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Statistical Treatment of

Acoustical Imaging

8.1Introduction

Acoustical imaging deals with the propagation of sound waves in solids and liquids in both homogeneous and heterogeneous media. Hence, the physical mechanism of sound propagations in random media is a key issue in acoustical imaging. There are various methods of treating this problem. For a homogeneous medium it is sufﬁcient to employ single scattering or ﬁrst-order multiple scattering (a Born approximation) or a Rytov approximation can be used, but for a heterogeneous medium or random medium, the multiple scattering effects become dominant in determining the ﬂuctuation characteristics of the sound wave. So far two theories have been used in dealing with multiple scattering problems: the analytical theory and the transport theory.

For the analytical theory, the treatment starts with the sound wave equation and there are two approaches for treating the medium of sound wave propagation. One is treating the medium as continuous. For sound propagation in a medium of random particles, we will include the scattering and absorption characteristics of the particles. This is mathematically rigorous as all the multiple scattering, diffraction and interference effects can be included. However, in practice, it is impossible to obtain a formulation that completely includes all these effects, and the various methods that yield useful solutions are all approximate, each being useful for a speciﬁc range of parameters. Examples are Twersky’s theory [1], the diagram method [2], the Bethe Salpeter equation and statistical treatment [3].

For the transport theory [4], the sound wave equation is not needed. It deals directly with the transport of energy through a medium containing particles. The development of the theory is heuristic and it lacks the mathematical rigour of the analytical theory. Even though diffraction and interference effects are included in the description of the scattering and absorption characteristics of a single particle, the transport theory itself does not include diffraction effects. It is assumed in transport theory that there is no correlation between ﬁelds and, therefore, the addition of powers rather than the addition of ﬁelds holds. The transport theory was devised by Schuster [4] in 1903. The basic differential equation used is the equation of transfer. It is equivalent to Boltzmann’s equation used in the kinetic theory of gases and

142	Acoustical Imaging: Techniques and Applications for Engineers

in neutron transport theory [5]. The formulation is ﬂexible and is capable of treating many physical phenomena.

Although the starting points of the analytical theory and the transport theory are different, both are dealing with the multiple scattering problem and there are therefore some fundamental relationships between them. For instance, the speciﬁc intensity used in transport theory, and the correlation function used in analytical theory, are related through a Fourier transform. This means that although the transport theory was developed on the basis of the addition of power, it contains information about the correlation of the ﬁelds. Also in transport theory, the polarization effects can be included through the Stokes’ matrix.

Our book will consider the analytical theory only because of its mathematical rigour. The statistical treatment will be used because this is currently the most authoritative method for the treatment of multiple scattering effects in sound propagation in random media due to its successful use by Kolmogorov [6] in the treatment of turbulence, which is a random medium. We will also consider the random medium as a continuum in a later section, where the moving particles and moving continuum of a medium will also be considered. The transformation of coordinates or symmetries will be used to deal with the relativistic in effect.

8.2Scattering by Inhomogeneities

Our work will be an extension of the work of Chernov [7] and Kolmogorov [6]. On the statistical treatment of general wave propagation in random media to sound wave propagation in random media – especially to propagation in solids with the use of the elastic wave equation – the basic wave equation will be used as the starting point.

Our work will pioneer the use of statistical treatment on the theory of acoustical imaging, that is, using a statistical approach on the problem of multiple scattering.

We will start with the case of a weakly inhomogeneous medium, or the method of small perturbations, and extend to large-scale inhomogeneities or a strongly inhomogeneous medium.

Ignoring the elastic properties of the medium, such as the involvement of the elastic constants, the acoustic wave equation for sound propagation in solids can be rewritten in terms of the acoustic pressure as

1 ∂2 p1		− 2 p1 = 4π Q	(8.1)
c2	∂t2	− 2 p1 = 4π Q	(8.1)
0

where p1 is the acoustic pressure of a secondary scattered wave, c0 is the mean value of sound velocity and Q is the density of elementary sources. For the condition of small perturbation, we assume that the density and sound velocity deviate only slightly from their mean values ρ0 and c0 as

ρ = ρ0 + ρ and c = c0 + c

(8.2)

where ρ ρ0 and c c0. We take as the zeroth approximation the plane wave p0 = A0 exp [−i (ωt − kx)]

where k = ω/c0 is the wavenumber in the medium with averaged characteristics.

Statistical Treatment of Acoustical Imaging

143

p1 is within a ﬁrst approximation under the inﬂuence of the primary wave, and each element of the inhomogeneous medium becomes a source of secondary scattered wave p. The total effect of the waves scattered by the volume V is given by the solutions of equation (8.1)

Q t − r

p1 =		c0	dV	(8.3)
	r

where r is the distance from the scattering element (ξ , η, ζ ) to the receiver point (x, y, z), that is,

r = (x − ξ )2 + (y − η)2 + (z − ζ )2 (8.4)

When sound wave propagates in an inhomogeneous random medium, ﬂuctuations of the characteristics of the waveﬁeld, due to the superposition of the scattered waves and the primary waves, are observed. There must be a dependence between the ﬂuctuation of the characteristics of the waveﬁeld and the ﬂuctuation of the refractive index. Our problem consists in ﬁnding this dependence, which can be used to draw conclusions about the statistical properties of the waveﬁeld, given the statistical properties of the medium.

The inverse problem is usually not unique and cannot be solved without additional assumptions. However, if we make reasonable assumptions about the form of the correlation coefﬁcient of the refractive index, then by measuring the ﬁeld ﬂuctuation, we can determine the mean value of the refractive index ﬂuctuation. Thus, the study of waveﬁeld ﬂuctuation enables us to study the properties of the medium in which the wave travels, and hence towards acoustical imaging of that medium.

8.3Study of the Statistical Properties of the Waveﬁeld

We begin the study of the statistical properties of the waveﬁeld by examining amplitude and phase ﬂuctuations of the sound waveﬁeld. We will ﬁrst consider small inhomogeneities or small perturbations. We assume that the random inhomogeneities occur only in the right half-space (x > 0) and that the left half-space (x < 0) contains no random inhomogeneities.

A plane sound wave

p0 = A0eiφ0 = A0e−i(ωt−kx)

advances from a homogeneous to an inhomogeneous medium. A receiver is located in the inhomogeneous medium at the point with coordinates x, y, z and the waves scattered by the inhomogeneities, as well as the wave p0, are incident on the receiver.

The total sum of the incident sound wave and the scattered wind wave is given by

p1 = A1eiφ1

Using the method of small perturbations, p can be found to a ﬁrst approximation if we take p0 as the zeroth approximation.

144	Acoustical Imaging: Techniques and Applications for Engineers

Ignoring elasticity effect, the acoustic wave equation can be written in terms of the acoustic pressure as [8]:

1 ∂ 2 p	− 2 p + log ρ · p = 0	(8.5)
c2 ∂t2

Using the method of small perturbations, we assume slight deviations in the values of the density of the medium and sound velocity from their mean values ρ0 and c0, that is,

ρ = ρ0 + ρ, c = c0 + c

(8.6)

where ρ ρ0 and c c0.

With the above conditions, equation (8.5) can be rewritten as [7],

1 ∂2 p

− 2 p =

2 c ∂2 p

( ρ ) · p

(8.7)

−

c2 ∂t2

∂t2

ρ0

Taking p0 as the zeroth approximation, the plane wave

p0 = A0 exp [−i (ωt − kx)]

(8.8)

where k is the wavenumber in the medium, is written as

1 ∂2 p1

− 2 p1

2 c ∂2 p0

( ρ ) · p0

(8.9)

−

∂t2

ρ0

for the ﬁrst approximation, p1, or

1 ∂2 p1

2 c

ik ∂ ( ρ )

i (ωt

kx)]

(8.10)

−

= −

−

c02

∂t2

ρ0

∂x

A0 exp [

Introducing the notations

(∂x

) A0 exp [−i (ωt − kx)]

4π Q = − 2k2

+ ρ0

∂

(8.11)

as an abbreviation, (8.10) can be rewritten as

1 ∂2 p1

− 2 p1 = 4π Q

(8.12)

∂t2

Substituting equation (8.11) into equation (8.3), we obtain

p1 = −

4π

2k2

ρ0

∂ξ

exp [ik (r + ξ )] dV

(8.13)

∂ ( ρ ) 1

Statistical Treatment of Acoustical Imaging

145

The ﬁrst term in the square brackets of equation (8.13) shows the scattering by ﬂuctuations of the sound velocity, and the second term in the square brackets shows the scattering by density ﬂuctuations.

Neglecting the density ﬂuctuations as compared with the velocity ﬂuctuations in equation (8.13), we obtain

	2A		eik(r	ξ )
p1 =	k 0	V	+		μ (ξ , η, ζ ) dV	(8.14)
	2π		r

where r is the distance from the scattering element dV with coordinates ξ , η, ζ to the observation point (x, y, z). The integration in (8.14) is to be taken over that part of space from which scattered waves arrive at the observation point.

If we denote the result of superimposing the primary and scattered wave by

p = Aeiφ = A0eiφ0 + A1eiφ1

then, dividing both sides by A0eiφ0 , we obtain
	A		A1
		ei(φ−φ0 ) = 1	+		ei(φ1	−φ0 )
	A0			A0

where, using equation (8.14), the last term is given by

(8.15)

(8.16)

ei(φ1−φ0 ) =

eik[r−(x−ξ )]μ (ξ , η, ζ ) dV

(8.17)

2π

Introducing the notation

ei(φ1

−φ0 ) = x + iy

(8.18)

we have

X =

cos k [r − (x − ξ )] μ (ξ , η, ζ ) dV

(8.19)

2π

Y =

sin k [r − (x − ξ )] μ (ξ , η, ζ ) dV

(8.20)

2π

Since the amplitude of the scattered wave A1 is small compared to the amplitude of the primary wave A0, the amplitude A = A0 + A and φ = φ0 + φ of the resulting wave differs only

slightly from the amplitude A0 and phase φ0 of the primary wave that is, A 1, φ 1.

Expanding the left-hand side of equation (8.16) in powers of small ﬂuctuation, and keeping only the leading term, we have

A + i φ = x + iy A0

146 Acoustical Imaging: Techniques and Applications for Engineers

Equating real and imaginary parts separately, and using equation (8.19) and (8.20), we obtain

		k2		1
φ =			V		sin k [r − (x − ξ )] μ (ξ , η, ζ ) dV	(8.21)
		2π		r
A		k2		1
	=		V		cos k [r − (x − ξ )] μ (ξ , η, ζ ) dV	(8.22)
A0		2π		r

for the phase ﬂuctuation φ and the amplitude ﬂuctuation A.

8.3.1Fresnal Approximation or Near-Field Approximation

In our work, we shall consider only the case of large-scale inhomogeneities, ka 1 where a is the correlation distance (Chernov [7, p. 8]). We can neglect wave reﬂections and limit the region of integration in equations (8.21) and (8.22) to ξ = x. An appreciable effect will be produced only by those inhomogeneities that are concentrated within a cone with its vertex at the receiving point and with an aperture angle of the order of 1/ka. Inside this cone,

the formula r = (x − ξ )2 + ρ2, where ρ2 = (y − η)2 + (z − ζ )2 , can be replaced by the approximation

r (x − ξ ) +	1	ρ2		(8.23)
r (x − ξ ) +	2 x	−	ξ	(8.23)
		−

Using (8.23) to replace r − (x − ξ ) in (8.21) and (8.22), and 1/r by 1/ (x – ξ ), we obtain

x ∞

Phase Fluctuation = S = k2

2π

0 −∞

Amplitude Fluctuation = log A (r) = k2

A0 2π

	(x − ξ )−		μ (ξ , η, ζ ) dξ dηdζ (8.24)
sin kρ2/2 (x			ξ )
x ∞	cos kρ2/2 (x				ξ )
		(x − ξ )		−	μ (ξ , η, ζ ) dξ dηdζ

0−∞

(8.25)

Equations (8.24) and (8.25) are equivalent to the Fresnal approximation in diffraction theory. Using the following abbreviation in the integrals of (8.24) and (8.25),

x − ξ

, ρ

≡

sin

kρ2

2π (x − ξ )

2 (x −

ξ )

x − ξ

, ρ

≡

cos

kρ2

2π (x − ξ )

2 (x − ξ )

Statistical Treatment of Acoustical Imaging

147

Moreover, dropping the prime in S and denoting the amplitude ﬂuctuation by B = log A/A0, (8.24) and (8.25) will become

	=	x	∞
	=			1	k
S (x, y, z)		k			x − ξ	, ρ	μ (ξ , η, ζ ) dξ dηdζ	(8.26)
S (x, y, z)		k				, ρ	μ (ξ , η, ζ ) dξ dηdζ	(8.26)
		0	−∞
	=	x	∞
	=			2	k
B (x, y, z)		k			x − ξ	, ρ	μ (ξ , η, ζ ) dξ dηdζ	(8.27)
B (x, y, z)		k				, ρ	μ (ξ , η, ζ ) dξ dηdζ	(8.27)
		0	−∞

Equations (8.26) and (8.27) were derived by Obukhov [9]. Introducing the dimensionless variables, x = kx, y = ky, z = kz, ξ = kξ , η = kη, ζ = kζ , ρ = kρ, equations (8.26) and

(8.27) can be rewritten as

S x

, y

, z

∞

x − ξ

, ρ

, η

, ζ

dξ dη dζ

(8.28)

−∞

B x

, y

, z

∞

x − ξ

, ρ

, η

, ζ

dξ dη dζ

(8.29)

0−∞

8.3.2Farﬁeld Imaging Condition (Fraunhofer Approximation)

Here we derive the mean square amplitude and phase ﬂuctuation [7]. We assume that the receiver is located at the point (L, 0, 0) or (L , 0, 0) where L = KL. Then equations (8.28) and (8.39) can be rewritten as

L	∞		L − ξ , ρ μ(ξ , η , ζ )dξ dη dζ
S L , 0, 0 =		1	L − ξ , ρ μ(ξ , η , ζ )dξ dη dζ	(8.30)
0	−∞
L	∞		L − ξ , ρ μ(ξ , η , ζ )dξ dη dζ
B L , 0, 0 =		2	L − ξ , ρ μ(ξ , η , ζ )dξ dη dζ	(8.31)

0−∞

where ρ

∞

2 + ζ 2. Squaring (8.30) and (8.31) and averaging, we obtain

S¯2 = μ¯ 2

L − ξ1

, ρ1

L − ξ2

, ρ2

N r

dξ1dξ2dη1

dη2dζ1dζ2

−∞

(8.32)

∞

B¯2 = μ¯ 2

L − ξ1

, ρ1

L − ξ2

, ρ2

N r

dξ1dξ2dη1

dη2dζ1dζ2

−∞

(8.33)

148 Acoustical Imaging: Techniques and Applications for Engineers

where N r = correlation coefﬁcient of the refractive index and

r = ξ1 − ξ2 2 + η1 − η2 2 + ζ1 − ζ2 2.

If we consider only statistically isotropic media, then the correlation coefﬁcient N depends only on the modules of r. Introducing relative coordinates

, ζ

−

(8.34)

1 −

and centre of mass coordinates

1 +

we obtain

(η

), z

(ζ

)

(8.35)

∞

L − ξ ,

2 + y

2 + z

S2 = μ2

0 −∞

× 1

L − ξ1,

N r dξ1dξ2dηdζ dydz (8.36)

− y

− z

∞

L − ξ1,

2 + y

+ z

B2 = μ2

00 −∞

× 2	L − ξ2,	2	− y	+	2	− z	2	N	r dξ1dξ2dηdζ dydz (8.37)
		η			ζ
		η		2	ζ

Performing integration with respect to the variables y and z in these equations, we can obtain

∞

1 L − ξ1, 2

+ y

2 + z

L − ξ2,

− y

2 − z

dydz

−∞

ξ1 − ξ2, ρ + 1

2L − ξ1 + ξ2 , ρ

(8.38)

∞

2 L − ξ1, 2

+ y

2 + z

L − ξ2,

− y

2 − z

dydz

−∞

ξ1 − ξ2, ρ − 1

2L − ξ1 + ξ2 , ρ

(8.39)

where ρ2 =

η2 + ζ 2

Statistical Treatment of Acoustical Imaging

149

Equations (8.38) and (8.39) can be further simpliﬁed to

S¯

(I1

I2 )

(8.40)

and

−

B¯2

μ2

(I1

I2 )

(8.41)

where

∞

I1 =

ξ1

− ξ2, ρ

dξ1dξ2dηdζ

(8.42)

−∞

∞

I2 =

2L − ξ1

+ ξ2

, ρ N

dξ1dξ2dηdξ

(8.43)

−∞

Further simpliﬁcation of I1 and I2 can be done if the distance a is large compared to the correlation distance L, that is, L a.

Introducing the relative coordinate ξ = ξ1 − ξ2 and the centre of mass coordinate x = 12 ξ1 + ξ2 , we are then justiﬁed to integrate ξ for the limits −∞ to ∞. Then equation (8.42) can be simpliﬁed to

L ∞

I1 = dx

1 (ξ , ρ ) N r dξ dηdζ

(8.44)

0−∞

and likewise equation (8.43) can be simpliﬁed to

L ∞

I2 = dx

2L − 2x, ρ N r dξ dηdζ

(8.45)

0−∞

Since the integrand in equation (8.44) does not depends on x, we have

	∞
I1 = L		1 (ξ , ρ ) N r					dξ dηdζ
	−∞
	∞	1			ρ2
= L				sin		N r dξ dηdζ		(8.46)
			2π ξ		2ξ

−∞

150	Acoustical Imaging: Techniques and Applications for Engineers

for equation (8.45). We can also integrate with respect to x, and since only one factor in the integrand, namely 1 2L − 2x, ρ , depends on x, the problem reduces to calculating the integral

ρ2

2L − 2x, ρ

dx =

sin

2π (2L

−

2x)

2 (2L

−

2x)

ρ2

Using the substitution, z =

, this can be further reduced to

2(2L −2x)

∞

sin z

ρ2

dz = −

4π 2

4π

/4L

si ν ≈ −

cos ν

−

sin ν

ν2

Thus,

∞

ρ2

N(r )dξ dηdζ

I2 = −

(8.47)

4π

−∞

Equations (8.46) and (8.47) can be transformed to polar coordinates (ρ, ) in the (η, ξ ) plane. Bearing in mind that the correlation coefﬁcient N(r ) is an even function of ξ , we can obtain

∞∞

	dξ	1		ρ2
I1 = 2L			sin		N(r )ρdρ	(8.48)
		ξ		2ξ

and

∞∞

I2 = −	dξ	si	ρ2	N(r )ρdρ	(8.49)
			4L

Equation (8.48) can be simpliﬁed further; instead of ρ, we introduce the new variable q = ρ2 ,

2ξ

then the second integral in equation (8.48) can be written as

∞

sin qN(r )dq

(8.50)

Integrating equation (8.50) by parts twice, we have

∞

∂

∂q

sin qN(r )dq = 0

− cos q · N(r ) + 0 cos q

)

N(r

∞

∂N

∞

∂

2N(r

)

= N (ξ , 0, 0) + 0

sin q

−

sin q

∂q

∂q2

∞

2N(r

)

= N (ξ , 0, 0) −

sin q

∂

(8.51)

∂q2

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