Cross-correlation function of acoustic fields generated by random high-frequency sourcesa) Oleg A. Godinb兲 CIRES, University of Colorado and NOAA/Earth System Research Laboratory, DSRC, Mail Code R/PSD99, 325 Broadway, Boulder, Colorado 80305-3328
共Received 9 February 2010; revised 29 May 2010; accepted 2 June 2010兲 Long-range correlations of noise fields in arbitrary inhomogeneous, moving or motionless fluids are studied in the ray approximation. Using the stationary phase method, two-point cross-correlation function of noise is shown to approximate the sum of the deterministic Green’s functions describing sound propagation in opposite directions between the two points. Explicit relations between amplitudes of respective ray arrivals in the noise cross-correlation function and the Green’s functions are obtained and verified against specific problems allowing an exact solution. Earlier results are extended by simultaneously accounting for sound absorption, arbitrary distribution of noise sources in a volume and on surfaces, and fluid inhomogeneity and motion. The information content of the noise cross-correlation function is discussed from the viewpoint of passive acoustic characterization of inhomogeneous flows. © 2010 Acoustical Society of America. 关DOI: 10.1121/1.3458815兴 PACS number共s兲: 43.30.Pc, 43.20.Bi, 43.30.Nb, 43.60.Rw 关RKS兴
I. INTRODUCTION
It is well known that, under certain restrictive assumptions about diffuse wave fields and their spatially distributed sources, an exact deterministic Green’s function can be retrieved from the two-point correlation function of diffuse wave fields.1–9 This property of diffuse wave fields suggests a passive approach to remote sensing of inhomogeneous media through noise interferometry,10–17 where use of controlled sources of probing signals is obviated and replaced by coherent processing of ambient noise, multiply scattered waves from a powerful localized source, or wave fields generated by incoherent sources of opportunity. However, with the exception of thermal noise,1–3,18–21 the restrictive conditions needed for exact retrieval of deterministic Green’s functions from cross-correlations of diffuse wave fields are hardly ever met in practice. In other words, in geophysical applications, fields of electromagnetic and mechanical waves are often diffuse but are never the “perfectly diffuse” wave fields implied in the theories. Moreover, to be practical, the noise interferometry should be capable of retrieving important physical parameters of inhomogeneous media without detailed knowledge of the properties of the noise field or its sources. Valuable information about the propagation medium can be obtained from wave fields without full knowledge of the deterministic Green’s function. In particular, acoustic travel times measured between a number of locations can be inverted for sound speed and flow velocity fields in fluids.22,23 Using stationary-point arguments, Snieder24 showed that, in a兲
Parts of this work have been previously reported at the 157th ASA Meeting 共Portland, OR, May 2009兲 and the 12th L. M. Brekhovskikh Conference on Ocean Acoustics 共Moscow, Russia, June 2009兲. b兲 Author to whom correspondence should be addressed. Electronic mail:
[email protected] 600
J. Acoust. Soc. Am. 128 共2兲, August 2010
Pages: 600–610
a homogeneous medium, the global requirement of the energy equipartitioning between normal modes can be relaxed and replaced by local conditions on noise sources distribution in the vicinity of the line connecting the receivers. Godin8 extended the results of Snieder24 to inhomogeneous and moving media and demonstrated that, when random sound sources are located on a surface in a lossless fluid, one can retrieve from two-point correlation function of noise the deterministic ray travel times for sound propagating in opposite directions between the receivers as long as 共unknown兲 spatial distribution of the noise sources is sufficiently smooth. In this paper, we apply the approach of Ref. 8 to more general propagation media and noise source distributions and show that deterministic travel times can be retrieved from noise fields under much less restrictive conditions than the exact Green’s functions. For noise sources distributed on a surface or in the volume of an inhomogeneous fluid, an explicit relation is established between contributions of ray arrivals to the noise cross-correlation function and the Green’s function. Earlier results obtained for homogeneous fluids,7,9,24 noise sources on a surface in a lossless medium,5,8,24 and for special distributions of noise sources7–9,24 are extended by simultaneously accounting for sound absorption, fluid inhomogeneity and motion, and arbitrary distribution of noise sources in a volume and on surfaces. The remainder of the paper is organized as follows. Section II introduces deterministic Green’s functions and provides a theoretical background for description of random acoustic fields and their correlations in motionless or moving inhomogeneous fluids. The ray approximation is discussed in Section III and applied in conjunction with the stationary phase method to obtain the dominant terms of highfrequency asymptotic expansion of the cross- and autocorre-
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lation functions of the noise field generated by random sources distributed on a surface. The high-frequency asymptotics of noise fields generated by random volumetric sources are derived in Section IV. In Section V, the asymptotic relations between the noise cross-correlations and the deterministic Green’s functions obtained in Sections III and IV are verified against known exact solutions to highly idealized problems. Properties of the noise cross-correlation function revealed by the asymptotic results and their implications for possible applications of the noise interferometry in underwater and atmospheric acoustics are discussed in Section VI. II. RANDOM SOUND FIELDS AND THEIR SOURCES
Consider acoustic waves generated by random sources in a stationary 共i.e., independent of time t兲, inhomogeneous, moving fluid with sound speed c共x兲, mass density 共x兲, and flow velocity u共x兲. The environmental parameters c, , and u are smooth functions of the position x. The fluid occupies a domain ⍀, which can be finite or infinite and can have ideal 共pressure-release and/or rigid兲 and impedance boundaries. Sources of volume velocity and external force are distributed in a ⍀ with volume densities dB / dt and F, respectively. Here d / dt = / t + u · ⵜ is the convective time derivative. In addition to volume sources, sound can be generated by sources of volume velocity and normal external force distributed on a surface S with surface densities db / dt and f, respectively. For continuous waves 共CW兲 we will assume and suppress the time dependence exp共−it兲 of acoustic pressure p, oscillatory displacement of fluid particles w, and other field quantities. The oscillatory displacement w is related to the oscillatory velocity v of fluid particles, i.e., the linear perturbation of fluid velocity due to an acoustic wave, by the equation25,26 v = dw / dt − 共w · ⵜ兲u. Let the volume and surface sources be statistically independent, the densities of the sources be delta-correlated in space and have zero statistical mean: 具B共x兲典 = 0, 具F共x兲典 = 0 , 具b共x兲典 = 0 , 具f共x兲典 = 0. In the frequency domain, 具B共x兲Bⴱ共y兲典 = Q共1兲共x兲␦共x − y兲,
共1兲 共2兲 = 1 , 2. Then, by definition, ␦S共x1 − x2兲 = ␦共共1兲 1 − 2 兲␦共1 共2兲 − 2 兲. As in Refs. 8 and 20, we introduce the Green’s function G共x , y , 兲 and vector g共x , y , 兲 as the spectra of, respectively, the acoustic pressure and the oscillatory displacement at point x due to a point source of volume velocity located at the point y, such that B = ␦共x − y兲␦⬘共t兲 and F = 0. In the case of a motionless fluid, the Green’s function so defined corresponds to the field of a monopole sound source. For the ˜ in the original medium and in a Green’s functions G and G medium with reversed flow, i.e., a medium with the same geometry and properties of boundaries and with parameters c共x兲, 共x兲, and −u共x兲, a simple symmetry relation25,26 holds:
˜ 共y,x, 兲. G共x,y, 兲 = G
The Green’s vector g does not necessarily have such a symmetry.25,26 The following identity20 holds in an arbitrary inhomogeneous, moving fluid: G共x2,x1, 兲 + Gⴱ共x1,x2, 兲 =i
冕
ds共x兲
⍀
+
共G1gⴱ2
ⴱ
具b共x兲f 共y兲典 = q 共x兲␦S共x − y兲, 具f共x兲f ⴱ共y兲典 = q共3兲共x兲␦S共x − y兲,
冕 冉 冊 d3x Im
⍀
1 关G1Gⴱ2 c2
· ⵜp0 + 共g1 · ⵜp0兲共gⴱ2 · ⵜp0兲兴,
冕 冕 ⍀
共3兲
i
˜ 共y,x, 兲 − F共y兲 · ˜g共y,x, 兲兴 d3y关B共y兲G ˜ 共y,x, 兲 − f共y兲N共y兲 · ˜g共y,x, 兲兴, ds共y兲关b共y兲G
S
共4兲
共1兲
where the asterisk designates complex conjugation. Dependence of the random functions B, F, b, f as well as of the deterministic functions Q共j兲 and q共j兲 on frequency is implied and not indicated explicitly for brevity. In Eq. 共1兲, ␦S is a two-dimensional delta-function defined on S. Let the surface be defined by the equation x = X共共1兲 , 共2兲兲, where 共1兲 and 共2兲 are curvilinear coordinates on S, the differential of 共1兲 共2兲 the surface area ds = d共1兲d共2兲, and xm = X共m , m 兲 , m J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
i +
共2兲
+
Gⴱ2g1兲
2
where parts of the boundary ⍀ of the domain ⍀ may lie at infinity, N is the external unit normal to ⍀, x1,2 苸 ⍀, the gradient of the background pressure ⵜp0 = −共u · ⵜ兲u, Im = Im u = 0, G j ⬅ G共x , x j , 兲, and g j ⬅ g共x , x j , 兲. The identity 共3兲, which is an acoustic counterpart of the Lorentz lemma for electromagnetic waves,3 is instrumental for the derivations in Sections III and IV below. Acoustic pressure generated by an arbitrary distribution of CW sources equals20,21
p共x, 兲 =
具b共x兲bⴱ共y兲典 = q共1兲共x兲␦S共x − y兲,
N · 关u共g1共u · ⵜ兲gⴱ2 − gⴱ2共u · ⵜ兲g1
+ 2ig1 · gⴱ2兲 − G1gⴱ2 + Gⴱ2g1兴 −
具B共x兲Fⴱ共y兲典 = Q共2兲共x兲␦共x − y兲, 共3兲 具Fm共x兲Fⴱj 共y兲典 = Qmj 共x兲␦共x − y兲,
共2兲
where N is a unit normal to S and the tilde “~” indicates quantities referring to a medium with reversed flow. Note that integration in Eq. 共4兲 is over receiver positions and involves Green’s functions and vectors in a medium with reversed flow. From Eqs. 共1兲 and 共4兲 it follows that the acoustic field generated by the random sources has zero mean, and the cross-correlation function C共x1 , x2 , 兲 ⬅ 具p共x1 , 兲pⴱ共x2 , 兲典 of the acoustic pressure is Oleg A. Godin: Cross-correlation of high-frequency noise
601
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C共x1,x2, 兲 = −2
冕
⍀
冋
plied by the large parameter in exponentials in Eqs. 共6兲 and 共7兲, the main effect of weak absorption on the acoustic field is via the eikonals. In the sound speed c = c1 + ic2, we consider c2 ⬅ Im c as perturbation. It follows from the Fermat principle27 that first-order eikonal corrections can be calculated as integrals of the perturbation along the unperturbed rays, which in our case are the rays in non-absorbing fluid. Using results of Ref. 27, we have
˜ 共y,x , 兲G ˜ ⴱ共y,x , 兲 d3y Q共1兲共y兲G 1 2
˜ 共y,x , 兲 − Q共2兲共y兲 · ˜gⴱ共y,x2, 兲G 1 ˜ ⴱ共y,x , 兲 − Q共2兲ⴱ共y兲 · ˜g共y,x1, 兲G 2 3
+
共3兲 ˜ m共y,x1, 兲g ˜ ⴱj 共y,x2, 兲 Qmj 共y兲g 兺 m,j=1
+ −2
冕
册
S
m共2兲共x,y兲 = −
˜ⴱ
˜ 共y,x , 兲N共y兲 · g 共y,x , 兲 − q 共y兲G 1 2
冕
e · dr , cg
x
hm Im c
y
˜ ⴱ共y,x , 兲N共y兲 · ˜g共y,x , 兲 + q共3兲共y兲 − q共2兲ⴱ共y兲G 2 1 ⫻共N共y兲 · ˜g共y,x1, 兲兲共N共y兲 · ˜gⴱ共y,x2, 兲兲兴.
x
y
˜ 共y,x , 兲G ˜ ⴱ共y,x , 兲 ds关q共1兲共y兲G 1 2
共2兲
冕
m共1兲共x,y兲 =
共5兲
The general expression 共5兲 for noise cross-correlation can be greatly simplified when the Green’s functions and vectors are replaced by their high-frequency asymptotics. III. HIGH-FREQUENCY NOISE SOURCES DISTRIBUTED ON A SURFACE A. The geometrical acoustics „ray… approximation
We assume that sound absorption in a fluid is small over distances of the order of the wavelength and will model the absorption by attributing a small negative imaginary part to sound speed c, which can depend on coordinates and wave frequency. In the ray approximation,26 G共x,y, 兲 = 兺 Am共x,y兲exp关im共x,y兲兴,
hm共x,y兲 =
e · dr , cg
共8兲
共1兲 共x,y兲, x m
where the integrals are taken along the m-th ray traveling from point y to point x, hm and cg = u + c1hm / hm are the slowness vector and the group velocity on the ray, and e = cg / cg is a unit vector tangent to the ray. 共In the motionless case, cg = c1 and hm = e / c1.兲 The real and imaginary parts of the eiko共1兲 共2兲 nal are non-negative, and m 共x , y兲 Ⰷ m 共x , y兲. Below, we will disregard the small differences between the ray amplitudes Am and the slowness vectors in media with and without absorption. Note that the ray, which travels from point y to point x in the original medium coincides with the ray traveling from point x to point y in the medium with reversed flow.26 It follows then from Eq. 共8兲 that
m共n兲共x,y兲 = ˜m共n兲共y,x兲,
n = 1,2.
共9兲
m
m = m共1兲 + im共2兲 ,
冋
m共x,y兲 i 1 − u共x兲 · g共x,y, 兲 = 兺 共x兲 m x ⫻exp关im共x,y兲兴
册
共6兲 −2
Am共x,y兲
m共x,y兲 . x
共7兲
B. Noise cross-correlation function
Consider cross-correlation of high-frequency noise fields generated in a finite volume ⍀ by random sources distributed on its boundary ⍀. In the ray approximation, substitution of Eqs. 共6兲 and 共7兲 into Eq. 共5兲 gives C共x1,x2, 兲 = 兺
m,n
Here Am and m are complex amplitudes and eikonals of 共1兲 共2兲 individual ray components of the field, m and m are real 共2兲 and imaginary parts of the eikonal. With m being multi-
冕
⍀
ds共y兲⌽1共y,x1,x2, 兲
共1兲 ˜m ˜ 共1兲 共y,x1兲 − i ⫻exp关i n 共y,x2兲兴,
共10兲
where
共2兲 ⌽1共y,x1,x2, 兲 = −2Am共x1,y兲Aⴱn共x2,y兲exp关− m 共x1,y兲 − 共2兲 n 共x2,y兲兴
再
⫻ q共1兲共y兲 + q共3兲共y兲 ⫻
602
冋
共N共y兲 · hm共x1,y兲兲共N共y兲 · hn共x2,y兲兲
2 2
2 共y兲c41共y兲hm 共x1,y兲h2n共x2,y兲
共2兲
−
q 共y兲 q共2兲ⴱ共y兲 N共y兲 · h N共y兲 · hm共x1,y兲 共x ,y兲 − n 2 2 h2n共x2,y兲 hm 共x1,y兲
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
i 共y兲c21共y兲
册冎
.
共11兲
Oleg A. Godin: Cross-correlation of high-frequency noise
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The dispersion equation of sound in moving fluid26 共12兲
hc1 + h · u = 1
and Eqs. 共2兲 and 共9兲 were used in the derivation of Eq. 共11兲. The integral in Eq. 共10兲 contains a rapidly oscillating exponential, and high-frequency asymptotic expansion of the integral is dominated by contributions of the stationary points of the exponent.26 The stationary points ys 苸 ⍀ satisfy the equation 关hm共x1,ys兲 − hn共x2,ys兲兴 ⫻ N共ys兲 = 0,
共13兲
i.e., tangential 共to ⍀兲 components of the slowness vectors hm and hn coincide at the stationary point. It was demonstrated in Ref. 8 that there exists a one-to-one correspondence between the stationary points ys and the intersections between ⍀ and rays passing through both points x1 and x2. Namely, every stationary point lies on an extension of such a ray in a direction opposite to the direction of sound propagation, and every solution to Eq. 共13兲 defines a ray, that originates on ⍀ and passes through both points x1 and x2.8,28 Solutions to Eq. 共13兲 exist only when n = m. It follows then from Eqs. 共8兲 and 共9兲 that contribution of a stationary point to high-frequency asymptotics of the integral in Eq. 共1兲 共10兲 has either the eikonal −m 共x2 , x1兲, when point x1 lies 共1兲 共x1 , x2兲, between ys and x2 on the ray, or the eikonal m when point x2 lies between ys and x1 on the ray. The amplitude of the stationary point contribution can be calculated using the stationary phase method. The cumbersome calculation can be avoided8 by making use of the identity 共3兲. Consider fluid without absorption and substitute Eqs. 共6兲 and 共7兲 for the Green’s functions and vectors into Eq. 共3兲. The volume integral in Eq. 共3兲 vanishes when there is no absorption. For the leading-order terms of highfrequency asymptotics, we have
兺m Am共x1,x2兲exp关im共1兲共x1,x2兲兴 + 兺n Aⴱn共x2,x1兲
m,n
冕
⍀
−
˜ 共1兲 i n 共y,x2兲兴,
共14兲
where ⌽2共y,x1,x2, 兲 =
2Am共x1,y兲Aⴱn共x2,y兲
2共y兲c21共y兲
N共y兲
· H共hm共x1,y兲,hn共x2,y兲兲, H=
C共x1,x2, 兲 = 兺 Am共x1,x2兲Sm共ys共m兲兲eim
共1兲
hn 共hm · hn兲共hm + hn兲u共y兲 hm + . 2 + 2 2 2hm 2h2n 2c1共y兲hm hn
共15兲
共1兲 共x1 , x2兲 in As discussed above, the term with the eikonal m the left-hand side of Eq. 共14兲 originates as the contribution of the stationary point ys共m兲 苸 ⍀ in the right-hand side such that a ray travels from ys共m兲 first through x2 and then through x1 共Fig. 1兲. Similarly, the term with the eikonal −共1兲 n 共x2 , x1兲 in the left-hand side of Eq. 共14兲 is the contribution of the stationary point ys共n兲 苸 ⍀ such that a ray travels from ys共n兲 first
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
共x1,x2兲
m
+ 兺 Aⴱn共x2,x1兲Sn共ys共n兲兲e−in
共1兲
共x2,x1兲
,
共16兲
n
where, according to Eqs. 共11兲 and 共15兲,
冋
⫻
ds共y兲⌽2共y,x1,x2, 兲
共1兲 ˜m 共y,x1兲 ⫻exp关i
through x1 and then through x2. Note that in inhomogeneous, moving media the rays originating at ys共m兲 and ys共n兲 are generally distinct, and even the number of summands in sums over m and n in the left-hand side of Eq. 共14兲 共i.e., the number of eigenrays propagating between x1 and x2 in opposite directions兲 may differ. The integrands in Eqs. 共10兲 and 共14兲 have the same rapidly oscillating exponential and differ only by slowly varying factors ⌽1 and ⌽2. The dominant term of a stationary point contribution to the asymptotic expansion of an integral is proportional to the value of the slowly varying factor at the stationary point.26 Hence, the leading order of the highfrequency asymptotic expansion of the noise crosscorrelation function 共10兲 differs from the left-hand side of Eq. 共14兲 by multiplying Am and Anⴱ by the ratio ⌽1 / ⌽2 evaluated at respective stationary points:
Sm共y兲 = q共1兲 +
⫻exp关− i共1兲 n 共x2,x1兲兴 =兺
FIG. 1. 共Color online兲 Stationary points and deterministic eigenrays when noise sources are located on a surface in a quiescent 共a兲 or moving 共b兲 inhomogeneous fluid.
2共N · hm兲Im q共2兲
c21hm2
+
共N · hm兲2q共3兲
22c41hm4
共2兲 共2兲 共x1,y兲 − m 共x2,y兲兴 exp关− m 2 2关共N · hm兲c1 + hm共N · u兲兴/c31hm
册
.
共17兲
Note that the quantity 共N · hm兲c1 + hm共N · u兲 in a denominator in the right-hand side of Eq. 共17兲 equals cghm cos ␥m, where ␥m is the angle the m-th ray makes with the normal N on ⍀; cos ␥m ⬍ 0 for convex surfaces ⍀ 共Fig. 1兲.30 It follows from Eqs. 共6兲 and 共16兲 that the correlation function C共x1 , x2 , 兲 reproduces the sum of the Green’s function G共x1 , x2 , 兲 and the complex-conjugated Green’s function Gⴱ共x2 , x1 , 兲 with the amplitude shading of individual eigenray contributions, 共2兲 共x1 , x2兲兴 and which equals Sm共ys共m兲兲exp关m 共n兲 共2兲 Sn共ys 兲exp关n 共x2 , x1兲兴, respectively. The shading factors include exponentials that describe sound absorption on its way from the stationary point on ⍀ to the nearest 共along the ray兲 of the two receivers. Note that, on a dB scale, the loss due to absorption in the cross-correlation function is twice the loss at propagation of a deterministic signal between the same points. Naturally, the amplitude shading is proportional to the strength of noise sources in the vicinity of the respective stationary points. If there are no noise sources in the vicinity of a certain stationary point, the respective ray comOleg A. Godin: Cross-correlation of high-frequency noise
603
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ponent in the sum of the Green’s functions is not reproduced in the noise cross-correlation. Equations 共16兲 and 共17兲 have been derived above for noise fields generated in a finite volume ⍀ by sources distributed on its boundary ⍀. The noise fields in a finite or an infinite domain outside a surface ⍀ due to sources distributed on the surface can be considered quite similarly. The results 共16兲 and 共17兲 remain unchanged, except N in Eq. 共17兲 now has the meaning of the unit internal 共rather than external兲 normal to ⍀. 共2兲 In the absence of absorption 共m ⬅ 0兲 and sources of 共2兲 共3兲 external force 共q = q = 0兲, Eqs. 共16兲 and 共17兲 reduce to the results obtained in Ref. 8. It should be emphasized that the functions q共1,2,3兲 have been assumed to vary smoothly along ⍀. When q共j兲 or its derivatives are discontinuous, the highfrequency asymptotic expansion of the integral in Eq. 共10兲 contains additional terms,26 which are associated with the discontinuities rather than the stationary points. Then, the noise cross-correlation function contains “spurious” arrivals, the travel times of which are distinct from travel times of the eigenrays connecting points x1 and x2. C. Noise autocorrelation function
In order for the stationary phase method we used in Section III B to be applicable, the factor ⌽1 in the integrand in Eq. 共10兲 should be a slowly varying function compared to 共1兲 the exponential exp关imn共y兲兴, where mn共y兲 = ˜m 共y , x1兲 共1兲 − ˜n 共y , x2兲. Specifically, it is required that, if a curve ⌫ is the boundary of the vicinity ⌫ 傺 ⍀ of a stationary point ys 苸 ⍀ such that 兩共y兲 − 共ys兲兩 ⱕ / 兩⌽1共y兲 − ⌽1共ys兲兩 Ⰶ 兩⌽1共ys兲兩
共19兲
for all y 苸 ⌫.26 ⌫ has the meaning of the 共first兲 Fresnel zone on the surface ⍀, where the sound sources are located. Condition 共19兲 implies that relative changes in the sound speed and mass density of the fluid as well as in the density of the noise sources, are small within the Fresnel zone, or equivalently, that the Fresnel zone size is much smaller than the representative spatial scales of variation of the functions 共y兲, c共y兲, u共y兲 and q共1,2,3兲共y兲. The Fresnel zone size generally increases with decreasing separation between the receivers. Moreover, the exponential exp关imm共y兲兴 is a rapidly oscillating function of y
兵Sm其⍀ ⬅
604
冕
⍀
冕
C共x,x, 兲 = −2 兺
再
⍀
m
共2兲 ds共y兲兩Am共x,y兲兩2exp关− 2m 共x,y兲兴
⫻ q共1兲共y兲 + +
2N共y兲 · hm共x,y兲Im q共2兲共y兲
共y兲c21共y兲hm2 共x,y兲
关N共y兲 · hm共x,y兲兴2q共3兲共y兲
22共y兲c41共y兲hm4 共x,y兲
冎
共20兲
.
The integral in the right-hand side of Eq. 共20兲 can be evaluated using the identity 共3兲, written for x1 = x2 = x and fluid without absorption. It follows from Eqs. 共14兲 and 共15兲 that, for high-frequency waves, the integrands in Eqs. 共3兲 and 共20兲 differ by the factor Sm共y兲 共17兲. Although the Green’s function G共x1 , x2 , 兲 diverges when 兩x1 − x2兩 → 0, the sum in the left-hand side of Eq. 共3兲 remains finite. When 兩x1 − x2兩 → 0, the high-frequency Green’s function can be replaced by the Green’s function in a homogeneous, moving fluid26 G共x,y, 兲 =
共18兲
for all y 苸 ⌫ and 兩共y兲 − 共ys兲兩 = / for all y 苸 ⌫, then
兺m
only when the distance between the receivers is large compared to the wavelength. For these reasons, application of the stationary phase method to the integral in Eq. 共10兲 is justified only for evaluation of long-range correlations.31 In the opposite case of autocorrelation, where x1 = x2, mm = 0 for any y, while exp关imn共y兲兴 with m ⫽ n is a rapidly oscillating function in the case of multipath propagation of high-frequency sound. Therefore, the terms with m ⫽ n do not contribute to the dominant term of the high-frequency asymptotic expansion, and from Eqs. 共10兲 and 共11兲 we find that
冉
i 3 iu 1+ · 4 x ⫻
再 冋
冊
2
c1R − u · 共x − y兲 1 exp i R c21 − u2
册冎
,
2 R = 冑兩x − y兩2 − c−2 1 关u ⫻ 共x − y兲兴 .
共21兲
From Eq. 共21兲, one finds that in inhomogeneous, moving media with smoothly varying parameters lim 关G共x,y, 兲 + Gⴱ共y,x, 兲兴 =
y→x
− 4c31共c21 + u2/3兲 2共c21 − u2兲3
,
共22兲
where , c1, and u should be evaluated at the point x. It follows from Eqs. 共3兲, 共20兲, and 共22兲 that C共x,x, 兲 = −
共x兲4c31共x兲关c21共x兲 + u2共x兲/3兴 兵Sm其⍀ , 共23兲 2关c21共x兲 − u2共x兲兴3
where
ds共y兲Sm共y兲兩x1=x2=x兩Am共x,y兲兩2cg共y兲cos ␥m共y兲/共y兲c31共y兲hm共x,y兲
兺m
冕
⍀
共24兲 ds共y兲兩Am共x,y兲兩 cg共y兲cos 2
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
␥m共y兲/共y兲c31共y兲hm共x,y兲
Oleg A. Godin: Cross-correlation of high-frequency noise
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is a weighted average over the surface ⍀ of the functions Sm given by Eq. 共17兲.32 Note that the autocorrelation function depends on an integral characteristic of the noise sources, while the long-range cross-correlation is controlled by local values of the noise source densities at the stationary points. In the particular case of motionless fluids with a single eigenray between x and every point on the surface ⍀ and in the absence of sources of external force, Eq. 共23兲 simplifies to C共x,x, 兲 =
再
共x兲4 q共1兲共y兲共y兲c1共y兲 −2共2兲共x,y兲 e 4c1共x兲 − cos ␥共y兲
冎
. ⍀
共25兲 共1兲
Here, q / 共−cos ␥兲 has a meaning of the density of the noise sources per unit area orthogonal to the eigenray, and 共2兲 e−2 q共1兲 / 共−cos ␥兲 is the effective noise sources density modified by sound absorption. Equations 共24兲 and 共25兲 allow one to estimate the frequency spectrum of noise sources from the autocorrelation
function of noise as measured by several receivers, and to apply the estimate to pre-whiten the source spectrum when processing the noise cross-correlations. This data-processing step is important for accurate retrieval of deterministic travel times from the two-point correlation function of noise.
IV. HIGH-FREQUENCY NOISE SOURCES DISTRIBUTED IN A VOLUME
Consider the two-point cross-correlation function of noise generated by random sources distributed in a simply connected volume ⍀. In the ray approximation, Eqs. 共5兲–共7兲 give C共x1,x2, 兲 = 兺
m,n
冕
−
2 hm 共x1,y兲
册
d3y⌽3共y,x1,x2, 兲
共1兲 ˜m ˜ 共1兲 共y,x1兲 − i ⫻exp关i n 共y,x2兲兴,
再
冋
Q共2兲共y兲 · hn共x2,y兲 h2n共x2,y兲
3
/共y兲c21共y兲
+
共26兲
where
共2兲 共1兲 ⌽3共y,x1,x2, 兲 = −2Am共x1,y兲Aⴱn共x2,y兲exp关− m 共x1,y兲 − 共2兲 n 共x2,y兲兴 Q 共y兲 − i
Q共2兲ⴱ共y兲 · hm共x1,y兲
⍀
2 2 4 2 2 兺 Q共3兲 lj 共y兲共hm共x1,y兲兲l共hn共x2,y兲兲 j/ 共y兲c1共y兲hm共x1,y兲hn共x2,y兲
l,j=1
冎
.
共27兲
Equations 共26兲 and 共27兲 are similar to Eqs. 共10兲 and 共11兲 for cross-correlation of noise due to surface sources but involve volume integration and a different slowly varying factor, ⌽3, in the integrand. As with the surface integral in Eq. 共10兲, the main contribution to the high-frequency asymptotic expansion of the volume integral in Eq. 共26兲 is due to vicinities of stationary points of the rapidly oscillating exponential in the integrand. The stationary points satisfy the equation ˜h 共y , x 兲 = ˜h 共y , x 兲 or, equivalently, m s 1 n s 2 hm共x1,ys兲 = hn共x2,ys兲.
共28兲
This means that the wave vectors on rays arriving at x1 and x2 from ys coincide. Since, away from caustics, only one ray can travel through a given point in any given direction, m = n and the three points ys, x1, and x2 lie on the same ray. Moreover, for any ray passing through both receivers, every point on the ray that precedes both x1 and x2 will satisfy Eq. 共28兲 and, hence, be a stationary point 共Fig. 2兲.33 If an eigenray connecting the points x1, and x2 passes first through the receiver at x1 and then through the receiver at x2, then, according to Eqs. 共8兲 and 共9兲, we have ˜共1兲 n 共ys , x1兲 共1兲 共y , x 兲 = − 共x , x 兲 in the exponent in Eq. 共27兲. On − ˜共1兲 2 1 s 2 n n 共1兲 共ys , x1兲 the contrary, if x2 precedes x1 on the eigenray, ˜m J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
共1兲 共1兲 − ˜m 共ys , x2兲 = m 共x1 , x2兲 in the exponent in Eq. 共27兲 共Fig. 2兲. Since the stationary points in Eq. 共26兲 are not isolated, the integral cannot be evaluated by the stationary phase method directly. Instead, to calculate the leading-order term of the high-frequency asymptotic expansion of the integral, it is convenient to utilize results obtained in Section III B in the case of surface sources. Let ⌿共兲, 0 ⱕ ⱕ 1 be a oneparameter family of spatial domains inside ⍀ such that
FIG. 2. 共Color online兲 Stationary points, deterministic eigenrays, and auxiliary surfaces S共兲 when noise sources are distributed in a volume in a moving inhomogeneous fluid. Thicker lines indicate those parts of the rays passing through both receivers, where the stationary points are located. Oleg A. Godin: Cross-correlation of high-frequency noise
605
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⌿共1兲 傺 ⌿共2兲 when 1 ⬍ 2, ⌿共1兲 = ⍀, and the volume of ⌿共0兲 equals zero. 共That is, the domain ⌿共兲 expands with increasing and fills ⍀ when = 1.兲 The family ⌿共兲 can be chosen in an infinite number of ways. E.g., when ⍀ is a convex surface defined by the equation x = X共共1兲 , 共2兲兲 and the origin of coordinates is chosen within ⍀, ⌿共兲 can be chosen as the domain inside the surface S共兲 defined by the equation x = X共共1兲 , 共2兲兲. It is convenient to cast the volume integral in Eq. 共26兲 as an iterated integral, namely, an integral over of a surface integral taken over the boundary S共兲 of the domain ⌿共兲: C共x1,x2, 兲 = 兺
m,n
冕 冕 冉 冊 1
d
0
N·
S共兲
共1兲 ⫻˜m 共y,x1兲
y ⌽3 exp关i v
˜ 共1兲 − i n 共y,x2兲兴ds共y兲.
共29兲
The leading term of the high-frequency asymptotic expansion of the surface integral in Eq. 共29兲 is given by the righthand side of Eq. 共16兲, where the factors Sm in the summands are now expressed in terms of values that the functions Q共j兲 in Eq. 共27兲 take at the stationary points on the surface S共兲. As discussed above, the stationary points ys are the intersections of S共兲 with rays which, after crossing the surface S共兲, will pass through both receivers at the points x1 and x2 共Fig. 2兲. Note that the exponentials in the right-hand side of Eq. 共16兲 are independent of the surface S共兲. By substituting the asymptotics 共16兲 of the surface integral into Eq. 共29兲, we obtain C共x1,x2, 兲 = 兺 Am共x1,x2兲Tm共x2兲eim
共1兲
共1兲
共x2,x1兲
,
共30兲
n
冋
⫻ Q
U␣dl;
To verify and illustrate the general asymptotic results obtained in Sections III and IV, it is instructive to consider several special cases.
U␣ = − 共1兲
+
A. Thermal noise in moving fluids
c31h␣ −共2兲共x ,y兲−共2兲共x ,y兲 1 2 ␣ ␣ 2cg
e
2共Im Q共2兲 · h␣兲
册
c21h␣2
+
1
3
兺 Q共3兲 lj 共y兲
22c41h␣4 l,j=1
⫻共h␣兲l共h␣兲 j .
共31兲
Integration in Eq. 共31兲 is along the rays that pass through both points x1 and x2. Here ys共m兲 is the origin on ⍀ of the m-th ray that passes first through x2 and then through x1, while ys共n兲 is the origin on ⍀ of the n-th ray that passes first through x1 and then through x2 共Fig. 2兲. In derivation of Eq. 共31兲 we took into account that, at stationary points, 共N · y / 兲d = cos ␥dl, where dl is an increment of the arc length of the respective eigenray, and the angle ␥ is defined after Eq. 共17兲. The high-frequency asymptotics 共30兲 of the crosscorrelation function of noise generated by volume sources is similar to the asymptotics 共16兲 obtained in the case of sur606
共32兲
x
ys共␣兲
␣ = m,n;
Tˆn = 关Sn共ys共n兲兲 + Tn兴exp关共2兲 n 共x2,x1兲兴. V. SPECIAL CASES
+ 兺 Aⴱn共x2,x1兲Tn共x1兲e−in
冕
共2兲 共x1,x2兲兴, Tˆm = 关Sm共ys共m兲兲 + Tm兴exp关m
共x1,x2兲
m
T␣共x兲 =
face sources and differs only by the amplitudes of individual ray components. As expected, the result 共30兲 is independent of the choice of auxiliary domains ⌿共兲. It follows from Eqs. 共6兲 and 共30兲 that the correlation function C共x1 , x2 , 兲 reproduces the sum of the Green’s function G共x1 , x2 , 兲 and the complex-conjugated Green’s function Gⴱ共x2 , x1 , 兲 with the amplitude shading of individual eigenray contributions, 共2兲 共x1 , x2兲兴 and Tn exp关 which equals Tm exp关m 共2兲 ⫻n 共x2 , x1兲兴, respectively. Here, the exponentials compensate for a part of the absorption losses described by the exponential factor in Eq. 共31兲. According to Eq. 共31兲, the amplitude of a particular ray arrival in the cross-correlation function is given by an integral of the density of noise sources located along the ray. Contributions of distant sources are decreased by twice 共on the dB scale兲 the absorption losses at propagation between the source location and the nearest receiver. When the noise field is generated by both volume and surface random sources described by Eq. 共1兲, it follows from Eqs. 共5兲, 共16兲, and 共30兲 that the correlation function C共x1 , x2 , 兲 again reproduces the sum of the Green’s function G共x1 , x2 , 兲 and the complex-conjugated Green’s function Gⴱ共x2 , x1 , 兲, with the amplitude shading of individual eigenray contributions now being, respectively,
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
Thermal acoustic noise in fluids, which is produced by chaotic thermal motion of microscopic particles, can be described macroscopically as the field generated by spatially delta-correlated, volumetric random sources of volume velocity and external force,20 such that Q共1兲 =
冉 冊
1 ⌰ Im , c2
Q共2兲 = −
共3兲 Qmj =
冉 冊 冉 冊
1 ⌰ ⵜ p0 Im , c2
共33兲
1 ⌰ p0 p0 Im xm x j c2
in Eq. 共1兲. Here ⌰ = 0.5ប coth共ប / 2T兲 , ⵜ p0 = −共u · ⵜ兲u, p0共x兲 is the pressure in fluid in the absence of acoustic waves, ⌰ is the mean energy of a quantum oscillator, T is temperature, and ប and are Planck’s and Boltzmann’s constants. Note that Q共1兲 , Q共2兲, and Q共3兲 lj have the same frequency dependence. It follows then from Eqs. 共31兲 and 共33兲 that, when considering the leading order of the high-frequency Oleg A. Godin: Cross-correlation of high-frequency noise
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asymptotic expansion of the cross-correlation function of thermal noise, one should disregard the sources of external force and let Q共2兲 = 0 , Q共3兲 lj = 0 in Eq. 共31兲. From Eqs. 共31兲 and 共33兲, one finds that T␣共x兲 =
−1 22
冕
x
ys共␣兲
⌰共y兲d exp关− ␣共2兲共x1,y兲 − ␣共2兲共x2,y兲兴.
absorption as long as is replaced by 1 in Eqs. 共11兲, 共15兲, 共17兲, 共27兲, and 共31兲. The change in the dissipation model affects the effective macroscopic sources of thermal noise. Instead of Eq. 共33兲, the volume densities of spatially deltacorrelated, random sources of volume velocity and external force are now given by20 Q共1兲 =
共34兲 In derivation of Eq. 共34兲 we used the relation d␣共2兲共x,y兲/dl共y兲
=
h␣共x,y兲c−1 g 共y兲Im
c共y兲,
共35兲
共3兲 Qmj =
⌰ ␦mj Im ,
where ␦mj is the Kronecker symbol. By substituting Eq. 共38兲 into Eq. 共31兲, one obtains T␣共x兲 =
共2兲 共x1,x2兲兴 Tm共x2兲 = − 共22兲−1⌰ exp关− m
C共x1,x2, 兲 = − 共22兲−1⌰关G共x1,x2, 兲 + Gⴱ共x2,x1, 兲兴 共37兲 is valid exactly 共rather than asymptotically兲. It follows from Eq. 共34兲 that, in the ray approximation, the assumption of thermal equilibrium can be replaced by a much weaker condition that the temperature changes are small in the vicinity 共2兲 共x2 , x兲兴 and of the receivers where the factors exp关−m 共2兲 exp关−n 共x1 , x兲兴 are of the order of unity. Moreover, the spatial dimensions of the fluid domain do not affect the relation 共37兲 between the deterministic Green’s functions and the thermal noise cross-correlation at sufficiently high fre共2兲 共x2 , ys共m兲兲兴 Ⰶ 1 and quencies, namely, when exp关−2m 共2兲 共n兲 exp关−2n 共x1 , ys 兲兴 Ⰶ 1. B. Thermal noise and other noise fields in quiescent fluids
Previously, we assumed that absorption of CW sound is described by the sound speed having an imaginary part, with the other environmental parameters being real-valued. When u ⬅ 0, the exact relation 共37兲 remains valid20 for thermal noise under more general assumptions about absorption in a fluid, namely, where both the sound speed and mass density are allowed to take complex, frequency-dependent values. As long as 0 ⱕ 兩Im 兩 Ⰶ Re ⬅ 1 and 0 ⬍ 兩c2兩 Ⰶ c1, absorption affects high-frequency sound fields through the eikonals of various ray arrivals, while its effect on the complex amplitude of the arrivals is negligible compared to its effect on the eikonal. Then, the derivations in Sections III and IV remain unchanged by the more general model of
−1 2
冕
x
ys共␣兲
共2兲
⫻e−␣
共36兲
In particular, when the fluid domain is unbounded, m共2兲共x1,2 , ys共m兲兲 → +⬁ and Tm共x2兲 = −共22兲−1⌰ exp关− 共2兲 ⫻m 共x1 , x2兲兴. Quite similarly, Tn共x1兲 = −共22兲−1⌰ exp关− ⫻共2兲 n 共x2 , x1兲兴 in this case. Thus, all ray arrivals in the sum G共x1 , x2 , 兲 + Gⴱ共x2 , x1 , 兲 are reproduced by the thermal noise cross-correlation function with the same “shading” factor −共22兲−1⌰. This finding agrees with the results of Refs. 20 and 21, where it is shown that, for thermal noise in a fluid at thermal equilibrium, the relation
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
Q共2兲 = 0,
共38兲
which follows from Eq. 共8兲. At thermal equilibrium, ⌰ is independent of position, and 共2兲 ⫻兵1 − exp关− 2m 共x2,ys共m兲兲兴其.
冉 冊
1 ⌰ Im , c2
冉
⌰1c31h␣ 1 Im Im 2 + 2 4 2 cg c 1c 1h ␣
共2兲 共x2,y兲 共x1,y兲−␣
冊
dl.
共39兲
Note that, unlike in the case considered in Section V A, the sources of external force contribute to the leading order of the high-frequency asymptotic expansion as long as Im ⫽ 0. Their contribution is represented by the second term in the first parenthesis in the integrand in Eq. 共39兲. Taking into account that c1 = cg and c1h␣ = 1 for sound in quiescent fluids, Eq. 共39兲 reduces to Eq. 共34兲. Hence, the agreement between the asymptotic results of Section IV and the exact relation 共37兲 as well as the other findings of Section V.A remain valid in the case of quiescent fluids with Im ⫽ 0. An examination shows that the results of Section IV also agree with the exact7,9,34 and asymptotic9,24 analyses of the cross-correlation of random sound fields generated by uniformly distributed, delta-correlated sources of volume velocity in an unbounded, homogeneous, quiescent fluid with dissipation. Snieder9 considered noise fields generated by deltacorrelated sources of volume velocity with the density Q共1兲 proportional to Im c in an inhomogeneous, quiescent fluid with Im = 0. When the fluid is unbounded, this problem 共as well as retrieval of the Green’s function of the diffusion equation36 from the response to uniformly distributed, deltacorrelated noise sources兲 can be viewed as a particular case of the thermal noise problem considered above,20 with the coefficient ⌰ in Eqs. 共33兲 and 共38兲 no longer determined by the temperature. When the fluid is bounded by a surface ⍀, the volume noise sources need to be complemented by surface noise sources in order for the undistorted Green’s functions to be retrieved from the noise cross-correlation function.9 It follows from Eqs. 共17兲, 共32兲, and 共36兲 that the amplitude shading of all ray arrivals will be identical, Tˆm = Tˆ = −共22兲−1⌰, if the density of surface sources of voln
ume velocity satisfies the condition 共1兲 共m兲 共ys 兲 = − ⌰cg cos ␥m/2c31hm . qm
共40兲
When the condition 共40兲 is met, the surface sources replace the volume noise sources outside of ⍀ that are needed for the cross-correlation function of thermal noise to reproduce the exact Green’s function. Equation 共40兲 reduces to the results of Ref. 9 in the special case considered therein. Oleg A. Godin: Cross-correlation of high-frequency noise
607
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The method of images was applied by Sabra et al.37 to study cross-correlation function of noise generated in a Pekeris waveguide by random sources of volume velocity uniformly distributed along a horizontal plane. Sound absorption was modeled by attributing a small imaginary part to the sound speed. An inspection shows that, in such an environment, Eqs. 共16兲 and 共17兲 reduce to the results of Ref. 37. This is expected since the method of images, as used in Ref. 37, is equivalent to the ray approximation, see p. 14 in Ref. 26. VI. DISCUSSION
We have demonstrated that the two-point correlation function of high-frequency, diffuse acoustic fields generated by smooth distributions of random sources located on a surface and/or in a volume approximates the sum of the deterministic Green’s functions, which describe sound propagation in opposite directions between the two points. In arbitrary moving or quiescent, absorbing fluids with smoothly varying sound speed, mass density, and flow velocity, the approximate Green’s functions contain the same ray arrivals with the same travel times as the exact Green’s functions, and differ by amplitudes of the ray arrivals. As far as effects of absorption are concerned, this conclusion is in agreement with results of recent laboratory experiments.38 The explicit expressions 共17兲, 共31兲, and 共32兲 for the amplitude “shading” of the individual ray arrivals in terms of the density of the noise sources are the main results of this paper. Although here we considered only delta-correlated, stationary 共as opposed to moving兲 noise sources, it was shown by Godin35 that all results obtained for both longrange cross-correlation and the autocorrelation function of noise apply equally to finite and moving noise sources, as long as the correlation radius of their volume or surface distributions remains small compared to the spatial dimensions of the Fresnel zone defined in Section III B. The two-point correlation function of noise contains ray arrivals that correspond to deterministic sound propagation in opposite directions between the receivers. Through a comparison of travel times 共or eikonals兲 of sound propagation in opposite directions, noise interferometry allows one to quantify acoustic non-reciprocity.8 This is critically important for acoustic measurements of flow velocities that are much smaller than the sound speed and can even be smaller than the errors in the sound-speed retrieval,22,23,39,40 a situation typical for geophysical flows. For a specific experimental setting with an anisotropic ambient noise, Eqs. 共17兲, 共31兲, and 共32兲 can be applied to design a distribution of receivers in space such that there is sufficient noise “illumination” of propagation paths in opposite directions between pairs of the receivers. As expected from physical considerations, for surface sources, the amplitude shading is proportional to the density of sources at such point ys on the surface that the difference of travel times from it to the receivers, where the noise is recorded, equals the travel time between the receivers. The point ys is the origination point of a ray that passes through both receivers 共Fig. 1兲. For volume sources, the amplitude shading is proportional to an integral of noise sources’ den608
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
sity over all the points, the difference of travel times from which to the receivers equals the travel time between the receivers. These points densely fill an extension of the eigenray connecting the receivers 共Fig. 2兲. There is no interference between contributions of noise sources located in the vicinity of the eigenray extension since all the contributions to the noise cross-correlation function have the same eikonal. In addition to these rather obvious properties, the expressions 共17兲 and 共31兲 have several interesting features, which one should bear in mind when planning experiments on passive remote sensing via noise interferometry. For a specific surface or volume noise source on an eigenray connecting two receivers, the source’s contribution to the eigenray amplitude depends on the source directionality and is proportional to the power it radiates in the direction of sound propagation along the eigenray. For a CW point source at point y, attenuation of acoustic pressure at point x due to sound absorption equals exp关−共2兲共x , y兲兴. When noise cross-correlation is treated as the probing signal, the effect of absorption is squared 共or, on a dB scale, doubled兲 for both volume and surface sources 关see Eqs. 共17兲 and 共31兲兴. This is a direct consequence of the cross-correlation function being a bilinear functional of the acoustic pressure. As a practical implication, we see that passive remote sensing is much more sensitive to sound absorption than remote sensing with controlled sources. In addition to absorption, contributions of distant point sources are attenuated due to geometrical spreading, i.e., the ray tube divergence. For a point monopole source, the geometrical spreading leads to the amplitude factor 1 / 兩x − y兩. Equations 共17兲 and 共31兲 indicate that, unlike point sources, contributions of distributed noise sources into the noise cross-correlation do not appear to be attenuated by geometrical spreading. The physics behind this paradoxical result is rather simple. It is sufficient to consider noise fields generated by sources on a surface ⍀, since in the case of volumetric sources the cross-correlation function can be obtained by summing up the results for various surfaces. Random sources within the Fresnel zone 共18兲 around a stationary point ys 苸 ⍀ make coherent contributions to the crosscorrelation function. The amplitude of the resulting ray arrival is proportional to the area of the Fresnel zone and the product of the amplitudes 兩Am共x1 , ys兲兩 and 兩Am共x2 , ys兲兩, which describe the geometric spreading at propagation from ys to receivers at points x1 and x2. In homogeneous, quiescent media, it follows from Eq. 共18兲 that the area of the Fresnel zone is proportional to the product 兩x1 − ys兩 · 兩x2 − ys兩. Hence, for more distant stationary points in a homogeneous fluid, an increase in the Fresnel zone area precisely compensates for the geometrical spreading and makes the stationary point contribution insensitive to the location of the surface ⍀. Equations 共17兲 and 共31兲 show that the exact compensation of the geometric spreading by the increase in the transverse dimensions of the Fresnel zone takes place in arbitrary moving or quiescent, inhomogeneous fluids. While the deterministic travel times retrievable from the noise cross-correlation function are independent of the origin of the diffuse noise, the amplitude shading factors are insensitive to certain details of the noise field. Very different disOleg A. Godin: Cross-correlation of high-frequency noise
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tributions of noise sources can produce the same noise crosscorrelations. The cross-correlation of fields does not depend on the details of the spatial distribution of volumetric sources as long as linear integrals along the rays in Eq. 共31兲 remain unchanged. When the noise sources are distributed on a surface outside the volume containing all the receivers, the position of the surface is not constrained by the observations and can be chosen arbitrarily. It follows from Eq. 共17兲 that the field cross-correlation function is unaffected by the change in the sources’ location, if the sources’ densities q共1,2,3兲 are modified to reflect the absorption of sound when it propagates between the surfaces. Furthermore, volume sources can be replaced by surface sources and vice versa, without changing the noise cross-correlations. Equations 共17兲, 共31兲, and 共32兲 provide the rules for calculating the equivalent noise sources. It should be noted, however, that the equivalence has been established only for the dominant term of the high-frequency asymptotics. Non-uniformity of noise sources distribution imposes limitations on the accuracy of retrieval of the acoustic travel times from noise cross-correlations.41,42 It remains to be investigated whether the estimates of the travel-time accuracy42 obtained for the noise due to surface sources hold for the noise generated by volumetric sources. ACKNOWLEDGMENTS
This work was supported in part by the Office of Naval Research through Grant No. N00014-08-1-0100. Discussions with M. Charnotskii, V. G. Irisov, and R. I. Weaver and helpful comments by anonymous referees are gratefully acknowledged. 1
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literature for homogeneous 共Refs. 9 and 24兲 and inhomogeneous 共Refs. 34 and 35兲 media. 34 J. Garnier and J. Papanicolaou, “Passive sensor imaging using cross correlations of noisy signals in a scattering medium,” SIAM J. Imaging Sci. 2, 396–437 共2009兲. 35 O. A. Godin, “Emergence of deterministic Green’s functions from noise generated by finite random sources,” Phys. Rev. E 80, 066605 共2009兲. 36 R. Snieder, “Recovering the Green’s function of the diffusion equation from the response to a random forcing,” Phys. Rev. E 74, 046620 共2006兲. 37 K. G. Sabra, P. Roux, and W. A. Kuperman, “Arrival-time structure of the time-averaged ambient noise cross-correlation function in an oceanic waveguide,” J. Acoust. Soc. Am. 117, 164–174 共2005兲. 38 D. Draganov, R. Ghose, E. Ruigrok, J. Thorbecke, and K. Wapenaar, “Seismic interferometry, intrinsic losses and Q-estimation,” Geophys. Prospect. 58, 361–373 共2010兲.
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Oleg A. Godin: Cross-correlation of high-frequency noise
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