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Wave scattering in discrete random media by the discontinuous stochastic field method, II: Contribution of the second order moment of the β-field
Journal of Sound
WAVE
and Vibration (1991) 147(2), 313-321
SCATTERING
IN DISCRETE
RANDOM
THE DISCONTINUOUS STOCHASTIC C?)NTRiBU _-_-_ 110~ _ 0j-j ,THE SECOND OF THE
MEDIA
BY
FIELD METHOD, II: _ _-_ _----UKUEK M(jMEN,I.
P-FIELD
K. C. LIU Northeast Research Institute, 3155 Giddings Rd, Auburn Hills, Michigan 48326, U.S.A. (Received 27 September 1986, and accepted in revised form 10 July 1990)
The general expression for the space correlation function of the multiply scattered field from a discrete random medium as estabiished in Part i is modified to a compact series form. The expression for the space correlation function of the discontinuous stochastic field P(r, o) is derived for spherical scatterers by means of stochastic geometry. From this, an expression for the first term of this series is obtained in terms of the sizes and volume density of the scatterers. In many practical problems, the first term accurately represents the whole space correlation function. For ultrasound scattering from blood cells, the results show better agreement with experimental results than those obtained from Twersky’s formula, especially for the case of a high volume density (higher than 1.5%) of scatterers.
1. INTRODUCTION In Part I [l], a general expression for the space correlation function of the multiply scattered field from a discrete random medium was established by means of introducing a discontinuous stochastic field /3(r,0). This general expression is in the form of a series, each term of which is an integral with a statistical moment of the discontinuous stochastic field /3(r,co)of corresponding order in its integrand. In order to obtain a specific expression for a specific random medium, the calculation of the statistical moments of /3(r, W) must first be carried out. In this second companion paper (Part II) the calculation of the second order moment (space correlation function) of P(r, w) for spherical scatterers is presented, as obtained hv ,,ri,,,, atnrhsat;o c,~n,,.,ntr., r3 21 E’rr\m +l.;m 0,. n~..lin:t a.r.s-amm:e.a. C-M .I.Ce.+ +--.a“J ..L.“‘6 OC”In.Lb~Cn~ ~-“‘L’bL’J LL, 2,. 1 1”l.l 1,115) all F;npuur G*p,GxiI”ll ,“I 111c 11131 Ctz“11,
the most important term of the series expression for the space correlation function of the scattered field, can be obtained. It can be shown that in many practical problems, the first term accurately represents the entire series: i.e., the sum of all other terms can be neglected. The theoretical results obtained are compared with some experimental results obtained by Shung et al. [4] in their experiments on ultrasound scattering from blood cells, and they are also compared with Twersky’s formula [5].
2. COMPACT
FORM OF GENERAL
EXPRESSION
The general expression for the space correlation function Kp of the multiply scattered field from a discrete random medium as given in Part I can be put into the compact form 313 0022-460X/91/110313+09%03.00/0
0
1991
Academic
Press
Limited
314
K. C. LIU &kl,*2,
b, t2)=
8Mrl,
+
fl,~Mr2,
C C &Al,
m=l
where the functions L,,
f2,~))
f2, b,
t2),
n=l
and K,,,, are defined as
A,
%‘{JI*(r,, t,, u)K,(r2,
r’, v, w)P(r’,
v, w)} eiv’2dr’dv,
(2.2) %‘{t,!~(r~, f2, w)K*,(r,,
r’, v, o)P*(r’,
v, w)} e-‘“‘I dr’ dv, (2.3)
K,,(r!,r2:
CT cc g{K;(r:;r;; t!: tz)=& (27# J J J J
~!~w)tJI*(r;~ v:~o)
9 -co
x
K,,(r2, r;, v2, w)P(r:,
v2, CO)}e-“l’l ei”zL2dr: dr: dv, dv2.
(2.4)
From these definitions, the following relations can be directly obtained: ~*m(r2,C1,fZttl)=L,(rl,~2,
t1, f2L
A,(r2,r,,t2,fl)=L*m(rl,r2,fl,
A,(r,,r~,tl,f2)=L*m(rz,r,,
f2,hL
A%rl,r2,
t2),
b, t2)=L(r2,rl,
(2.5,2.6) (2.7,2.8)
~2, 0
Therefore, the functions A, can be obtained from the functions L,, and hence it is sufficient to calculate the functions L, and the functions K,,,,. Using the expressions for the functions J/(r, t, w), W(r, v, a), K(r, r’, v, w) and c__.-- -___-r!-~--,~ *\ I_ .\ ~~ _*_A_* I.. ra’TT.7 TI I- _I v,w) \ in ~~(r,r, rdrt I 11~,1rom equauons (L.LJ-(L.~~ one omainstheiniegraiexpressions +oO
1 LAr,,
r2,&9
G*(r,, r’, v’)G(r2, r”, v)G(r”, r”‘, v) $
tz)=(2T)2 a
0
Y2 2 2
-m
x Pg(r’, v’)ZJ,,(f”, v)Kb))(r’, r”, r”‘) e-‘““I ei”‘zdr’ dr” dr”’dv dv’,
(2.9)
G*(r,, r’, v’)G(r2, r”, v) Y fxr” ,,IC/r” I. V,’ ,’WJ” ,“ ,V\_ ).P), p) . . . qp+y
d2
y2
p+3,
2)
m+1
(>
P$(r’, v’)P,,(r(“‘+‘), v)Kkm+‘)(r’, r”, . . . , r(“‘+‘)) co co X e-i”“I eiv’zdr’ dr” . . . &(m+2) d,, dv’, m=l,2,3 ,..., xz
1
(2.10)
where the functions KF’, KF’, . . . , Kb”’ are the higher order statistical moments of the stochastic field P(r, o), defined as follows: Kr)(r’, r”, f”) = %{/3(r’, w)p(r”, w)p(f”, w)},
(2.11)
Kg)(r’, r”, r”‘, rc4)) = EP{P(r’,w)P(r”, w)j?(r”‘, o)p(rc4’, co)},
(2.12)
KhmJ”)(r’ r”, , r”‘, . . . , P))
= %{/3(r’, w)@(r”, w) - - * p(rtm), w)},
m=l,2,3
,....
(2.13)
SCATTERING
IN DISCRETE
RANDOM
MEDIA,
315
II
There are expressions similar to equations (2.9) and (2.10) for A,(r,, K,,(r,, r2, t,, tr) in terms of the higher order moments of P(r, 0).
r2, t,, f2) and
3. SPACE CORRELATION FUNCTION OF /3(r, w) FOR SPHERICAL SCATTERERS From the expression for &(r,, rz, t, , t2), equation (4.10) and equation (5.2) of Part I [ 11, one can see that the space correlation function Kp (I), or the corresponding space spectrum density K@(k), plays an important role in the calculation of the statistical moments of the scattered field. K,(1) depends on the shape and size of each scatterer and the concentration u (relative volume density) of scatterers. The calculation of Kp (l) or K,(k) is an exercise in geometrical probability [2,3]. In view of the complexity and difficulty of this task, in this paper, it is carried out only for spherical scatterers. Obviously, the stochastic field /3(r, w) is ergodic [2]. Therefore, for any sample (realization) p (r, wi), one has Kp(I)=
1
lim V(D)-m V(D)
P(r, w,)P(r-I,
wi) dr,
(3.1)
where D is a region with any shape in ge,, and V(D) is the volume of D. Assume that the distance from any sphere to its nearest sphere is a random variable s(w), and that s(w) has the eXpOneniiai prOb&iiity distribution density
(3.2) where &, is the mean value of t(w); i.e., 50 = 8(5(w)).
(3.3)
Then, using equation (2.8) of Part I [l] and equations (3.1) and (3.2), from geometry one obtains (see Figures 1 and 2) (3.4)
Figure
1. Two different
configurations
of the random
system and their region
of overlap.
316
K. C. LIU
Figure
2. Nearest
neighbours
and the random
distance
[+2a
between
them.
where j, = int {(1/2a) - l},
(3.5)
in which int {A} denotes the integer part of A,
+[I_2(j-l)a]
eA(‘-*JO)/Q
jc (1/2a) - 1,
(3.6) I( I, a, 0) =
1 E [O, 2a]
2a-1,
lS2a
{ 0,
(3.7)
I ’
Substituting equations (3.6) and (3.7) into equation (3.4), one obtains an approximate expression for Ka (I) as K,(I)=be-Y’{1+cos(2rr1/1,)}, where b, y are constants which are independent
(3.8)
of 1, and
lo = 2a + & = (2/&)a.
(3.9)
On the other hand, for a small value of /, (I G 2a), from geometry, one can obtain an accurate expression for Kp (I):
Kp(~)=r~P:l(l-a)21[(1-a)-t(~la)+~(~3/a3)l,
lS2a.
(3.10)
From equation (3.8), dK,(WdL,=
K,(O)=2b.
-2yb,
(3.11)
From equation (3.10), dK,(l)/dll,=,
= -3@:/4(
1-
~)*a,
Kp(0)=&(l-a).
(3.12)
Comparing equations (3.11) and (3.12) gives b=#/(l-a), Therefore,
y=;[l/(l-a)a].
(3.13)
substituting expression (3.13) into equation (3.8), one obtains KP(I)=f[a/(l-o)]~~e-a”“-~)a[l+cos(~&I/a)].
(3.14)
SCATTERING
4. SCATTERING
IN
DISCRETE
FORMULA
RANDOM
MEDIA,
FOR SPHERICAL
317
II
SCATTERERS
Substituting equation (3.14) into equation (5.16) of Part I [ 11, integrating substituting the results into equation (6.6) in Part I [1], one obtains K,(2k,)
= ($)’ 3
vo
a2a( 1 - u)~@: 4k,,a +
and again
2k,a + TT~ [I + @‘( 1 - u)2(2koa + rfi)‘]’
+
(4.1)
and M’,Z’(r, t) =
2k,a + PG
2A2viR2ra2 ci(c: - c:)’ 27r2 (CIC214
4koa+[l+(~)2(l-(r)2(2koa+~~)2]2
2koa -ITS%
(4.2)
+[l+(~)2(l-~)2(2koa-~fi)2]2 For the case of wavelengths A >>a, equation (4.2) can be simplified to M’,Z’(r, t) = ($)’
A27R2a3vico( c: - c:)’
a(1 - a)4
(c1c2)4r2
For the case of a short wavelength expression (4.2) can be simplified to Mf’(r,
t) = (3)”
’+
[1
+
1 (4,r/3)‘a’/‘(
1 a(1-a)4
c2J4r2
.
-(r)2(2koa)2]2
‘+[I+($)‘(1
(4.3)
such that 2koa >>~6,
or small concentration
A27R2a3v~cO(c~-~:)2 (c,
1 _ #]2
(4.4)
By using equation (2.6) of Part I [1], equation (4.3) can also be written as M’,)(r,
t) = (3)’
A2rR2a3vi(c:-cf)2
(c,c2)3r2
a(l-a)4 &c;+(l
1
1+ -a)cf
[1+ (4Tr/3)2a2’3( 1 - C7)2]’ *
In formula (4.3), the part that depends on the concentration fO(cr)=C(l-V)4 In comparison
{
1+
u can be expressed as
1
1
[ 1+ (4T/3)‘C7”3( 1 -a)‘]’ 1 v&c:+ (1 - o,c:’
(4.5)
with Twersky’s formula [5] Rhc= [ Wo(l - w;j”/(i +2 wo)z](~bsj v),
where W, is the concentration
a, the factor a( 1 -a)”
(4.6)
is the same, but the factor
l/(l+2wo)‘.
(4.7 )
in Twersky’s formula is now replaced by 1 [ 1+ (4?r/3)2#3(
1 1 - C)2]’
ac:+(l-a)c:’
(4.8
1
318
K. C. LIU 5. COMPARISON
Ilr
53.
WITH
EXPERIMENT BY BLOOD
ON ULTRASOUND CELLS
SCATTERING
For red blood cells, the density p, is l-099 g/cm3, and the compressibility I. .,--I2 2, I I x IV cm-jayne. Thus, the sound speed in biood ceiis is c, = l/G
= 1596 m/s.
For salt water, the density p2 is l-005 g/cm3, lo-l2 cm2/dyne, and the sound speed is thus c2 = l/G
K1
is
(5.1)
the compressibility
= 1499 m/s.
~2
is
44.3 x (5.2)
Therefore c,-c,=97m/scc
c, and c2.
(5.3)
Hence p, CC1, p2<< 1, /3(r, w)c 1, Vrc Se3, o E R. Therefore one can neglect those terms which consist of the higher order moments of P(r, w), and obtain &,(rl,r21
cl,
t2)=Wr,,r2,
cl,
&,(r,
f2L
A4’ * (r Pt2’(r5 t) = kC2’
3.
1) =
&(r,
0,
t)
(5.4)
(5.5)
Therefore, the mean square value of scattered field ME’(r, t) can be calculated from formulas (4.2) and (4.3), or (4.4), and equation (4.3) can be rewritten as Mg’(r,
t) = ($)3[A2dZ2a3v~(~:- ~f)~/(c,c~)~r~]f~(a),
(5.6)
where &(a) is given by equation (4.5). In Figure 3, Jo(a) is plotted as a function of a. This curve also shows the dependence of the mean square value of the scattered field on the concentration u. “L...._ er_. UI. _I ~41 r.7 cdrrleu _.-_-Z-d out _.-I 8ii ejrptTiiTieili mung Ori uiir%xxirid scaiieririg from biood ceiis. measured the mean square values of the scattered fields for different concentrations (a changes from 0% to 60%). They compared their measured results with Twersky’s theoretical results [5] and found that in the low concentration region, they agreed well, but in the high concentration region there is an obvious and systematic deviation. Our
They
O.‘Oc
I
0
I
0.1
1
1
0.2
I
I
0.3
0.4
0.5
0.6
0.7
0.8
Figure 3. Graph of mean square value of scattered field versus volume concentration of scatterers. c,, Wave speed of scatterer maieriai; c2, wave speed of surrounding medium; 0, voiume conceniraiion of scaiierers; ,&(o), function of c defined by equation (4.5).
SCA-ITERING
IN DISCRETE
RANDOM
Figure 5. Back-scattering coefficient versus concentration.
results agree with the experimental Fio~ww - ‘0~---
MEDIA,
II
319
Cl, Experimental results [4].
results for both low and high concentration
(see
4-?).
6. CONCLUSIONS The general formula for the space correlation function of a multiply scattered field from a discrete random medium can be put into a compact series form. The first term of this series represents the contribution of the second order moment of the discontinuous stochastic field /3(r, 0). The second term (L, + A,) of this series represents the contribution of the third-order moment of /3(r, 0). The third term (L,+ A,+ K,,) represents that of the fourth order moment of P(r, w), and so on. Since the centralized dimensionless stochastic field j3(r, w ) is proportional to the square A+Af tk.¶ PwaPA.2 if ~nad thr &,‘W IUIVP .._.I uywwu Ai(FPrmnrr U‘II”I~X.VW hrtwrm “1...V”‘S C..V “1 +h.p cut rPl”*:.,P LU1QI1.W A;u~r~n,-.a Ul“~l~..“~ “a &I.” ,,,a-,“Gs.b ayrruc?, n1 tha materials of the scatterers and the host medium is relatively small (the case of weak
320
K. C. LIU
0
10
20
30
40
50
1
Qwe) Figure
7. Back-scattering
coefficient
versus concentration.
0, Experimental
results
[4].
fluctuation), then the first term plays the dominant role, and this is precisely the Born approximation [6]. But if the wave speed difference is not relatively small (the case of strong fluctuation), then the other terms, the contributions of the higher order moments of P(r, w), must be taken into account. For spherical scatterers, the second order moment (space correlation function) of p(r, w) can be obtained as an explicit expression approximately derived from stochastic geometry. Therefore, the spatial spectrum density of P(r, CO)as well as the first term of the space correlation function and the mean square value of the scattered field can be obtained in terms of the wave speeds of the two materials, the radius of each scatterer and the concentration (or volume density) of the scatterers. The specific formula for this case shows that (1) the first term, or the intensity of the scattered field in the weak fluctuation case, is proportional to the square of the difference of the wave speeds (this is physically reasonable) and (2) it is also proportional to the fourth power of the frequency of the incident wave, which agrees with the well known Rayleigh relation [7].
SCATTERING
IN DISCRETE
RANDOM
MEDIA, II
321
The most important result is the functional dependence of the first term, or the intensity of the scattered field for the weak fluctuation case, on the concentration of the scatterers. This result has been plotted in Figure 3. The dependence formed is different from that obtained by Twersky [5], and it agrees with experimental results obtained for ultrasound scattering from blood cells [4], which is a weak fluctuation case. This agreement explains the deviation of these experimental results from the theoretical results obtained by Twersky [5].
ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr Leon H. Sibul, Dr George Adomian and Dr Alexander Wasiljeff for their valuable suggestions, and to Dr Tung H. Weng, Dr Vijay K. Varadan, Dr Vasundara V. Varadan and Dr Sabih I. Hayek for their review of the manuscript.
REFERENCES 1. K. C. LIU 1991 Journal ofSound and Vibration 147, 301-311. Wave scattering in discrete random media by the discontinuous stochastic field method, I: basic method and general theory. 3_. __. R “. 1 ~_I Anr II.. FIN lOR1 7% nf J~)lfi Wiley. _._-._- Gmrn~tt-~ --‘I...--., _~ _&n&a,% _Fip& &Jew Y&: 3. E. F. HARDING and D. G. FENDALL 1974 Stochastic Geometry. New York: John Wiley. 4. K. K. SHUNG, Y. W. YUAN, D. Y. FEI and J. M. TARBELL 1984 Journalofthe Acousrical Society of America 75(4), 1265-1272. Effect of flow disturbance on ultrasonic backscatter from blood. 5. V. TWERSKY 1978 Journal of the Acoustical Society of America 64, 1710-1719. Acoustic bulk parameters in distribution of pair-correlated scatterers. 6. M. BORN and E. WOLF 1959 Principles of Optics. New York: Pergamon Press. 7. LORD RAYLEIGH 1945 Theory of Sound. New York: Dover.
WAVE
and Vibration (1991) 147(2), 313-321
SCATTERING
IN DISCRETE
RANDOM
THE DISCONTINUOUS STOCHASTIC C?)NTRiBU _-_-_ 110~ _ 0j-j ,THE SECOND OF THE
MEDIA
BY
FIELD METHOD, II: _ _-_ _----UKUEK M(jMEN,I.
P-FIELD
K. C. LIU Northeast Research Institute, 3155 Giddings Rd, Auburn Hills, Michigan 48326, U.S.A. (Received 27 September 1986, and accepted in revised form 10 July 1990)
The general expression for the space correlation function of the multiply scattered field from a discrete random medium as estabiished in Part i is modified to a compact series form. The expression for the space correlation function of the discontinuous stochastic field P(r, o) is derived for spherical scatterers by means of stochastic geometry. From this, an expression for the first term of this series is obtained in terms of the sizes and volume density of the scatterers. In many practical problems, the first term accurately represents the whole space correlation function. For ultrasound scattering from blood cells, the results show better agreement with experimental results than those obtained from Twersky’s formula, especially for the case of a high volume density (higher than 1.5%) of scatterers.
1. INTRODUCTION In Part I [l], a general expression for the space correlation function of the multiply scattered field from a discrete random medium was established by means of introducing a discontinuous stochastic field /3(r,0). This general expression is in the form of a series, each term of which is an integral with a statistical moment of the discontinuous stochastic field /3(r,co)of corresponding order in its integrand. In order to obtain a specific expression for a specific random medium, the calculation of the statistical moments of /3(r, W) must first be carried out. In this second companion paper (Part II) the calculation of the second order moment (space correlation function) of P(r, w) for spherical scatterers is presented, as obtained hv ,,ri,,,, atnrhsat;o c,~n,,.,ntr., r3 21 E’rr\m +l.;m 0,. n~..lin:t a.r.s-amm:e.a. C-M .I.Ce.+ +--.a“J ..L.“‘6 OC”In.Lb~Cn~ ~-“‘L’bL’J LL, 2,. 1 1”l.l 1,115) all F;npuur G*p,GxiI”ll ,“I 111c 11131 Ctz“11,
the most important term of the series expression for the space correlation function of the scattered field, can be obtained. It can be shown that in many practical problems, the first term accurately represents the entire series: i.e., the sum of all other terms can be neglected. The theoretical results obtained are compared with some experimental results obtained by Shung et al. [4] in their experiments on ultrasound scattering from blood cells, and they are also compared with Twersky’s formula [5].
2. COMPACT
FORM OF GENERAL
EXPRESSION
The general expression for the space correlation function Kp of the multiply scattered field from a discrete random medium as given in Part I can be put into the compact form 313 0022-460X/91/110313+09%03.00/0
0
1991
Academic
Press
Limited
314
K. C. LIU &kl,*2,
b, t2)=
8Mrl,
+
fl,~Mr2,
C C &Al,
m=l
where the functions L,,
f2,~))
f2, b,
t2),
n=l
and K,,,, are defined as
A,
%‘{JI*(r,, t,, u)K,(r2,
r’, v, w)P(r’,
v, w)} eiv’2dr’dv,
(2.2) %‘{t,!~(r~, f2, w)K*,(r,,
r’, v, o)P*(r’,
v, w)} e-‘“‘I dr’ dv, (2.3)
K,,(r!,r2:
CT cc g{K;(r:;r;; t!: tz)=& (27# J J J J
~!~w)tJI*(r;~ v:~o)
9 -co
x
K,,(r2, r;, v2, w)P(r:,
v2, CO)}e-“l’l ei”zL2dr: dr: dv, dv2.
(2.4)
From these definitions, the following relations can be directly obtained: ~*m(r2,C1,fZttl)=L,(rl,~2,
t1, f2L
A,(r2,r,,t2,fl)=L*m(rl,r2,fl,
A,(r,,r~,tl,f2)=L*m(rz,r,,
f2,hL
A%rl,r2,
t2),
b, t2)=L(r2,rl,
(2.5,2.6) (2.7,2.8)
~2, 0
Therefore, the functions A, can be obtained from the functions L,, and hence it is sufficient to calculate the functions L, and the functions K,,,,. Using the expressions for the functions J/(r, t, w), W(r, v, a), K(r, r’, v, w) and c__.-- -___-r!-~--,~ *\ I_ .\ ~~ _*_A_* I.. ra’TT.7 TI I- _I v,w) \ in ~~(r,r, rdrt I 11~,1rom equauons (L.LJ-(L.~~ one omainstheiniegraiexpressions +oO
1 LAr,,
r2,&9
G*(r,, r’, v’)G(r2, r”, v)G(r”, r”‘, v) $
tz)=(2T)2 a
0
Y2 2 2
-m
x Pg(r’, v’)ZJ,,(f”, v)Kb))(r’, r”, r”‘) e-‘““I ei”‘zdr’ dr” dr”’dv dv’,
(2.9)
G*(r,, r’, v’)G(r2, r”, v) Y fxr” ,,IC/r” I. V,’ ,’WJ” ,“ ,V\_ ).P), p) . . . qp+y
d2
y2
p+3,
2)
m+1
(>
P$(r’, v’)P,,(r(“‘+‘), v)Kkm+‘)(r’, r”, . . . , r(“‘+‘)) co co X e-i”“I eiv’zdr’ dr” . . . &(m+2) d,, dv’, m=l,2,3 ,..., xz
1
(2.10)
where the functions KF’, KF’, . . . , Kb”’ are the higher order statistical moments of the stochastic field P(r, o), defined as follows: Kr)(r’, r”, f”) = %{/3(r’, w)p(r”, w)p(f”, w)},
(2.11)
Kg)(r’, r”, r”‘, rc4)) = EP{P(r’,w)P(r”, w)j?(r”‘, o)p(rc4’, co)},
(2.12)
KhmJ”)(r’ r”, , r”‘, . . . , P))
= %{/3(r’, w)@(r”, w) - - * p(rtm), w)},
m=l,2,3
,....
(2.13)
SCATTERING
IN DISCRETE
RANDOM
MEDIA,
315
II
There are expressions similar to equations (2.9) and (2.10) for A,(r,, K,,(r,, r2, t,, tr) in terms of the higher order moments of P(r, 0).
r2, t,, f2) and
3. SPACE CORRELATION FUNCTION OF /3(r, w) FOR SPHERICAL SCATTERERS From the expression for &(r,, rz, t, , t2), equation (4.10) and equation (5.2) of Part I [ 11, one can see that the space correlation function Kp (I), or the corresponding space spectrum density K@(k), plays an important role in the calculation of the statistical moments of the scattered field. K,(1) depends on the shape and size of each scatterer and the concentration u (relative volume density) of scatterers. The calculation of Kp (l) or K,(k) is an exercise in geometrical probability [2,3]. In view of the complexity and difficulty of this task, in this paper, it is carried out only for spherical scatterers. Obviously, the stochastic field /3(r, w) is ergodic [2]. Therefore, for any sample (realization) p (r, wi), one has Kp(I)=
1
lim V(D)-m V(D)
P(r, w,)P(r-I,
wi) dr,
(3.1)
where D is a region with any shape in ge,, and V(D) is the volume of D. Assume that the distance from any sphere to its nearest sphere is a random variable s(w), and that s(w) has the eXpOneniiai prOb&iiity distribution density
(3.2) where &, is the mean value of t(w); i.e., 50 = 8(5(w)).
(3.3)
Then, using equation (2.8) of Part I [l] and equations (3.1) and (3.2), from geometry one obtains (see Figures 1 and 2) (3.4)
Figure
1. Two different
configurations
of the random
system and their region
of overlap.
316
K. C. LIU
Figure
2. Nearest
neighbours
and the random
distance
[+2a
between
them.
where j, = int {(1/2a) - l},
(3.5)
in which int {A} denotes the integer part of A,
+[I_2(j-l)a]
eA(‘-*JO)/Q
jc (1/2a) - 1,
(3.6) I( I, a, 0) =
1 E [O, 2a]
2a-1,
lS2a
{ 0,
(3.7)
I ’
Substituting equations (3.6) and (3.7) into equation (3.4), one obtains an approximate expression for Ka (I) as K,(I)=be-Y’{1+cos(2rr1/1,)}, where b, y are constants which are independent
(3.8)
of 1, and
lo = 2a + & = (2/&)a.
(3.9)
On the other hand, for a small value of /, (I G 2a), from geometry, one can obtain an accurate expression for Kp (I):
Kp(~)=r~P:l(l-a)21[(1-a)-t(~la)+~(~3/a3)l,
lS2a.
(3.10)
From equation (3.8), dK,(WdL,=
K,(O)=2b.
-2yb,
(3.11)
From equation (3.10), dK,(l)/dll,=,
= -3@:/4(
1-
~)*a,
Kp(0)=&(l-a).
(3.12)
Comparing equations (3.11) and (3.12) gives b=#/(l-a), Therefore,
y=;[l/(l-a)a].
(3.13)
substituting expression (3.13) into equation (3.8), one obtains KP(I)=f[a/(l-o)]~~e-a”“-~)a[l+cos(~&I/a)].
(3.14)
SCATTERING
4. SCATTERING
IN
DISCRETE
FORMULA
RANDOM
MEDIA,
FOR SPHERICAL
317
II
SCATTERERS
Substituting equation (3.14) into equation (5.16) of Part I [ 11, integrating substituting the results into equation (6.6) in Part I [1], one obtains K,(2k,)
= ($)’ 3
vo
a2a( 1 - u)~@: 4k,,a +
and again
2k,a + TT~ [I + @‘( 1 - u)2(2koa + rfi)‘]’
+
(4.1)
and M’,Z’(r, t) =
2k,a + PG
2A2viR2ra2 ci(c: - c:)’ 27r2 (CIC214
4koa+[l+(~)2(l-(r)2(2koa+~~)2]2
2koa -ITS%
(4.2)
+[l+(~)2(l-~)2(2koa-~fi)2]2 For the case of wavelengths A >>a, equation (4.2) can be simplified to M’,Z’(r, t) = ($)’
A27R2a3vico( c: - c:)’
a(1 - a)4
(c1c2)4r2
For the case of a short wavelength expression (4.2) can be simplified to Mf’(r,
t) = (3)”
’+
[1
+
1 (4,r/3)‘a’/‘(
1 a(1-a)4
c2J4r2
.
-(r)2(2koa)2]2
‘+[I+($)‘(1
(4.3)
such that 2koa >>~6,
or small concentration
A27R2a3v~cO(c~-~:)2 (c,
1 _ #]2
(4.4)
By using equation (2.6) of Part I [1], equation (4.3) can also be written as M’,)(r,
t) = (3)’
A2rR2a3vi(c:-cf)2
(c,c2)3r2
a(l-a)4 &c;+(l
1
1+ -a)cf
[1+ (4Tr/3)2a2’3( 1 - C7)2]’ *
In formula (4.3), the part that depends on the concentration fO(cr)=C(l-V)4 In comparison
{
1+
u can be expressed as
1
1
[ 1+ (4T/3)‘C7”3( 1 -a)‘]’ 1 v&c:+ (1 - o,c:’
(4.5)
with Twersky’s formula [5] Rhc= [ Wo(l - w;j”/(i +2 wo)z](~bsj v),
where W, is the concentration
a, the factor a( 1 -a)”
(4.6)
is the same, but the factor
l/(l+2wo)‘.
(4.7 )
in Twersky’s formula is now replaced by 1 [ 1+ (4?r/3)2#3(
1 1 - C)2]’
ac:+(l-a)c:’
(4.8
1
318
K. C. LIU 5. COMPARISON
Ilr
53.
WITH
EXPERIMENT BY BLOOD
ON ULTRASOUND CELLS
SCATTERING
For red blood cells, the density p, is l-099 g/cm3, and the compressibility I. .,--I2 2, I I x IV cm-jayne. Thus, the sound speed in biood ceiis is c, = l/G
= 1596 m/s.
For salt water, the density p2 is l-005 g/cm3, lo-l2 cm2/dyne, and the sound speed is thus c2 = l/G
K1
is
(5.1)
the compressibility
= 1499 m/s.
~2
is
44.3 x (5.2)
Therefore c,-c,=97m/scc
c, and c2.
(5.3)
Hence p, CC1, p2<< 1, /3(r, w)c 1, Vrc Se3, o E R. Therefore one can neglect those terms which consist of the higher order moments of P(r, w), and obtain &,(rl,r21
cl,
t2)=Wr,,r2,
cl,
&,(r,
f2L
A4’ * (r Pt2’(r5 t) = kC2’
3.
1) =
&(r,
0,
t)
(5.4)
(5.5)
Therefore, the mean square value of scattered field ME’(r, t) can be calculated from formulas (4.2) and (4.3), or (4.4), and equation (4.3) can be rewritten as Mg’(r,
t) = ($)3[A2dZ2a3v~(~:- ~f)~/(c,c~)~r~]f~(a),
(5.6)
where &(a) is given by equation (4.5). In Figure 3, Jo(a) is plotted as a function of a. This curve also shows the dependence of the mean square value of the scattered field on the concentration u. “L...._ er_. UI. _I ~41 r.7 cdrrleu _.-_-Z-d out _.-I 8ii ejrptTiiTieili mung Ori uiir%xxirid scaiieririg from biood ceiis. measured the mean square values of the scattered fields for different concentrations (a changes from 0% to 60%). They compared their measured results with Twersky’s theoretical results [5] and found that in the low concentration region, they agreed well, but in the high concentration region there is an obvious and systematic deviation. Our
They
O.‘Oc
I
0
I
0.1
1
1
0.2
I
I
0.3
0.4
0.5
0.6
0.7
0.8
Figure 3. Graph of mean square value of scattered field versus volume concentration of scatterers. c,, Wave speed of scatterer maieriai; c2, wave speed of surrounding medium; 0, voiume conceniraiion of scaiierers; ,&(o), function of c defined by equation (4.5).
SCA-ITERING
IN DISCRETE
RANDOM
Figure 5. Back-scattering coefficient versus concentration.
results agree with the experimental Fio~ww - ‘0~---
MEDIA,
II
319
Cl, Experimental results [4].
results for both low and high concentration
(see
4-?).
6. CONCLUSIONS The general formula for the space correlation function of a multiply scattered field from a discrete random medium can be put into a compact series form. The first term of this series represents the contribution of the second order moment of the discontinuous stochastic field /3(r, 0). The second term (L, + A,) of this series represents the contribution of the third-order moment of /3(r, 0). The third term (L,+ A,+ K,,) represents that of the fourth order moment of P(r, w), and so on. Since the centralized dimensionless stochastic field j3(r, w ) is proportional to the square A+Af tk.¶ PwaPA.2 if ~nad thr &,‘W IUIVP .._.I uywwu Ai(FPrmnrr U‘II”I~X.VW hrtwrm “1...V”‘S C..V “1 +h.p cut rPl”*:.,P LU1QI1.W A;u~r~n,-.a Ul“~l~..“~ “a &I.” ,,,a-,“Gs.b ayrruc?, n1 tha materials of the scatterers and the host medium is relatively small (the case of weak
320
K. C. LIU
0
10
20
30
40
50
1
Qwe) Figure
7. Back-scattering
coefficient
versus concentration.
0, Experimental
results
[4].
fluctuation), then the first term plays the dominant role, and this is precisely the Born approximation [6]. But if the wave speed difference is not relatively small (the case of strong fluctuation), then the other terms, the contributions of the higher order moments of P(r, w), must be taken into account. For spherical scatterers, the second order moment (space correlation function) of p(r, w) can be obtained as an explicit expression approximately derived from stochastic geometry. Therefore, the spatial spectrum density of P(r, CO)as well as the first term of the space correlation function and the mean square value of the scattered field can be obtained in terms of the wave speeds of the two materials, the radius of each scatterer and the concentration (or volume density) of the scatterers. The specific formula for this case shows that (1) the first term, or the intensity of the scattered field in the weak fluctuation case, is proportional to the square of the difference of the wave speeds (this is physically reasonable) and (2) it is also proportional to the fourth power of the frequency of the incident wave, which agrees with the well known Rayleigh relation [7].
SCATTERING
IN DISCRETE
RANDOM
MEDIA, II
321
The most important result is the functional dependence of the first term, or the intensity of the scattered field for the weak fluctuation case, on the concentration of the scatterers. This result has been plotted in Figure 3. The dependence formed is different from that obtained by Twersky [5], and it agrees with experimental results obtained for ultrasound scattering from blood cells [4], which is a weak fluctuation case. This agreement explains the deviation of these experimental results from the theoretical results obtained by Twersky [5].
ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr Leon H. Sibul, Dr George Adomian and Dr Alexander Wasiljeff for their valuable suggestions, and to Dr Tung H. Weng, Dr Vijay K. Varadan, Dr Vasundara V. Varadan and Dr Sabih I. Hayek for their review of the manuscript.
REFERENCES 1. K. C. LIU 1991 Journal ofSound and Vibration 147, 301-311. Wave scattering in discrete random media by the discontinuous stochastic field method, I: basic method and general theory. 3_. __. R “. 1 ~_I Anr II.. FIN lOR1 7% nf J~)lfi Wiley. _._-._- Gmrn~tt-~ --‘I...--., _~ _&n&a,% _Fip& &Jew Y&: 3. E. F. HARDING and D. G. FENDALL 1974 Stochastic Geometry. New York: John Wiley. 4. K. K. SHUNG, Y. W. YUAN, D. Y. FEI and J. M. TARBELL 1984 Journalofthe Acousrical Society of America 75(4), 1265-1272. Effect of flow disturbance on ultrasonic backscatter from blood. 5. V. TWERSKY 1978 Journal of the Acoustical Society of America 64, 1710-1719. Acoustic bulk parameters in distribution of pair-correlated scatterers. 6. M. BORN and E. WOLF 1959 Principles of Optics. New York: Pergamon Press. 7. LORD RAYLEIGH 1945 Theory of Sound. New York: Dover.