акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en
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Statistical Treatment of Acoustical Imaging |
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22 30 40 45 50 |
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80 L, m |
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√ ¯2 ◦ =
Figure 8.2 Dependence of B on L: , measurements of 1 August, frequency ν 5 kcps; , 31 July, ν = 5 kcps; +, 31 August, ν = 3 kcps; •, 9 August, ν = 3 kcps; , 10 August, ν = 3 kcps (Krasilnikov and Ivanov-Shyts [12])
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, whereas the ray theory leads |
the phase fluctuations, grow with distance in proportion to L 2 |
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of the amplitude fluctuation on distance. To explain the |
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to a stronger dependence |
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discrepancy between theory and experiment, Krasilnikov [7] (and others) suggested that the ray approach was not suitable under the conditions in the experiment; that is, the condition that the wavelength be small compared to the scale of the inhomogeneities was not met. The inner dimension of the turbulence in the layer of the atmosphere near the earth is in the order of 1 cm, which is much smaller than the length of the wave used in the experiment, namely 6.6 cm (5000 kcps). On the other hand, the diffraction theory gives the dependence of the amplitude fluctuations on distance, which was experimentally observed, as
2 ≈ 1/2
B L
The experiments on sound propagation in the ocean were performed by Sheehy [15]. In this experiment, a series of sound pulses was sent at each fixed distance and the size of the relative fluctuations of the pressure amplitude was determined and expressed as a percentage of the mean amplitude. The experimental line (line 1 in Figure 8.3) shows that the size of the fluctuation grows as the square root of the distance, although the scattering of the experimental points is large. Using Lieberman’s data (μ¯ 2 = 5 × 10−9, a = 60 cm) and a correlation coefficient of the form e−r2/a2 , Mintzer [11] represented on the same graph the dependence
V = π 1/2μ¯ 2k2aL
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Acoustical Imaging: Techniques and Applications for Engineers |
V, % |
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Figure 8.3 Dependence of the size of the relative fluctuation of the pressure amplitude of the direct signal on distance (Sheehy [15] and Mintzer [16])
between the relative pressure fluctuation V and the distance L (line 3) and also the dependence V L3/2 which follows from the theory (line 2). As can be seen from the graph, the latter dependence is in strong contradiction with experiment. This is explained by the fact that Sheehy’s data pertains to the Fraunhofer zone. In fact, for distances in the range 50 m to 2500 m for a frequency v = 24 kHz and for a scale of inhomogeneities a = 60 cm, the wave parameter varies from 6 to 300, that is, it is much larger than 1. The results of the diffraction theory (line 3) are in good agreement with the experimental data. We note that between the dashed boundary lines, containing 90% of the data, the quantity μ¯ 2a varies from 1.5 × 10−6 to 1.5 × 10−7 (the experimental line 1 correspond to the work 5 × 10−7).
8.3.7Application of Fluctuation Theory to the Diffraction Image of a Focusing System
We will consider the influence of the fluctuation theory on the focusing effect of the acoustic lens of an acoustical imagery system. The acoustic lens converts a plane wave into a spherical wave. The field near the focus cannot be calculated using ray theory since that theory gives an infinite intensity at the focus. Using wave considerations, we can determine the intensity distribution area as the focus, that is the diffraction image. Fluctuations of the amplitude and phase in the incident wave give rise to fluctuations of the diffraction image – that is, the image quivers. When this happens, not only do we observe deviations of the intensity from the mean distribution, but also the mean distribution itself depends in an essential way on the fluctuations in the incident wave. In this aspect, there are two problems to be addressed in the theory of a focusing system:
1.Finding the mean distribution in the diffraction image.
2.Finding the distribution of fluctuations in the diffraction image.
Statistical Treatment of Acoustical Imaging |
163 |
8.3.8Conclusion
This is the first application of a statistical approach and relativity to acoustical imaging. It is capable of handling the case of large-scale inhomogeneities and multiple scattering in the presence of diffraction. The next stage would be simulation to obtain acoustical images for practical applications to, for example, nondestructive evaluation and ultrasound medical imaging.
8.4Continuum Medium Approach of Statistical Treatment
8.4.1Introduction
So far our statistical treatment of large inhomogeneities in solids has been based on a medium consisting of particles (Chernov’s method [7]). In this section, we also present an alternative statistical approach (Ishimaru’s method [15]) by considering a continuous medium. This is of relevance to the acoustical imaging of biological tissues.
For the treatment of a continuous medium, we consider refractive index fluctuation and use the Gaussian model or the Gaussian correlation function that is used in the treatment of continuum media such as turbulence. The Gaussian correlation function is given as
Bn (rd ) = n12 |
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exp − l |
(8.101) |
where n1 = fluctuation, rd = r1 − r2 and l = correlation distance.
8.4.2Parabolic Equation Theory
The parabolic equation method is used here, but so far exact solutions have been obtained only up to the second moment. Finding solutions for higher moments constitutes one of the most important unsolved problems in strong fluctuation theory.
For acoustic waves, first-order approximation gives the following equation [16]:
2 + k2 u (r) = −k2[n2 (r) − 1]u (r) |
co |
(8.102) |
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where n is the complex refractive index at position r and is equal to n (r) = |
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the sound propagation velocity in the medium in which the object is immersed, and c (r) =the sound propagation velocity at location r in the object.
To derive the parabolic equation, one has to consider that, as a wave u (r) propagates in the x direction, its phase propagates substantially as ikx.
If we write |
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u (r) = U (r) eikx |
(8.103) |
then u (r) will be a slowly varying function of x. Substituting (8.103) into (8.102), we obtain
2ik |
∂U (r) |
+ 2U (r) + k2 n2 (r) − 1 U (r) = 0 |
(8.104) |
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∂x
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Acoustical Imaging: Techniques and Applications for Engineers |
Since U (r) is a slowly varying function of x and varies only over the distance of the scale size l of the random medium,
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∂U |
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∂ 2U |
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(8.105) |
∂x |
∂x2 |
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as long as l λ.
Therefore, we can replace 2 in (8.104) by the transverse Laplacian t2 = obtain the following parabolic equation for U (r):
2ik ∂U (r) + t2U (r) + k2[n2 (r) − 1]U (r) = 0
∂x
∂2 + ∂2 and
∂y2 ∂z2
(8.106)
8.4.3Assumption for the Refractive Index Fluctuation
In our analysis of strong fluctuation, we shall assume that
(a)The fluctuation in the dielectric constant ε(r) is a Gaussian field and therefore its characteristics are completely described by the correlation function S.
Bε r − r = ε1 (r) ε1 |
r |
(8.107) |
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(b) n1 (r) is delta correlated in the direction of sound propagation (the x direction). |
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ε1 (x, ρ ) ε1 |
x , ρ = δ x − x A(ρ − ρ ) |
(8.108) |
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The assumption expressed in (8.108) is based on the fact that although the correlation of the refractive index in the transverse direction ρ has a direct bearing on the transverse correlation of the field, the correlation of the refractive index in the direction of the wave propagation has little effect on the fluctuation characteristics of the wave.
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The relation between A(ρ ρ ) and the spectrum of the refractive index fluctuation n (k) is given by the Wiener–Khinchine theorem. The correlation function is a Fourier transform of the spectral density:
Bε |
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= ε |
k exp i |
ik |
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where φε k |
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k and k = dk1dk2dk3 = dk1dk. |
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Taking the inverse transform of (8.109) and using (8.108) we obtain
dK (8.109)
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d ρ − ρ |
(8.110) |
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This is a two-dimensional Fourier transform. Taking the inverse transform, we obtain
A (ρ ) = 2π |
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k expik |
·ρ dk |
(8.111) |
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Statistical Treatment of Acoustical Imaging |
165 |
8.4.4Equation for the Average Field and General Solution
Taking the ensemble average of the parabolic equation (8.106)
∂ |
U (r) + t2 U (r) + k2 n2 (r |
− 1) U (r) = 0 |
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(8.112) |
Expressing the last term of equation (8.112) in terms of the average field in the following form:
ε1 (r) U (r) = g (r) U (r) |
(8.113) |
we obtain the differential equation for U (r) . In order to obtain g (r) in equation (8.113) we note that U (r) is a functional on ε1 (r), and making use of the following relationship, valid for a Gaussian random field ε1 (r) and a functional U (r) on ε1 (r):
ε1 (r) U (r) = |
dV < ε1 (r) ε1 r |
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(r) " |
(8.114) |
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δU |
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where δU is the functional derivative or variational derivative.
δε1
A functional is a quantity that depends on a function ε1 (r). In contrast, a function is a quantity that depends on a variable. Using the delta-correlated assumption (8.108), equation (8.114) becomes
ε1 (r) U (r) = |
dρA(ρ − ρ ) |
#∂ε1 |
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∂U (x, ρ ) |
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To calculate ∂U , we first obtain U (r) by integrating equation (8.106) with respect to x from
x = 0 to x:
∂ε1
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2iU (x, ρ ) − 2ikU (0, ρ ) + t2 U (ξ , ρ ) dξ + k |
ε1 (ξ , ρ ) U (ξ , ρ ) dξ = 0 (8.116) |
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Now, taking the functional derivative of equation (8.116)
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(x , ρ ) |
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(8.117) |
and let x → x.
It is evident from equation (8.116), that the field at (x, ρ ) depends on the inhomogeneity
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in the range x < x. Physically this means that the backscattering from ε1 |
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∂ε1 x , ρ = |
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166 Acoustical Imaging: Techniques and Applications for Engineers
Therefore
δε1 |
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δU (ξ , ρ ) |
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which becomes zero as x → x. The last term in equation (8.117) needs some additional consideration because of ε1 (ξ , ρ ) in the integrand. We write this term with the unit step function H(ξ ):
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and the delta function − :
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U |
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Now the functional derivative of a functional of the form |
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I (x) = a |
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is given by |
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Using this, equation (8.121) becomes |
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δξ , ρ
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x
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As x → x, the second term of (8.124) vanishes as a result of assumption equation (8.118), and the first term becomes
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δ(ρ − ρ )U x , ρ |
(8.125) |
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Statistical Treatment of Acoustical Imaging |
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Then, substituting equation (8.125) into equation (8.117), we obtain
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Substituting equation (8.126) into equation (8.115), we obtain |
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Therefore, equation (8.112) becomes
(8.126)
(8.127)
(8.128)
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together with the boundary condition at x = 0,
U (0, ρ ) = U0 (ρ )
Equation (8.129) completely determines the coherent field U (x, ρ ) . The solution to equation (8.129) can be easily obtained by writing
U (x, ρ ) = f (x, ρ ) exp(−α0x)
where
∞
αo = k2A(0) = 2π 2k2 n (k) kdk
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0
Substituting equation (8.131) into equation (8.129) we obtain
(8.129)
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∂ |
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However, equation (8.133) is the parabolic differential equation for the field in free space (n1 = 0) and, therefore, f (x, ρ ) is the field in free space in the absence of randomness. Letting the free-space field be f (x, ρ ) = U0 (x, ρ ), we obtain the final solution:
U (x, ρ ) = U0 (x, ρ ) exp(−α0x) |
(8.134) |
The coherent integral is therefore given by
| U (x, ρ ) |2 = |U0 (x, ρ )|2 exp(−2α0x) |
(8.135) |
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References
[1]Twersky, V. (1973) Multiple scattering of sound by a periodic line of obstacles. J. Acoust. Doc. Am., 53, 96–112.
[2]Frisch, V. (1968) Wave propagation in random media, in Probabilistic Methods in Applied Mathematics, vol. 1 (ed. A.T. Barucha-Reid), Academic Press, New York, pp. 76–198.
[3]Barabanenkov, Y.N., Kravtsov, Y.A., Rytov, S.M. and Tatarski, V.I. (1971) Status of the theory of propagation of waves in a randomly inhomogeneous medium. Dov. Phys. Usp., 13, 551–580.
[4]Schuster, A. (1905) Radiation through a foggy atmosphere. Astrophys. J., 21, 1–22.
[5]Davison, B. (1958) Neutron Transport Theory, Oxford Univ. Press, London and New York.
[6]Kolmogorov, A.N. (1938) Uspekhi Matem. Nauk, 5, 34.
[7]Chernov, L.A. (1960) Wave Propagation in a Random Media, Dover, New York.
[8]Krasilnikov, V.A. and Tatarski, V.I. (1953) Doklady Akad. Nauk SSSR, 88, 435.
[9]Obukhov, A.M. (1949) Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz., 13, 58.
[10]Boyev, G.P. (1950) Probability Theory. Gos. Tekh. Teor. Izdat., Moscow-Leningrad, pp. 208.
[11]Mintzer, D. (1954) J. Acoust. Soc. Am., 26, 186.
[12]Krasilnikov, V.A. and Ivanov-Shyts, K.M. (1949) Doklady Akad. Nauk SSSR, 67, 639.
[13]Krasilnikov, V.A. and Ivanov-Shyts, K.M. (1950) Vestnik Mosk. Univ., Fiz. ( 2).
[14]Krasilnikov, V.A. (1949) Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz., 13, 33.
[15]Sheehy, J. (1950) Acoust. So. Am., 22, 24.
[16]Mintzer, D. (1953) J. Acoust. Soc. Am, 25, 992.
[17]Ryzhik, I.M. and Gradshtein, I.S. (1951) Tables of Integrals, Sums, Series and Products, Gos. Tekh. Izdat., Moscow-Leningrad.
9
Nondestructive Testing
9.1Defects Characterization
In nondestructive evaluation it is more interesting to evaluate the defect type and shape, rather than its size, especially when it is necessary to differentiate The characterization and classification of defects according to their type, shape and size, is more difficult and time consuming than the detection of defects from the ultrasonic echoes and images. This is fundamentally due to the basic handicap of ultrasonic testing in having a poor lateral resolution due to the relatively long wavelength of ultrasound used. Even with the most modern complex imaging method, a precise defect image cannot be obtained.
Destructive testing of similar specimens can provide some information on the defect and need not be ignored. From the defect position one can conclude with certainty that it is of a certain type. Often the existence of any defect is a sufficient cause for rejection, without precise information about its type and size. Thus a clear specification of the testing task in advance can simplify a problem and save the unnecessary time spent on the characterization of the defects.
Ultrasound nondestructive testing consists of two steps: (1) defect detection; and (2) defect characterization. The defect echo, even when maximized, cannot be related to the defect size without further information about the transducer data and the gain employed.
Recently many ultrasound testing standards refer to quantitative measurement methods instead of just a qualitative defect indication.
Another important defect evaluation case is to deal with shallow incipient cracks. In acoustical imaging the method mainly used is the ALOK, which is based on a scanning system. Using one or more probes at each point of the scanning line, the transit times/position curves of the reflector echo are recorded so that after using electronic interference suppression methods, a B-scan image of the reflector in the plane of incidence is obtained. Wide-angled beams are preferred for this method and have been proved successful with natural but not-too-complicated defects. So far, however, they have only been used with system that evaluates the size of the defect. As mentioned above, for castings and welds, the defect type and shape are often more interesting than the defect size.
Acoustical Imaging: Techniques and Applications for Engineers, First Edition. Woon Siong Gan. © 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.
