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Traveltimes and amplitudes of seismic waves: a re-assessment

Guust Nolet, F.A. Dahlen and Raffaella MontelliDepartment of Geosciences, Princeton University, Princeton NJ.

Abstract

In this paper we give a simplified derivation of the sensitivity of travel time measurements bycross-correlation and of amplitudes of body waves to the seismic velocity structure in the Earth,taking into account the effect of finite frequencies. We introduce a new technique to computekernels in 3D media, using graph theory and ray bending. We show that the finite-frequencysensitivity kernels (or ‘banana-doughnut’ kernels) are sizeable even for local studies done at veryhigh frequencies, e.g. in refraction surveys. We conclude that it is advisable to applyfinite-frequency theory to most, if not all, modern seismic surveys.

A. Levander and G. Nolet (eds.), A rray analysis of broadband seismograms, AGU Monograph Series, in press,



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1. Introduction

Travel time measurements have been the backboneof seismology since the earliest seismographs showedP and S waves preceding the more dominant surfacewaves near the end of the 19th century (Bates et al.,1982). Since then, generations of seismologists havebeen occupied with picking arrival times from smokedpaper or photographic records.

In recent times, the proliferation of digital record-ings has changed that. Cross-correlation methods(VanDecar and Crosson, 1990) have replaced manualpicking, and enable us to measure travel times accu-rately even from long period seismograms (Woodwardand Masters, 1991). Because of their excellent signal-noise ratio, long period records often yield more sta-ble travel time measurements, and offer an attractiveaddition to short-period arrival times.

Theory, on the other hand, has been slower to catchup. The onset of a body wave from a short-periodseismogram represents a travel time at fairly high fre-

quency, usually well within the range of validity of the approximation of geometrical optics. This ap-proximation assumes that the seismic energy travelsalong ‘rays’ that are very much thinner than the sizeof heterogeneities in the medium.

The limitations of the ray approximation gainedattention when broadband sensors became more preva-lent (Wielandt, 1987; Nolet, 1992; Cerveny and Soares,1992; Nolet et al., 1994; Woodward, 1993; Yomogida,1992), but much of the interpretation of seismic ar-rival times, be it for earthquake location or for imag-ing, is still beholden to the assumptions that thewavelength of the seismic waves is small both with re-

spect to the scale length of heterogeneities and withrespect to the length of the ray itself. These limi-tations have become restrictive as the number andthe quality of broadband data have quickly improvedand measurements were more generally done at longerperiods. A compressional or P wave with a period of 20 seconds has a wavelength of 260 km in the lowermantle - far larger than the thickness of a slab frag-ment and probably larger than the typical width of aplume. Such small sized heterogeneities may have avery much reduced influence on the time of the firstarriving seismic energy, even though they influencethe waveshape.

One could try to invert for waveform distortions

rather than for travel time anomalies when interpret-ing long period seismograms. This, however, intro-duces its own complications. Early waveform inver-

sion techniques relied on the summation of surfacewave modes to generate body waves such as SS andSSS (Nolet et al., 1986; Nolet 1990). But classicalsurface wave theory assumes a layer-cake structure forthe Earth, and to take full account of lateral hetero-geneity one must incorporate mode coupling effects(Li and Tanimoto, 1993; Li and Romanowicz, 1995;Marquering et al., 1996). This makes waveform in-versions computationally intensive.

Marquering et al. (1999) recognized that the de-lay times obtained by cross-correlation are the moresuitable, discrete data to invert for. They still usedsurface-wave mode summation to calculate the changein waveform that affects the cross-correlation, whichmade an application to large data sets infeasible.Dahlen et al. (2000) abandoned the mode summa-tion approach and computed the change in waveformusing rays that reflect from heterogeneities (Figure 1).In its most efficient variant, paraxial rays are substi-tuted for exactly traced rays. Thus, we obtain thenecessary computational efficiency using ray theory

to circumvent the limitations of ray theory: the as-sumption of infinite frequency has been replaced bythe less drastic approximation of single scattering .

In this paper we shall closely follow Dahlen et al.(2000), though we shall attempt to give this papera more tutorial flavour, concentrating on conceptsrather than a full development of the theory. A newelement in the present paper is the use of graph the-ory for the computation of the travel times and am-plitudes of scattered waves, rather than the paraxialapproximation. This makes the method more eas-ily applicable in media where the background modelis already laterally heterogeneous, since it avoids the

problems of two-point ray tracing.

2. Heuristic introduction of

banana-doughnut kernels

We develop our ideas using the simple cartoon of Figure 2. In this figure, the direct wave from sources to receiver r arrives together with a scattered wavethat arrives with a slight delay. The sign of the de-layed wave is determined by the sign of the velocityanomaly that causes the scattered wave: fast anoma-lies generate scattered waves with a negative polar-ity. Figure 2 shows the two possible effects of thisin the seismogram. u(t) denotes the seismogram inan unperturbed medium. If the seismogram is com-posed of the direct wave plus a scattered wave δu(t)from a slow velocity anomaly (generating a scatterer

A. Levander and G. Nolet (eds.), Analysis of broadband seismograms, AGU Monograph Series, in press, 2005.



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with a positive polarity as in the bottom of Figure 2),the maximum of the cross-correlation C (t) is delayed.Clearly, the onset of the observed wavelet u(t) +δu(t)is not affected, in agreement with the common as-sumption that onsets are dominated by very high fre-quencies, such that ray theory is valid. The situationin the top part of Figure 2 is the reverse: the scattereris now fast, the polarity of the scattered wave nega-tive, and the cross-correlation shows a time advance.

These two figures show the basic ideas of finite fre-quency seismology: scatterers off the ray path maynot influence the onset of the wave, but they do ad-vance or delay the full waveform. Cross-correlationmethods that use all or part of the waveform to de-termine the travel time delay, are sensitive to thesewaveform effects. To be consistent, travel time dataderived from cross correlation should therefore not beinverted with methods based on ray theory.

A second phenomenon, which is rather counter-intuitive, is that a small scatterer on the raypath doesnot change the shape of the waveform but only per-

turbs its amplitude. This situation is depticted inFigure 3. The reason is that the scattered wave δu(t)in this case is not delayed (the sum of its paths to andfrom the scatterer equals the original raypath). Thedelay incurred by the scattered wave is essential to in-fluence the travel time of the wave. Thus, scatteringobjects must be located away from the actual ray tobe seen in a tomographic inversion. Of course, largeobjects always extend beyond the direct vicinity of aray and therefore do influence the travel time, con-forming to our ray-theoretical intuition. But a smallpebble will only be seen if it is not on the ray!

In Figures 2 and 3 we assumed that δu(t) preservesthe shape of u(t), i.e. there is no phase shift. But oncethe wave has passed a focal point or caustic, the waveenergy defocuses again. Theory tells us that a waveacquires a π/2 phase shift upon passage through acaustic and a shift of π upon passage through a three-dimensional focal point. The latter case reverses thesign of the added scattered wave, and may turn adelay into an advance. Focal points are rare if notentirely absent in seismology, but a π/2 shift is quitecommon in a layered Earth, or for surface reflectionslike PP waves. We shall see that such a phase shifthas a complicated effect on the resulting delay of the

wave.The upshot of this all is that the delay time mea-sured by cross-correlation is not just a function of the anomalies encountered on the geometrical ray be-tween source and receiver, but in a volume around

it. The volume is finite, because a wave that arrivestoo late to be included in the cross-correlation win-dow clearly has no influence on the measured delaytime. If the ray is straight, as in the homogeneousmedium depicted in Figure 1, the volume includingall timely scatterers is given by an ellipse. In morerealistic media where the geometrical ray is bent, thevolume resembles that of a banana. Since the sensi-tivity is zero on the geometrical ray itself if there isno caustic phase shift, the sensitivity kernel (‘Frechetkernel’ in the mathematical jargon) has a hole in thecenter. Marquering et al. (1999) therefore namedthese kernels ‘banana-doughnut kernels’. For an ex-ample of such a kernel in global seismology, take alook at Figure 4.

The net effect of wavefront healing is that delaysspread out over the wavefront, and diminish in mag-nitude as the wave progresses along the raypath (No-let and Dahlen, 2000; Hung et al., 2001). This phe-nomenon is frequency and wavelength dependent asshown in Figure 5. For infinite frequencies, or in-

finitely small wavelength, it will be negligible at fi-nite distance from the anomaly. As the wavelengthgrows to a length comparable to the dimension of theanomaly, the wavefront healing becomes significanteven at short distances from the anomaly.

One last caveat: the theory as described here is asingle scattering theory that is fully linear. As suchit predicts that the magnitude of a delay is propor-tional to the amplitude of the anomaly. In particu-lar, fast and slow anomalies of equal magnitude arepredicted to have delay times that are equal in mag-nitude, they only differ in sign. While correct to firstorder, there is a second order effect that does intro-

duce asymmetry, as pointed out by Wielandt (1987).A fast anomaly will advance the wavefront, and thisadvanced wavefront will propagate without further in-terference. Thus, negative delay times do not ‘heal’.A slow anomaly, on the other hand, creates a de-layed wavefront with an unperturbed zone that maybe filled in by energy radiating from the sides (us-ing Huygens’ Principle). Nolet and Dahlen (2000)showed that the situation is more complicated thanthat, however: the advanced wavefront loses energyquickly, and a negative delay will at first retain itsmagnitude (or even grow!), but as it spreads its en-ergy too thinly, at some point the wave amplitude willbecome very small and in practice drown in the noise.




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3. Travel time delays

It is now time to translate the heuristic ideas of the previous section into a quantitative theory. Inthe limit of weak scattering, we may use the ray ap-proximation to build a theory that extends beyondthe limitations of ray theory itself. This may seemcircular or magical, but it actually works, as was con-vincingly shown in numerical simulations by Hung et

al. (2000).We assume that the velocity in the Earth can be

represented by a smoothly varying background modelfor which ray theory is valid, and to which heterogene-ity is added that acts to scatter a (small) fraction of the wave energy. As in Figure 1, the signal in the re-ceiver consists of a direct ray arrival u(t) and a scat-tered ray arrival δu(t). For a point scatterer (i.e. ascatterer of a size very much smaller than the dom-inant wavelength of the wave), the waveshape of thescattered wave will be identical to that of the directwave, but it will be delayed by a delay τ , and it willonly have a fraction of the original amplitude:

g(t,x) = (x)u(t− τ ) (1)

where we used the notation g to indicate the scattereris a point response or Green’s function. (1) assumesthere are no caustic phase shifts along the paths trav-elled to and from the scatterer. Such shifts couldeasily be handled by adding a Hilbert transform or(−) sign to (1), but we shall in this paper only dealwith the simple case of first arrivals which never havecaustic phase shifts. It also assumes that we can ne-glect P→S and S→P scattering, which is appropriatebecause the converted waves arrive with a very largetime advance or delay unless the scatterer is near thesource.

For scatterers of realistic dimension, we can com-pute δu using ray theory by a summation of pointscatterers:

δu(t) =

(x)u(t− τ )d3x (2)

How does δu influence the travel time? The au-tocorrelation of the unperturbed seismogram u(t) isgiven by:

γ (t) =

u(t − t)u(t) dt (3)

We define the travel time delay by the maximum of the cross-correlation function of the observed signal

u + δu with the unperturbed wave u:

γ (t) + δγ (t) =

u(t − t)[u(t) + δu(t)] dt (4)

For the unperturbed wave, the cross-correlation reachesits maximum at zero lag, so:

γ (0) = 0 (5)

and for the perturbed wave the maximum is reachedafter a delay δT :

γ (δT ) + δγ (δT ) = 0 (6)

where the dot denotes time differentiation. Develop-ing γ to first order, we find (Luo and Schuster 1991;Marquering et al., 1999):

γ (δT ) + δγ (δT ) = γ (0) + γ (0)δT + δγ (0)+O(δ2) = 0(7)

or using (5) and (3):

δT = −δγ (0)γ (0)

= − ∞−∞ u(t)δu(t)dt ∞

−∞u(t)u(t)dt

(8)

We get u by computing a synthetic signal for the back-ground model. This introduces a number of compli-cations — such as uncertainties in the source locationand excitation as well as in the attenuation along thepath — that we shall ignore here. Equation (8) isin the time domain, and could in principle be usedto compute the sensitivity kernel for velocity pertur-bations δc by expressing (x) in (2) in terms of theδc. However, to avoid that we have to perform both avolume integration over x and a convolution over scat-

terer delays τ , it is more convenient to express (8) inthe frequency domain. Using Parseval’s theorem andthe spectral property of real signals u(−ω) = u(ω)∗,where an asterisk denotes the complex conjugate:

δT =Re

∞

0iωu(ω)∗δu(ω)dω

ω2u(ω)∗u(ω)dω(9)

4. Born theory for seismic waves

In this section we shall express δu in terms of theP and S velocity perturbations δα and δβ . We shalldenote the signed amplitude of δu by δu (δu is the

wave stripped of its phase delay due to propagation,but retaining any sign changes acquired upon scat-tering). This amplitude is defined by the scatteringstrength (x) in (1) and (2). For a full development of the theory of single scatterers, we refer to Dahlen et




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al. (2000) or the references therein, especially Wu andAki (1985). For this paper, we shall take a simplified,more tutorial approach. In particular, we assume thatthe coefficient for forward scattering at an angle θ = 0is valid for all angles. The reason is that scatteringat large angles almost always implies a large delayof the scattered wave, making an interference withina narrow time window such as sketched in Figure 2impossible. In particular we neglect:

• mode conversions upon scattering

• scattering from density perturbations

• the angle dependence of scattering

For example, the amplitude of a scattered wave from asmall volume dV with perturbations in density δρ andin Lame parameters δλ and δµ, for P→P scatteringfrom an incoming wave of amplitude 1 is given by (Wuand Aki, 1985):

δuPP (ω) = (10)

ω2

4πα2

1

r

δρ

ρcos θ − δλ

λ + 2µ− 2

δµ

λ + 2µcos θ2

dV

where θ is the scattering angle, ρ the density, α the Pvelocity and λ and µ Lame’s elastic parameters. Forpurely forward scattering (θ = 0) this gives:

δuPP (ω) =ω2

4πα2

1

r

δρ

ρ− δλ

λ + 2µ− 2

δµ

λ + 2µ

dV

= − ω2

2πα2

1

r

δα

αdV (11)

because

δρρ − δλ + 2δµλ + 2µ = −2δαα (12)

The factor −ω2/2πα2 is denoted as the scattering co-efficient S PP α , where the subscript indicates the typeof heterogeneity. Note that the power |δuPP |2of thescattered wave is proportional to ω4 (‘Rayleigh scat-tering’). This high-pass filtering effect of a point-likescatterer compensates for the low-pass filtering effectof any integration over space to build the responseof a large object. Table 1 in Dahlen et al. (2000)provides a convenient summary of all scattering coef-ficients. For this paper it is sufficient to note that, inthe approximation of forward scattering:

P → P : S PP α = − ω2

2πα2(13)

S → S : S SSβ = − ω2

2πβ 2(14)

all other scattering coefficients being 0. Althoughscattering of shear waves depends on the polariza-tion of the wave, the expressions reduce to the samesimple constant for forward scattering.

5. Sensitivity kernels at finite

frequency

So far, we have made no assumptions about thenature of the zero order field u(ω). Clearly, the moreaccurate and complete the zero order field is, the moreaccurate our estimate of δu(ω). The method to use forthe computation of δu(ω) is less critical since approx-imation errors will be of second order in u(ω)+δu(ω).

As we discussed in the heuristic introduction, Zhaoet al. (2000) construct both δu(t) and u(t) by sum-mation of discrete normal modes nS

m and nT

m of the

Earth, each with its own frequency nωml , a technique

that finds its roots in the formalism developed for verylow frequency waveforms by Woodhouse and Girnius(1982). More efficient — but still too cumbersome

for large scale inversions — is the approach adoptedby Marquering et al. (1999), who summed surfacewaves in the frequency domain for a equidistant setof frequencies of interest. Summation of modes hasthe advantage that every possible wave arrival in thetime window of interest is represented in u(ω), evenif it is a diffracted or evanescent wave. Dahlen etal. (2000) used what is probably the most efficientmethod and constructed u using ray theory. More-over, they use a paraxial approximation to estimatethe geometrical spreading R as well as the travel timeto the scattering point.

In this paper we shall also use ray theory, though

we shall refrain from a paraxial approximation for thescattering detour time. Our simplified developmentuses several results from Aki and Richards (1980),Chapter 4, and we shall refer to equation numbers inthat chapter using an indication ‘AR’.

The spectrum of a far field P wave from a momenttensor source with time behaviour m(t) in a homo-geneous medium with velocity α0 and density ρ0 is(using AR 4.29):

uP (ω) =F P m(ω)

4πρ0α30rrse−iωrrs/α0 (15)

where the geometrical spreading is given by the dis-tance rrs from the source s to the receiver r, andwhere F P denotes an amplitude factor that includesthe radiation pattern. ω = 2πf is the frequency of the wave; m(t) is the derivative of m(t) and m(ω)




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is the Fourier transform of m(t). In a heterogeneousmedium, the ‘ray approximation’ involves modelingthe amplitude by a local factor (αrρr)−

1

2R−1rs , where

Rrs replaces rrs as the geometrical spreading factorand where (αrρr)−

1

2 models the effect of changes inimpedance, such that the energy flux is conserved; inaddition, the travel time is generalized from r/α0 toT rs. Fitting the remaining constants in this Ansatzto (15) near the source, where the two should agree,results in the generalized expression (cf. AR 4.88):

u(ω) =F P m(ω)

4πα2sRrs

√ρsρrαsαr

e−iωT rs (16)

Thus, the effect of gradual changes in the mediumis obtained by replacing:

ρα3r→ α2sRrs√ρsρrαsαr (17)

Replacing the source s with the scatterer x, we mayuse (17) to generalize the expression for the scatteredwave (11) to heterogeneous media by adding the ap-propriate geometrical decay and a phase delay:

δuPP (ω) = − ω2ρxdV

2πRrxαx√αxαrρxρr

δα

α

x

e−iωT rx

(18)

Computationally, it is very inefficient to have tocalculate the geometrical spreading Rrx for each pos-sible scattering location x. However, if we computethe geometrical spreading from the receiver to everypoint in the model, the reciprocity principle gives usthe spreading factor Rrx from (Dahlen and Tromp,1998, eq. 12.30):

αrRxr = αxRrx (19)

If we replace the receiver r by a scatterer x, (16)can also be used to find the wavefield that impactson a point scatterer. This gives us the necessary am-plitude for the wave that impacts on the scatterer.Multiplying the scattered wave (11) with this ampli-tude, and applying (19) results in:

δuPP (ω) = (20)

− F P m(ω)ω2

8π2α5/2s ρ1/2s α3/2r ρ1/2r

(δα/α)xdV

αxRxsRxre−iω(T rs+T rx)

We find the time delay induced by such a pointscatterer if we insert (16) and (20) into the expression

(9) for the cross-correlation time delay:

δT = (21)

− (δα/α)xdV

2παrαx

Rrs

RxrRxs

Re ∞

0iω3|m(ω)|2e−iω∆T dω ∞

0ω2|m(ω)|2dω

where ∆T is the extra time needed for the ray to visitthe scatterer x at x:

∆T (x) = T rs − T xr − T xs (22)

For a more general heterogeneity, we integrate overall point scatterers:

δT =

K α(x)

δα

αdV (23)

where K α(x) is the Frechet kernel (often namedbanana-doughnut kernel):

K α(x) = (24)

− 1

2παrαx

Rrs

RxrRxs

∞

0ω3|m(ω)|2 sin[ω∆T (x)]dω

∞

0ω2|m(ω)|2dω

The sensitivity of an arrival time may deviate sig-nificantly from that predicted by geometrical ray the-ory, as is evident from their shape and width such asdepicted in Figure 4. Montelli et al. (2004a) indeeddemonstrate that the use of finite-frequency kernels

has a significant influence on the amplitude of veloc-ity anomalies in tomographic images from long perioddata, which can easily be amplified by as much as 50%or more. In fact, even ISC-derived travel time de-lays, often presumed to be representative of the traveltimes for very high frequency waves, may be in needof finite-frequency treatment, as a careful compari-son of short- and long period delay times by Montelli(2003) has shown.

The use of banana-doughnut kernels in an inversionof teleseismic data enabled Montelli et al. (2004b) toinvert jointly the short-period ISC delay times with

long period travel times obtained by cross-correlation(Bolton and Masters, 2001), which was the first studyto show convincingly that a large number of mantleplumes originate in the lowermost mantle.

We note that the simple expressions for amplitudeand delay time inversion (24) and (21) have been de-rived using the far-field expressions for the seismicwavefield. This gives problems in the case the scat-terer is located near the source or directly beneaththe receiver. Though the singularity in the banana-doughnut kernels is integrable and may be handled byassuming a small region of constant velocity pertur-bation around it, (24) and (21) do not contain near-

and intermediate field terms that may be importantespecially at low frequency. In an important contri-bution, Favier et al. (2004) develop finite frequencykernels for travel time perturbations in the near field,as well as for S-wave splitting observations.






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end nodes are fixed), and similarly for yk and zk:

∂T

∂xk=

∂Lk

vk∂xk+

∂Lk+1

vk+1∂xk− Lk

v2k

∂ vk∂xk

− Lk+1

v2k+1

∂ vk+1∂xk

(31)where

∂Lk

∂xk=

xk − xk−1Lk

(32)

∂Lk+1

∂xk =xk−xk+1

Lk+1 (33)

and∂ vk∂xk

=∂ vk+1∂xk

=1

2

∂vk∂xk

(34)

This gives a tri-diagonal system of equations of theform:

akxk−1 + bkxk + ckxk+1 = rk (35)

which, after combining it with similar equations foryk and zk is extremely fast and efficient to solve. (31)is not exactly linear, since a relocation of the nodesmay change the average velocity vk, but we find that

it iterates quickly to a minimum.An appropriate local velocity model to test these

results is a model based on refraction seismic experi-ments around Mount Vesuvius (Lomax, pers. comm.,1999). A slightly smoothed version of this model isshown in Figure 6. Note that at shallow depth the ve-locity variations approach 100%. The structure below4 km depth is much more homogeneous.

The algorithm to solve the ray tracing equationshas no difficulty with this extreme velocity structure.Figure 7 shows the travel time field in the plane z = 0for a source at the surface, west of Vesuvius.

As can be seen from (22), the computation of thetravel time field from every source and every receivergives us the required delay times to use in (24). Of course, we also need the geometrical spreading. Forthis, we used the expression (Dahlen and Tromp,1998, eq. 12.27):

Rxs =

|dΣx|dΩs

(36)

where dΩs is the solid angle at the source s spannedby four rays next to the central ray in orthogonaldirections, and where |dΣx| is the cross-sectional area

spanned by these rays at the model point x of interest.A careful balance must be struck between choosinga large solid angle dΩ with a stable estimate of Rthat averages out many of the spatial fluctuations inamplitude, or a more localized estimate that suffers a

larger numerical error. Since wavefront healing affectswave amplitudes as well we are permitted to choosewider angles, but if the ray tubes for different centralrays overlap we may violate energy conservation. Weinvestigated the accuracy of R by testing reciprocityfor the Vesuvius model and conclude that errors of the order of 10% are common. Such errors, whilelarge, lead only to a second order error in the Frechetkernels and are considered acceptable for our purpose,but efforts to improve on the efficient computation of R are ongoing (Nolet and Virieux, work in prep.).

Figures 8–9 show Frechet kernels for the Vesuviusmodel. The dominant frequency here is 10 Hz. Thisis slightly optimistic, since the observed frequenciesduring the TOMOVES refraction experiment wheregenerally not higher than about 7 Hz (J. Virieux,pers. comm. 2003). Yet, we see that the size of the traveltime and amplitude kernels are large, with awidth of 2–3 km: larger than many of the features onewould hope to resolve. What the banana-doughnutkernels allow one to do is correct for the wavefront

healing induced by the large width of the kernels.There is one caveat, though. If traveltime anoma-lies heal towards zero at large distances (i.e., towardsthe right in Figure 5), attempts to interpret them us-ing the banana-dougnut kernels are akin to invertinga diffraction equation, which is notoriously unstable(Nolet and Dahlen, 2000). One can understand thisintuitively, because at large distance, a strong velocityanomaly gives only a small travel time delay. Invert-ing this involves multiplying the small delay by largenumbers, running the risk that errors in the obser-vation are magnified at the same time. Proper reg-ularization is therefore an important prerequisite for

finite-frequency inversions.

Acknowledgments. The authors gratefully acknowl-edge support for the research reported in this paper fromNSF grants EAR0105387 (TD and RM) and EAR0309298(GN and RM).

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Received XX; revised XX; accepted XX.

This preprint was prepared with AGU’s LATEX macros

v5.01. File paper3p formatted February 23, 2005.






0 ˚

6 0 1 2 0 ˚

1 8 0 ˚

2 4 0 3 0 0 ˚

Figure 4: Two examples of kernels for long period (20 s) teleseismic P waves in the Earth’s mantle. The

kernel in the Northern hemisphere is for a surface reflected PP wave at ∆ = 120, the shorter kernel in

the Southern hemisphere for a P wave at 60. Darker colours indicate more negative values of the kernel,

the whitish regions have a positive value for the kernel, implying a positive delay for a positive velocity

perturbation. Such ‘reverse’ sensitivities are located in the second Fresnel zone. Note the region of reduced

sensitivity at the center of the kernels, except near the reflection point of PP. The extra complexity of the PP

kernel is caused by a 90 phase shift at the caustic, as well as by the fact that scattered waves may also reflect

from the surface. The dark shading of the Earth’s core does not indicate a sensitivity.

Figure 5: Evolution of body-wave delay times along a ray traversing the center of a spherical anomaly, as

a function of the distance x traveled past the anomaly, scaled by the halfwidth L of the anomaly. The three

curves are for different values of the halfwidth-to-wavelength ratio L/λ. The initial delay in each case is

τ max. From Nolet and Dahlen [2000].






0

5

k m

-10 -5 0 5 10

km

-3 -2 -1 0

Figure 8: Travel time Frechet kernel for a 10 Hz wave passing beneath Mount Vesuvius. Units are s/km3

0

5

k m

-10 -5 0 5 10

km

-5 0 5 10

Figure 9: Amplitude Frechet kernel for a 10 Hz wave passing beneath Mount Vesuvius. Units are km−3


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