sensitivity kernels for seismic fresenl volume tomography – liu et al - 2009

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GEOPHYSICS, VOL. 74, NO. 5 �SEPTEMBER-OCTOBER 2009�; P. U35–U46, 12 FIGS., 3 TABLES.10.1190/1.3169600

ensitivity kernels for seismic Fresnel volume tomography

uzhu Liu1, Liangguo Dong1, Yuwei Wang2, Jinping Zhu2, and Zaitian Ma1

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ABSTRACT

Fresnel volume tomography �FVT� offers higher resolu-tion and better accuracy than conventional seismic raypathtomography. A key problem in FVT is the sensitivity kernel.We propose amplitude and traveltime sensitivity kernels ex-pressed directly with Green’s functions for transmittedwaves for 2D/3D homogeneous/heterogeneous media. TheGreen’s functions are calculated with a finite-difference op-erator of the full wave equation in the frequency-space do-main. In the special case of homogeneous media, we analyzethe properties of the sensitivity kernels extensively and gainnew insight into these properties. According to the construc-tive interference of waves, the spatial distribution ranges ofthe monochromatic sensitivity kernels in FVT differ fromeach other greatly and are 1 /8, 2 /8, 3 /8 and 4 /8 periods ofseismic waves, respectively, for 2D amplitude, 3D ampli-tude, 2D traveltime, and 3D traveltime conditions. We alsohave a new understanding of the relationship between ray-path tomography and FVT. Within the first Fresnel volume ofthe dominant frequency, the band-limited sensitivity kernelsof FVT in homogeneous media or smoothly heterogeneousmedia are very close to those of the dominant frequency.Thus, it is practical to replace the band-limited sensitivitykernel with a few selected frequencies or even the singledominant frequency to save computation when performingband-limited FVT. The numerical experiment proves thatFVT using our sensitivity kernels can achieve more accurateresults than traditional raypath tomography.

INTRODUCTION

Seismic raypath tomography based on the asymptotic high-requency solution of the wave equation has been used widely ineismology, geodynamics, engineering, and petroleum exploration

Manuscript received by the Editor 17 July 2008; revised manuscript receiv1Tongji University, State Key Laboratory of Marine Geology, Shangha

du.cn.2Tongji University, School of Ocean and Earth Science, Shanghai, China. E2009 Society of Exploration Geophysicists.All rights reserved.

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Sheriff and Geldart, 1982; Dziewonski, 1984; Nolet, 1987; Harrist al., 1992; Pulliam et al., 1993; Billette and Lambaré, 1998; Akind Richards, 2002�. However, it cannot resolve the fine structuresf the media and is insensitive to high spatial wavenumber variationsf the media because of its infinite frequency approximation. Wave-quation tomography also has been studied since the 1980s �Dev-ney, 1984; Wu and Toksöz, 1987; Pratt and Goulty, 1991; Wood-ard, 1992; Schuster and Quintus-Bosz, 1993�. Theoretically,ave-equation tomography should give higher resolution results

han raypath tomography. However, it has not been widely used inractice until now because of very high computational cost, low sig-al-to-noise ratio, and imprecise seismic wavelets.

In seismic experiments, the recorded transmitted signals containroadband frequency waves. For a single source-receiver pair, notnly the points on the raypath but also those outside the ray affectave propagation �Woodward, 1992; Snieder and Lomax, 1996;arquering et al., 1999; Spetzler and Snieder, 2001, 2004�. In ray-

ath tomography, the time delay or the amplitude variation is onlyrojected to the raypath. It does not consider the influence of the off-ay points on wave propagation; therefore, the inversion result is in-ccurate. To take advantage of raypath tomography and considerave propagation, fat-ray tomography has been proposed �Michel-

na and Harris, 1991; Vasco and Majer, 1993; Watanabe et al., 1999;u et al., 2006�. However, fat-ray tomography just fattens the ray on

ertain principles without considering the physics of seismic waveropagation.

The fat ray constructed on the basis of the wave equation is theensitivity kernel, also called the Fréchet kernel �Tarantola, 1987�. Iteflects the sensitivity of the recorded seismic signals �including am-litude and traveltime� to the perturbation of the media. The regionith the highest sensitivity corresponds to the first Fresnel volume,

hrough which the main energy of the wave travels to the receiverround the geometric ray �Çervený and Soares, 1992; Woodward,992�. This kind of tomography related to the Fresnel volume isalled Fresnel volume tomography �FVT� �Yomogida, 1992;nieder and Lomax, 1996; Marquering et al., 1998, 1999�. In FVT,

he time delay or the amplitude variation is mapped to the first

pril 2009; published online 14 September 2009.. E-mail: [email protected]; [email protected]; mazaitian@tongji.

[email protected]; huayi�[email protected].

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resnel volume according to the sensitivity kernel. For simplicity,e ue the term Fresnel volume to denote the first Fresnel volume un-

ess noted otherwise.Since Slaney et al. �1984� and Wu and Toksöz �1987� applied

orn and Rytov approximations to diffraction tomography and stud-ed their inversion abilities, diffraction tomography based on Bornr Rytov approximation has undergone extensive seismic investiga-ion. Woodward �1992� provides a uniform expression on raypathomography and diffraction tomography, and he proposes the con-epts of wavepath and band-limited raypath. Çervený and Soares1992� present a modeling method for the Fresnel volume. Yo-ogida �1992� calculates the band-limited raypath according to the

raditional Fresnel volume and applies it to diffraction tomography.nieder and Lomax �1996� derive the sensitivity kernels of a veloci-

y perturbation on phase and amplitude according to Born and Rytovpproximations.

Following Snieder and Lomax �1996�, Spetzler and Snieder2001, 2004� and Jocker et al. �2006� propose band-limited ampli-ude and traveltime sensitivity kernels for homogeneous media andummarize some of their properties. Spetzler et al. �2007, 2008� pro-ose the sensitivity kernels for smoothly heterogeneous media withJacobi determinant and apply them to time-lapse crosswell tomog-

aphy. Tian et al. �2007� also propose a method to calculate the sensi-ivity kernels by using dynamic ray tracing. Marquering et al. �1998,999� derive sensitivity kernels for finite-frequency seismic travel-imes measured by crosscorrelating broadband waveforms. Follow-ng the method of Marquering et al. �1998, 1999�, many studiesDahlen et al., 2000; Zhao et al., 2000; Hung et al., 2001; Dahlen,005; Zhao and Jordan, 2006; Zhang et al., 2007� investigate theroperties of band-limited traveltime sensitivity kernels in homoge-eous media and analyze the influence of abnormal factors on waveropagation in seismology and geodynamics.

In this paper, on the basis of previous work �Woodward, 1992;nieder and Lomax, 1996; Spetzler and Snieder, 2001, 2004; Jockert al., 2006; Spetzler et al., 2007, 2008�, we present the amplitudend traveltime sensitivity kernels expressed directly with Green’sunctions. They are suitable for homogeneous and heterogeneousedia. In the special case of homogeneous media, we extensively

nalyze the properties of the monochromatic and band-limited sen-itivity kernels by using the constructive interference of waves. Theumerical experiments indicate that the properties of the homoge-eous case can also be used approximately for smoothly heteroge-eous media. In the last part, the near-surface velocity structure of aheoretical model is inverted using FVT with sensitivity kernels andhe corresponding properties proposed in this paper.

METHODS

For a recorded seismic signal, amplitude perturbation �A is an in-egral of the amplitude sensitivity kernels KA�r� multiplied by thelowness perturbation field �s�r� over all points r in volume v be-ween the source and the receiver. It can be expressed as

�A��vKA�r��s�r�dr . �1a�

imilarly, traveltime perturbation �� can be expressed as


�� vKT�r��s�r�dr, �1b�

here KT�r� is the traveltime sensitivity kernel �Woodward, 1992;nieder and Lomax, 1996; Dahlen et al., 2000; Spetzler and Snieder,001, 2004�.

These functions can be generalized to a form ��vK�r��s�r�dr for all kinds of seismic tomography. How to cal-

ulate K�r� is the key problem in tomography. In fact, K�r� is the sen-itivity of observations to media perturbation. In equations 1a andb, K�r� is the sensitivity of amplitude or traveltime variations to thelowness perturbation per unit area in two dimensions or per unitolume in three dimensions. In FVT, it is restricted in the Fresnelolume.

In wave-equation tomography, K�r� is called the wavepathWoodward, 1992�. According to seismic information used in to-ography, K�r� corresponds to different sensitivity kernels, such as

he amplitude sensitivity kernel KA�r�, traveltime sensitivity kernelT�r�, and waveform sensitivity kernel KU�r�. The symbol �� cor-

esponds to amplitude variation �A, traveltime delay �� , or wave-eld residual �u, respectively. These residuals can be calculated us-

ng different theories, such as ray theory, FVT, or seismic wave theo-y, according to how the sensitivity kernel is constructed. In thisork, we only study KA�r� and KT�r� based on FVT.Following Jocker et al. �2006�, we substitute equation 3 into equa-

ions 2; comparing with equations 1a and 1b, we get the monochro-atic sensitivity kernels �equations 4� expressed with Green’s func-

ions directly:

�A�A0 Re�u1

u0�, �2a�

�� Im�u1

u0�

�; �2b�

u1�g,s��v

2�2�s�r�V0�r�

·G0�g,r�u0�r,s�dr; �3�

KA�r,��2�2A0

V0�r�·Re�G0�g,r�u0�r,s�

u0�g,s� �, �4a�

KT�r,��2�

V0�r�· Im�G0�g,r�u0�r,s�

u0�g,s� � . �4b�

n equations 2–4, u0 denotes the unperturbed wavefield, u1 is therst-order Born scattering wavefield, A0 is the unperturbed ampli-

ude, � is the angular frequency, G0�g,r� is the Green’s function ofoint g from a point source in r for the unperturbed media V0�r�, and0�r,s� is the unperturbed field in r from a point source in s.Equations 4 are the general forms of the sensitivity kernels for ho-ogeneous and heterogeneous media. They involve calculating thereen’s function G0 and synthetic wavefields u0, which can be calcu-

ated with wave-equation modeling or dynamic ray tracing. We cal-ulate them directly through a finite-difference operator of the fullave equation in the frequency-space domain �Jo et al., 1996�.Considering that the recorded transmitted signals contain broad-

and waves, we get the corresponding band-limited sensitivity ker-


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els by adding the normalization integral over frequency �Spetzlernd Snieder, 2001, 2004�:

KA�r��1

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ere, P�� is the weight coefficient calculated from the amplitudepectrum or a given Gauss function. It satisfies the relationship

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�2P��d� �1. We express a Gaussian distribution in equations 6,elected for the weight function:

P�f��w�f�

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f2

w�f�df

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n equations 6, f is the circular frequency and � denotes the band-idth of the frequencies with the highest energy near the central cir-

ular frequency f0.By crosscorrelating broadband waveforms �Marquering et al.,

998, 1999�, the band-limited sensitivity kernels also can be derivedZhang et al., 2007�. The resulting expressions are different fromurs, but choosing a special weight function can equalize them.

PROPERTIES OF SENSITIVITYKERNELS FOR FVT

To perform FVT, the spatial distribution and characteristics of theensitivity kernels should be studied in detail. The kernels for hetero-eneous media are complicated, and it is not easy to summarize theirroperties. However, they should share some properties with theensitivity kernels for homogeneous media. So, we first analyze thepecial case of a point source in homogeneous media.

In homogeneous media, the unperturbed wavefield u0 in equationsequals the Green’s function. According to the analytical formulasf the Green’s functions in 2D �equation 7a� and 3D �equation 7b�

able 1. Amplitude and traveltime sensitivity kernels of finiteomography in homogeneous media.

imension KA�r,��

2DA0��,g s� �sg�3

2�V�rs�rgcos��t�

�

4 � �sg

2�V�

3DA0��,g s�

�sg�2

2�V�rs�rgcos��t�

�s

2�V


omogeneous media, the monochromatic sensitivity kernels can bexpressed by the equations listed in Table 1:

G02D�g,s��

i

4H0

�1��k0s�g�, �7a�

G03D�g,s��

eik0s�g

4� s�g. �7b�

n equations 7, s and g denote the source and receiver positions, re-pectively; H0

�1� is the zeroth-order Hankel function of the first kind,nd k0 is a spatial wavenumber in unperturbed media. In the equa-ions in Table 1, �rs, �rg, and �sg are distances from spatial point r toource s, from r to receiver g, and from s to g, respectively. The valuet is the traveltime delay between the detour path s→r→g and theirect wavepath s→g, that is, �t� ��rs��rg��sg� /V.The sensitivity kernels in Table 1 are almost the same as those de-

ived by Spetzler and Snieder �2004�, except that the approximations/�rs �1 /x and 1 /�rg �1 /�sg�x are made during their deri-ation �Figure 1�.Although the geometric spreading factors after thepproximations are accurate enough, we do not follow them in thistudy because �rs, �rg, and �sg are easy to obtain.

Fresnel volume can be interpreted physically as a region wherecattered waves interfere constructively with the direct wave. It issually around the corresponding central ray. Slowness perturba-ions inside the Fresnel volume have the greatest effect on seismicave propagation from the source to the receiver. The distribution

ange of the Fresnel volume can be defined by the maximum detourraveltime delay �tmax �Spetzler and Snieder, 2001, 2004�. Accord-ng to the sensitivity kernels listed in Table 1, the distribution rangesf the Fresnel volumes corresponding to different conditions can bealculated easily.

The results are listed in Table 2, where T is the period of the mono-hromatic wave. From Table 2, we can see that �tmax is not alwaysqual to T /2, as some tomographic inverse schemes use �Michelenand Harris, 1991; Vasco and Majer, 1993; Watanabe et al., 1999; Xut al., 2006�. It varies with dimensions and seismic information �buts independent of the source type, according to Liu and Dong2008��. Only for 3D traveltime tomography �tmax�4T /8 is theresnel volume the broadest. Liu and Dong �2008� propose the samexpressions and spatial distributions of the sensitivity kernels, buthey use the far-field approximation and stationary-phase approxi-

ation following Snieder and Lomax �1996� during the derivation.Fresnel volume usually is defined geometrically as a special range

nd is suitable only for monochromatic signals. Now, by combiningwith the sensitivity kernels �equations 5� derivedfrom wave theory, we can define the Fresnel vol-ume for the band-limited signal.

Considering the frequency band in seismic ex-ploration, we set the frequency f to vary from0 to 100 Hz with a frequency interval of 2 Hzand a dominant frequency of 50 Hz. As an exam-ple, we set � as 20 and f0 as 50 Hz in this experi-ment. The weighting curve without normaliza-tion is shown in Figure 2. The band-limited sensi-tivity kernels are illustrated in Figure 3. The colorscale indicates the relative sensitivity of ampli-tude or traveltime variation to slowness perturba-tion in the Fresnel volume. To highlight the prop-erties of the sensitivity kernels, different gains are

ency

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U38 Liu et al.


pplied to the plots. The white dashed line in every plot denotes theistribution range of the corresponding Fresnel volume of the domi-ant frequency. To exhibit the characteristics of the sensitivity ker-els clearly, the vertical cross sections at position x�500 m �Figure� are shown in Figure 4.

To analyze the properties of the sensitivity kernels in smoothlyeterogeneous media, the band-limited sensitivity kernels in mediaith a constant velocity gradient are calculated in the same way. Theernels are illustrated in Figure 5, and the corresponding cross sec-ions are shown in Figure 6.

From the equations in Table 1 and from Figures 3–6, we come toour major conclusions. First, the side lobes of the band-limited sen-itivity kernels cancel out because of the destructive interference ofhe monochromatic kernels when integrating over broadbandrequencies. Meanwhile, the main lobe in the Fresnel volume istrengthened by the constructive interference. This means the band-imited sensitivity kernels and Fresnel volumes can be used to wellescribe the seismic wave propagation between two points.

Second, we see in Figures 4 and 6 that the green lines coincideith the red lines very well in the Fresnel volumes. Thus, it is reason-

ble to restrict the band-limited sensitivity kernel within the range ofhe Fresnel volume of the dominant frequency. That is, when theackground of the media is supposed to be homogeneous or smooth-y heterogeneous, it is practical to replace the band -limited sensitiv-ty kernel with several frequencies or even the single dominant oneor less computation when performing band-limited FVT. Certainly,hen the background is complex, it is necessary to perform band-

imited FVT with some discrete frequencies uniformly sampled inhe frequency band.

Third, we note that under the stationary-phase approximation,petzler and Snieder �2001� prove the equivalence between FVTnd ray theory in homogeneous media. The higher the frequency, thearrower the Fresnel volume. So when the frequency is infinite, theresnel volume reduces to a ray with infinitely small width. In thisase, the Fresnel volume theory converges to the ray theory; hence,

e, and �d� 3D traveltime sensitivity kernels excited by a point sourceand the receiver is located at �800,250� m. In �b� and �d�, the central200,250� m, and the receiver is located at �800,250� m.

able 2. Distribution ranges of Fresnel volumes in differentonditions.

imensionAmplitude

Fresnel volumeTraveltime

Fresnel volume

D �tmax�T /8 �tmax�3T /8

D �tmax�2T /8 �tmax�4T /8

lrs

lsg

lrg

s g

x

z

rx

igure 1. The 2D definition of the geometry for a point source in ho-

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Frequency (Hz)

igure 2. Weight curve without normalization, calculated from

a) c)

d)b)

igure 3. �a� Two-dimensional amplitude, �b� 3D amplitude, �c� 2D traveltimn homogeneous media. In �a� and �c�, the source is located at �200,250� mlices of the 3D sensitivity kernels in the x-z-plane; the source is located at �


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VT is equal to raypath tomography.Additionally, the integral of theraveltime sensitivity kernel along a line perpendicular to the centralay equals one �refer to Appendix A�. So for a single shot-receiverair, the FVT updates the points in the Fresnel volume with the sensi-ivity kernel as a weight function.

Fourth, we conclude that, commonly, the distribution ranges ofhe Fresnel volumes for amplitude tomography are smaller thanhose for traveltime tomography, and the distribution ranges of theD Fresnel volumes are smaller than those for 3D conditions. Fur-hermore, considering a cross section perpendicular to the centralay, the amplitude sensitivity kernel has its maximum value on theeometric ray and decreases off the ray. However, the traveltimeensitivity kernel has its minimum value on the geometric ray and in-reases away from the raypath. After reaching a peak at a certainoint, it gradually decreases to zero at the boundaries of the Fresnelolume.

igure 4. Vertical cross sections of the �a� 2D amplitude, �b� 3D ampleneous media shown in Figure 3, which are the functions of sensitiith thin blue lines are monochromatic components starting from 0

he monochromatic cross sections. The thick green lines are the cross

a)

igure 5. �a� Two-dimensional amplitude and �b� 2D traveltime sens


It is important to note that, for the 3D traveltime sensitivity kernel,he values on the geometric ray are zero. Marquering et al. �1999�all this form of sensitivity kernel a banana-doughnut. Obviously,here is a contradiction between ray theory and the finite-frequencyave theory. According to finite-frequency wave theory, the weight

oefficients on the central ray are zero. However, tomography basedn ray theory updates the points on the central ray with an invariableeight coefficient of one, and zero elsewhere. Jocker et al. �2006�

est the validity of finite-frequency wave theory through physical ex-eriments and explain the banana-doughnut as the interferential re-ult of monochromatic waves. So ray theory is just a special case ofnite-frequency theory.As mentioned, finite-frequency theory equals ray theory when the

requency tends to infinity. However, the frequencies of seismic sig-als are finite. Does this mean traditional traveltime tomographyoes not work? Actually, the points with large sensitivity generally

c� 2D traveltime, and �d� 3D traveltime sensitivity kernels in homo-lues versus distance perpendicular to the geometric ray. The curvesHz, with a step of 2 Hz. The thick red lines are the stacked results ofs of the 50-Hz dominant frequency.

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kernels excited by a point source in slowness media with a constant

itude, �vity vato 100section

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U40 Liu et al.

re close to the geometric ray. That is, the velocity-updated points forraditional tomography are very close to the points whose velocityhould be updated. That is why we can still obtain the large-scaletructure of the media by traditional traveltime tomography. Obvi-usly, it decreases the inversion resolution. This is also why we can-ot resolve fine structures by traditional traveltime tomography.petzler and Snieder �2004� try to reconcile the two theories through

he Fresnel volume, i.e., the ray theory is valid when the slownesserturbation has a geometric size greater than the width of theresnel volume. Otherwise, the finite-frequency wave theory shoulde used to describe seismic wave propagation.

Sensitivity kernels are related to the positions of the source and re-eiver, the frequency band of the wavelet, source type, wave style,nd velocity distribution. Sensitivity kernels may be very complexn some models. Multiple paths may exist in some conditions. Figurea and b shows band-limited sensitivity kernels in a smoothed Mar-ousi model. The sensitivity kernels with high values are distribut-

d complicatedly between the source and receiver and are surround-d by a small part with low values. If we restrict the sensitivity ker-els within the corresponding Fresnel volumes according to Table 2,he blue parts and the surrounding parts with low values in Figure 7and b disappear and we see the multipath phenomenon �Figure 7cnd d�. Although the Fresnel volumes are complicated in the Mar-ousi model, we still find they roughly obey the principles de-

cribed above.

NUMERICAL TESTS OF FVT

The sensitivity kernel is the key problem in seismic modeling andnversion using Fresnel volume theory. Theoretically, the inverse re-ult of FVT should be more accurate than that of raypath tomogra-hy because the Fresnel volume describes seismic wave propagationore accurately than ray theory. To test the validity of FVT with sen-

itivity kernels, we apply 2D traveltime FVT and 2D traveltime ray-ath tomography to invert the near-surface velocity structure of aheoretical model. Because of practical problems such as attenua-ion, noise, and wavelets, amplitude FVT is not practical yet. So onlyraveltime kernel is actually used in this experiment.

The complex 2D theoretical model is shown in Figure 8a. It con-ists of 4001�75 grid points; the spatial interval is 10�10 m. Theelocities vary from 800 to 4300 m /s. The FVT of the numericalxperiment consists of six steps.

0.06

0.04

0.02

–0.02

–0.04

–0.06

0

Sen

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plitu

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pert

urba

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1)

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igure 6. Vertical cross sections of the �a� 2D amplitude and �b� 2D trensitivity values versus depth. The curves with thin blue lines are mohick red lines are the stacked results of the monochromatic cross sectuency.


tep 1: Pick first-arrival traveltimes

We simulate 640 shots �shot interval of 40 m� with staggered-rid, high-order, finite-difference elastic wave modeling. The firsthot is located at the surface at 5000 m; 201 receivers are locatedymmetrically at both sides of the excited point, with a trace intervalf 20 m. Thus, the maximum offset is 2000 m and the minimum off-et is zero. Figure 9 shows the vertical components of two syntheticecords excited at the surface at 9 and 26 km. The first-break travel-ime is picked on the synthetic records.

tep 2: Construct the initial model

To invert the near-surface velocity, an initial model is needed. Wepecify a constant gradient model as the initial model. The velocitiesre calculated according to the equation v�z��500.0�3.0*z.

tep 3: Calculate the traveltime residual

Traveltime residual of the ray is usually similar to that of theresnel volume in the near offset, so we use a traveltime residual cal-ulated by kinematic ray tracing to reduce calculation costs.

tep 4: Calculate the kernels

Equation 4b is used to calculate the monochromatic traveltimeensitivity kernel. Green’s function and synthetic wavefields are cal-ulated with a finite-difference operator of the full wave equation inhe frequency-space domain �Jo et al., 1996�. To reduce the calcula-ion cost of FVT, only the dominant frequency of 50 Hz is used. Asn example, the monochromatic traveltime sensitivity kernel on theircled part �Figure 8b� of the smoothed initial model �Figure 8a� ishown in Figure 10b, which is restricted within the range of theresnel volume of the dominant frequency based on Table 2. Foromparison, the amplitude sensitivity kernel is also shown �Figure0a�, although it is not used.

tep 5: Construct the FVT equation

Substituting equation 4b into equation 1b, we obtain the originalonochromatic FVT equation 8a. In case of instability, we performVT using equation 8b:

�r

2�

V0�r�· Im�G0�g,r�u0�r,s�

u0�g,s� � ·�s�r� ·ds�� , �8a�

0.005

0

–0.005

Sen

sitiv

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plitu

deto

slow

ness

pert

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(m–

1)

)

e sensitivity kernels shown in Figures 5, which are the functions ofomatic components starting from 0 to 100 Hz, with 2-Hz steps. Thehe thick green lines are the cross sections of the 50-Hz dominant fre-

b

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� �u0�g,s��2�� . �8b�

ere, ds is the size of the area element. Considering that the Fresnelolume is much wider than the corresponding ray, the SIRT methodLiu and Dong, 2007� is used directly to solve tomographic equationb without regularization.

tep 6: Update model

After solving the tomographic equation, the model is updated.teps 3–6 are repeated until the stop criteria are satisfied.The final inverted results of raypath tomography and FVT are

hown in Figure 11. Figure 12 shows the velocity sections of the the-retical model and the inverted results at different depths fromto 160 m beneath the surface, with a depth interval of 40 m. We

an see from Figures 11 and 12 that the result of FVT is much moreccurate than that of raypath tomography. Raypath tomography canbtain only the smooth background variations of the media. It is in-ensitive to the structural variations with high spatial wavenumber.

0

1

2Dep

th(k

m)

0

1

2Dep

th(k

m)

5500

4511

3521

2532

Velocity (m/s)

0 2 4

0 2 4

Distance (km)a)

b)

igure 7. The 2D band-limited �a� amplitude and �b� traveltime sensensitivity kernels excited by a point source in a smoothed Marmous5000,40� m.

0

0.25

0.5

Dep

th(km)

Velocity (m/s)

5500

4210

2920

1630

340

a)

10 20

Distance (km)

igure 8. �a� The 2D complex near-surface theoretical model with top


owever, FVT reveals the smooth background and the high-wave-umber perturbations more accurately.

DISCUSSION

From the red curves in Figures 4 and 6, we see that the first sideobe of the stacked cross section remains large in spite of cancelinghe monochromatic sensitivity kernels as a result of destructive in-erference. This means the perturbation inside the first side lobe alsoffects the wave propagation significantly. The sign of the sensitivityernel has different physical meanings. For example, according toquation 1b, a positive value corresponds to a delay in traveltime be-ause of increasing slowness; a negative value corresponds to an ad-ance in traveltime becaue of increasing slowness. So both the mainobe and the first side lobe should be used to define the band-limitedresnel volume.Figures 4 and 6 also show that the extended band-limited Fresnel

olume has a good coincidence with the second Fresnel volume ofhe dominant frequency. The distribution ranges of the secondresnel volumes of the dominant frequency are listed in Table 3.owever, for the experimental model �Figure 8�, FVT with this sec-nd Fresnel volume does not obtain a better tomographic result than

0

1

2Dep

th(k

m)

0

1

2Dep

th(k

m)

0

1

2Dep

th(k

m)

0

1

2Dep

th(k

m)

6 0 2 4 6

6 0 2 4 6

Distance (km)c)

d)

kernels and restricted band-limited �c� amplitude and �d� traveltimel. The source is located at �1275,40� m, and the receiver is located at

28

0

0.25

0.5

Dep

th(km)

Distance (km)b)

3029

y and �b� the zoomed view of the circled part of �a� after smoothing.

itivityi mode

30

ograph


Fpr

rhrrtst

Ffihtptt

dl

F surface

Ft

F

U42 Liu et al.

VT using the first Fresnel volume. This failure may be a result of re-lacing traveltime delays of the Fresnel volumes with those of theays.

In numerical experiments to compare the traveltime delays of theays and Fresnel volumes, we find they are similar in near offsets butave larger differences in far offsets. The traveltime delays of theays and the first Fresnel volumes are similar, but the delays of theays and the second Fresnel volumes have large differences. Maybehe traveltime delays of the rays cause the failure of FVT using theecond Fresnel volume. On the other hand, this failure might be at-ributed to replacing the band-limited Fresnel volume by the second

Position (km)

Tim

e(m

s)

igure 9. Vertical components of two synthetic records excited at the

0

0.25

0.5

Dep

th(k

m)

28 29a) Distance (km)

Velocity (m/s)

5500

4210

2920

1630

340

igure 10. �a�Amplitude and �b� traveltime monochromatic sensitivhe receiver located at the surface at 30 km.

0

0.25

0.5

Dep

th(k

m)

10 20a) Distance (km)

Velocity (m/s)

5500

4210

2920

1630

340

igure 11. Tomographic results of �a� raypath tomography and �b� FV


resnel volume of the dominant frequency.After all, the misfit of therst side lobe is larger than that of the main lobe in homogeneous andeterogeneous media �Figures 4 and 6�. The integral of the travel-ime kernel over a line perpendicular to the ray is equal to one �Ap-endix A�, which means the positive region has a stronger effect thanhe negative region. So, this misfit is especially large for the travel-ime sensitivity kernels.

For the amplitude kernels, the misfits are less. According to theerivation inAppendix A, the integral of the amplitude kernel over aine perpendicular to the ray is equal to zero, which means the posi-

Position (km)

at 9 and 26 km.

0

0.25

0.5

Dep

th(k

m)

28 29 30b) Distance (km)

els of 50 Hz between the source located at the surface at 28 km and

0

0.25

0.5

Dep

th(k

m)

b) Distance (km)10 20 30

d on the picked traveltime of the first arrivals.

30

ity kern

30

T base



a)

b)

c)

d)

e)

Figure 12. Velocity sections of the theoretical mod-el �red�, the inverted results of raypath tomography�green�, and FVT �blue� in different depths from0 to 160 m beneath the surface.

Downloaded 24 Mar 2010 to 222.66.175.192. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

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U44 Liu et al.

ive and negative regions have the same important effect on the ob-ervation. So, as for the amplitude FVT, the second Fresnel volumef the dominant frequency should be considered.

CONCLUSIONS

Sensitivity kernels are important for seismic modeling and inver-ion with Fresnel volume theory. On the basis of Born and Rytov ap-roximations, we directly express the amplitude and traveltime sen-itivity kernels for 2D and 3D FVT with Green’s functions. Thesequations are suitable for homogeneous and heterogeneous media.he final tomographic experiment verifies the validity of the sensi-

ivity kernels we propose. Our results show that FVT using theseensitivity kernels can achieve much more accurate inversion thanonventional raypath tomography.

Through the analytical expressions of the sensitivity kernels listedn Table 1, we summarize their properties. Although they are for ho-

ogenous media, the numerical experiments show they also can besed approximately in heterogeneous media �Figures 2–7�.

According to constructive interference of monochromatic waves,e find that the spatial distributions of the sensitivity kernels inonochromatic FVT vary widely for different conditions. They are/8, 2 /8, 3 /8, and 4 /8 periods, respectively, for 2D amplitude, 3Dmplitude, 2D traveltime, and 3D traveltime conditions.

The amplitude sensitivity kernel has its maximum value on theeometric ray and decreases off the ray. The traveltime sensitivityernel has its minimum value on the geometric ray and increases offhe ray.After reaching a peak at a certain point, it gradually decreaseso zero at the Fresnel volume boundary.

The band-limited sensitivity kernel can be calculated by stackinghe monochromatic kernels with a weight function similar to the am-litude spectrum of the wavelet. Under two preconditions, only aew selected frequencies or even a single dominant frequency maye necessary to perform FVT. First, the sensitivity kernels must beestricted within the first Fresnel volume of the dominant frequency.econd, the background media must be homogeneous or smoothlyeterogeneous. This replacement is a preliminary substitute. Whenhe media are complex or calculation cost is not a problem, a restrict-d band-limited kernel is preferable.

Another trade-off performed to make the FVT application practi-al is that, in near offset, the traveltime residuals of the rays replacehose of the Fresnel volumes. This approximation is based on numer-cal experiments that are not presented in this paper. For higher accu-acy, the traveltime residuals should be estimated through crosscor-elation or dynamic ray tracing.

ACKNOWLEDGMENTS

This study is supported by the National Natural Science Founda-ion of China �grant 40804023�, the New Century Excellent Talents

able 3. Distribution ranges of band-limited Fresnel volumesn different conditions.

imension

AmplitudeFresnelvolume

TraveltimeFresnelvolume

D �tmax�T /4 �tmax�3T /4

D �tmax�2T /4 �tmax�4T /4


rogram of the University of the Education Ministry of China �grantCET-05-0384�, China’s Ongoing National Keystone Basic Re-

earch program �grant 2006CB202402�, and the State Key Libraryf Marine Geology of China project �grant MG20080205�. Most ofhe figures were created using the Seismic Unix software packagerom Colorado School of Mines. We are grateful to Hua-Zhong

ang, Jiu-Bing Cheng, Kai Yang, Mei Xue, and Chun-Feng Li fromongji University for their constructive comments on the manu-cript. We are also grateful to Bill Harlan for arranging reviews ofhis paper. Jesper Spetzler’s, Moritz Fliedner’s, and Reinaldo

ichelena’s detailed reviews and comments greatly improved theanuscript. Meanwhile, we thank the other two anonymous review-

rs for comments.

APPENDIX A

RELATIONSHIP BETWEEN FRESNELVOLUME AND RAY

Spetzler and Snieder �2001� prove that in homogeneous media,he Fresnel volume theory converges to ray theory, and FVT is equalo raypath tomography when the frequency tends to infinity. In theame way, we give a further relationship between the two theories.

The integral of the 2D monochromatic traveltime sensitivity ker-el �Table 1� over a line perpendicular to the ray can be written asFigure 1�:

��

��

KT2D�x,z,��dz

��

��

�sg�

2�V�rs�rgsin��t�

�

4�dz

��

��

L�

2�Vx2�z2�L�x�2�z2

�sin ��x2�z2��L�x�2�z2�L

V�

�

4�dz .

�A-1�

ere, L is the distance between source and receiver, which equals �sg

s Figure 1 shows. We assume that z /x��1, z / �L�x��1; we setx2�z2, �L�x�2�z2 to the first-order Taylor approximation and/x2�z2, 1 /�L�x�2�z2 to the zeroth-order Taylor approxima-

ion to get equation A-2:

��

��

KT2D�x,z,��dz��

��

��

L�

2�Vx�L�x�

�sin � L�z2

2x�L�x�V�

�

4�dz�1.

�A-2�

The above characteristic is independent of frequency, which


mnItv

ne

U

mnIiv

A

B

Ç

D

D

D

D

H

H

J

J

L

—

M

M

M

N

P

P

S

S

S

S

S

S

—

S

S

TT

V

W

W

W

X


eans the integral of the 2D band-limited traveltime sensitivity ker-el �equation 5b� over a line perpendicular to the central ray is one.n the same way, we can get the integral of the 3D traveltime sensi-ivity kernel over a plane perpendicular to the ray. It is not difficult toerify that the integral also equals one.

The integral of the 2D monochromatic amplitude sensitivity ker-el �Table 1� over a line perpendicular to the ray can be written asquation A-3 �Figure 1�:

��

��

KA2D�x,z,��dz

��

��

A0��,gs� �sg�3

2�V�rs�rg

�cos ��t��

4�dz

��

��

L�3

2�Vx2�z2�L�x�2�z2

�cos ��x2�z2��L�x�2�z2�L

V�

�

4�dz .

�A-3�

nder the same approximations as above, we get equation A-4:

��

��

KA2D�x,z,��dz

��

��

L�3

2�Vx�L�x�cos� L�z2

2x�L�x�V�

�

4�dz�0

�A-4�

The above characteristic is also independent of frequency, whicheans the integral of the 2D band-limited amplitude sensitivity ker-

el �equation 5a� over a line perpendicular to the central ray is zero.n the same way, we can get the integral of the 3D amplitude sensitiv-ty kernel over a plane perpendicular to the ray. It is not difficult toerify that the integral also equals zero.

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