Contribution of double scattering in diffuse ultrasonic backscatter measurements Ping Hu and Joseph A. Turnera) Mechanical and Materials Engineering, University of Nebraska-Lincoln, W342 Nebraska Hall, Lincoln, Nebraska 68588-0526

(Received 8 May 2014; revised 7 November 2014; accepted 22 November 2014) Diffuse ultrasonic backscatter measurements are used to describe the effective grain scattering present during high frequency ultrasonic inspections. Accurate modeling of the backscatter is important for both flaw detection and microstructural characterization. Previous models have been derived under the assumption of single scattering for which the ultrasound is assumed to scatter only once in the time between excitation and detection. This assumption has been shown to be valid in many experiments for which the time scales are short or the frequency is sufficiently low. However, there are also many instances (e.g., for strongly scattering materials, unfocused beams, or long propagation paths) for which the single scattering assumption appears to break down. In this article, a model for the double scatter is developed within the previous formalism based on Wigner distribution functions. The final expression allows the effect of double scattering to be estimated for any combination of experimental parameters. The improved proposed model is anticipated to increase the capabilities of ultrasonic microstructural evaluation, especially in terms of probability C 2015 Acoustical Society of America. of detection estimates. V [http://dx.doi.org/10.1121/1.4904920] [ANN]

Pages: 321–334

I. INTRODUCTION

The propagation of waves through polycrystalline materials is known to be affected by the grain structure of the medium, an effect that is especially important when the wavelength is on the order of the grain size.1–7 Scattering of incident energy occurs at the grain boundaries and part of this energy can return to the source transducer. This type of ultrasonic backscatter has been used in the last 30 years to quantify grain noise associated with ultrasonic inspections of metals and other heterogeneous materials typically by analyzing the spatial variance of an ensemble of pulse-echo signals.8–15 Models of such measurements have been developed primarily under the assumption that incoming energy scatters only once before it returns to the transducer. This work has been very successful for quantifying the probability that a defect is present within the base material and for quantifying grain size information.16–19 Because most of these scattering models are based on a single-scattering ansatz, it is not possible to estimate their range of validity with any accuracy.4,9–11 Such an estimate requires information about the magnitude of the second term in a multiple scattering expansion and it must be formulated to include all aspects of the measurement system such as the transducer properties (e.g., frequency, focal length), focal depth, grain size, grain properties, etc.5,14,20–24 Knowledge of this second term would give researchers the ability to assess the uncertainty of their measurements very precisely. Without it, they must do everything possible to reduce the

scattering to the smallest level possible with the hope that the single-scattering assumption is applicable.10,11 Recently, a model for the spatial variance was derived within a multiple scattering framework that includes all aspects of the measurement system.14 This model includes an expansion in the order of scattering that is clearly defined. When the expansion was limited to the first term only (i.e., single scattering), the model was shown to reduce to the results derived previously using an explicit single-scattering approach. Experimental corroboration of this model has subsequently been presented using a pulse-echo measurement at normal incidence15 and a pitch-catch normal-to-oblique incidence (longitudinal-to-shear) measurement25 with good agreement between the estimated grain sizes in comparison with optical micrographs. Thus, this approach serves as the foundation for additional research regarding higher-order scattering. In this article, the diffuse ultrasonic backscatter response for a pulse-echo type of experiment is derived with inclusion of single and double scattering effects. First, the formulation and full description to this problem are reviewed for clarity. Then the multiple scattering expansion that arises within the derivation is truncated at two terms and the derivation is completed. The results presented allow the uncertainty of the single-scattering approximation to be determined explicitly with respect to the complete measurement system. In addition, this work provides an important step into the understanding of the full multiple scattering problem such that it can be posed rigorously. II. REVIEW OF THE APPROACH

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 137 (1), January 2015

The propagation of ultrasound through polycrystalline materials is complicated by the heterogeneous structure of

0001-4966/2015/137(1)/321/14/$30.00

C 2015 Acoustical Society of America V

321

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the medium. The key approach to solve such problems is to use statistical methods to analyze the wave propagation by considering the elastic material properties to vary spatially and this approach is now reviewed. The primary governing partial differential equation for an elastic wave generated by an impulsive disturbance is given by3 " # @2 @ @ Gia ðx; y; tÞ dli 2 þ Cklij ðxÞ @t @xk @xj ¼ d3 ðx  yÞdðtÞdla :

(1)

In Eq. (1), the density of the medium has been assumed to be constant and is set to unity. Gia ðx; y; tÞ is the Green’s function, which defines the displacement response at position x in the i direction due to a unit impulse applied at position y in the a direction at time t. The position dependent modulus is of the form Cklij ðxÞ ¼ C0klij þ dCklij ðxÞ, in which C0klij ¼ hCklij ðxÞi is the averaged moduli of the medium and dCklij ðxÞ is the fluctuation away from average modulus (the angle brackets denote an ensemble average). The Green’s function G reduces to the bare Green’s function G0 when dC ¼ 0. With the definitions of the temporal Fourier transform pair (t $ x) of the Green’s function ð þ1 Gðx; y; xÞ ¼ Gðx; y; tÞexpðixtÞdt; 1 (2) ð 1 þ1 ð Þ Gðx; y; tÞ ¼ Gðx; y; xÞexp ixt dt; 2p 1

J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

where the introduction of delta functions is representative of the statistical homogeneity of the medium. The spatial Fourier transforms of G and m are defined analogously, and ~ and r ~ , respectively. With these definitions, are denoted as G the Dyson equation can be solved using the spatial Fourier transform of Eq. (4), resulting in14 ~ ~ ðpÞ1 : hGðpÞi ¼ ½Iðx þ ieÞ2  p  C0  p þ r

(7)

Here, if the medium is assumed to be infinite, the mean ~ Green’s function, hGðpÞi can be further reduced to its longitudinal and transverse components,3 ~ ~ L ðpÞi þ hG ~ T ðpÞi; hGðpÞi ¼ hG

(8)

^p ^ p and ~ L ð pÞ x2  p2 c2L þ r ^p ^ Ip ~ T ðp Þi ¼ hG : 2 2 2 ~ T ð pÞ x  p cT þ r L

(3)

(4)

where m is the “self-energy”(“mass”) operator. Equation (4) defines the mean Green’s function and Eq. (4) is commonly called the Dyson equation. With the first-order smoothing approximation (FOSA), m can be approximated as3  @ @ 0 mbj ðz; z0 Þ  dCabcd ðzÞ G ðz; z0 Þ @za @zd ck  @ @ (5)  0 dCikjl ðzÞ 0 : @zi @zl 322

(6)

~ ðp Þi ¼ hG

where the factor ie is included to guarantee the infinitesimal positive imaginary part of x. This addition ensures that Gðx; y; xÞ is analytic in the upper half complex x plane. In Eq. (3), the x dependence has been suppressed on G but will be reinvoked when necessary. According to diagrammatic methods, the solution of Eq. (3) can be written as the mean Green’s function with respect to an integrable function3,26 ðð 0 hGia ðx; yÞi ¼ Gia ðx; yÞ þ G0ib ðx; zÞmbj ðz; z0 Þ  hGja ðz0 ; yÞid 3 zd 3 z0 ;

0  G~ ia ðpÞd3 ðp  qÞexpðiq  yÞ;

where

Eq. (1) can be rewritten in the frequency (x) domain as   @ @ 2 0 ðx þ ieÞ dli þ Cklij @k @j þ dCklij ðxÞ @xk @xj  Gia ðx; yÞ ¼ d3 ðx  yÞdla ;

It has been shown that the Dyson equation can be solved easily for a statistically homogeneous medium after applying a spatial Fourier transform (x $ p and y $ q), where wave numbers p and q are the transform variables of x and y, respectively. The spatial Fourier transform pair of G0 is thus defined as ðð 1 0 d3 xd3 y expðip  xÞ G~ ia ðpÞd3 ðp  qÞ  ð2pÞ3  G0ia ðx; yÞexpðiq  yÞ; ðð 1 0 Gia ðx; yÞ ¼ d 3 pd 3 q expðip  xÞ ð2pÞ3

The infinitesimal positive quantity e can now be ignored ~ . Similarly, r ~ L ðpÞ and due to the finite imaginary parts of r T ~ ðpÞ are the spatial Fourier transforms of the longitudinal r ^ ~ ðpÞ ¼ r ~ L ðpÞ^ pp and transverse self-energies, i.e., r T ^p ^ Þ. cL and cT are the longitudinal and ~ ðpÞðI  p þr transverse wave speeds, respectively, of a medium with ^ ) is the stiffness C0klij . p (with amplitude p and direction p effective wave number that is defined from the dispersion relations,3 ( ~ L ðpÞ ¼ 0; x2  p2 c2L þ r (9) ~ T ðpÞ ¼ 0: x2  p2 c2T þ r ~ In Eq. (9), p can be found directly if the value of p within r is assumed to equal the bare wave number x=c.27 In this case Leff

p

"  #1=2   x 1 L x x 1 x ~ ~L ¼ 1þ 2r  þ r ; cL x cL cL 2xcL cL (10a) P. Hu and J. A. Turner: Double scattering in polycrystals

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Teff

p

"  #1=2   x 1 T x x 1 L x ~ ~ ¼ 1þ 2r  þ r ; cT x cT cT 2xcT cT

Just as the Dyson equation governs the mean field, the Bethe-Salpeter equation governs the covariance. This equation is written in the transform domain as3,26

(10b) L

2

~ ðx=cL Þ=x  1 where the approximations are based on r ~ T ðx=cT Þ=x2  1. As shown in Eq. (10), the real parts and r of the effective wave numbers define the dispersion relations in the scattering medium, pL ¼ x=cL and pT ¼ x=cT , while the imaginary parts define the attenuations respectively, aL and ~ L ðx=cL Þ=2xcL and aT ¼ Im r ~ T ðx=cT Þ= aT , i.e., aL ¼ Im r 2xcT . The inverse Fourier transform of Eq. (8) gives hGðx; y; tÞi ¼ hGL ðx; y; tÞi þ hGT ðx; y; tÞi;

(11)

where   expðaL jx  yjÞ jx  yj L ^p ^; d t p hG ðx; y; tÞi ¼  cL 4pc2L jx  yj expðaT jx  yjÞ hGT ðx; y; tÞi ¼  4pc2 jx  yj  T  jx  yj ^p ^ Þ: ðI  p d t cT Note that the bare Green’s functions may be deduced from the mean Green’s function when aL ¼ aT ¼ 0 in Eq. (11). With the statistical homogeneity of the bare Green’s function, G0 ðx; yÞ ¼ G0 ðx  yÞ, the temporal Fourier transform of the bare Green’s function can be written with respect to u ¼ x  y, G0 ðu; xÞ ¼ G0L ðu; xÞ þ G0T ðu; xÞ;

(12)

where   ^p ^ p x exp i j uj and cL 4pc2L juj   ^p ^Þ ðI  p x j u j : exp i G0T ðu; xÞ ¼  cT 4pc2T j u j G0L ðu; xÞ ¼ 

Typical diffuse field measurements are performed by analyzing an average of the squared signals. The mean Green’s function, hGi, governed by the Dyson function is insufficient for these purposes. The mean square of the Green’s function is needed and thus the Green’s function covariance must be examined.3,5,14,26 The Green’s function covariance is defined as hGab ðx; x0 ; xÞG ij ðy; y0 ; x þ XÞi;

(13)

where denotes the complex conjugate. Again, x and x þ X are suppressed in the following discussions. Similar to Eq. (6), the inverse spatial Fourier transform of hGG* i including four wave vectors is given as  p a bq 3 0 0 p0 i Hj q0 d p þ q  q  p ð  1  d 3 xd 3 x0 d 3 yd 3 y0 hGab ðx; x0 ÞG ij y; y0 i 6 ð2pÞ  (14)  exp ip  x þ iq  x0 þ ip0  y  iq0  y0 : J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

p a bq pþD i Hj qþD

¼ ai Cbj qqþD d3 ðp  qÞ ð þ d 3 s ai Cck ppþD ppþD ck Kld ssþD ssþD dl Hjb qqþD ; (15)

where C is the double-mean-field Green’s function defined by3 a bq i Cj qþD

¼ hGab ðq; xÞi hG ij ðq þ D; x þ XÞi:

(16)

In Eq. (15), K is the second-order intensity operator. It can be defined using an analogous approximation to the FOSA (in this case, called the ladder approximation) as3,7,14 p c dq pþD k Kl qþD

¼ pa qb ðpi þ Di Þðqj þ Dj Þ  ~g ðp^ p  q^ q Þ p^a q^b p^i q^j Nabcd ijkl :

(17)

The intensity operator defines the degree of the scattering within the medium and consists of the spatial Fourier transform of the spatial correlation function ~g and the elastic moduli covariance function N. g defines the probability that two randomly chosen points fall within an average length scale given by correlation length L and N describes the elastic moduli fluctuations of the medium due to anisotropic constituents. Both are defined in more detail in the following analysis. The covariance of ppþD ai Hjb qqþD can be expanded to arbitrary order. Ghoshal et al.14 derived the two term expansion which included single scattering effects only. Here, this formalism is extended one order higher such that a second term of the series in terms of the intensity operator K is included. The formal expansion of H to second-order is given by p a bq pþD i Hj qþD ¼ ai Cbj qqþD d3 ðp

ð

 qÞ þ ai Cck ppþD ppþD ck Kld qqþD dl Cbj qqþD

þ d 3 s ai Cck ppþD ppþD ck Kld ssþD  dl Csm ssþD ssþD sm Knp qqþD pn Cbj qqþD :

(18)

The first two terms of Eq. (18) have been used previously for the derivation of the singly-scattered response in which the ultrasound is assumed to scatter only once in the time between excitation and detection.14 A multiple scattering formalism can be generated from the Green’s function covariance.3,26 This main idea can be understood by examining the three terms on the right side of Eq. (18). With the definition of the double mean field Green’s function C, the first term represents a coherent propagation from source to receiver. The second term gives the contribution of a propagation from wave vector q to wave vector p after a scattering event, in which the intensity operator K alters the direction P. Hu and J. A. Turner: Double scattering in polycrystals

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of wave vector from q to p. The third term including five items is more complicated but follows a similar logic. The propagation begins from the source with wave vector q and encounters the first scattering event with the intensity operator K which changes the direction of wave vector from q to s. Then, the propagator with wave vector s reaches the second scattering location, in which the direction is modified once again to the final wave vector p. Finally, the wave vector p propagates to the receiver. In this article, an improved model of the double scattered expansion is formulated to include the full ultrasonic system model consisting of source and receiver distributions. It is the emphasis of Sec. III. III. DOUBLY-SCATTERED RESPONSE (DSR)

In a real ultrasonic measurement, three basic elements exist. They are the source, the heterogeneous sample, and the receiver. First, the source is assumed to transmit a coherent wave and is modeled using a time-space distribution function Ba ðxÞSðtÞ. The field in the material from the source is given as14 ð WSb ðx; tÞ ¼ Gba ðx; y; tÞBa ðyÞ SðtÞd 3 y; (19) where denotes a temporal convolution. Similarly, the receiver can be characterized by a time-space distribution function RðtÞAd ðxÞ and the received field is again given by a convolution with the Green’s function,14 ð WRc ðx; tÞ ¼ RðtÞ Ad ðyÞGdc ðy; x; tÞd3 y: (20) Then the formal signal /ðtÞ in a ultrasonic measurement can be specified as14

U

ð2Þ ð

ð

dx 3 3 3 ~ d pd qd DA a ðpÞA~i ðp þ DÞB~b ðqÞB~j ðq þ DÞ XÞ ¼ 2p  RðxÞR ðx þ XÞSðxÞS ðx þ XÞ:

/ðtÞ ¼

ðð

RðtÞ Ab ðyÞGba ðy; x; tÞBa ðxÞ SðtÞd 3 yd 3 x: (21)

The mean signal h/ðtÞi is easily obtained by replacing the Green’s function G with the mean Green’s function hGi (the source and receiver distributions are nonstochastic) and is given as ðð h/ðtÞi ¼ RðtÞ Ab ðyÞhGba ðy; x; tÞi  Ba ðxÞ SðtÞd 3 yd 3 x:

(22)

Generally, the square of the signal, h/2 ðtÞi, is utilized to analyze the scattered field. Ghoshal et al. obtained the Fourier transform of the square of a signal as14 ð dx 3 3 3 ~ d pd qd DA a ðpÞA~i ðp þ DÞB~b ðqÞ UðXÞ ¼ 2p  B~j ðq þ DÞp a Hjb q pþD i

qþD

 RðxÞR ðx þ XÞSðxÞS ðx þ XÞ:

(23)

Substituting Eq. (18) into Eq. (23), the Fourier transform of the square signal is found to be a summation of the zerothorder Uð0Þ ðXÞ, the first-order Uð1Þ ðXÞ, and the second-order Uð2Þ ðXÞ scattering responses such that UðXÞ ¼ Uð0Þ ðXÞ þUð1Þ ðXÞ þ Uð2Þ ðXÞ. The zeroth-order and the first-order terms have already been discussed previously.14,15 By analogy with the first-order scattering, the second-order scattering Uð2Þ ðXÞ includes all contributions in the signal that scatter twice in the time between excitation and detection. Following a similar formalism, the second-order scattering response is obtained by considering the last component on the right side of Eq. (18) giving



d 3 s ai Cck ppþD ppþD ck Kld ssþD dl Csm ssþD ssþD sm Knp qqþD pn Cbj qqþD

 (24)

Expanding the three double-mean-field Green’s functions in Eq. (24), ai Cck ppþD , dl Csm ssþD , and pn Cbj qqþD leads to Uð2Þ ðXÞ ¼

ð

dx 3 3 3 3 ~ d pd qd sd D½A a ðpÞhGac ðp; xÞi RðxÞ A~i ðp þ DÞhG ik ðp þ D; x þ XÞi R ðx þ XÞ 2p

 ssþD sm Knp qqþD hGds ðs; xÞi hG lm ðs þ D; x þ XÞi  ½hGpb ðq; xÞi B~b ðqÞSðxÞ ½hG nj ðq þ D; x þ XÞi B~j ðq þ DÞS ðx þ XÞ:

p c ds pþD k Kl sþD

(25)

Note that the values in square brackets ½  are the Fourier transforms of the source and receiver fields as shown in Eqs. (19) and (20). For instance, hWSp i ¼ hGpb ðq; xÞi B~b ðqÞSðxÞ and hWRc i ¼ A~a ðpÞhGac ðp; xÞi RðxÞ:

(26)

Taking the inverse Fourier transforms of the source fields, the receiver fields, and the mean Green’s functions, Eq. (25) becomes 324

J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

P. Hu and J. A. Turner: Double scattering in polycrystals

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"ð # dx 3 3 3 3 p c d s s s p q d 3 xdt R d pd qd sd D pþD k Kl sþD sþD m Kn qþD U ðX Þ ¼ hWc ðx; tÞiexpðþip  x þ ixtÞ 2p ð2pÞ3=2 "ð #  d 3 ydu R hWk ðy; uÞiexp iðp þ DÞ  y  iðx þ XÞu  ð2pÞ3=2 ð  ð   00 00 3 00 00 00 00 00 00 Þ 3 00 00 00 00 ð ð Þ ð Þ ð Þ d y du hGlm y ; u iexp þi s þ D  y  i x þ X u  d x dt hGds x ; t iexp is  x þ ixt "ð # "ð #   d 3 x0 dt0 d3 y0 du0 S  0 0 S 0 0 0 0 0 0 hWp ðx ; t Þiexp iq  x þ ixt hWn y ; u iexp þiðp þ DÞ  y  iðx þ XÞu :  ð2pÞ3=2 ð2pÞ3=2 ð

ð 2Þ

(27)

Using a similar procedure as the previously derived singlyscattered response (SSR) theory, the second-order scattering response shown in Eq. (27) can be simplified by changing variables properly and then individually introducing the Wigner distribution functions for the source field hWS i, the receiver field hWR i, and the mean Green’s function hGi. First introduced by Eugene Wigner in 1932,28 the Wigner distribution function is widely used in signal processing. Here, the Wigner distribution function WðX; T; k; xÞ represents the signal in the space-time (X; T) and the wave vector-frequency (k; x) domains simultaneously. First, the following change of variables is made: x ¼ X þ n=2; y ¼ X  n=2; t ¼ T þ s=2; u ¼ T  s=2; x0 ¼ X0 þ n0 =2; y0 ¼ X0  n0 =2; t0 ¼ T 0 þ s0 =2; 00

00

00

y ¼ X  n =2; u00 ¼ T 00  s00 =2:

u0 ¼ T 0  s0 =2; 00

00

x00 ¼ X00 þ n00 =2;

00

t ¼ T þ s =2; (28)

Then, it is noted that K is independent of X as shown in Eq. (17). Additionally, the Bethe-Salpeter equation is based on the approximation of D  j k j. Therefore, it is appropriate to ignore the dependence of small D and X in K. The corresponding inverse Fourier transform of Uð2Þ ðXÞ can be obtained after changing variables with a shift of X=2 on x and a shift of D=2 on both p and q. The second-order scattering as shown in Eq. (25) can then be rewritten with respect to t as ð dx 3 3 3 3 3 0 0 3 00 d pd qd sd Xd X dT d T Uð2Þ ðtÞ ¼ ð2pÞ4  R  Wck X; t  T 0  T 00 ; p; x

S Wpn ðX;

ð

T; k; xÞ ¼ d3 ndshWSp ðX þ n=2; T þ s=2Þi  hW n S ðX  n=2; T  s=2Þi

 exp ðin  k þ isxÞ; (30b) ð G ðX; T; k; xÞ ¼ d 3 ndshGds ðX þ n=2; T þ s=2Þi Wdslm  hG lm ðX  n=2; T  s=2Þi  exp ðin  k þ isxÞ:

(30c)

Equation (29) is the first main result of this article. The second-order scattering, similar to the first-order scattering, is a convolution of the inner product of three Wigner distribution tensors (WR , WG , and WS ) and two intensity operators (pp Kss and ss Kqq ). The new quantity introduced here, WG , governs the energy propagation within the heterogeneous medium. Equation (29) also illuminates the route of a coherent wave as it propagates from the source to the receiver. The route begins with the coherent source field with wave vector q that is scattered to wave vector s. The energy then propagates in the material until it is scattered to wave vector p by the second scattering event and then returns to the receiver. The double scattering process compared with SSR is illustrated in Fig. 1. It should be noted that all

G ðX  X0 ; T 00 ; s; xÞ ss sm Knp qq  pp ck Kld ss Wdslm  S  Wpn X0 ; T 0 ; q; x : (29)

In Eq. (29), WR , WS , and WG are the Wigner distribution functions of the receiver field, the source field, and the mean Green’s function, which are defined as14 ð R Wck ðX; T; k; xÞ ¼ d3 ndshWRc ðX þ n=2; T þ s=2Þi  hW k R ðX  n=2; T  s=2Þi  exp ðþin  k þ isxÞ; J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

(30a)

FIG. 1. (Color online) Schematic diagram of the first-order scattering (a) and the second-order scattering (b). P. Hu and J. A. Turner: Double scattering in polycrystals

325

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possible paths in this scattering expansion are assumed to arrive at the receiver incoherently (diffusely). Combining the SSR from Ghoshal et al.,14,15 ð dx 3 3 3 R U ð 1 Þ ðt Þ ¼ d pd qd XdTWck ðX; t  T; p; xÞ ð2pÞ4  pp ck Kjd qq WdjS ðX; T; q; xÞ;

(31)

the DSR is the summation of the SSR and the second-order scattering. Therefore, the DSR is given by UðtÞ ¼ Uð1Þ ðtÞ þ Uð2Þ ðtÞ ð  0 dx 3 3 3 0 R ¼ d pd qd X dTWck X ; t  T; p; x 4 ð2pÞ   pp ck Kjd qq WdjS X0 ; T; q; x ð dx 3 3 3 3 3 0 0 3 00 þ d pd qd sd Xd X dT d T ð2pÞ4  R  ½Wck X; t  T 0  T 00 ; p; x G ðX  X0 ; T 00 ; s; xÞ ss sm Knp qq  pp ck Kld ss Wdslm  S  Wpn X0 ; T 0 ; q; x :

(32)

L

G G Wdslm ðX; T; k; xÞ ¼ Wdslm ðX; T; k; xÞ

L

G ðX; T; k; xÞ ¼ Wdslm

ð

d 3 nds exp ðin  k þ isxÞ

 hG L lm ðX  n=2; T  s=2Þi; ð GT ðX; T; k; xÞ ¼ d 3 nds exp ðin  k þ isxÞ Wdslm  hGTds ðX þ n=2; T þ s=2Þi  hG T lm ðX  n=2; T  s=2Þi: Using the basic property of the delta function, f ðxÞdðx  aÞ L T ¼ f ðaÞdðx  aÞ, the attenuation terms inside WG and WG can be factored, giving J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

where WG and WG are used to denote the Wigner distribution functions of longitudinal and transverse bare Green’s functions. The decomposition of the Wigner distribution functions into longitudinal and shear components simplifies the following procedure and allows for the contribution of each to be evaluated. According to the high-frequency asymptotic approximation, the Wigner distribution function is approximated as29,30

(35)

As shown in Eq. (35), the amplitude A and phase H need to be extracted from the individual longitudinal and transverse components of the bare Green’s function shown in Eq. (12) [A0L ðu; xÞ and A0T ðu; xÞ denote the amplitudes of longitudinal and transverse parts of the bare Green’s function G0 , respectively, and H0L ðu; xÞ and H0T ðu; xÞ are their corresponding phases]. Individually, these are given as

1 and A0L ðu; xÞ ¼ 4qpc2L j u j

1 0T 2 A ðu; xÞ ¼ 4qpcT j u j ; x x (36) H0L ðu; xÞ ¼ j u j and H0T ðu; xÞ ¼ j u j: cL cT The two Wigner distribution functions of the bare Green’s function are then given by 0L

G Wdslm ðX; T; k; xÞ     p x^ jXj ^ ^ ^ ^ 3 d k X d T  kdksklkm ¼ cL cL 2q2 c2L jXj2

and 0T

G Wdslm ðX; T; k; xÞ ¼

(33)

 hGLds ðX þ n=2; T þ s=2Þi

326

0T

3

The mean Green’s function is a sum of longitudinal and transverse parts as given by Eq. (11). According to the definition of the Wigner distribution function in Eq. (30), WG can be simplified after substituting Eq. (11) into Eq. (30c). First, the cross-terms hG L i hGT i and hGL i hG T i are neglected because the difference in wave speed does not allow for coherent energy propagation. In this case,

where

(34) 0L

 ð2pÞ jAðX; xÞj2 d3 ðk  rX HðX; xÞÞ   @ HðX; xÞ : d T @x

IV. WIGNER DISTRIBUTION FUNCTION OF THE MEAN GREEN’S FUNCTION

T

0T

G þ exp ð2aT cT TÞ Wdslm ðX; T; k; xÞ;

WðX; T; k; xÞ

The DSR can be determined after obtaining the needed components within the integrand and performing the various integrations. The Wigner distributions of the source and receiver have already been discussed.14,15 Here, the new quantity of interest is WG and is derived in Sec. IV.

G þ Wdslm ðX; T; k; xÞ;

0L

G G Wdslm ðX; T; k; xÞ ¼ exp ð2aL cL TÞ Wdslm ðX; T; k; xÞ

  x^ 3 d k  X cT 2q2 c2T j X j2    jXj ^dk ^l d T ddl  k cT  ^m : ^sk  dsm  k

(37)

p

(38)

G ðX; T; k; xÞ can be obtained by substituting Finally, Wdslm Eqs. (37) and (38) into Eq. (34). The full expansion for WG is written as G Wdslm ðX; T; k; xÞ

  x^ d k X ¼ expð2aL cL T Þ cL 2q2 c2L j X j2   jXj ^ ^ ^ ^ d T k d k s k l k m þ expð2aT cT T Þ cL     p x ^ jXj 3  d k X d T cT cT 2q2 c2T j X j2   ^ l dsm  k ^m : ^dk ^sk  ddl  k p

3

(39)

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Following the decomposition of WG into longitudinal and transverse parts, the second-order scattering that begins and ends in longitudinal form can be written in terms of two parts Uð2Þ ðtÞ ¼ ULLL ðtÞ þ ULTL ðtÞ;

(40)

where ULLL ðtÞ and ULTL ðtÞ are the longitudinal and transverse components of the second-order scattering, respectively. They indicate the mode-conversion occurring after the first scattering event. Specifically, they are given by ð dx 3 3 3 3 3 0 0 3 00 d pd qd sd Xd X dT d T ULLL ðtÞ ¼ ð2pÞ4  R  Wck X; t  T 0  T 00 ; p; x

In Eqs. (41) and (42), there are two intensity operators, K, which are defined in Eq. (17). Compared with the magnitudes of the wave numbers, x=cL and x=cT , Eq. (17) can be reduced for the two expressions needed in Eqs. (41) and (42) as D is assumed small such that pc ds p k Kl s

¼ p2 s2 ~g ðp^ p  s^s Þ p^a s^b p^i s^j Nabcd ijkl

(43)

ss pq s m Kn q

lsp ¼ s2 q2 ~g ðs^s  q^ q Þ s^l q^ s^e q^f Nef mn :

(44)

and

In Sec. V, the integration of the second-order scattering in Eqs. (41) and (42) is presented.

L

G ðX  X0 ; T 00 ; s; xÞ ss sm Knp qq  pp ck Kld ss Wdslm  S  Wpn X0 ; T 0 ; q; x (41)

and ULTL ðtÞ ¼

ð

dx ð2pÞ

d3 pd 3 qd 3 sd3 Xd 3 X0 dT 0 d3 T 00 4

 R  Wck X; t  T 0  T 00 ; p; x T

G ðX  X0 ; T 00 ; s; xÞ ss sm Knp qq  pp ck Kld ss Wdslm  S  Wpn X0 ; T 0 ; q; x : (42)

S Wpn



V. SECOND-ORDER SCATTERING: PISTON TRANSDUCER

The longitudinal component of the second-order scattering ULLL ðtÞ in Eq. (41) is examined first in detail. The integration of ULTL ðtÞ follows by analogy. The Wigner distribution functions of the source and receiver, WS and WR , were derived by Ghoshal and Turner for a pulse-echo configuration15

" # w20 2X02 2Y 02 X ; t; q; x ¼ exp  2 0  2 0 w 1 ðZ 0 Þw 2 ðZ 0 Þ w 1 ðZ Þ w 2 ðZ Þ "   2 # 2Z0 Z0  2tcL 1 t ^ Þ2 q^p q^n ^n d3 ðq  q0 Þðq  2aL Z0  r2 ðx  x0 Þ2  2  exp  2 r r2 c2L

0

3 A20 Tf2L ð2pÞ

pffiffiffiffiffiffi 2pr

(45)

and R Wck ðX;

t; p; xÞ

" # w20 2X2 2Y 2 exp  2  w 1 ð Z Þw 2 ð Z Þ w1 ðZ Þ w22 ðZ Þ "  2 # 2Z ðZ  2tcL Þ t ^ Þ2 p^c p^k : ^n d 3 ðp  p 0 Þd ðx  x 0 Þðp  2aL Z  2  exp  r r2 c2L

2 ð Þ4 ¼ A20 TLf 2p

In Eqs. (45) and (46), A0 is an amplitude correction for ultrasound propagation in a coupling pffiffiffi fluid and given by A0 ¼ expðaf zf Þ=ð4pw20 qf c2f rkf pÞ:31 The properties of the fluid are the density qf , the wave speed cf , and the attenuation af at the center frequency x0 with the wave number defined as kf ¼ x0 =cf . The variable zf is used to represent the distance between the surface of the source transducer and the sample, usually called the water path in an ultrasound measurement, and r is the temporal width of the incident pulse. The transducer is represented by a Gaussian distribution where w0 is the effective initial beam width. Specifically, w0 ¼ 0:7517  a0 =2 for which a0 is the nominal diameter of J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

(46)

the transducer.32 Tf L and TLf are the transmission coefficients of the longitudinal wave from the coupling fluid to the sample and from the sample back to the coupling fluid, respectively. For a pulse-echo configuration, Tf L ¼ 2qf cf =ðqcL þ qf cf Þ and TLf ¼ qf cf Tf L =qcL .31 cL is the longitudinal wave speed in the sample with density q, longitudinal attenuation aL , and transverse attenuation aT . w1 ðZÞ (w1 ðZ 0 Þ) and w2 ðZÞ (w2 ðZ 0 Þ) are the Gaussian profile widths along the propagation axis Z (or Z0 ). In the case of a planar surface, the curvatures on the sample surface are equal to zero such that w1 ðZÞ ¼ w2 ðZÞ and w1 ðZ 0 Þ ¼ w2 ðZ 0 Þ. Using a single-order Gaussian beam model and assuming a flat sample surface, wðZÞ is defined by14,33 P. Hu and J. A. Turner: Double scattering in polycrystals

327

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1 1 2 ¼ i ; qð Z Þ Rð Z Þ kf w 2 ð Z Þ cL qðZ Þ ¼ qð0Þ þ zf þ Z: cf

w2 ð Z Þ ¼

2  kf Im 1=qðZ Þ

and

(47)

In Eq. (47), Rð0Þ ¼ F is the initial radius of the wavefront, where F is the focal length of the transducer in the fluid. The ratio of wave speeds, cL =cf , is utilized to convert the axial propagation distance Z with respect to the fluid into the sample. The physical meanings of the complex parameters qðZÞ can be individually explained through its real and imaginary parts. The imaginary parts of qðZÞ are found to be the widths of the single Gaussian beam and real parts define the radii of curvature of the wavefronts, which are written as

ULLL ðtÞ ¼c2 ðTf L TLf Þ2 q2 A40 ð2pÞ

3 pffiffiffiffiffiffi

2pr



Rð Z Þ ¼

1  : Re 1=qðZ Þ (48)

The analysis that follows is based on a planar sample surface, but the more general model for a curved surface is still expected to hold. The longitudinal part of the second-order scattering [Eq. (41)] can be obtained by substituting Eqs. (37) and (43)–(46) into Eq. (41). ^r denotes the unit vector of X  X0 , i.e., X  X0 ¼ jX  X0 j^r . According to the sampling property of the delta function, the integrations over p, s, q, and x are trivial. Then, Eq. (41) can be expressed as

ð   p x80 3 3 0 0 00 x0 x0 ^ ^ ~ d Xd X dT dT  g p r p^0a r^b p^0i r^j Nabcd ijkl cL 0 cL 2c4L c8L

 x0 x0 w40 lsp ^ ^  ~g r q 0 r^l q^0 r^e q^0f Nef mn 2 cL cL w ð Z Þw 2 ðZ 0 Þ " #  2X2 2Y 2 2Z Z  2cL ðt  T 0  T 00 Þ    exp  2  2aL Z r2 c2L w ð Z Þ w2 ð Z Þ "  2 # t  T 0  T 00 expð2aL cL T 00 Þ ^ Þ2 p^0c p^0k ^0  n  exp 2 ðp r jX  X0 j2 " #   02 02 jX  X0 j 2X 2Y d  T 00 r^d r^s r^l r^m exp  2 0  2 0 cL w ðZ Þ w ðZ Þ "  0 2 # 2Z0 ðZ0  2cL T 0 Þ T 0 ^ Þ2 q^0p q^0n ; ^0  n  2aL Z  2 ðq  exp  r r2 c2L

(49)

where c is an experimental calibration parameter that acts as an intermediate variable between the displacement field and ^0, q ^ 0 , ^r , and the transducer voltage.15 The relations among p ^ are shown in Fig. 2. The angle between p ^ 0 and ^r is h while n ^ ¼ 1 and ^ 0 is p  h. Then, p ^0  n the angle between ^r and q ^ ¼ 1. The Fourier transform of the correlation func^0  n q tions can be written as   x0 x0 ^  ^r ¼ ~g LL ðhÞ and ~ p g cL 0 cL   x0 x0 ^ ^ ¼ ~g LL ðp  hÞ: ~ (50) g r q cL cL 0 The inner products on the covariance tensor with the incoming and scattered wave directions can be described in terms of the angle between these wave vectors, ^ p p^ r^r^

^0a r^b p^0i r^j p^0c p^0k r^d r^l ¼ N^p 0 p^ 0 r^r^ ðhÞ; Nabcd ijkl p 0 0 ^ r r^q^ q^

lsp ^l q^0 r^e q^0f r^s r^m q^0p q^0n ¼ N^r r^q^ 0 q^ 0 ðp  hÞ: Nef mn r 0 0

(51)

The functions in Eqs. (50) and (51) all depend on h through factors of cos h. Because of this dependence and the 328

J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

FIG. 2. (Color online) Schematic diagram of the incoming and scattering wave directions. P. Hu and J. A. Turner: Double scattering in polycrystals

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assumption regarding focused transducers, we assume h ¼ 0 as a reasonable approximation for the remaining derivation (a similar assumption is often made for the SSR derivation). Furthermore, the temporal integration over T 00 is easy to complete through the sampling property of its delta function. The temporal integration over T 0 is based on the general integral (for positive a)34   rffiffiffi ð þ1 p b2 2 : (52) exp expðax  bxÞ ¼ 4a a 1 Then, ð þ1



4T 02 4tT 0 4T 0 j X  X0 j 4ZT 0 4Z 0 T 0 dT exp  2 þ 2   2 þ 2 r r r cL r cL r2 cL 1 0



" 2 # pffiffiffi tcL þ j X  X0 j þ Z  Z 0 r p exp : ¼ 2 r2 c2L

(53)

With the result of the two temporal integrations and the simplified notations of covariance and correlation functions, Eq. (49) is reduced to   pffiffiffi p x80 LL r p t2 3 pffiffiffiffiffiffi ^ p 0 p^ 0 r^r^ ^ r r^q^ 0 q^ 0 2 2 2 4ð LL Þ ð Þ ð Þ ð Þ exp  2 g 0 N^p 0 p^ 0 r^r^ 0 ~g ðpÞN^r r^q^ 0 q^ 0 ðpÞ 2pr 4 8 ~ ULLL t ¼ c ðTf L TLf Þ q A0 2p r 2 2cL cL " # ð 0 2 0 4 2 2 1 w 2X 2Y j X  X j 2tj X  X j 0    d 3 Xd 3 X0 þ exp  2 r2 cL r2 c2L w ð Z Þ w2 ð Z Þ j X  X0 j2 w2 ðZÞw2 ðZ 0 Þ " # Z2 2Zt 2Zj X  X0 j 2X02 2Y 02 Z 02 0  2aL Z2aL j X  X j  2 0  2 0  2 2  exp  2 2 þ 2  r cL r cL r2 c2L w ðZ Þ w ðZ Þ r c L " # 2ZZ0 2tZ 0 2Z 0 j X  X0 j : (54)  exp 2aL Z 0  2 2 þ 2  r cL r cL r2 c2L A change of variable is used by defining X ¼ X0 þ r, and using spherical coordinates on r and Cartesian coordinates on X0 . In this case, Eq. (54) becomes   pffiffiffi p x8 LL r p t2 3 pffiffiffiffiffiffi ^ p p^ r^r^ ^ r r^q^ q^ exp g ð0ÞN^p 00 p^ 00 r^r^ ð0Þ~g LL ðpÞN^r r^q^ 00 q^ 00 ðpÞ ULLL ðtÞ ¼ c2 ðTf L TLf Þ2 q2 A40 ð2pÞ 2pr 4 80 ~ r2 2 2cL cL ð 1 w40  d 3 X0 r 2 dr sin / d/ d# 2 2 0 r w ðZ þ r cos /Þw2 ðZ0 Þ " # 2 2 2ðX0 þ r sin / cos #Þ 2ðY 0 þ r sin / sin #Þ r2 2tr   2 2þ 2  exp  w2 ðZ 0 þ r cos /Þ w2 ðZ0 þ r cos /Þ r cL r cL " # ðZ 0 þ r cos /Þ2 2tðZ 0 þ r cos /Þ 2rðZ 0 þ r cos /Þ þ   exp  r2 cL r2 c2L r2 c2L " # 2X02 2Y 02 Z 02 0  exp 2aL ðZ þ r cos /Þ  2aL r  2 0  2 0  2 2 w ðZ Þ w ðZ Þ r c L " # 2Z 0 ðZ 0 þ r cos /Þ 2tZ0 2Z 0 r 0 þ 2  2 2 :  exp 2aL Z  (55) r cL r cL r2 c2L The integration over X0 and Y 0 in Eq. (55) can be performed in a similar fashion as the integration over T 0 in Eq. (53). Reorganization of Eq. (55) gives   pffiffiffi p x80 LL r p  2 4 t2 3 pffiffiffiffiffiffi ^ p 0 p^ 0 r^r^ ^ r r^q^ 0 q^ 0 2 2 2 4ð LL p w0 exp ULLL ðtÞ ¼ c ðTf L TLf Þ q A0 2pÞ 2pr 4 8 ~g ð0ÞN^p 0 p^ 0 r^r^ ð0Þ~g ðpÞN^r r^q^ 0 q^ 0 ðpÞ r2 2 2cL cL " # ð þ1 ð p ð þ1 sin / 2r 2 sin2 / r2 2tr 0  dZ dr d/ 2 0 þ exp  2 0  w ðZ þ r cos /Þ þ w2 ðZ0 Þ w ðZ þ r cos /Þ þ w2 ðZ 0 Þ r2 c2L r2 cL 0 0 0 " # 02 ðZ 0 þ r cos /Þ2 2tðZ 0 þ r cos /Þ 2rðZ 0 þ r cos /Þ Z  exp  þ  2aL r  2aL ðZ 0 þ r cos /Þ  2 2 r 2 cL r2 c2L r cL r2 c2L " # 0ð 0 0 0 2Z Z þ r cos /Þ 2tZ 2Z r þ 2  2 2 :  exp 2aL Z 0  (56) 2 2 r cL r cL r cL The experimental parameter c can be acquired from the calibration of the transducer15 and expressed as J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

P. Hu and J. A. Turner: Double scattering in polycrystals

329

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rffiffiffi 2 wðzF Þ  2 2 ð2af zF Þ c ¼ Vmax pw0 e : Rf f Dðx0 Þ p w20 2qf kf2 rc2f

(57)

In Eq. (57), all of the corresponding parameters are related to the calibration procedure. Vmax denotes the maximum voltage amplitude of a reflected signal from a planar surface for a water path of zF , where zF is the specific water path which is considered to be the focal length of the transducer. The reflection coefficient is given by Rf f ¼ ðqcL  qf cf Þ=ðqcL þ qf cf Þ, and diffraction coefficient can be written as Dðx0 Þ ¼ j 1  eð2pi=sÞ ½J0 ð2p=sÞ þ iJ1 ð2p=sÞ j with s ¼ 4pcf zF =x0 w20 .15,35,36 Then, Eq. (56) can be reorganized after substituting the physical expressions of A0 and c, giving ( ) #2  2 " 2 4 w z T T p qc p x ð Þ F ^ ^ ^ f L Lf p r r ^ p 2 L 0 LL ð Þ pffiffiffi ~g 0 N^p 0 p^ 0 r^r^ ð0Þ expð4af zF  4af zf Þ ULLL ðtÞ ¼ Vmax 0 0 w0 2 c8L qf c2f Rf f Dðx0 Þ 8 2 ( )  ð ð ðp p x40 LL t2 þ1 0 þ1 4pw20 sin / ^ r r^q^ 0 q^ 0 ~ exp g  p p dZ dr d/ 2 0 ð ÞN ð Þ ^ r r^q^ 0 q^ 0 8 2 r 2 cL w ðZ þ r cos /Þ þ w2 ðZ 0 Þ 0 0 0 " # 2r 2 sin2 / r2 2tr þ 2  2aL r   exp  2 0 2 2 2 0 ð Þ cL r r cL w ðZ þ r cos /Þ þ w Z " # 4Z 02 4Z 0 r cos / r2 cos2 / 0   exp 2aL r cos /  2 2  4aL Z  r2 c2L r2 c2L r cL " # 4rZ 0 2r 2 cos / 4tZ 0 2tr cos / : (58) þ 2 þ  exp  2 2  r cL r2 cL r2 c2L r cL This result may be understood physically by the three integrals. The integral over Z 0 accounts for the depth of the first scattering event. The integrations in cylindrical coordinates over r and / account for all possible locations of the second scattering event. Due to the similarity of the longitudinal second-order scattering, the transverse second-order scattering can be obtained following an analogous formalism. It is given by ULTL ðtÞ

( ) #2  2 " 2 qcL TfL TLf p wðzF Þ p x40 LT ^ p 0 p^ 0 r^ r^ pffiffiffi ~g ð0ÞN^p p^ r^ ðI r^ Þð0Þ expð4af zF  4af zf Þ 0 0 w0 2 c8L qf c2f Rff Dðx0 Þ 8 2 ( )  ð þ1 ð þ1 ð p p x40 LT t2 4pw20 sin/ r^ r^q^ 0 q^ 0  0 ~g ðpÞN ðI r^ Þr^q^ q^ ðpÞ exp  dZ dr d/ 8 0 0 r2 2 cL w2 ðZ0 þ rcos/Þ þ w2 ðZ0 Þ 0 0 0 " # 2r2 sin2 / r2 2tr  2 2 þ 2  2aT r  exp  2 0 2 0 w ðZ þ rcos/Þ þ w ðZ Þ r cT r cT " # 4Z 02 4Z 0 rcos/ r 2 cos2 / 0  2 2  exp 2aL rcos/  2 2  4aL Z  r cL r2 c2L r cL " # 4rZ 0 2r2 cos/ 4tZ0 2trcos/  2 þ 2 þ 2 ;  exp  2 r cL c T r cL r cL cT r cL

2 ¼ Vmax

(59)

where ~ g and N are the correlation and covariance functions of transverse second-order scattering given as     x0 x x x ^ 0  0 ^r ¼ ~g LT ðhÞ and ~g 0 ^r  0 q ^ 0 ¼ ~g LT ðp  hÞ ~ p g cL cT cT cL

(60)

and ^ p p^ r^

^0a r^b p^0i r^j p^0c p^0k ðddl  r^d r^l Þ ¼ N^p 0 p^ 0 r^ ðI rr^^ ÞðhÞ; Nabcd ijkl p 0 0

r^q^ q^

lsp r^ 0 0 r^l q^0 r^e q^0f ðdsm  r^s r^m Þ q^0p q^0n ¼ N Nefmn  ðI r^ Þr^q^ 0 q^ 0 ðp  hÞ:

(61) Similar to the SSR, both the longitudinal and transverse second-order scattering responses consist of the experimental calibration parameters, two diffuse backscattering coefficients (in brackets ½ ), and the remaining input wave and beam characterization terms including the triple integral of second-order scattering over the depth Z0 , the radius of r, and inclination angle /. The diffuse backscattering coefficient, which is the product of the covariance and correlation function, is determined 330

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by the microstructural properties of the sample as shown in Eqs. (63) and (64). The microstructural information can be extracted with the assistance of experimental measurements as demonstrated previously.15 Along with the singly-scattered response, the DSR in Eq. (32) can be written in final form as #2    2 " 2 qc T T p wðz Þ t2 fL Lf F 2 L UðtÞ ¼Vmax pffiffiffi exp 4a z  4a z exp ð Þ f F f f r2 qf c2f Rff Dðx0 Þ 8 2 w0 8" #ð " # < p x4 þ1 2 4Z 0 ðZ 0  tcL Þ ^ p 0 p^ 0 q^ 0 q^ 0 0 w0 0 0 LL ~g ð0ÞN^p p^ q^ q^ ð0Þ  dZ 2 0 exp 4aL Z  0 0 0 0 : 2 c8L w ðZ Þ r2 c2L 0 " #" #ð ð þ1 ð p þ1 p x40 LL p x40 LL 4pw20 sin/ ^ p 0 p^ 0 r^r^ ^ r r^q^ 0 q^ 0 0 ~ ~ þ g g ð0ÞN ð0Þ ðpÞN ðpÞ dZ dr d/ 2 0 ^ r r^q^ 0 q^ 0 ^ p 0 p^ 0 r^r^ 8 8 2 cL 2 cL w ðZ þ rcos/Þ þ w2 ðZ 0 Þ 0 0 0 " # 2r2 sin2 / r2 2tr  exp  2 0  þ  2aL r w ðZ þ rcos/Þ þ w2 ðZ0 Þ r2 c2L r2 cL " # 4Z02 4Z0 rcos/ r 2 cos2 / 4rZ 0 2r2 cos/ 4tZ0 2trcos/ 0  exp 2aL rcos/  2 2  4aL Z   2 2  2 2 2 2 þ 2 þ 2 r cL r cL r2 c2L r cL r cL r cL r cL " #" #ð ð ð þ1 þ1 p p x40 LT p x40 LT 4pw20 sin/ ^ p p^ r^ 0 r^ r^q^ 0 q^ 0 ~g ð0ÞN^p 0 p^ 0 r^ ðIrr^^ Þð0Þ ~g ðpÞN ðI Þ ðpÞ dZ dr d/ þ ^  r ^ ^ ^ r q q 8 8 2 0 0 0 0 0 2 cL 2 cT w ðZ þ rcos/Þ þ w2 ðZ0 Þ 0 0 0 " # 2r2 sin2 / r2 2tr   exp  2 0 þ  2aT r w ðZ þ rcos/Þ þ w2 ðZ0 Þ r2 c2T r2 cT " ) 4Z02 4Z0 rcos/ r 2 cos2 / 4rZ 0 2r 2 cos/ 4tZ0 2trcos/ 0  2 2  2  þ þ 2 : (62)  exp 2aL rcos/  2 2  4aL Z  r cL cT r2 cL cT r2 cL r cL r cL r2 c2L r cL

Equation (62) is the final result of the theoretical model of the doubly-scattered response based on the Wigner distribution function. The DSR is dependent on the experimental conditions, the transducer parameters, and the material properties as expected. It is important to note that the SSR is proportional to the single-crystal covariance of the material N while the second scattering is proportional to the square of this quantity. Thus, the relative importance of the double scattering is connected directly to this quantity as will be shown subsequently. In addition, it is of interest to examine the relative contributions of the second scattering in terms of the LLL and LTL components. The LTL term is much more important because of the large amount of longitudinal energy that converts to shear energy at the first scattering event. Although the ratio of LTL/LLL is dependent on many factors including time, it is roughly on the order of the wave speed ratio to the eighth power, such that LTL/LLL > 300 for aluminum and >130 for iron. In Sec. VI, numerical results are presented to compare directly the SSR and DSR expressions by varying parameters typical of experiments.

The integrations given in Eq. (62) are evaluated at discrete points of time in order to create the complete temporal scattering curve. The integrands are well behaved such that the infinite integrals (for z and r) are stopped after the value of the integration has converged at each time point. The discretization used for / was found by reducing it until convergence was attained. The single crystal properties of the two materials of interest are given in Table I. The properties used for the coupling fluid, water in this case, are qf ¼ 998:2 kg=m3 , wave speed cf ¼ 1486 m/s, and attenuation af ¼ 0:0253  ðx=2pÞ2 Np/m (the unit for frequency f0 ¼ x0 =2p is MHz).31 The nominal element diameter a0 ¼ 9:525 mm and the pulse width of transducer r ¼ 0:158 ls are selected as constant values. The water path for the transducer is assumed to be zf ¼ 20:5 mm, and the material path in the sample is obtained through z ¼ ðF  zf Þ  cf =cL (where F is the focal length of the transducer). The longitudinal and

TABLE I. Single crystal material properties for numerical analysis (Refs. 37 and 38).

VI. NUMERICAL RESULTS

In order to explore the DSR in detail, especially with respect to the SSR, polycrystalline iron and aluminum are used for the following numerical analysis because they represent strongly and weakly scattering materials, respectively. J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

Material Type

c11 (GPa)

c12 (GPa)

c44 (GPa)

 (GPa)

q ðkg=m3 Þ

cL (m/s)

cT (m/s)

Iron Aluminum

219.2 106.7

136.8 60.41

109.2 28.34

136 10.39

7860 2700

5900 6408

3230 3119

P. Hu and J. A. Turner: Double scattering in polycrystals

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FIG. 3. (Color online) Comparisons between normalized SSR and DSR with varied kL for aluminum and iron sample. Aluminum: (a) kL ¼ 0.0735, (b) kL ¼ 0.1471, (c) kL ¼ 0.2942. Iron: (d) kL ¼ 0.08, (e) kL ¼ 0.16, (f) kL ¼ 0.32. Note: ——, DSR;   , SSR.

transverse material attenuations for these calculations are based on the theory developed by Weaver.3 With further assumptions on the single crystal properties, the grains of the sample are cubic, equiaxed, and single phase, which leads to the correlation functions, Eqs. (50) and (60), and covariance functions, Eqs. (51) and (61), for the longitudinal and transverse components as3,5,39 " #2 2 1 1 2x ~ g LL ðhÞ ¼ 2 þ 2 0 ð1  coshÞ ; p L L2 cL ^ p p^ r^r^

^ r r^q^ q^

N^p 00 p^ 00 r^r^ ðhÞ ¼ N^r r^q^ 00 q^ 00 ðp  hÞ   2 9 6 1 þ cos2 h þ cos4 h ¼ 2 q 525 525 525 (63)

and " #2 1 1 x20 x20 2x20 g~ ðhÞ¼ 2 þ þ  cosh ; p L L2 c2T c2L cT cL LT

^ p p^ r^

r^q^ q^

r^ 0 0 N^p 00 p^ 00 r^ ðI rr^^ ÞðhÞ¼N  ðI r^ Þr^q^ 0 q^ 0 ðphÞ    2 15 6 1 2 4 þ cos h cos h ; ¼ 2 q 525 525 525 (64)

where L is the correlation length of the material, often as a measure of the grain size, and  ¼ c11  c12  2c44 is the anisotropy coefficient for cubic crystal symmetry. The first example for aluminum and iron is a comparison between SSR and DSR as a function of dimensionless

FIG. 4. (Color online) Comparisons between normalized SSR and DSR with varied focal length F at 10 MHz for iron sample (L ¼ 15 mm). (a) F ¼ 50.8 mm, (b) F ¼ 101.6 mm, (c) F ¼ 152.4 mm. Note: ——, DSR;   , SSR. 332

J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

P. Hu and J. A. Turner: Double scattering in polycrystals

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FIG. 5. (Color online) Comparisons between normalized SSR and DSR with varied focal length F at 10 MHz for iron sample (L ¼ 15 mm) fitted by SSR. (a) F ¼ 50.8 mm fitted by SSR (L ¼ 16.08 mm), (b) F ¼ 101.6 mm fitted by SSR (L ¼ 17.38 mm), (c) F ¼ 152.4 mm fitted by SSR (L ¼ 18.53 mm). Note:   , SSR (L ¼ 15 mm); ——, DSR (L ¼ 15 mm);     , Fitted SSR (varied L).

frequency kL, which relates the center frequency of the transducer and the microstructural grain dimensions of the material, where wave number is k ¼ x0 =cL . The center frequencies used to generate Fig. 3 are f0 ¼ 7.5, 10, and 15 MHz, and the corresponding correlation lengths are L ¼ 10, 15, and 20 lm. With fixed water path zf ¼ 20:5 mm and constant focal length F ¼ 50:8 mm for each case, their material paths are similar, i.e., z ¼ 7:0 mm for aluminum and z ¼ 7:6 mm for iron, respectively. In Figs. 3(a), 3(b), and 3(c), it is apparent that the difference between the SSR and DSR for aluminum is negligible under these three distinct conditions. However, in Figs. 3(d), 3(e), and 3(f), the changes between SSR and DSR for iron increase as kL increases. Thus, for the highest level of scattering [Fig. 3(f)], the importance of the higher-order scattering is substantial. The second example depicts the dependence of SSR and DSR on the focal length F of the transducer, i.e., F ¼ 50.8, 101.6, and 152.4 mm. For a constant water path, a larger value of F implies that the first scattering event will occur deeper in the material. This larger depth for the first scattering event increases the opportunity for double scattering to be relevant within the time window of interest. Thus, the expectation for DSR to be more important for larger values of F was explored. The correlation lengths of aluminum and iron are chosen the same, L ¼ 15 lm, and the center frequency of transducer is uniform, f0 ¼ 10 MHz. On the basis of the relation among the focal length, the water path, and the material path, z ¼ ðF  zf Þ  cf =cL , the material paths for each case are altered accordingly. Similar to the first example, the results for aluminum do not show significant differences between the SSR and DSR because the scattering is so weak. Thus, these results are not presented. For iron, the results are shown in Fig. 4. It is also clear that the widths of the scattering responses increase with larger focal lengths as expected. Modeling the backscatter or grain “noise” to extract the microstructural parameters in heterogeneous materials is one application of diffuse ultrasonic backscatter techniques. Ghoshal and Turner successfully attained the correlation length of the sample by combining the theoretical SSR model and experimental measurements.15 However, the validity of that SSR could not be tested to understand the change in the predicted L with higher-order scattering. The correlation length will be determined by means of fitting the peak of the DSR using an SSR model for an iron sample J. Acoust. Soc. Am., Vol. 137, No. 1, January 2015

with L ¼ 15 mm, and f0 ¼ 10 MHz for varied focal lengths corresponding to results in Fig. 4. These results are shown in Fig. 5. It is observed that the first fitted SSR curve fits most of the DSR curve except the “tail” region. The difference between DSR for L ¼ 15 lm and the fitted SSR curve of L ¼ 16:08 lm is 7.2% shown in Fig. 5(a). The largest discrepancy exists for the third case of F ¼ 152:4 mm, which is shown in Fig. 5(c), with 23.5% between difference of the correlation length and 0.54 ls separation of the arrival times of the DSR and the fitted SSR peaks caused by the additional traveling time of longitudinal and transverse waves during the second scattering. From the above three examples, it is noted that aluminum displays negligible changes between SSR and DSR than the corresponding cases for iron, which generally possesses stronger scattering for the same microstructural length scale. The results also suggest that the portion of the second-order scattering should be taken into account for measurements involving strong scattering materials or long propagation path inspections. The influence of the factors that contribute to stronger scattering is still under investigation. It is anticipated that this work will ultimately lead to the full multiple-scattering form.

VII. CONCLUSIONS

In this article, a theoretical diffuse ultrasonic backscatter model was derived which includes first and second scattering events. Similar to the SSR, the new doubly-scattered model is a convolution of WS , WG , and WR with the scattering operator K. Unlike the Wigner distribution functions of the source and receiver transducers (WS and WR ), the quantity WG is related to energy propagation in the heterogeneous medium. WG was determined using the high-frequency asymptotic assumption on the Wigner distribution function of the mean Green’s function. With the specific expressions for each item in the formulation of DSR, the second-order scattering of a piston transducer was determined for a pulseecho type of measurement. The solutions for polycrystalline aluminum and iron, which represent weak and strong scattering samples, were used to verify the developed DSR model and the results were compared with the SSR model. The results show that the portion of second-order scattering can be neglected for weakly scattering materials when focused P. Hu and J. A. Turner: Double scattering in polycrystals

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beams are used due to limited differences between SSR and DSR. The DSR model also can be utilized to extract microstructural information by combining with the experimental variance results in the ultrasonic backscatter measurements, such as estimation of the correlation length of the sample.15 It will be more complicated due to two correlation functions in the expression of the DSR model than directly from the SSR model, but the accuracy of the correlation length would be increased by the consideration of the contribution of the second-order scattering. The only decision needed to make is to inspect the necessity of the DSR model, especially for the strong scattering situations. It will be the topic of future research as well as the full multiple scattering expansion. ACKNOWLEDGMENTS

The support of the Federal Railroad Administration and the China Scholarship Council for this research are gratefully acknowledged. Lucas W. Koester and Christopher M. Kube are to be thanked for the suggestions on the manuscript. 1

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Contribution of double scattering in diffuse ultrasonic backscatter measurements.

Diffuse ultrasonic backscatter measurements are used to describe the effective grain scattering present during high frequency ultrasonic inspections. ...
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