ADVANCES IN GEOPHYSICS, VOL. 50, CHAPTER 2

THEORY OF TRANSMISSIONFLUCTUATIONS IN RANDOM MEDIA

WITH A DEPTH-DEPENDENTBACKGROUND VELOCITY STRUCTURE

Yingcai Zheng and Ru-Shan Wu

Abstract

An extended theory on the coherence function of log amplitude and phase for waves passing

through random media is developed for a depth-dependent background medium using the WKBJ-

approximated Green’s function, the Rytov approximation, and the stochastic theory of the random

velocity field. The new theory overcomes the limitation of the existing theory that can only deal

with constant background media. Our extended coherence functions depend jointly on the angle

separation between two incident plane waves and the spatial lag between receivers. The theory is

verified through numerical simulations using the iasp91 background velocity model with two layers

of random media. The current theory has the potential to be used to invert for the depth-dependent

spectrum of heterogeneities in the Earth.

KEY WORDS: Transmission fluctuation, coherence function, random media, heterogeneity spec-

trum, Phase, amplitude, WKBJ, Rytov � 2008 Elsevier Inc.

1. Introduction

The statistical approach, complementary to the deterministic method such as seismictomography, is indispensable in probing the small-scale inhomogeneities in the Earthusing scattered seismic waves. Its characterization of heterogeneities using statisticalparameters has yielded many important physical constraints and interpretations for thelithosphere to the core-mantle boundary region, and led to key implications regarding theEarth’s compositional constituents, thermal state, dynamic mixing, and others.

In 1970s, many attempts (Capon, 1974; Capon and Berteussen, 1974; Berteussen,1975; Berteussen et al., 1975, 1977) had been made to infer the spatial spectrum ofthe velocity heterogeneity using observed fluctuations of the logarithmic amplitude(logA) and the phase f across a seismic array since the pioneering work by Aki(1973). Aki essentially employed the Chernov theory (Chernov, 1960) to study thetransverse coherence function (TCF) using the data from the Large Aperture SeismicArray (LASA), Montana. He found that a 60-km random layer in the lithospherewith ~10 km scale length for the heterogeneities and 4% root-mean-square (rms)P-wave velocity fluctuation could explain the data. The Chernov theory (1960)studies wave propagation in a single layer of stationary random velocity heterogene-ities with a Gaussian correlation function and with a constant background velocity

21 # 2008 Elsevier Inc. All rights reserved.ISSN: 0065-2687

DOI: 10.1016/S0065-2687(08)00002-2

and as such, the TCF has a nice closed mathematical form. By the word stationary,we mean the spatially translational invariance of the statistic (e.g., correlationfunction). The TCF for random media of general spectral type can be found inTatarskii (1971) and Ishimaru (1978) and it does not possess a simple analyticalexpression in general. It is interesting to note that this stochastic approach predatedthe deterministic tomography method (Aki and Lee, 1976; Aki et al., 1976, 1977).

The heterogeneity spectrum of the Earth is fractal based on well-logging data (Wuet al., 1994b; Jones et al., 1997; Goff and Holliger, 1999) and it may not be Gaussian.Capon and Berteussen (1974) found that the Chernov theory was not applicable tofluctuations of logA and phase data under the Norwegian Seismic Array (NORSAR);however, they attributed this to the validity of the Born approximation at high frequen-cies involved in the theory. The Earth’s large-scale structures are largely stratified indepth and the herteogeneity spectrum can be slowly varying with depth. Flatte and Wu(1988) introduced a new kind of coherence function, called the angular coherencefunction (ACF), to resolve the depth-dependent spectra under the NORSAR. A two-layer model with power-law type medium was favored over a one-layer stationaryGaussian medium. Wu and Flatte (1990) also formulated the joint transverse and angularcoherence function (JTACF) in which both the spatial lag between two seismic stationsand the angular separation between two plane waves were taken into account. Theirformulation was based on wave perturbation theory and assumed the Rytov and theparabolic approximations for the wave equation. Chen and Aki (1991) independentlyobtained the same JTACF result using the Born approximation. Parametric (Flatte et al.,1991a) and nonparametric (Wu and Xie, 1991) inversions have been carried out toprocess real data or to study the depth resolution in the spectral inversion. Wu and Xie(1991) called such inversion “stochastic tomography” and they found that the JTACF hasthe best depth resolution, that the ACF has limited depth resolution close to the surface,and that the TCF has no depth resolution at all. To invert for the spectrum of a single-layer stationary random heterogeneity, Zheng et al. (2007) proposed a new scheme usingonly the TCF data for logA and phase, in which a Fourier transform was establishedbetween the heterogeneity spectrum and the combination of the logA and phase TCFdata. There was some concern on the discrepancy on the phase coherence functionbetween the theory and the numerical simulation (Line et al., 1998a; Hong et al.,2005). However, recent investigation has shown that this discrepancy was caused bythe incorrect phase picking method used in the numerical and field experiments. Appli-cation of the correct phase measuring method has resulted in excellent agreementbetween numerical tests and the theory (Zheng and Wu, 2005). The coherence functionformation using combined data from arrays with different apertures was investigated byFlatte et al. (1991b) and Flatte and Xie (1992). The theory of transmission fluctuationwas also applied to seismic reflection data to obtain heterogeneities in the upper crust(Line et al., 1998b). The theory of JTACF has been applied to the NORSAR data(Wu et al., 1994a) and the Southern California Seismic Network data (Liu et al., 1994;Wu et al., 1995). For earlier reviews on the subject, see Wu (2002) and Sato et al. (2002).

Besides the coherence function study, many other seismological methods have beendevised to characterize the small-scale heterogeneities. The seismic coda envelopeanalysis (Sato and Fehler, 1998) has been widely used to investigate the scatteringstrength in the Earth. Through studying the radiated power carried by precursors to thePKIKP phase (Cleary and Haddon, 1972), the spectrum for P-wave volumetric scatters inthe D00 region, or for the core-mantle boundary topographical relief (Bataille and Flatte,1988; Bataille et al., 1990), or for the whole mantle (Hedlin et al., 1997) and mid-mantle(Hedlin and Shearer, 2002) has been constrained.

22 ZHENG AND WU

Despite significant progress that has been made using the coherence function tocharacterize the Earth’s small-scale heterogeneities, all these theoretical treatment ofthe problem is based on wave propagation in a homogeneous background medium. Thisis inadequate for the real Earth, which has a depth-dependent velocity profile to firstorder. In this chapter, we show how we can generalize the theory of coherence functions(logA and phase f) to arbitrary random medium superimposed on general depth-dependent background medium. This generalized theory can be directly utilized for thereal Earth.

2. Acoustic Waves in Stratified Media and WKBJ Green Function

The linearized wave equation for pressure in the frequency domain is

r▽� 1

r▽p

� �þ o2

c2p ¼ 0: ð1Þ

Let o be the angular frequency, p ¼ p x; y; z;oð Þ the pressure field, and ▽ the spatialgradient operator. Define z as depth variable. r and c are density and wave propagationspeed, respectively. For a stratified medium, c ¼ c zð Þ; r ¼ r zð Þ, the 2-D Fourier trans-form can be performed to Eq. (1) with respect to spatial coordinates, x and y, to obtain

k2z zð Þpþ @2p

@z2� r�1 @r

@z

@p

@z¼ 0; ð2Þ

where kz is the vertical wave number defined as

kz zð Þ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c2zð Þ � k2x � k2y

r: ð3Þ

kx and ky are horizontal wave numbers corresponding to x and y, respectively. The plussign in Eq. (3) corresponds to the downgoing wave and the minus sign the upgoing wave.We use the same symbol p to denote the pressure before and after the Fourier transformand this should not cause confusion. Plugging the trial solution in the form ofp ¼ A zð Þexp if zð Þ½ � into Eq. (2) and the real part is

k2z Aþ A00 � f02A

� �� r�1 @r

@zA0 ¼ 0; ð4Þ

and the imaginary part reads

2f0A0 þ Af00 � r�1 @r@z

Af0 ¼ 0: ð5Þ

The prime represents the partial derivative with respect to depth z. If the amplitude isslowly varying with depth, we can use the WKBJ approximation (i.e., get rid of all termsthat involve derivatives of A) in Eq. (4) and it reduces to

k2z � f02 ¼ 0 ) f0 ¼ kz: ð6Þ

23THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

Therefore, the phase function is solved as

f zð Þ ¼ðzzs

kz z0ð Þdz0; ð7Þ

where zs is a reference depth, commonly taken as the source depth. The validity of theWKBJ approximation is assured if the wavelength is shorter than the characteristic scaleof the background velocity model. The WBKJ also implies energy conservation forthe transmitted wave, which means that no reflected waves are produced (Wu and Cao,2005). Substituting Eq. (6) in Eq. (5), we have

2d ln A

dzþ d ln kz

dz� d ln r

dz¼ 0; ð8Þ

Rearranging Eq. (8), we get

d ln A ¼ d ln

ffiffiffiffiffiffiffiffiffiffir zð Þkz zð Þ

s; ð9Þ

So the solution for the amplitude in Eq. (9) is

A zð Þ ¼ C

ffiffiffiffiffiffiffiffiffiffir zð Þkz zð Þ

s; ð10Þ

where C is a constant. The final solution to Eq. (2) is

p kx; ky; z;o� � ¼ C

ffiffiffiffiffiffiffiffiffiffir zð Þkz zð Þ

sexp i

ðzzs

kz z0ð Þdz0

: ð11Þ

Once we have obtained Eq. (11), the solution for Eq. (1) is just the inverse Fouriertransform

p x; y; z;oð Þ ¼ 2pð Þ�2

ðþ1

�1

ðþ1

�1p kx; ky; z;o� �

exp i kxxþ kyy� �� �

dkxdky: ð12Þ

Next, let us investigate the solution for a point source. Under this case, the Eq. (1) has anadditional source term

r▽� 1

r▽p

� �þ o2

c2p ¼ �d x� xsð Þd y� ysð Þd z� zsð Þ: ð13Þ

xs; ys and zs are source position coordinates. For simplicity, the source is placed at theorigin. In the homogeneous case, the solution in frequency domain is

p x; y; z;oð Þ ¼ 1

4prexp

iorc

� �; r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p: ð14Þ

24 ZHENG AND WU

When a receiver is sufficiently close to the source, the solution (12) should coincide with(14). In order to compare both solutions in the wave number domain, we first expand thepoint source solution (14) into plane wave components using the Weyl integral (e.g., Akiand Richards, 2002 p. 190). The coefficient to component kx; ky

� �is 1=2ikz zsð Þ and the

corresponding coefficient in Eq. (12) is C=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir zsð Þ=kz zsð Þp

. Clearly, these two should beequal. Thus, this constant C is

C ¼ 1

2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir zsð Þkz zsð Þp : ð15Þ

Therefore, the complete Green’s function for both 3-D and 2-D cases can be obtained.Let us rewrite the Green’s function in 3-D case

G x; y; z;oð Þ ¼ 2pð Þ�2

ðþ1

�1

ðþ1

�1

1

2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikz zsð Þkz zð Þp

ffiffiffiffiffiffiffiffiffiffir zð Þr zsð Þ

s

exp i kxxþ kyyþðzzs

kz z0ð Þdz0 �

dkx dky: ð16Þ

In 2-D case, just drop all terms pertaining to ky and the inverse Fourier transform constantis 2pð Þ�1

. Also remember that the 2-D Fourier transform becomes a 1-D transform.

3. Rytov Solution to the Wave Equation in a

Heterogeneous Medium

For convenience, we rewrite the monochromatic wave equation

r▽� r�1▽p� �þ o2

c2p ¼ 0: ð17Þ

The solution for the pressure wavefield is sought in the form of

p ¼ p0ec: ð18Þ

c ¼ c x!0;o� �

is the complex phase and p0 ¼ p0 o; x!0� �

is the background incidentwavefield. In what follows we suppress the explicit dependence on o for the wavefieldfor simplicity in notation. Assuming no lateral density perturbation and substituting (18)into (17), we have

r▽�▽r�1 þ k2� �

p0cð Þ ¼ �2k2gp0 � p0▽c�▽c; ð19Þwhere g ¼ c20=c

2 � 1� �

=2 � �dc=c0 is the scattering potential, dc ¼ c� c0. The back-ground wave number k is defined as k ¼ o=c0 z0ð Þ. The Rytov approximation assumes▽cð Þ2 << 2k2g. The right-hand side of Eq. (19) is a spatially distributed source term. Thecomplex phase can be solved as

c x!� �

¼ 2

p0 x!ð ÞðV

k2 x!0ð ÞG x

!; x!0;o

� �g x

!0ð Þp0 x!0ð Þd3x!0: ð20Þ

From Eq. (20), it can be seen that contributions of scattering potentials g at differentlocations to the complex phase are independent from each other. However, this linear

25THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

relationship is to the complex phase, not to the field itself, which is different from theBorn approximation. Note that this complex phase is naturally “unwrapped.” It can bedirectly used to obtain the phase velocity. We see that the Rytov approximation is asingle-scattering approximation in terms of complex phase. A local Rytov approximationhas been developed (Huang et al., 1999) to calculate wavefields, in which case themultiple forward scattering is included.

4. Complex Phase c Due to a Plane Wave Incidence

To obtain the complex phase at depth z, we can rewrite Eq. (20) in an explicit form

cðx!Þ ¼ 2

p0ðx!ÞðLz

k2ðz0Þdz0ð ð

Gðx!T � x!0T; z; z

0;oÞgðx!0T; z

0Þ

� p0ðx!0Þd2x!0T; ð21Þ

where x! ¼ x!T; z� �

is the receiver location and x!0 ¼ x!0T; z

0ð Þ is the location for theheterogeneity. In the case of an upgoing plane wave (negative vertical wave number),incidence with ray parameter q!,

p0 x!0ð Þp0 x!ð Þ ¼

ffiffiffiffiffiffiffiffiffiffir z0ð Þr zð Þ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikz q; zð Þkz q; z0ð Þ

sexp ioq

!� x!0T � x

!T

� �� i

ðz0z

kz q; sð Þds" #

; ð22Þ

where q ¼ q!j j and kz q!; zð Þ ¼ offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=c2 zð Þ � q2

p. We always choose a nonnegative o � 0

because of the complex conjugate symmetry p o; x!ð Þ ¼ p �o; x!ð Þ. Substituting Eqs. (16)and (22) into Eq. (21) and recognizing that it represents a convolution, we can easily putdown the complex phase in the wave number domain:

cðq!; x!Þ ¼ 2ð2pÞ�2

ðLz

dz0ð ð

k

k2ðz0Þ2i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikzðk! þo q!; z0Þkzðk! þ o q!; zÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikzðq; zÞkzðq; z0Þ

s

� exp

(i

ðLz

½kzðk!þo q!; z00Þ � kzðq; z00Þ�dz00 þ i k!� x!gdnÞðk!; z0Þ; ð23Þ

where k! is the horizontal wave number of the heterogeneity spectrum and dn k!; z0ð Þ is theFourier-Stieltjes spectral density (Yaglom, 1962). Such a spectral representation is verytechnical and it seems harmless to replace dn k!; z0ð Þ by n k!; z0ð Þd2k!, where n k!; z0ð Þ is theFourier spectrum of the velocity perturbation field g x!T; z

0� �in the transverse plane at

depth z0. If one wants to pursue the mathematical exactness of the Green function (16), allwave numbers should be integrated, including both propagating (real kz) and evanescentcomponents (imaginary kz). However, the evanescent components are not used in ourcase. For high-frequency wave propagation in random media, most scattered energy isin forward direction within an angle that spans ~1/(ka), with a being the scale length ofthe heterogeneity. The forward direction is understood as the incoming direction of the

26 ZHENG AND WU

incident wave. If we consider high-frequency wave propagation, Eq. (23) can besimplified as

c q!; x!� �

� 2pð Þ�2

ðL0

dz0ðð

k

k2 z0ð Þikz q; z0ð Þ e

ik! � x! þiÐ z0

0kz k! þo q

!;zð Þ�kz q;zð Þ½ �dz

dn k!; z0� �

;

ð24Þwhere L is the depth of the lower boundary of the heterogeneous layer and receivers

are placed at zero depth (Fig. 1). Here we define

D k!; z0� �

¼ 1

o

ðz00

kz k!þ o q!; z

� �� kz q; zð Þ

h idz: ð25Þ

Under the forward scattering approximation (ka > 1, k is the wave number and a is thecharacteristic scale length of the heterogeneity), only wave numbers around q! areintegrated. We can expand the D function [Eq. (25)] around k! ¼ 0 using the Taylorexpansion,

D k!; z0� �

� � k!� r! q!; z0

� � 1

oþ y k!; q!;o; z0

� �þ :::: ð26Þ

r! q!; z0ð Þ is the transverse vector between the piercing points at depths 0 and z0 for a rayincidence from below with slowness vector q!. The quadratic term of k is

y k!; q!;o; z0� �

¼ � 1

2

I1 q; z0ð Þk2o2

� 1

2

I2 q; z0ð Þ k! � q!ð Þ2o2

; ð27Þ

where

I1 q; z0ð Þ ¼ðz00

c�2 zð Þ � q2� ��1=2

dz; ð28Þ

and

0X1

X2

Ray1 Ray2r1(x) r2(x)

Plane waves

L

Z

X

x

FIG. 1. Schematic geometry used in the theoretical derivation. The heterogeneous layer is

bounded between depth 0 and L. Seismic receivers (triangles) are placed on depth zero. Ray 1

connects to station x1! and it has the same slowness as plane wave 1. r1

!ðxÞ is the transverse distancebetween ray 1 at depth x to station x1

!. Similar meaning applies to ray 2 and r2!ðxÞ.

27THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

I2 q; z0ð Þ ¼ðz00

c�2 zð Þ � q2� ��3=2

dz: ð29Þ

Obviously, we have I1; I2 > 0. The following replacement

k!� q!� �2

! k2q2 ð30Þ

in (27) will result in a more rapidly oscillatory eioy with respect to k. We can use theapproximated phase [Eq. (26)] with replacement of Eq. (30) in (24).

Before we proceed further, let us introduce a t function

t p; z0ð Þ ¼ðz00

B z; pð Þdz; ð31Þ

where the vertical slowness at depth z is

B z; pð Þ ¼ c�2 zð Þ � p2� �1=2

: ð32Þ

This function has all the kinematic information we want (Buland and Chapman, 1983).For example, the horizontal distance traveled by a ray with ray parameter q from depth z0

to depth zero is t0p p¼q ¼ �@t=@p p¼q

���� . The second-order derivative of t with respect top contains information on the geometrical spreading of the wave front. The t function isthe Legendre transform of the travel time function. So the travel time can be easilycomputed using this function. Using (30) in (27), we obtain

yðk!; q!;o; z0Þ ¼ 1

2t0pp

����p¼q

k2

o2: ð33Þ

The subscript “ 0pp ” denotes second-order derivative with respect to the argument p, thatis, t0pp ¼ @2t=@p2. The derivative of y with respect to depth is

@y k!; q!;o; z0ð Þ@z0

¼ 1

2B0pp

k2

o2: ð34Þ

The complex phase (24) can be expressed as

c q!; x!� �

� � 2pð Þ�2

ðL0

ia q; z0ð Þdz0ððkeioy�ik! � r! þik! � x!dn k!; z0

� �; ð35Þ

where

a q; z0ð Þ ¼ k2 z0ð Þkz q; z0ð Þ : ð36Þ

28 ZHENG AND WU

5. Coherence Function Between Two Plane Waves

Considering two plane waves with slowness vectors, q!1 and q!2, we can express thecorresponding complex phases c1 and c2 at depth zero as:

c1 ¼ c q!1; x

!1

� � � � 2pð Þ�2

ðL0

ia1 z0ð Þdz0ðð

keioy1 z0ð Þ�ik!

1�r!1z0ð Þþik!

1�x!1dn k!1; z0� �;

ð37Þ

and

c2 ¼ c q!2; x

!2

� � � � 2pð Þ�2

ðL0

ia2 z00ð Þdz00ðð

keioy2 z00ð Þ�ik!2�r!2 z00ð Þþik!2�x!2dn k!2; z

00� �:

ð38ÞThe symbols are

a1 z0ð Þ ¼ a q1; z0ð Þ; a2 z00ð Þ ¼ a q2; z

00ð Þ; ð39Þ

y1 z0ð Þ ¼ y k!1; q!1;o; z

0� � ¼ 1

2t0ppjp¼q1

k21o2

; ð40Þ

y2 z00ð Þ ¼ y k!2; q!2;o; z

00� � ¼ 1

2t0ppjp¼q2

k22o2

; ð41Þ

and

r!1 z0ð Þ ¼ r

!q!1; z

0� �; r

!2 z00ð Þ ¼ r

!q!2; z

00� �: ð42Þ

The coherence function between c1 and c2 is

hc1c2i ¼ 2pð Þ�4

ðL0

ðL0

a1 z0ð Þa2 z00ð Þdz0dz00

�ðð

k!2

ððk!1

eio y1 z0ð Þ�y2 z00ð Þ½ �þik!2� r

!2z00ð Þ�x

!2½ ��ik!

1� r!1z0ð Þ�x

!1½ �

� �dn k!1; z

0� �dn k!2; z

00� ��; ð43Þ

where h�i is the ensemble average from multiple realizations of the random medium. Fora brief introduction on random variables and random function, and several usefulcorrelation functions and their spectral representations, see Appendix. It can be shownthat in the 3-D case (Tatarskii, 1971) that

�dn k!1; z

0� �dn k!2; z

00� �� ¼ 2pð Þ2W k!1; z0; z00

� �d k!1 � k!2� �

d2k!1 d2k!2: ð44Þ

In the 2-D case, we need replace 2pð Þ2 by 2p at the right-hand side of Eq. (44). In view ofthe simple substitution of dn by nd2k! in Section 4, we see that W k!1; z

0; z00� �

is the

29THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

correlation function of the two spectral fields n k!1; z0� �and n k!1; z

00� �at depths z0 and z00,

respectively. Applying the coordinate transformation

� ¼ z0 � z00; x ¼ z0 þ z00

2ð45Þ

in (43) and if the correlation function of the heterogeneity is slowly varying with depth,we can approximate W as

W k!; z0; z00

� �� W x; k

!; �

� �: ð46Þ

Equation (43) can be simplified as

hc1c2i ¼ 2pð Þ�2

ðL0

ðL0

a1 xþ �

2

� �a2 x� �

2

� �dx d�

�ððk!eio½y1ðxþ

�2Þ�y2ðx��

2Þ�þi k! �½r!2ðx��

2Þ�x!

2��i k! �½r!1ðxþ�

2Þ�x!

1�Wðx; k!; �Þd2 k!

ð47ÞBecause the correlation function W decreases rapidly with the vertical separation dis-tance �j j, we can extend the integration limit of � from �1 to þ1 without introducingmuch error. We also make following approximations:

a1 xþ �

2

� �� a1 xð Þ; a2 x� �

2

� �� a2 xð Þ; ð48Þ

r!1 xþ �

2

� �� r

!1 xð Þ; r

!2 x� �

2

� �� r

!2 xð Þ: ð49Þ

Using Eqs. (48) and (49) in (47), we obtain

hc1c2i ¼ ð2pÞ�2

ðL0

dxðþ1

�1a1ðxÞa2ðxÞd�

�ðð

k!eio½y1ðxþð�=2ÞÞ�y2ðx�ð�=2ÞÞ�þik! �½r!2ðxÞ�x!

2��i k! �½r!1ðxÞ�x!

1�Wðx; k!; �Þd2 k! :

ð50ÞThe integral value (50) is nonnegligible only when �j j is small, so we can have thefollowing expansion

y1 xþ �

2

� �� y2 x� �

2

� �� y1 xð Þ � y2 xð Þ þ y�1 þ y�2

2�; ð51Þ

where

y�1 ¼ @y1 zð Þ@z

����z¼x

; y�2 ¼ @y2 zð Þ@z

����z¼x

: ð52Þ

30 ZHENG AND WU

Taking into account Eq. (51), Eq. (50) can be written as

hc1c2i ¼ 2pð Þ�2

ðL0

a1 xð Þa2 xð Þdxðð

k!d2 k! eio y1 xð Þ�y2 xð Þ½ �þik!� r

!2 xð Þ�x

!2½ ��i k! r

!1 xð Þ�x

!1½ �

�ðþ1

�1d�ei o=2ð Þ y�1þy�2ð Þ�W x; k!; �

� �: ð53Þ

The depth-dependent power spectrum P and correlation functionW are Fourier transformpairs, which can be shown as

W x; k!; �� �

¼ 1

2p

ðþ1

�1P x; k!; kz� �

eikz�dkz: ð54Þ

Combination of Eqs. (53) and (54) yields

hc1c2i � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞððð

k!d2 k! ei k

! �½r!2ðxÞ�x!2��ik! �½r!1ðxÞ�x!

1�

� eio½y1ðxÞ�y2ðxÞ�P x; k!;oð _y1 þ _y2Þ

2

" #:

ð55Þ

We can also derive

hc1c2i � � 2pð Þ�2

ðL0

dxa1 xð Þa2 xð Þððð

k!d2 k! eik

! � r!2xð Þ�x

!2½ ��ik!� r

!1xð Þ�x

!1½ �

� eio y1 xð Þþy2 xð Þ½ �P x; k!;o _y1 � _y2� �

2

24

35: ð56Þ

The Log amplitude u and phase f can be expressed by the complex phase

u ¼ cþ c

2and f ¼ c� c

2i: ð57Þ

Using this relation to the two plane waves, 1 and 2, we have log A coherence function

hu1u2i ¼ 1

2Rehc1c

2i þ

1

2Rehc1c2i; ð58Þ

and the phase coherence function

hf1f2i ¼1

2Rehc1c

2i �

1

2Rehc1c2i: ð59Þ

31THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

We can also form the logA-phase coherence function

hu1f2i ¼1

2Imhc1c2i �

1

2Imhc1c

2i: ð60Þ

If q!1 ¼ q!2 , the two plane waves coming from same direction, we obtain TCFs and theydepend only on the spatial lag x!1 � x!2. Note that these waves are not necessarily verticalincidences as in the Chernov theory. If x!1 ¼ x!2, the ACFs are obtained for plane waves,q!1 and q!2. Because the background velocity profile is depth dependent, these ACFsdepend on q!1 and q!2 independently, not necessarily the difference q!1 � q!2. The mostgeneral case is JTACF, x!1 6¼ x!2; q

!1 6¼ q!2.

For 3-D isotropic heterogeneous media, the coherence functions can be simplifiedusing the following identity (Abramowitz and Stegun, 1965)

2pJ0 kRð Þ ¼ð2p0

e�ikRcosada; ð61Þ

where J0 is the 0th order Bessel function. Therefore, we arrive at

hc1c2i � 2pð Þ�1

ðL0

dxa1 xð Þa2 xð Þ �ð10

kdkJ0 kR xð Þ½ �eio y1 xð Þ�y2 xð Þ½ �

� P x; k;o _y1 þ _y2� �

2

24

35; ð62Þ

where R xð Þ ¼ r!2 xð Þ � x!2 � r!1 xð Þ þ x!1�� �� has an obvious meaning, the transverse distance

between two rays at depth x, with slownesses q!1 and q!2, respectively (see Fig. 1).Likewise,

hc1c2i � � 2pð Þ�1

ðL0

dxa1 xð Þa2 xð Þ �ð10

kdkJ0 kR xð Þ½ �eio y1 xð Þþy2 xð Þ½ �

� P x; k;o _y1 � _y2� �

2

24

35: ð63Þ

For 2-D case, Eqs. (55) and (56) are

hc1c2i � 2pð Þ�1

ðL0

dxa1 xð Þa2 xð Þðkdkeik r2 xð Þ�x2½ ��ik r1 xð Þ�x1½ � � eio y1 xð Þ�y2 xð Þ½ �

� P x; k;o _y1 þ _y2� �

2

24

35; ð64Þ

hc1c2i � � 2pð Þ�1

ðL0

dxa1 xð Þa2 xð Þðkdkeik r2 xð Þ�x2½ ��ik r1 xð Þ�x1½ �

� eio y1 xð Þþy2 xð Þ½ �P x; k;o _y1 � _y2� �

2

24

35: ð65Þ

32 ZHENG AND WU

6. Coherence Functions Using Delta-Correlated Assumption

The delta-correlated assumption between two depths is often invoked to simplifycomputation for wave propagation in stochastic media (Tatarskii, 1971; Ishimaru, 1978;Wu and Flatte, 1990)

�dn k!1; z

0� �dn k!2; z

00� �� ¼ 2pð ÞmW k!1; z0; z00

� �d k!1 � k!2� �

d z0 � z00ð Þdk!1dk!2; ð66Þ

in which m ¼ 1 for 2-D case and m ¼ 2 for 3-D case. Under this assumption, the verticalwave number of the power spectrum P is zero, thus Eqs. (55) and (56) reduce to

hc1c2i � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞðð

k!d2 k! eik½r2ðxÞ�x2��ik½r1ðxÞ�x1�

� eio½y1ðxÞ�y2ðxÞ�Pðx; k!; 0Þ; ð67Þ

and

hc1c2i � � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞðð

k!d2 k! eik½r2ðxÞ�x2��ik½r1ðxÞ�x1�

� eio½y1ðxÞþy2ðxÞ�Pðx; k!; 0Þ: ð68Þ

Various coherence functions according to Eqs. (58)–(60) can be formed as the following:

hu1u2i � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞ

�ðð

k!ei k

! �½r!2ðxÞ�x!2��ik! �½r!1ðxÞ�x

!1� sin½oy1ðxÞ� sin½oy2ðxÞ�Pðx; k!; 0Þd2 k!;

ð69Þ

hf1f2i � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞ

�ðð

k!ei k

! �½ r!2ðxÞ�x!

2��i k! �½r!1

ðxÞ�x!1� cos½oy1ðxÞ�cos½oy2ðxÞ�Pðx; k!; 0Þd2 k!;

ð70Þ

hu1f2i � � ð2pÞ�2

ðL0

dxa1ðxÞa2ðxÞ

�ðð

k!eik

! �½r!2ðxÞ�x

!2��i k! �½r!1

ðxÞ�x!1� sin½oy1ðxÞ�cos½oy2ðxÞ�Pðx; k!; 0Þd2 k! :

ð71ÞUsing identity (61), we can explicitly obtain three types of coherence functions in 3-D:

33THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

hu1u2i � ð2pÞ�1

ðL0

dxa1ðxÞa2ðxÞð10

J0½kRðxÞ�sin½oy1ðxÞ�sin½oy2ðxÞ�Pðx; k!; 0Þkdkð72Þ

hf1f2i � ð2pÞ�1

ðL0

dxa1ðxÞa2ðxÞð10

J0½kRðxÞ�cos½oy1ðxÞ�cos½oy2ðxÞ�Pðx; k!; 0Þkdkð73Þ

hu1f2i � �ð2pÞ�1

ðL0

dxa1ðxÞa2ðxÞð10

J0½kRðxÞ�sin½oy1ðxÞ�cos½oy2ðxÞ�Pðx; k!; 0Þkdk:ð74Þ

7. Coherence Functions in a Constant Background Medium

To compare our results with those in Wu and Flatte (1990), we assume that thebackground velocity model is homogeneous and the incident angle is small. Under theseconditions, the t function can be approximated as

t x; pð Þ ¼ðx0

c�2 � p2� �1=2

dz � c�1 1� 1

2p2c2

� �x: ð75Þ

Phase functions y1 and y2 can be explicitly solved

oy1 ¼ � 1

2cx

k2

o¼ � x

2kk2: ð76Þ

The amplitudes can be approximated as

a1 xð Þa2 xð Þ � k2: ð77ÞSubstituting Eqs. (75)–(77) into Eqs. (72)–(74), the generalized results in this chapterreduce to those contained in Wu and Flatte (1990) for a homogeneous backgroundmedium.

8. Numerical Examples

To demonstrate the validity of the theory proposed in this chapter, we compared thecoherence functions from numerical simulations and theoretical predictions. We con-structed a 2-D random model (Fig. 2a) with a background velocity profile same as theiasp91 model (Fig. 2b; Kennett and Engdahl, 1991). Two random layers are super-imposed on the background model. The top layer is from 0 to 120 km in depth and ithas a Gussian correlation function with correlation length 10 km in both horizontal andvertical directions, and rms 1% of the background velocity for those random velocityperturbations. The bottom layer extending from 120 to 310 km depth also has a Gaussian

34 ZHENG AND WU

correlation function with correlation length 20 km in both directions and rms 1% of thebackground velocity. For both layers, we constrain the velocity perturbations notexceeding �3% of the background velocity. Below 310 km depth, the medium ishomogeneous and has no random perturbations and the velocity is same as the one atthe 310 km depth. We used a full-wave finite difference code (Xie, 1988) to simulate theacoustic plane wave propagation in the random medium. The spectral amplitude andphase fluctuations at a given frequency are extracted from the waveforms recorded at thesurface (Zheng and Wu, 2005). Then coherence functions are formed using thosefluctuations and averaged over an ensemble of 100 different stochastic model realiza-tions. The logA (Fig. 3) and phase (Fig. 4) coherence functions at 0.5 Hz from thenumerical simulations are well predicted by our formula for four different incidencegeometries between two plane waves. The incidence angle is measured at the surface forconsistency. The logA coherence functions in Fig. 3 are sensitive to the angular separa-tion between two incident plane waves. However, phase coherence functions are rela-tively simple (Fig. 4) in shape. Another salient feature is that the maximum amplitude ofthe coherence function (for both logA and phase) is decreasing with increasing angularseparation between the two plane waves, which is expected. It is also interesting to notice

500

(a)

0

400

200

0

1000X (km)

Dep

th (

km)

−0.2

0 km/s

0.2

(b)

5.5500

450

400

350

300

250

200

150

100

50

0

6 6.5 7 7.5

Velocity (km/s)

8 8.5 9

Dep

th (

km)

FIG. 2. (a) Random velocity model used in the 2-D numerical modeling; (b) 1-D iasp91

background velocity model (solid line) and a vertical velocity profile (dash line) at location X ¼600 km, to indicate the randomness of the velocity fluctuation.

35THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

that there are several velocity discontinuities in the iasp91 background model and thisseems to pose difficulty to using the WKBJ Green’s function as discontinuities canproduce reflected waves. However, because we are using plane wave incidence, thosediscontinuities affect all receivers in a similar fashion and such effect is removed duringcoherence function formation.

9. Validity of the Delta-Correlated Assumption

In Section 6, we see that the delta-correlated assumption significantly simplifies themathematical derivation of the coherence functions. This assumption is equivalent tocertain conditions under which the result of the coherence function is same to thatobtained as if the medium is delta-correlated along the depth direction. The conditionis easily found to be k2�=2k 1 in view of Eqs. (53) and (76) for a homogeneousbackground medium. We assume that L is the smallest correlation length at whichW x; k!; �ð Þ is only slightly different from zero. We also assume that P x; kð Þ~0 ifk > km ¼ 2p=‘0, with ‘0 being the inner scale of the random medium. Therefore, ifcondition lL ‘0 is satisfied, the delta-correlated assumption will hold. Here l is thewavelength. Of course, we generally further require ‘0 < L. To assess the error

0.06

0.04

0.02

0

−0.02−200

(a) (b)

(c) (d)

0 200

0.04

0.02

0

−0.04

−0.02

−200 0 200

0.02

0.01

0

−0.02

−0.01

−200 0 200x (km)

<u 1

u 2>

<u 1

u 2>

x (km)

0.015

0.01

0.005

0

−0.01

−0.005

−200 0 200

FIG. 3. log A coherence functions hu1u2i for two plane waves with incidence angles, 0� and

0� (a), 0� and 5� (b), 0� and 10� (c), and 0� and 15� (d). Numerical results are indicated as dash lines

and the theoretical predictions are the solid lines.

36 ZHENG AND WU

introduced by this assumption, we need numerical verification for a given randommodel.For a homogeneous background medium, Zheng et al. (2007) numerically found thatthe delta-correlated assumption will cause little error if the layer thickness of therandom medium is large for short wave propagation. Using the model presented inFig. 2, we computed coherence functions with (Section 6) and without (Section 5) thedelta-correlated assumption and we found the results are basically the same. The delta-correlated assumption will not significantly simplify the coherence function calculation.However, it does provide convenience for us to invert for the depth-dependent randomspectrum. With the delta-correlated assumption, we can specify the unknowns easilywithout concerning the vertical wave number in the spectrum P.

10. Discussions and Conclusions

Theory of the coherence function for both log A and phase in a depth-dependentbackground velocity model is important to draw correct inference on the randommediumproperty. In the past, coherence functions have been theorized in the context of using aconstant background medium, thus a simple Green’s function. In this chapter, the naturalspectral representation of the Green’s function in the wave number domain allows us toextend the previous theory to a scenario where the backgroundmedium is depth-dependent.

0−200−0.05

0

0.05

<f 1

f 2>

<f 1

f 2>

0.1

0.15(a) (b)

(c) (d)

200 0−200−0.02

0

0.02

0.04

0.06

0.08

200

0−200−0.02

0

0.02

0.04

0.06

200x (km)

0−200−0.02

0

0.02

0.04

0.06

200x (km)

FIG. 4. Phase coherence functions hf1f2i for two plane waves with incidence angles, 0� and

0� (a), 0� and 5� (b), 0� and 10� (c), and 0� and 15� (d). Numerical results are indicated as dash lines

and the theoretical predictions are the solid lines.

37THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA

The agreement between the numerical simulation and the theoretical prediction showsthe correctness of the theory and the potential to apply the theory to real seismic data. Weformulated the coherence function both in 3-D case and in 2-D case, with and without thedelta-correlated assumption. The delta-correlation does not lead to discernable differ-ences for the model we used to do the simulation. However, with the delta-correlatedassumption, the parameterization for the inverse problem can be easier.

The previous theory on the coherence function C for the log A or for the phase dependson the spatial lag between stations, and the angle separation between the two planewaves, C ¼ C x

!2 � x

!1; q

!2 � q

!1

� �. However, this is valid only when the slowness vectors

of the two plane waves are close, that is q!1 � q

!2. This condition cannot always be true

and it also limits the depth resolution of the coherence function, resulting the spectralsmearing along the depth direction. Our current theory is capable of dealing with muchlarger angular separation between the two plane waves and the mathematical formulationfor the coherence function depends on q

!1 and q

!2 individually, and it takes a function form

of C ¼ C x!2 � x

!1; q

!2; q

!1

� �.

Acknowledgments

Comments from Professor Haruo Sato, Dr. Tatsuhiko Saito, and an anonymous reviewer have

greatly improved the clarity of the chapter. We also thank Dr. Xiao-bi Xie for reading our

manuscript and discussions. Y. Zheng is also grateful to Professor Thorne Lay for his encourage-

ment. We thank the W.M. Keck Foundation for facility support. This work is funded in part by

DOE/Basic Sciences, NSF EAR0635570, and the Wavelet Transform on Propagation and Imaging

Consortium/UCSC. This is Contribution Number 495 of CSIDE, IGPP, University of California,

Santa Cruz.

Appendix: Random Variables, Random Functions

The concept of random variables (RVs) occupied a central role in stochastic analysis.Excellent books on this topic are by Papoulis (1965) and Yaglom (1962). Here we give abrief overview of this concept. In probability theory, the outcomes of experimentsconducted under identical condition form a set, denoted by O ¼ z1; z2; � � �f g. O0 can bedenumerable or not. If we associate each event zi a numerical value v zið Þ, the statistic likethe mean hv zið Þi and the variance can be computed. The values of v can be either discreteor continuous. Quite often, people have suppressed the explicit dependence of v on theexperiment and its possible outcomes zi 0s. This simplified notation frequently causedconfusion. A random function is just a collection of RVs typically varying with time orlocation, f x

!� �. If we choose one element from the event set O at each location x

!, we

obtain a realization. We can produce many realizations and calculate the statistic (mostinterestingly the coherence function to us) using the ensemble average.

The spectral theory of the random field involves stochastic Fourier–Stieltjes integraltheory to overcome some theoretical problems, like the absolute integrability of therandom functions. In practice, this is not a serious problem if we assume that the randommedium has spatial periodicity. Usually the correlation function is used to characterize arandom medium. We assume that the mean of the ensemble is zero. If the mean is notvanishing, we can first subtract the mean. The correlation function B r

!1; r

!2

� �is defined as

38 ZHENG AND WU

B r!1 � r

!2

� � ¼ h f r!1

� �f r

!2

� �i. Symbol h�i is used to denote the ensemble average. Bystationary random medium we mean that the statistic does not change by a translation d

!,

for example, Bðr!1; r!2Þ ¼ Bðr!1 þ d!; r!2 þ d

!Þ. If the correlation function only depends onthe Euclid distance between r

!1 and r

!2, the medium is called isotropic, that is,

B r!1; r

!2

� � ¼ Bðjx!1 � x!2jÞ. To obtain the isotropic spectrum P kð Þ, we use a Fourier

transform of the correlation function, that is, P kð Þ ¼ ÐB rð Þe�ikrdr. In numerical studies,

it is practical to produce realizations of random media of certain correlation function andthis can be done in the spectral domain (Shapiro and Kneib, 1993). We first assign thespectral amplitude at each wave number then generate random phases in range �p; p½ �.Of course, the complex conjugate symmetry has to be used if a real random mediumis desired. We list two most common correlation functions and their spectra. TheGaussian correlation function reads B rð Þ ¼ e2exp �r2=r20

� �. e2 is the perturbation

strength and r0 is the correlation length. Its Fourier transform in N dimension isP kð Þ ¼ e2rN0 p

N=2exp �k2r20=4� �

. The exponential correlation function can be expressedas B rð Þ ¼ e2e� r=r0j j. Its Fourier transforms are P kð Þ ¼ 2e2r0= 1þ k2r20

� �in the 1-D case

and P kð Þ ¼ 8p3e2r30= 1þ k2r20� �2

in 3-D case. These results can be easily generalized toanisotropic cases, in which the correlation lengths in different directions (with correla-tion lengths, r0x; r0y, and r0z, in x, y, and z directions, respectively) can be different,r0x 6¼ r0y 6¼ r0z. The simplest way is to do the following replacement, k2 � k2x þ k2y þ k2z ;r20 � r0xr0y in 2-D and r30 � r0xr0yr0z in 3-D case.

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41THEORY OF TRANSMISSION FLUCTUATIONS IN RANDOM MEDIA