**Roel Snieder***

Center for Wave Phenomena and Department of Geophysics, Colorado School of
Mines, Golden, Colorado 80401, USA

(Received 5 July 2006; revised manuscript received 9 August 2006; published 27
October 2006)

It is known that the Green's function for nondissipative
acoustic or elastic wave propagation can be extracted

by correlating noise recorded at different receivers. This property is often
related to the invariance for time

reversal of the acoustic or elastic wave equations. The diffusion equation is
not invariant for time reversal. It

is shown in this work that the Green's function of the diffusion equation can
also be retrieved by correlating

solutions of the diffusion equation that are excited randomly and are recorded
at different locations. This

property can be used to retrieve the Green's function for diffusive systems from
ambient fluctuations. Potential

applications include the fluid pressure in porous media, electromagnetic fields
in conducting media, the diffusive

transport of contaminants, and the intensity of multiply scattered waves.

**I. INTRODUCTION**

The Green's function for acoustic or elastic waves can be

extracted by cross correlating recorded waves that are excited

by a random excitation; see Ref. [1], for a tutorial.

Derivations of this principle have been presented based on

normal modes [2], on representation theorems [3-5], on

time-reversal invariance [6,7], and on the principle of stationary

phase [8-10]. This technique has found applications

in ultrasound [11-13], crustal seismology [14-18], exploration

seismology [19,20], structural engineering [21,22], and

numerical modeling [23]. Recently, the extraction of the

Green's function by cross correlation has been derived for

general coupled systems of linear equations [24].

The principle of extracting the Green's function of a system

from ambient fluctuations creates the possibility to retrieve

the impulse response of a system without using controlled

point sources. This impulse response can be used for

imaging, tomography, or other methods to determine the

properties of the medium. For example, models of the crust

in California have been constructed using surface wave tomography

based on microseismic noise [15,18]. The autocorrelation

of ambient seismic noise has been used for daily

monitoring of fault zones [25] and volcanoes [26]. The

Green's function extracted from ambient noise can also be

used to model the response of a system to a prescribed excitation

without knowing the in situ properties of the system.

The extraction of the Green's function from ambient noise

has been described extensively for wave propagation of

acoustic or elastic waves without intrinsic attenuation (e.g.,

[1-10]). In the absence of intrinsic attenuation, the wave

equation is invariant for time reversal, and several derivations

of the reconstruction of the Green's function are indeed

based on time-reversal invariance [6,7,20].

Many physical systems are not invariant under time reversal.

Intrinsic attenuation breaks the symmetry for time reversal

for acoustic and elastic wave propagation. Electrical conductivity

breaks the time-reversal symmetry of Maxwell's

equations. It has been shown theoretically [27] and observationally

observa-tionally
[21,22] that the impulse response of attenuating

waves can be retrieved from ambient fluctuations.

Time-reversal invariance is, however, not essential for retrieving

the Green's function from ambient noise. This can

be seen by considering the diffusion equation

where the diffusion constant D may depend on position r.

The diffusion equation is not invariant for time reversal because

the operation t→ -t changes the sign of the first term.

Equation (1) is of practical importance because it describes

conductive heat transport, diffusive transport of tracers and

contaminants, fluid flow in porous media [28], electromagnetic

waves in conducting media [29], and the energy transport

of multiply scattered waves, e.g., [30].

The derivation in this work is applicable to the frequency

domain, and the following Fourier convention is used:

With this convention, the diffusion

equation is, in the frequency domain, given by

Time reversal corresponds, in the frequency domain, to
complex

conjugation. The time-reversed diffusion equation is

thus given by

where the asterisk denotes complex conjugation. The sign

difference in the first terms of expressions (2) and (3) is due

to the lack of time-reversal invariance of the diffusion equation.

It is shown here that the Green's function for the diffusion

equation can be retrieved by correlating noise recorded at

several locations in a diffusive system. One application of

this technique is monitoring flow in porous media. We therefore

refer to the solution of the diffusion equation as pressure,

but the results are equally valid for other diffusive systems.

In the following, all expressions are given in the

frequency domain, and the frequency dependence is not

shown explicitly.

**II. REPRESENTATION THEOREMS
OF THE CONVOLUTION AND
CORRELATION TYPE**

Following Fokkema and van den Berg [31,32], we consider

representation theorems of the convolution and correlation

types by using expressions (2) and (3) for two solutions

p

The representation theorem of the convolution type is obtained

by computing where E

Eq. (2) for state A, and where denotes an integration

over volume V. This gives

Note that in the subtraction the
terms cancel.
Applying

an integration by parts to the left-hand side of expression (4)

and using Gauss's theorem gives

where the integral
is over the surface that bounds

volume V.

A representation theorem of the correlation type can be

obtained by evaluating
where Q_{B} denotes

Eq. (3) for state B. Carrying out an integration by parts gives

Note that now the iω terms in expressions (2) and (3) do
not

cancel, but combine to give the volume integral in the lefthand

side. The presence of this term results from the lack of

invariance of time-reversal of the diffusion equation.

In the following integration over all space is used. The

contribution of the surface integral vanishes because the solution

of the diffusion equation p(r,ω) vanishes exponentially

as r→∞. Therefore, the representation theorems (5)

and (6) reduce to

and

The contribution of the surface integral also vanishes for
a

finite volume in case the solution satisfies either Dirichlet

conditions (p=0), Neumann conditions
or mixed

boundary conditions
at the surface that

bounds the volume.

**III. REPRESENTATION THEOREMS
AND GREEN'S FUNCTIONS**

The Green's function for the diffusion equation is the solution

to Eq. (2) when the forcing is a function,

Setting
in expression (7) implies that p_{A}(r)

=G(r,r_{0}). For this choice of q_{A}, expression (7) reduces to

where the subscripts B are dropped. Alternatively, setting

implies that

For this choice of the states A and B, expression (7)
reduces

to the reciprocity relation,

Inserting the states (12) into expression (8) gives

Using reciprocity, this expression can also be written as

The left-hand side of Eq. (15) corresponds, in the time
domain,

to the superposition of the Green's function and the

time-reversed Green's function. In Sec. IV, we consider how

this superposition can be retrieved from the cross correlation

of the pressure generated by uncorrelated sources.

**IV. RETRIEVING THE GREEN'S FUNCTION**

In order to show how the Green's function can be extracted

from the correlation of solutions generated by random

sources, let us consider spatially uncorrelated sources

with power spectrum
that does not depend on location,

where the angular brackets denote the expectation value.
In

practical applications, this expectation value is usually replaced

by using several nonoverlapping time windows (e.g.,

[33,26]). Multiplying Eq. (15) with
, the volume integral

in that expression can be written as

RETRIEVING THE GREEN'S FUNCTION OF THE&[8230;

When we use this result and expression (10), Eq. (15)
after

multiplication with
is given by

where p(r,ω) is the pressure at location r due to the
random

forcing q(r,ω).

Equation (18) states that the superposition of the Green's

function G(r_{A},r_{B},ω) and its time-reversed version is, after

multiplication with the power spectrum of the excitation,

equal to the correlation of the random fields at locations r_{A}

and r_{B}, respectively. The prefactor 2iω corresponds, in the

time domain, with -2d/dt. Since multiplication in the frequency

domain corresponds, in the time domain, to convolution

expression (19), in the time domain, is given by

where the asterisk denotes convolution,
denotes
correlation,

and C_{q}(t) is the autocorrelation of _{q}(t).

**V. DISCUSSION**

The Green's function of the diffusion equation can be

retrieved by cross correlating measurements of a diffusive

system that is excited by random noise. Since the diffusion

equation is not invariant for time reversal, this shows that

invariance for time reversal is not essential for the retrieval

of the Green's function by cross correlation.

For elastic and acoustic waves, the Green's function can

be extracted from waves that are excited randomly at the

surface that surrounds the volume [3,5]. This is not the case

for the diffusion equation. For a volume of radius R, the

surface area grows with R^2, but for a homogeneous medium

the solution of the diffusion equation varies with the radius

as The contribution of the surface integral

therefore depends on the radius of the volume, and the

derivation shown here holds for an infinite volume (R→∞),

or for a finite volume when Dirichlet, Neumann, or mixed

boundary conditions hold at the surface that bounds the vol-ume.

In contrast to the retrieval of the Green's function for

nonattenuating acoustic or elastic waves, where one needs

random sources on a surface that bounds the volume, one

needs random sources throughout the volume for the retrieval

of the Green's function for the diffusion equation.

The theory presented here provides an example that time-reversal

invariance is not required for the extraction of the

Green's function from ambient fluctuations. The diffusion

equation governs physical systems of practical importance,

and the derivation presented here makes it possible to retrieve

the impulse response of diffusive systems from measured

fluctuations.

In this work, the phrase pressure is used for the solution

of the diffusion equation because the pore pressure in a porous

medium follows the diffusion equation. The theory of

this work makes it possible to retrieve the Green's function

for fluid flow in an aquifer or hydrocarbon reservoir from

recorded pressure fluctuations. This Green's function can be

used to estimate parameters such as hydraulic conductivity

and to model the fluid transport in the subsurface without

explicit knowledge of the in situ hydraulic conductivity.

Similarly, the impulse response for the diffusive transport of

contaminants can be retrieved from observations of ambient

fluctuations in the concentration. The Green's function thus

obtained can then be used to predict the diffusive transport of

a localized release of the contaminant.

Electromagnetic fields in a conducting media satisfy the

diffusion equation. This has been used in the magnetotelluric

method where the ambient fluctuations in the electric and

magnetic fields observed at one location are used to determine

the electrical conductivity [34]. The theory presented

here makes it possible to retrieve the Green's function for

electromagnetic fields for noncoincident points from observed

electromagnetic fluctuations.

The intensity of multiply scattered waves satisfies, for late

times, the diffusion equation. Controlled intensity fluctuations

of multiply scattered waves have been used to create

images of the spatial distribution of the diffusion constant.

This has found application in medical imaging, e.g., [35].

Instead of using controlled, spatially localized, sources for

the intensity of scattered waves, one may use the theory of

this work to use random spatially distributed sources instead.

As always, the application of the theory to these, and

other, applications faces implementation issues. The assumption

that the sources of the ambient fluctuations have a homogeneous

spatial distribution may not be satisfied in practical

applications. For applications where this condition is

satisfied, the theory can be used to extract the impulse response

of diffusive systems without using a controlled, localized,

source.

**ACKNOWLEDGMENTS**

The author appreciates the comments of Ken Larner. This

research was supported by the Gamechanger Program of

Shell International Exploration and Production Inc.

**ROEL SNIEDER**

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ROEL SNIEDER PHYSICAL REVIEW E 74, 046620 (2006)