00 0.03 0.04 0.05 0.06 0.07 0.08

2 x standard error km/s0.09 1.00

1.00

Plate 2. (a) Rayleigh wave phase velocity Bo at 0.035 Hzassuming velocity varies only with distance from the ridgeaxis (Model 3. Table I). Contour interval at om km/s,color interval at 0,02 km/s (b) Uncertainty of the solutionrepresentedin terms of contours of twice the standarddevia-tion of Bo.Significant variations in velocity in (a) correspondto approximately these values. (c) Rayleigh wave phasevelocity at 0.035 Hz assuming BI and B2 terms representingazimuthal anisotropy vary only with distance from the ridgeaxis (Model 4, Table I). Bo is free to vary in two-dimensions.Contour interval for Bo at 0.01 km/s, color interval at 0.02km/s. Direction of short lines indicates fast direction of

propagation, length is proportional to degree of azimuthalanisotropy. (d) Uncertainty of the solution represented interms of contours of twice the standard deviation of Bo,Note that this measureof model variance is closely relatedto density of crossing lines (Figure 2). (e) Rayleigh wavephasevelocity at0.035 Hz assuming no azimuthal anisotropyand that Bo is free to vary in two-dimensions (Model 5,Table I). Contour interval for Bo at 0.0I km/s, color intervalat 0.02 km/s. In comparison to Model 4 in Plate 2c, theaveraging length is shorter and the damping is relaxed byremoving the smoothing condition, (f) Uncertainty of thesolution represented in terms of contours of twice the stan-dard deviation of Bo. Note the increase in model variancecompared to Model 4.


model 4. Reducing Lwto 50 km and removing the smoothingconstraint yields a 2-D model (Plate 2e) with nearly identicalmeasures of misfit (modelS, Table 1). As is typical of suchsolutions, more parameters are required for the isotropic fitto match the anisotropic fit; the rank is increased by 32.

Averaging over smaller areas increases the variance ofthe mapped phase velocities (compare Plate 2f to Plate 2d)and introduces many short-wavelength oscillations betweenstations. If the array had been optimally designed for surfacewave analysis with stations more randomly distributed inthe study area, the scale of heterogeneity needed to matchthe anisotropic inversion could have been further reduced,but it would still have been possible to generate such amodel. The scale of the heterogeneities required to mimic theeffects of anisotropy in this study (Plate 2e) are substantiallysmaller than the typical width of the Fresnel zone for thewavelength ofthe Rayleigh waves and the scale of the exper-iment. Thus, it is highly unlikely that these short-wavelengthfluctuations are real, and, applying Occam's razor, westrongly prefer the azimuthally anisotropic solution and con-clude that there is an average of about 4% variation in phasevelocity with direction in the vicinity of the East Pacific Rise.

7. CONCLUSIONS

Variations in surface wave amplitudes and phases causedby multipath propagation through heterogeneities often ex-hibit a relative simple pattern across an array of stations thatcan be satisfactorily described by the interference of twoplane waves. We have developed a tomographic imagingmethod for phase velocities in the vicinity of an array ofstations in which the velocity field is represented by aGaussian weighting function of values at a set of grid pointsand the incoming wavefield from each source event is repre-sented by the amplitude, phase and direction of two planewaves. For simple paths that lie primarily within the Pacificbasin, the apparent azimuth of the larger of the two planewaves is typically within a few degrees of the great circlepath. There is a wide variation in apparent relative ampli-tudes of the two plane waves, although usually interferenceeffects are stronger and the amplitudes more nearly equalat shorter periods. Typically, the characteristic wavelengthof the interference pattern is on the order of several hundredkilometers and the pattern is approximately aligned alongthe great circle path. Near the East Pacific Rise, the phasevelocities of Rayleigh waves vary rapidly with distance fromthe ridge axis and azimuthal anisotropy reaches a maximumof 5 to 6%.

Acknowledgments. Most figures were prepared using the Ge-neric Mapping Tools program of Wessel and Smith [1995]. Karen

Fischer gave a helpful review. Forsyth was partially supported asa Green Scholar at the Institute for Geophysics and PlanetaryPhysics, Scripps Institution of Oceanography. This work is part of

the Ridge Inter-Disciplinary Global Experiments (RIDGE) programand was supported by the National Science Foundation grants OCE-9402375 and OCE-98122208.

REFERENCES

Alsina, D., R. Snieder, and V. Maupin, A test of the great circleapproximation in the analysis of surface waves, Geophys. Res.Lett., 20, 915-918, 1993.

Blackman, D.K., J.-M. Kendall, P.R. Dawson, H.-R. Wenk, D.Boyce, and J. Phipps Morgan, Teleseismic imaging of subaxialflow at mid-oceanridge: traveltime effects of anisotropicmineraltexture in the mantle, Geophys. J. Int., 127, 415-426, 1996.

Capon, J., Analysis of Rayleigh-wave multipath propagation atLASA, Bull. Seism. Soc. Am., 60, 1701-1731, 1970.

Christensen, N.I., and M.H. Salisbury, Seismic anisotropy in theupper mantle: Evidence from the Bay of Islands ophiolite com-plex, J. Geophys. Res., 84,4601-4510, 1979.

Clifford, A.A., Multivariate Error Analysis, Wiley, New York,1975.

Constable, S.e., R.L. Parker, e.G. Constable, Occam's inversion:a practical algorithm for generating smooth models from electro-magnetic sounding data, Geophys., 52, 279-288, 1987.

Cox, W.e., S.e. Webb, and T.K. Deaton, A deep sea differentialpressure gauge, J. Atmos. Oceanic Technol., 1,237-246, 1984.

Ekstrom, G., J. Tromp, and E.W.F. Larson, Measurements andglobal models of surface wave propagation, J. Geophys. Res.,102, 8137-8157, 1997.

Evemden, J.F., Directionof approachof Rayleighwaves and relatedproblems, Part I, Bull.Seism. Soc. Am., 43, 335-374, 1953.

Forsyth, D.W., The early structural evolution and anisotropy ofthe oceanic upper mantle, Geophys.J. R. Astron. Soc., 43, 103-162, 1975.

Forsyth, D.W., S.e. Webb, L.M. Dorman, and Y. Shen, Phasevelocities of Rayleigh waves in the MELT experiment on theEast Pacific Rise, Science, 280, 1235-1238, 1998.

Friederich, W., and E. Wielandt, Intepretation of seismic surfacewaves in regional networks:joint estimation of wavefield geome-try and local phase velocity - Method and tests, Geophys. J. Int.,/20,731-744,1995.

Friederich, W., E. Wielandt, and S. Stange, Non-plane geometriesof seismic surface wavefields and their implications for regionalsurface-wave tomography, Geophys. J.Int., 119,931-948,1994.

Friederich, W., Wave-theoretical inversion of teleseismic surfacewaves in a regional network: Phase velocity maps and a three-dimensional upper-mantle shear-wave velocity model for south-ern Germany, Geophys. J. Int., 132,203-225, 1998.

Larson, E.W.F., J. Tromp, and G. Ekstrom, Effects of slight anisot-ropy on surface waves, Geophys. J. Int., 132, 654-666, 1998.

Laske, G., Global observation of off-great circle propagation oflong-period surface waves, Geophys.J. Int., 123,245-259, 1995.

Laske, G. and G. Masters, Surface-wave polarization data andglobalanisotropic structure, Geophys.J. Int., 132,508-520, 1998.

Lerner-Lam, A.L., and J.J. Park, Frequency-dependent refractionand multipathing of 10-100 second surface waves in the westernPacific, Geophys. Res. Lett., 16,527-530, 1989.

Li, A, D.W. Forsyth, and K.M. Fischer, Evidence for shallowisostatic compensation of the southern Rocky Mountains fromRayleigh wave tomography, Geology, 30, 683-686, 2002.

Li, A, D.W. Forsyth, and K.M. Fischer, Shear wave structureand azimuthal anisotropy beneath eastern North America fromRayleigh wave tomography, J. Geophys. Res., in press, 2003.

Marquardt, D.W., Generalized inverse, ridge regression, biasedlinear estimation, and nonlinear estimation, Technometrics, 12,591-612, 1970.

Maurer, H., K. Holliger, and D.E. Boerner, Stochastic regulariza-tion: smoothness or similarity?, Geophys. Res. Lett., 25, 2889-2892, 1998.

MELT Seismic Team, Imaging the deep seismic structure beneatha mid-ocean ridge: The MELT Experiment, Science, 280, 1215-1218,1998.

Menke, W., Geophysical data analysis: discrete inverse theory,Acad. Press, San Diego, 1989.

Montagner, J.P., and T. Tanimoto, Global anisotropy in the uppermantle inferred from the regionalization of phase velocities, J.Geophys. Res., 95, 4797-4819, 1990.

Nishimura,C.E., and D.W. Forsyth, Rayleigh wave phase velocitiesin the Pacific with implications for azimuthal anisotropy andlateral heterogeneities, Geophys. J., 94, 479-501, 1988.

Pollitz, F.F., Regional velocity structure in northern Californiafrom inversion of scattered seismic surface waves, J. Geophys.Res., 104, 15043-15072,1999.

Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery,

FORSYTH AND LI 97

Numerical Recipes in FORTRAN: The Art of Scientific Comput-ing (2nd edition), Cambridge Univ. Press., Cambridge,963pp., 1992.

Raitt, R.W., G.G. Shor, T.J.G. Francis, and G.B. Morris, Anisotropyof the Pacific upper mantle, J. Geophys. Res., 74, 3095-3109,1969.

Smith, M.L. and F.A. Dahlen, The azimuthal dependence of Loveand Rayleigh wave propagation in a slightly anisotropic medium,J. Geophys. Res., 78, 3321-3333, 1973.

Smith, W.H.F. and D.T. Sandwell, Global seafloor topographyfrom satellite altimetry and ship depth soundings, Science, 277,1956-1962, 1997.

Snieder, R. and G. Nolet, Linearized scattering of surface waveson a spherical Earth, J. Geophys., 61, 55-63, 1987.

Tarantola, A., and B. Valette, Generalized non-linear problemssolved using the least-squares criterion, Rev. Geophys. Sp. Phys.,20, 219-232, 1982.

Webb, S.c., Broadband seismology and noise under the ocean,Rev. Geophysics, 36, 105-142, 1998.

Wessel, P. and W.H.F. Smith, New version of the Generic MappingTools released, Eos Trans. AGU, 76, 329, 1995.

Wielandt, E., Propagation and structural interpretationof non-planewaves, Geophys. J. Int., 113,45-53, 1993.

Wolfe, C.J. and S.C. Solomon, Shear-wave splitting and implica-tions for mantle flow beneath the MELT region of the EastPacific Rise, Science, 280, 1230-1232, 1998.

D.W. Forsyth, Dept. of Geological Sciences, Brown University,Providence, RI 02912-1846

A Li, Dept. of Geosciences, University of Houston, 4800 Cal-houn Road, Houston, TX 77204-5503

array analysis of two-dimensional variations in surface...

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