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GEOPHYSICS, VOL. 71, NO. 6 �NOVEMBER-DECEMBER 2006�; P. T167–T177, 13 FIGS.10.1190/1.2358412

alidation of first-order diffraction theory forhe traveltimes and amplitudes of propagating waves

eroen Jocker1, Jesper Spetzler2, David Smeulders2, and Jeannot Trampert3

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ABSTRACT

Ultrasonic measurements of acoustic wavefields scatteredby single spheres placed in a homogenous background medi-um �water� are presented. The dimensions of the spheres arecomparable to the wavelength and the wavelength and repre-sent both positive �rubber� and negative �teflon� velocityanomalies with respect to the background medium. The sen-sitivity of the recorded wavefield to scattering in terms oftraveltime delay and amplitude variation is investigated. Theresults validate a linear �first-order� diffraction theory forwavefields propagating in heterogeneous media with anoma-ly contrasts on the order of ±15%. The diffraction theory iscompared further with the exact results known from literaturefor scattering from an elastic sphere, formulated in terms ofLegendre polynomials. To investigate the 2D case, the first-order scattering theory is tested against 2D elastic finite-dif-ference calculations. As the presented theory involves a vol-ume integral, it is applicable to any geometric shape, and thescattering object does not need to be spherical or any otherspecific symmetrical shape. Furthermore, it can be imple-mented easily in seismic data inversion schemes, which is il-lustrated with examples from seismic crosswell tomographyand a reflection experiment.

INTRODUCTION

The characteristics of finite-frequency waves propagating in het-rogeneous media are fundamentally different from what is predict-d by standard ray theory. Ray theory is a high-frequency solutionor the wave equation, which means that energy travels along lines inpace joining the source and receiver. It is correct to use ray theory toodel high-frequency wave propagation in simple media consisting

Manuscript received by the Editor June 8, 2005; revised manuscript receiv1Formerly Delft University of Technology, Department of Geotechnolog

idgefield, Connecticut 06877. E-mail: jjocker@ridgefield.oilfield.slb.com.2Delft University of Technology, Department of Geotechnology, P.O

.m.j.smeulders@tudelft.nl.3Utrecht University, Faculty of Geosciences, P.O. Box 80021, 3508 TAUtr2006 Society of Exploration Geophysicists.All rights reserved.

T167

f large-scale velocity structures. On the other hand, the high-fre-uency approximation in ray theory breaks down for finite-frequen-y waves propagating in complex media with velocity structuresmaller than the wavelength and the Fresnel volume. The Fresnelolume is the region between the source and receiver in which thecattered wavefield contributes constructively at the receiver posi-ion; i.e., arrival times of scattered waves differ by less than half a pe-iod from the direct wave. In such heterogeneous media, diffractionffects become important so that the phase and amplitude variationf propagating waves behave so that cannot be explained by the geo-etric ray approximation.We present an overview of a first-order finite-frequency wave the-

ry for heterogeneous media for which ray theory is invalid. In brief,frequency-dependent term for the single-scattering process �i.e., arst-order approach� is added to the standard point-source solutionor a transmitted wavefield. New ultrasonic waveform data are re-orded in an experimental setting where a sphere of teflon or rubbers located in a reference water medium. The ultrasonic experimentllows us to clearly illustrate the fundamental differences between anite-frequency wave theory and ray theory. The considered theory

s limited to contrasts in the compressional speed of sound only, ando density contrasts are considered. With respect to the ultrasoniceasurements, this means that density contrasts as well as shearave effects are neglected. The influence of these omissions is stud-

ed through comparison with the exact solution for scattering of anncident field because of the presence of an elastic sphere �Hickling,962�.

In the frequency-dependent single-scattering approach, the maxi-um traveltime sensitivity to velocity anomalies is off the geometric

ay, whereas the amplitude has a significant sensitivity along the rayath. In ray theory, these sensitivities appear to be quite different, butn the high-frequency limit, the scattering kernels converge to the rayheoretical ones. Hung et al. �2001� performed a 3D numerical wavexperiment with a smooth velocity anomaly. They used a pseu-ospectral method to solve the 3D acoustic wave equation, and the

28, 2006; published online November 3, 2006.Netherlands; presently Schlumberger-Doll Research, 36 Old Quarry Road,

5028, 2600 GA Delft, The Netherlands. E-mail: j.spetzler@tudelft.nl;

e Netherlands. E-mail: jeannot@geo.uu.nl.

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T168 Jocker et al.

moothness of the velocity perturbation was a numerical require-ent. In our experiment, the effect of sharp discontinuities on real

ata is investigated. As real experiments are inherently 3D, the 2Dase, which has fundamentally different finite-frequency kernels, isnvestigated numerically.

The applied finite-frequency wave theory is well documented bye Wolf �1967�, Aki and Richards �1980�, Rytov et al., �1989�,oodward �1992�, Snieder and Lomax �1996�, and Spetzler and

nieder �2001; 2004�. The latter used the Rytov approximation to in-lude the effect of heterogeneous small-scale structures on the phasef propagating waves. Born and Wolf �1959�, Marquering et al.1999�, Zhao et al. �2000�, and Hung et al. �2001� chose to work withhe Born approximation to account for single-scattering effects onhe traveltime of propagating waves. Aki and Richards �1980�,

oodward �1992�, and Dahlen and Baig �2002� focused on the fi-ite-frequency wave theory for amplitude variations. De Hoop1985� showed an elegant derivation of the first Born approximationor plane waves where the geometric shape of the contrast term is ex-licitly included. Beydoun and Tarantola �1988� tested the validityf the Born and Rytov approximation for a 1D layered medium andound that the Rytov approach is most suitable for transmittedaves, whereas for reflected waves, the Born approximation is rec-mmended.

The finite-frequency wave theory we use for traveltimes and am-litudes takes first-order scattering effects into account, and it is thusinear with respect to velocity perturbations. However, depending onhe complexity in terms of shape and strength of velocity anomalies,onlinear ray-bending effects become increasingly important whenriplications occur in the wavefield. We have recorded 3D real ultra-onic waveform data with �teflon� and without �rubber� triplications.e supplement the 3D real ultrasonic data experiment with 2D nu-erical simulations. The numerical study yields the possibility to es-

imate the range of validity of the linear finite-frequency wave theo-y in the regime of weakly developed triplications.

In this paper, we present and validate a simple first-order scatter-ng theory for wave propagation in the presence of objects with di-

ensions comparable to the wavelength, but smaller in size than theresnel zone. Because of the linearity between the traveltimes andmplitude attributes and the velocity perturbation field, it is easy tomplement the presented wave theory in seismic imaging experi-

ents.Anomalies do not need to be a specific shape, as the theory in-olves a general volume integral over the velocity perturbation field.o illustrate this, two applications of the tested wave-scattering the-ry in exploration seismic experiments are discussed. The first ex-mple is crosswell tomography, and the second example is from re-ection seismic.In the following, we give first an overview of the theory for travel-

ime and amplitude variations of finite-frequency wavefields. Next,he 3D ultrasonic wave experiment and theory validation section areresented, followed by a section on the 2D numerical simulations.e conclude after discussing the two possible seismic applications

or the proposed wave theory.

THEORY

In this section, the theory is presented for finite-frequency wavesropagating in heterogeneous media with anomalies smaller thanhe Fresnel zone. The wavefield is modeled using the first-order Ry-ov approximation. The Rytov approximation only describes con-rasts in sound speed �not in density� and neglects any shear-wave ef-

ects. The importance of these approximations will be investigatedy comparing the exact solution for a point-source wavefield inci-ent upon an elastic sphere in a fluid. First, we derive expressions forraveltime shifts and amplitude variations for the 2D case, followedy a generalization to three dimensions.

In equation 1, the Rytov wavefield PR�rr,rs,�� at the angular fre-uency � = 2�f emitted from the source position rs and recorded athe receiver position rr is given by

PR�rr,rs,�� = P0�rr,rs,��exp�PB

P0�rr,rs,��� , �1�

hich accounts for the single-scattering process of a propagatingavefield in heterogeneous media �Born and Wolf, 1959; Aki andichards, 1980; Snieder and Lomax, 1996�. In equation 1 and in all

ubsequent derivations, P is defined as a pressure field. The refer-nce wavefield inherent to the reference velocity model v0�r� is de-oted as P0�rr,rs,��, while the first-order Born wavefield is writtens PB�rr,rs,��.

The phase and amplitude perturbations of the scattered field withespect to the reference field are derived by separation of the real andmaginary parts of the exponential function in equation 1. Hence, thehase delay resulting from the presence of a velocity anomaly is giv-n by

���rr,rs,�� = I�PB

P0�rr,rs,��� , �2�

hich yields for the traveltime residual �t

�t�rr,rs,�� =1

�I�PB

P0�rr,rs,��� , �3�

here we have used the linear expression �� = ��t.Similarly, the amplitude AR�rr,rs,�� of the Rytov wavefield with

espect to the amplitude A0�rr,rs,�� of the reference wavefield isiven by

ln�AR

A0�rr,rs,��� = R�PB

P0�rr,rs,��� . �4�

et the amplitude difference be defined as �A = AR − A0. For smallerturbations of the amplitude of the Rytov wavefield, the logarithmo leading order is ln�1 + x��x for x�1. The amplitude variationith respect to the reference wavefield is approximated then as

�A

A0�rr,rs,�� � R�PB

P0�rr,rs,��� �5�

or �A/A0�1.Aki and Richards �1980� and Snieder and Lomax �1996� derived

he first Born wavefield �from now on, simply called the Born wave-eld�, which is a solution to the acoustic-wave equation for a con-tant density and in the presence of a velocity perturbation. The Bornavefield is given by

PB�rr,rs,�� = �V

2�v�r��2

v03�r�

P0�r,rs,��G�rr,r,��dV,

�6�

here the velocity perturbation with respect to a homogeneous ref-rence velocity v �r� = v is denoted as �v�r�, and the 2D integra-

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Validation of first-order diffraction theory T169

ion is indicated by V. The 2D far-field Green’s function for theomogeneous reference medium is given by �Snieder and Lomax,996�

�rr,r,�� = − � v0

8���rr − r�exp i���rr − r�

v0+

�

4� . �7�

s shown previously �Spetzler and Snieder, 2001�, equation 7 can besed to express the Born wavefield in equation 6 explicitly in the sys-em coordinates. For a point source, this yields

B�rr,rs,�� =�

4�v02 exp i��L

v0+

�

4�

� �−�

�

�0

L

�v�r�exp i� �Lr2

2v0z�L − z�+

�

4�

�z�L − z�dzdr .

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he paraxial approximation that r�z,L − z is used to obtain theorn approximation above. A sketch of the system coordinates is

hown in Figure 1. Let the point source be located at the origin; theeceiver position is placed on the z-axis at z = L, and the position ofhe scatterer point is �z,r�. The Born wavefield in equation 8 is thenivided by the point source solution

P0�rr,rs,�� = − � v0

8���rr − rs�exp i���rr − rs�

v0

+�

4� , �9�

here �rr − rs� = L, in order to obtain the monochromatic traveltimeelay in equation 3 and the monochromatic amplitude variation inquation 5.

Recorded waves are never monochromatic, but have broadbandrequency properties. The monochromatic traveltime delay and rela-ive amplitude variation are therefore integrated over the frequencyand f0 − �f ; f0 + �f. If the broadband frequency characteristicsf recorded wavefields with a normalized amplitude spectrum —hat is, � f0−�f

f0+�fA� f�df = 1 — are taken into account, the broadbandraveltime shift for source position rs and receiver position rr is giv-n by

�t�rr,rs� = �−�

�

�0

L

�v�z,r�K�t2D�z,r�dzdr , �10�

ith the frequency-averaged sensitivity function �also known as theréchet kernel� for traveltime residuals defined as

K�t2D�z,r� = − � L

v05z�L − z�

� �f0−�f

f0+�f

A�f��f sin� f�Lr2

v0z�L − z�+

�

4�df ,

�11�

hich was originally derived by Spetzler and Snieder �2001�. Identi-ally, the broadband relative amplitude variation is given by

�A

A0�rr,rs� = �

−�

,�

�0

L

�v�z,r�K�A2D�z,r�dzdr , �12�

ith the Fréchet kernel for amplitude variations defined as

K�A2D�z,r� = − 2�� L

v05z�L − z�

� �f0−�f

f0+�f

A�f�f3/2 cos� f�Lr2

v0z�L − z�+

�

4�df .

�13�

The frequency-averaged traveltime shifts and amplitude varia-ions for the Rytov wavefield in a 3D medium can be derived similar-y. In three dimensions, the monochromatic Green’s function for thecoustic wave equation with constant density and velocity is giveny

G�rr,r,�� = −

exp i��rr − r�

v0

4��rr − r�. �14�

quation 14 is used to express the direct wavefield between theource position rs and the receiver position rr, and the scatteredavefield from the point source to the scatterer point r to the receiverosition �see Figure 1�.

For the broadband traveltime shift �t, this leads eventually to

�t�rr,rs� = �−�

�

�−�

�

�0

L

�v�r,�,z�K�t3D�r,�,z�rdzd�dr .

�15�

he Fréchet kernel for traveltime residuals in equation 15 is given by

K�t3D�r,�,z� = −

L

v03z�L − z�

� �f0−�f

f0+�f

A�f�f sin� f�Lr2

v0z�L − z��df , �16�

igure 1. Illustration of the geometric variables to describe the sin-le-scattering process.

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Hung et al., 2001; Spetzler et al., 2002�. In addition, the broadbandelative amplitude variation equals

�A

A0�rr,rs� = �

−�

�

�−�

�

�0

L

�v�r,�,z�K�A3D�r,�,z�rdzd�dr ,

�17�

ith the sensitivity function for relative amplitude variations writtens

K�A3D�r,�,z� = −

2�L

v03z�L − z�

� �f0−�f

f0+�f

A�f�f2 cos� f�Lr2

v0z�L − z��df .

�18�

n expression similar to equation 18 is derived by Dahlen and Baig2002� from the Born wavefield.

roperties of the 2D and 3D wave-diffraction theoryThe presented single-scattering wave theory for traveltime delays

s a natural extension of the well-known ray theory that provides aigh-frequency solution for the elastic wave equation, e.g., Červený2001�. It follows for both 2D and 3D scattering theory that

limf→�

�tscat�rr,rs� � − �0

L

�v�z�v0

2 dz = �tray�rr,rs� , �19�

here the coordinate system in Figure 1 is used. A derivation of aimilar expression can be found in Spetzler and Snieder �2001, equa-

igure 2. Cross sections of normalized 2D and 3D Fréchet kernelsor the traveltime delay and amplitude variations. �a� 2D sensitivityernel for traveltime shifts; �b� 3D sensitivity kernel for traveltimeesiduals; �c� 2D Fréchet kernel for amplitude variations; �d� 3D

ion 29�. Ray theory predicts that the amplitude of a propagatingavefield is modified by transmission/reflection effects, geometric

preading of the wavefield, and attenuation resulting from energyoss. Following an approach similar to the one taken to arrive atquation 19, one finds that the single-scattering wave theory �equa-ions 12 and 17 for the amplitude attribute� yields the ray theoreticalesult for infinite-frequency waves. Hence, again using the coordi-ate system in Figure 1, it follows that the amplitude ratio in theigh-frequency limit is given by

limf→�

��A

A0�rr,rs�

scat�

1

2Lv0�0

L

z�L − z��2�v�z�

�r2 dz , �20�

here �2�v�z�/�r2 is the second derivative of the velocity perturba-ion field perpendicular to the reference raypath — in this case, atraight line joining the source and receiver points because of theonstant reference velocity. Expression 20 resembles equation 30 ofeele et al. �1993� who worked with ray-perturbation theory, as well

s equation 170 of Dahlen and Baig �2002� who used a higher-orderxpansion of the velocity-perturbation field to deduce the high-fre-uency behavior of the amplitude part of finite-frequency waves.otice that equation 20 is derived using ray-perturbation theory sim-

lar to Neele et al. �1993�. This implies that formally, the infinite fre-uency amplitude formulation in equation 20, is valid only inmooth velocity-perturbation media.

Ray theory �Červený, 2001� is valid in media with anomalies of aize a larger than the wavelength and the width LF of the first Fresnelolume �from now on, simply called the Fresnel volume�; hence,

�/a 1 and LF/a 1. �21�

hysically speaking, the Fresnel volume is the region between theource and receiver for which the scattered wavefield contributesonstructively at the receiver position. According to Spetzler andnieder �2001�, at a distance z from a point source, the width LF�z� of

he 2D Fresnel volume for a source-receiver distance L �see Figure� is given by

LF2D�z� = �3�z�L − z�

L, �22�

hereas for 3D media,

LF3D�z� = �4�z�L − z�

L. �23�

otice that LF3D�z = L/2�LF

2D�z = L/2�. As a result of the paraxialpproximation, expressions 22 and 23 are approximately accurateway from the source and receiver. However, the single-scatteringave theory does not introduce any limitation on the size of the

nomaly relative to the Fresnel volume.The cross sections perpendicular to the direction of propagation

f the normalized Fréchet kernels for traveltime delays in equations1 and 13 and for relative amplitude variations in equations 16 and8 are illustrated in Figure 2. The source-receiver distance= 20 cm; the constant reference velocity v0 = 1480 m/s. The fre-

uency integration is performed analytically over the frequencyand f − �f = 300 kHz and f + �f = 600 kHz, for which it is as-umed that A� f� = 1/�2�f� �Spetzler et al., 2002�. The crossections of the sensitivity kernels are computed at the midpoint

réchet kernel for amplitude variations.

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Validation of first-order diffraction theory T171

etween the source and receiver �i.e., z = L/2� where the sensitivityernels have the maximum width. The figures show normalized sen-itivity to velocity perturbations with respect to traveltime and am-litude variations as a function of the perpendicular distance fromhe source-receiver line which, in the ray theoretical sense, is equalo the geometric ray path between the source and receiver. Wave dif-raction theory predicts that the frequency-averaged traveltime shifts maximum for scatterers located off the geometric ray. The sensi-ivity to slowness perturbations decays away from the geometric rayath because of the integration over a broad frequency band. Therea of most significant sensitivity is equivalent to the Fresnel vol-me �Spetzler and Snieder, 2001�.

For wave propagation in a 3D medium, the traveltime delay is in-ensitive to velocity perturbations on the source-receiver line �Fig-re 2b�. This differs from ray theory that predicts a nonzero sensitiv-ty only for velocity anomalies intersecting with the geometric rayath. For 2D wave propagation, the ray theoretical prediction is alsoifferent from the finite-frequency wave result because the sensitivi-y is maximally away from the geometric ray �Figure 2a�. Both theD and 3D Fréchet kernels for amplitude variations have a maxi-um sensitivity to a velocity anomaly on the geometric ray, whereas

he sidelobes outside the Fresnel volume decay away from the ray,gain, because of broadband frequency integration �Figure 2c and d�.

THE 3D ULTRASONIC WAVE EXPERIMENT

The experimental setup used for the velocity perturbation mea-urements is shown schematically in Figure 3. Both source and re-eiver are suspended in a stiff frame. A positioning system is used tonable translation of the receiver perpendicular to the z-axis. Thetationary source is a piezoelectric, spherically focused 500-kHzransducer �Panametrics, type V389� with a focal distance of ap-roximately 6 cm in water. The focal point acts as a point source.he receiver �indicated by triangles in Figure 3� is a broadband nee-le hydrophone with a tip diameter of 1 mm �SEA, type PZT-Z44-000�. Source, receiver, and sample are placed in a 1-cubic-meterank filled with tap water.

To act as a perturbation of the homogeneous velocity field, smallpheres with various diameters d made from synthetic rubberd = 10 mm� and teflon �d = 7.5 mm� are used. For the spheres toemain stationary during the measurements, they are suspended in alexiglas frame using 0.25-mm-thick fishing line. Teflon has a nega-

ive velocity contrast �1275 m/s�, and rubber has a positive velocityontrast �1514 m/s�, compared to water at room temperature±1485 m/s�. The propagation velocities of teflon and rubber wereetermined during separate traveltime measurements on cylindricalamples from which the spheres were manufactured at a later stage.

For the velocity sensitivity measurements, the following proce-ure is applied. First, the sphere is placed on the axis between theource �i.e., the focal point� and central receiver position. The centerf the sphere is 10 cm from both the source and central receiver.ext, the field is scanned along the receiver line as indicated in Fig-re 3. The receiver separation is 1 mm, whereas the receiver arrayxtends to ±5 cm on both sides of the central receiver. The proce-ure is repeated in absence of the sphere so that the traveltime differ-nces and the relative amplitude variation between the two data setsan be determined for each individual receiver location.

The recorded �perturbed and reference� data sets are bandpass fil-ered between 232 and 720 kHz, as the S/N ratio is the highest forhis range. The traveltime residuals and the relative amplitude varia-ion are determined from the perturbed and reference traces using apectral-ratio approach. Let the reference waveform in the frequen-y domain be given by

dref��� = Aref���exp�i�tref�Sref��� , �24�

nd the perturbed waveform is written as

dpert��� = Apert���exp�i�tpert�Spert��� . �25�

ote that the source wavelets Sref��� = Spert��� in the ultrasonic ex-eriment. The amplitude factor A��� includes transmission effects,he geometric spreading factor, and attenuation inherent to energyosses. The ratio of the perturbed and reference wavefield splits inton amplitude part and a traveltime delay part:

dpert

dref��� =

Apert

Aref���exp�i��t� , �26�

hich is a complex number. The angle of the wavefield ratio in theomplex plane divided by the angular frequency gives the observedraveltime shift:

�t��� =1

�� �dpert

dref���� . �27�

he relative amplitude variation �A/Aref��� = Apert/Aref��� − 1where Apert��� = Aref��� + �A���� is retrieved from the norm ofhe ratio between the perturbed and reference waveform. Hence,

�A

Aref��� = �dpert

dref���� − 1. �28�

he broadband frequency characteristics of the two waveforms arencluded by averaging the monochromatic traveltime delay in equa-ion 27 and the relative amplitude variation in equation 28 over thessociated frequency interval.

Equations 15 and 17 are used to compare the experimental resultsith 3D theory.Again, the condition � f0−�f

f0+�fA� f�df = 1 is used to nor-alize the experimentally determined spectra in equations 16 and 18here A� f� corresponds to the spectrum of the trace recorded at the

eceiver position under investigation. In the ray theoretical case, theraveltime delay is computed from linear ray theory in equation 19,

igure 3. Schematic �top� view of the setup used for the sensitivityests. For each test, two complementary scans are performed: oneith and one without an anomaly present between source and

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T172 Jocker et al.

s well as from nonlinear ray theory, including ray-bending effects inhe perturbed velocity field. The ray theoretical relative-amplitudeariation inherent to the geometric spreading factor is computed us-ng equation 20.

Figure 4 shows the recorded wavefields in the presence of the te-on and rubber spheres. For the teflon sphere �Figure 4a�, a clear

riplication �i.e., area of high acoustic energy� occurs in the observedaveform data. The presence of the triplication indicates that strong

ay-bending effects are encountered. In Figures 5–7, the traveltimeelay and amplitude variation for the wavefields in Figure 4 arehown. We plotted the equivalent results based on linear and nonlin-ar ray theory, together with the exact results for a point-sourceavefield propagating in a homogeneous background medium and

ncident upon an elastic sphere.The exact results were calculated using the theory presented by

ickling �1962�. Using a spherical coordinate system centered in thenomaly, the wavefield Pi��� emanating from a point source at a dis-ance r0 can be expressed as

Pi��� = ikP0�n=0

�

�2n + 1��− 1�nPn�cos ��jn�kr�hn�kr0� , �29�

here Pn is the nth order Legendre polynomial; jn and hn are spheri-al Bessel and Neumann functions, respectively; k is the fluid wave-umber; and r is the distance between the origin of the sphere and theeceiver location. The scattered field resulting from the point-source

igure 4. The recorded wavefields in the presence of �a� negative-teflon� velocity perturbation and �b� positive-�rubber� velocity per-urbation. Notice the triplication resulting from the negative velocityifference at the midpoint of the horizontal axis in Figure 4a.

igure 5. Comparison between the experimentally observed travel-ime delays and the predicted ones using finite-frequency wave theo-y and ray theory. �a� �/a = 0.4, LF

3D/a = 3.2, and �v/v = −13.5%i.e., teflon�. �b� �/a = 0.3, LF

3D/a = 2.4, and �v/v = 2.7% �i.e., rub-

igure 6. �a� Similar to Figure 5 except for the relative amplitudeariation in the case of teflon. �b�Aclose-up view of �a�.

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Validation of first-order diffraction theory T173

avefield incident upon an elastic sphere is expressed as

Ps��� = P0�n=0

�

cnhn�kr�Pn�cos �� , �30�

here coefficients cn follow from the appropriate boundary condi-ions at the surface of the sphere, i.e., �1� continuity of fluid pressurend normal component of the stress in the solid at the interface; �2�ontinuity of normal displacement of solid and fluid at the interface;nd �3� vanishing of shear stress in the solid at the interface. Amongther parameters, coefficients cn contain information on the elasticphere, such as density � and Poisson ratio �through which thehear wave velocity is determined�. The and �, both absent in theinear-scattering theory, are determined independently from the ul-rasonic measurements. For teflon, we found � = 2157 kg/m3 and

= 0.413 �i.e., cs = 491 m/s�, whereas for rubber, � = 1020 kg/m3

nd = 0.23 �i.e., cs = 896 m/s�. The total field external to thephere Ptot��� is given by the sum of the incident field and the scat-ered field:

Ptot��� = Pi��� + Ps��� . �31�

pplying and averaging equation 31 over the frequency interval32–720 kHz, the results can be compared immediately with theurves in Figures 5–7 to study the influence of the limitations of lin-ar scattering theory in relation to the absence of shear-wave andensity contributions.

For the teflon anomaly, the traveltime delay and relative ampli-ude variations resulting from the velocity contrast between thephere and water are shown in Figures 5a and 6a and 6b. In general,he time delay and fractional amplitude distortion predicted by rayheory are several times larger in magnitude than the experimentallybserved values. Also, the width of the ray theoretical curves ismaller than what is observed in the ultrasonic experiment. The 3Dnite-frequency wave theory yields better results than ray theoryhen predicting the observed traveltime shift and amplitude ratio,oth in magnitude and shape of the experimental curves. However,here is still a clear mismatch between the 3D finite-frequency waveheory and the experimental observations as a result of the triplica-ion present in the recorded waveform data. The 3D finite-frequencyave theory is linear and, in turn, does not include strong nonlinear

ay-bending and multipathing effects that are associated with theormation of triplications.

For the rubber measurements �see Figures 5b and 7�, the agree-ent between linear theory and experiment has improved, compared

o the teflon results. Again, ray theory predicts values too large forhe traveltime shifts and relative amplitude variation. On the otherand, the linear scattering theory underestimates the traveltime de-ay and, to a lesser extent, the amplitude variation as observed duringhe experiments. This is because of the small difference in compres-ional wave velocity between fluid and rubber, in addition to themission of shear-wave effects in linear scattering theory. Compar-ng the scattering theoretical result for rubber with the exact result inhe absence of a shear wave �i.e., cs �0 m/s or �0.5�, it is ob-erved that both curves are very similar. However, when a shearave is introduced into the exact solution by decreasing the value

or the Poisson ratio to the experimentally determined value of= 0.23, the exact result increasingly agrees with the experimental

esults, including the characteristic uprise in the relative amplitudeariation around the central receiver position �Figure 7b�. In conclu-ion, in the case of rubber, the P-wave velocity perturbation between

uid and elastic solid is so small that the shear wave in the sphere ishe dominating event. Because shear waves are unaccounted for inhe presented linear theory, traveltime and amplitude effects are un-erestimated compared to the experimental observations �thoughtill predicted considerably better than by linear and nonlinear rayheory�. In practice, this will rarely pose a problem, for a small con-rast in compressional wave velocities between two elastic mediasually will go together with a small contrast in shear-wave velocity,ontrary to the case discussed above for an elastic sphere in ancoustic �i.e., shearless� background medium.

Finally, regarding Figures 6 and 7, a simple computation of the to-al transmission coefficient

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t the interfaces between the spheres �the density and P-wave veloci-y are denoted as �sph and vsph� and the surrounding fluid ��w and vw

re the density and sound speed of water� of the incident and outgo-ng wavefield for the center receiver position reveals that the ampli-ude loss is on the order of 15% for teflon and 0.5% for rubber be-ause of the impedance differences between background and anom-ly. However, Figures 6 and 7 show that the ray theoretical resultsre several times larger than the observed values. This is caused byhe presence of the second-derivative operator in equation 20 thatcts to make effects of small-scale structures on ray-theoretical am-litudes very prominent. In reality, the wavefield is subject to ampli-ude variations on the order of 15% for both the teflon and rubbernomaly because of the effect of finite-frequency waves.

igure 7. �a� Similar to Figure 5 except for the relative amplitudeariation for the rubber anomaly. �b�Aclose-up view of �a�.

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2D NUMERICAL SIMULATIONS OF THEULTRASONIC WAVE EXPERIMENT

The finite-frequency wave theory tests in the ultrasonic laboratoryxperiment are restricted to a few case studies of different velocityerturbations with respect to the velocity of sound in water. Beyondeflon and rubber, we are aware of no other appropriate materials.hat is why a numerical finite-difference �FD� solution of the elas-

ic-wave equation is used to mimic the ultrasonic laboratory experi-ent for a large range of different velocity contrasts. The FD simula-

ions are similar to the ultrasonic wave experiment with the excep-ion of the dimensionality �2D instead of 3D�. The source waveletas a broadband characteristic with the central frequency fixed at00 kHz. The source is modeled as a Ricker wavelet. The geometryor this numerical test is identical to the sketch in Figure 3, and theD ultrasonic experimental parameters are repeated. Hence, a 2D,pherical �cylindrical� wavefield is emitted from a point source, andhe total wavefield is recorded along an array of receivers perpendic-lar to the line going through the source position and the center of thenomaly. Sets of perturbed and reference data are obtained by re-pectively including and omitting the spherical anomaly in the ve-ocity model. Numerical results are obtained for anomaly velocities

pert equal to 1260 m/s, 1360 m/s, 1600 m/s, and 1700 m/s, where-s the reference velocity vref is fixed at 1480 m/s. The velocity con-rasts between the anomaly and the reference velocity are then15%, −8%, 8%, and 15%, in which the case of the −15% velocitynomaly is comparable to the teflon experiment. For all computa-ions, the diameter of the spherical anomaly is 7.5 mm.

Similar to the experimental data processing, the traveltime shiftsnd the amplitude variation inherent to the velocity anomaly are ob-ained using the previously discussed spectral-ratio approach. Theheoretical traveltime delays and amplitude fluctuations are comput-d with 2D finite-frequency wave theory in accordance with equa-ions 10 and 12, as well as with linear and nonlinear ray theory,

igure 8. Numerical FD results for a range of velocity perturba-ions. �a� �v/v = −15%; �b� �v/v = −8%; �c� �v/v = 8%; and �d�v/v = 15%. In �a�, a single caustic is generated at the central

eceiver position because of a strong negative-velocity perturbationi.e., −15%�. In �d�, two triplications resulting from a large positive-elocity difference �i.e., 15%� are seen clearly on both sides of theentral receiver position.

hich takes into account ray bending resulting from velocity con-rasts.

Gaussian beam theory �Červený, 2001� offers an alternative wayo account for the finite-frequency effects of waves. We have testedhe Gaussian beam theory as described in Hill �1990; 2001� in the 2Dumerical experiment. The time delays computed with Gaussianeam theory do not differ significantly from the time shifts obtainedith the nonlinear two-point ray tracer. For the relative

mplitude variation computation, Gaussian beam theory returnsaximum values of ±0.5%−1% of amplitude variation for the15% velocity-contrast experiments.In Figure 8, synthetic FD results are shown for a range of velocity

ontrasts between −15% and 15% with respect to the propagationelocity in water �i.e., vref = 1480 m/s�. The triplication observedreviously in Figure 4a is clearly visible also in Figure 8a. The otherase of caustic formation is illustrated in Figure 8d, where two caus-ic points away from the center receiver line are observed. Figure 8bnd c shows less significant signs of triplications because of theeaker velocity perturbations.Similar to the 3D case, the 2D diffraction theory leads to signifi-

antly better results compared to linear and nonlinear ray theory �seeigures 9–11�. In general, the traveltime shifts and amplitude ratiosalculated using ray theory are too large in magnitude, and theidths of the curves are too narrow, compared to the numerically ob-

erved values. The high-frequency oscillations in the nonlinear trav-ltime delays are caused again by strong ray-bending and multipath-ng effects. The 2D wave-diffraction theory predicts relative ampli-ude changes in the order of magnitude of ±40%. To a lesser extent,his is observed also in the experimental 3D data �Figures 6 and 7�.otice the small discrepancy between the traveltime residual and

elative amplitude variation predicted by wave-scattering theory andhe numerically observed ones in Figures 9b, 9c, and 11b and 11c. Inhese figures, nonlinear ray-bending effects resulting from the for-

ation of triplications are still weak, justifying the use of finite-fre-uency wave theory. In Figures 9a, 9d, 11a, and 11d, there might be

igure 9. Comparison between the numerically observed traveltimeelays and the predicted ones compiled with finite-frequency waveheory and ray theory. The experimental parameters are �/a = 0.4nd LF

2D/a = 2.8. The range of velocity contrasts are �a� �v/= −15%; �b� �v/v = −8%; �c� �v/v = 8%; and �d� �v/v = 15%.

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everely bent rays going through caustics that are not included in theheoretical formulation. Consequently, we are at the limit of the va-idity of the single-scattering wave approach.

SEISMIC APPLICATIONS

Although not perfect, finite-frequency kernels have the potentialo improve resolution of seismic imaging experiments at low com-utational costs. While in seismology, these benefits are still over-hadowed by incomplete data coverage �Van der Hilst and De Hoop,005; Trampert and Spetzler, 2006�, the seismic applications clearlyuffer less from lack of data. This is illustrated by two examples of fi-ite-frequency wave theory in seismic imaging experiments. Thewo applications that we discuss are �1� traveltime tomography us-ng transmission data and �2� amplitude recovery of reflected datarior toAVO/AVAinversion.

In the first example, waveform data from the Durham laboratoryxperiment are used �Legget et al., 1993; Pratt, 1999�. In this experi-ent, ultrasonic transmitted waves are recorded in a crosswell con-guration with two vertical wells. The velocity structure is manufac-

ured with epoxy. Figure 12a illustrates schematically the velocityodel that consists of small and large structures, compared to theaximum Fresnel width �i.e., LF�. However, the separate layers in

he physical model are not that homogeneous, as indicated in the fig-re, nor is the contact surface between the layers completely free ofiny air bubbles because the mixing of the resins had to be doneuickly �Neil Goulty, personal communication, 2003�. According toquation 21, the requirements for ray theory are not satisfied for theow-velocity half-cylinder. On the contrary, the conditions for thepplication of the finite-frequency wave theory are valid for thehole velocity model in the Durham experiment. In addition, the ul-

rasonic transmission experiment includes realistic noise contribu-ions because of uncertainties in source-receiver positions, in theraveltime estimation, and in the estimation of the reference velocityLegget et al., 1993�.

In the ultrasonic experiment, 500-kHz piezoelectric transducersre used to emit and record waveform data. Fifty-one sources and re-eivers give a total of 2601 measured broadband waveforms. Thehysical dimensions for the crosswell velocity structure are6.5 mm in the offset direction and 125 mm in the depth direction.

igure 10. Similar to Figure 9 except for relative amplitude varia-

o simulate a realistic crosswell measurement with frequencies be-ween 200 and 500 Hz, all distances, times, and frequencies arecaled by a factor 1000. In turn, the lateral length and depth of therea between the two wells have the dimensions 46.5�125 m. Thearget zone for the inversion is the half-cylinder-shaped, low-veloci-y anomaly and the vertical fault below. The reference velocity mod-l is a layered velocity structure without the fault between the lowestwo layers. A detailed description of the tomographic inversion ofhe acoustic data using both finite-frequency scattering theory anday theory can be found in Spetzler �2003�.

The inverted velocity model, compiled with the finite-frequencyave theory and ray theory, is presented in Figure 12b and c, respec-

ively. The solid lines indicate the interfaces between the homoge-eous layers of resin. The low-velocity half-cylinder between 60 and0 m depth is resolved best in the velocity model inverted with therst-order finite-frequency wave theory. The strength of the low-ve-

ocity contrast of the half-cylinder is underestimated as well asmoothed out in Figure 12c. This is a consequence of the limitationsf the high-frequency approximation in ray theory. In both inversion

igure 11. Aclose-up view of Figure 10.

igure 12. Crosswell tomography. �a� Schematic velocity structure.b� Velocity cross section compiled with the finite-frequency waveheory. �c� Velocity cross section generated with ray theory.

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esults, there seems to be an imprint of the small, vertical fault at ap-roximately 80 m depth.

The finite-frequency wave tomographic inversion does not re-uire extra hardware or memory than is necessary for a similar inver-ion approach based on ray theory. The two inversions in this exam-le were carried out on a laptop with an Intel Pentium IV, 3.2-Hz processor with 512 mb memory running on Suse linux 9.1. For

he construction of the forward-modeling matrix for a grid parame-rization of 24�64 = 1536 grid cells and 2601 data points, it takes0 CPU minutes for the finite-frequency wave approach and 1 CPUinute for ray theory. The tomography inversion for both approach-

s takes 5 CPU minutes.The second example is a numerical-reflection seismic experi-ent. A gas-saturated overburden anomaly �also known as a gas

himney� is located over an interface dipping 5°. The P-wave veloci-y model is illustrated in Figure 13a. The seismic parameters in theop layer are vp = 2000 m/s, vs = 1200 m/s, and � = 1100 kg/m3,hereas in the bottom layer, vp = 3000 m/s, vs = 1300 m/s, and= 1500 kg/m3. The gas-saturated overburden perturbation isodeled with a Gaussian function. The maximum velocity contrast

s 130 m/s with respect to the homogeneous reference velocity in theop layer, and the standard deviation of the overburden anomaly is00 m in both the horizontal and vertical directions.As illustrated onhe Schlumberger seismic glossary web page �i.e., www.glossary.ilfield.slb.com�, a gas chimney may have a significant attenuatingffect on P-P stacks. An additional example of transmission distor-ions as large as 70% on prestack seismic reflected waves inherent to

igure 13. Propagation effect resulting from a gas-saturated overburmplitude of reflection seismic data. �a� Velocity structure; �b� Fréected wavefield; �c� CMP gather, and �d� near-offset gather.

ault structures in the Gulf of Mexico is found in Hatchell �2000�.his numerical example shows that the attenuation of the amplitudef a P-P reflection below a gas-saturated overburden anomaly can bexplained by single-scattering wave theory.

The finite-frequency sensitivity kernel for a reflected wave withhe frequency content between 20–60 Hz is shown in Figure 13b.wo-way wavefield theory is used to decompose the Fréchet kernel

nto down- and upgoing parts which are computed with first-orderiffraction theory. The gray-scale color illustrates the sensitivity toelocity perturbations. The straight white line indicates the raypath,nd the white dashed line marks the boundary for the Fresnel volumeor the central frequency of 40 Hz. The width of the Fresnel volumebove the interface is on the order of 500 m, which is several timesarger in size than the overburden anomaly.

A full, elastic, finite-difference modeling experiment was appliedo compute reference and perturbed reflection data, respectively,ithout and including the gas-saturated overburden anomaly. The

pectrum of the reflection data is dominant between 20 and 60 Hz.Aommon midpoint �CMP� gather for the CMP at 1250 m �below theas chimney� and a near-offset gather were constructed from theimulated reflection data. The relative amplitude variation resultingrom propagation through the overburden perturbation was obtainedrom the spectral ratio of the perturbed and reference gathers. Theheoretical relative amplitude was computed with 2D wave diffrac-ion theory for reflected waves. In Figure 13c and d, the observed andomputed values are compared. For both gathers, the amplitude is at-enuated by 80%–90%, solely as a result of the presence of the gas-

saturated anomaly. Hence, the finite-frequencywave theory offers a way to correct for amplitudelosses resulting from overburden anomalies inCMP gathers and near-offset gathers. Afterward,the amplitude-corrected gathers can be used forfurther estimation of structure and seismic pa-rameters.

CONCLUSIONS

Finite-frequency wave propagation in mediawith structures smaller in size than the Fresnelvolume leads to the counterintuitive result thatthe sensitivity to traveltime shifts is maximallyaway from the geometric ray and that the sensitiv-ity for the amplitude is maximum on the geomet-ric ray for both 2D and 3D. Remarkably, for wavepropagation in 3D, the sensitivity for traveltimeshifts is zero on the ray path because of wave-scattering effects. Those findings are in contra-diction with standard ray theory. By means of ul-trasonic wavefield experiments, these results arevalidated successfully for a relevant range ofanomaly sizes and velocities for which the regimeof ray theory is invalid. The findings in the ultra-sonic experiment are supported by numericalsimulations. The counterintuitive results from fi-nite-frequency, wave-scattering theory and theassociated laboratory experiments can be ex-plained physically by the principle of wavefieldinterference. Monochromatic waves diffractedby velocity contrasts interfere with other mono-chromatic components in the field. Adding up

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onstructively at the receiver, these waves compose the recordedroadband wavefield.

First-order, finite-frequency wave theory works well for anomalyiameters which are approximately one-third of the Fresnel volumeidth and for velocity differences up to ±15% in the reference veloc-

ty. This is concluded from 2D numerical simulations in which therere no constraints on the magnitude of the velocity contrast. Fortronger velocity contrasts, nonlinear effects related to the formationf caustics dominate the problem.

To improve the resolution of seismic imaging experiments, ade-uate wave-propagation theory should be applied. The finite-fre-uency wave theory has immediate use in several such experiments,nd its regime goes beyond the standard ray theory. We have dis-ussed two possible applications to illustrate how to incorporate thenite-frequency wave theory in seismic imaging experiments. An-ther possible application of the first-order diffraction theory is toompute the phase and amplitude operators in the wavefield-extrap-lation part of migration experiments.

ACKNOWLEDGMENTS

This manuscript is sponsored by the European Communityhrough the Atlass project NNE5-1999-20211 and by the Dutchechnology Foundation STW through project DAR.6293. Theanuscript has been improved because of the thorough work of an

nonymous reviewer. We appreciate the time and effort that this re-iewer invested in our study of finite-frequency wave theory. Weratefully acknowledge Neil Goulty and Gerhard Pratt for permis-ion to use the Durham University laboratory data.

REFERENCES

ki, K., and P. G. Richards, 1980, Quantitative seismology, theory and meth-ods: Freeman and Company.

eydoun, W. B., and A. Tarantola, 1988, First Born and Rytov approxima-tions: Modeling and inversion conditions in a canonical example: Journalof the Optical Society ofAmerica, 83, 1045–1055.

orn, M., and E. Wolf, 1959, Principles of optics: Pergamon Press.ervený, V., 2001, Seismic ray theory: Cambridge University Press.ahlen, F., and A. Baig, 2002, Fréchet kernels for body-wave amplitudes:

Geophysical Journal International, 150, 440–466.

e Hoop, A. T., 1985, Time-domain far-field scattering of planer scalarwaves in the Born approximation: Journal of the Optical Society ofAmeri-ca, 2, 1961–1964.

e Wolf, D. A., 1967, Validity of Rytov’s approximation: Journal of the Opti-cal Society ofAmerica, 57, 1057–1058.

atchell, P. J., 2000, Fault whispers: Transmission distortions on prestackseismic reflection data: Geophysics, 65, 377–389.

ickling, R., 1962, Analysis of echoes from a solid elastic sphere in water:Journal of theAcoustical Society ofAmerica, 34, 1582–1592.

ill, N. R., 1990, Gaussian beam migration: Geophysics, 55, 1416–1428.—–, 2001, Prestack Gaussian-beam depth migration: Geophysics, 66,1240–1250.

ung, S., F. Dahlen, and G. Nolet, 2001, Wavefront healing: A banana-doughnut perspective: Geophysical Journal International, 146, 289–312.

egget, M., N. R. Goulty, and J. E. Kragh, 1993, Study of traveltime and am-plitude time-lapse tomography using physical model data: GeophysicalProspecting, 415, 599–619.arquering, H., F. A. Dahlen, and G. Nolet, 1999, Three-dimensional sensi-tivity kernels for finite-frequency traveltimes: The banana-doughnut para-dox: Geophysical Journal International, 137, 805–815.

eele, F., J. C. VanDecar, and R. Snieder, 1993, A formalism for includingamplitude data in tomographic inversions: Geophysical Journal Interna-tional, 115, 482–496.

ratt, G. R., 1999, Seismic waveform inversion in the frequency domain, Part1: Theory and verification in a physical scale model: Geophysics, 64, 888–901.

ytov, S. M., Y. A. Kravtsov, and V. I. Tatarskii, 1989, Principles of statisticalradiophysics 4: Wave propagation through random media: Springer Ver-lag.

nieder, R., and A. Lomax, 1996, Wavefield smoothing and the effect ofrough velocity perturbations on arrival times and amplitudes: GeophysicalJournal International, 125, 796–812.

petzler, J., 2003, Comparison of ray theory and finite-frequency wave theo-ry in crosswell tomography: 65th Annual Meeting EAGE, Extended Ab-stracts, F40.

petzler, J., and R. Snieder, 2001, The effect of small-scale heterogeneity onthe arrival time of waves: Geophysical Journal International, 145,786–796.—–, 2004, Tutorial: The Fresnel volume and transmitted waves: Geophys-ics, 169, 653–663.

petzler, J., C. Sivaji, O. Nishizawa, and Y. Fukushima, 2002, A test of raytheory and scattering theory based on a laboratory experiment using ultra-sonic waves and numerical simulations by finite-difference method: Geo-physical Journal International, 148, 165–178.

rampert, J., and J. Spetzler, 2006, Surface wave tomography: Finite fre-quency effects lost in the null space: Geophysical Journal International,164, 394–400.

an der Hilst, R. D., and M. V. De Hoop, 2005, Banana-doughnut kernels andmantle tomography: Geophysical Journal International, 163, 956–961.oodward, M. J., 1992, Wave-equation tomography: Geophysics, 57,15–26.

hao, L., T. Jordan, and C. Chapman, 2000, Three-dimensional Fréchet dif-ferential kernels for seismic delay times: Geophysical Journal Internation-

al, 141, 558–576.