Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga

INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 4 (4) 499 5 WAVES IN RANDOMMEDIA PII: S959-774(4)789- Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga Department of Electrical Engineering, Box 355, University of Washington, Seattle, WA 9895, USA E-mail: ishimaru@ee.washington.edu Received 4 March 4, in final form 8 May 4 Published 4 June 4 Online at stacks.iop.org/wrm/4/499 doi:.88/959-774/4/4/ Abstract This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second-order moments by using the circular complex Gaussian assumption. The stochastic Green s functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.. Introduction There has been extensive research on the radar cross section of rough surfaces and the atmosphere [ 4]. However, if an object is located within a random medium, its cross section is affected by multiple scattering in the medium. The wave incident upon the object consists of coherent and incoherent waves, and the scattered wave also experiences multiple scattering. In addition, the induced current on the object has coherent and incoherent components. To 959-774/4/4499+3$3. 4 IOP Publishing Ltd Printed in the UK 499

5 A Ishimaru et al r r t G(r,r ) Random medium G(r s,r t ) T r Object r s Figure. The Dirichlet object is illuminated by a transmitter at r t through a random medium and the scattered wave is observed at r. obtain the stochastic induced current, we need to solve the stochastic surface integral equation for the object [5, 6]. In this paper, however, we assume that the object size and surface radius of curvature are much greater than the wavelength, and therefore, the Kirchhoff approximation is applicable to obtain the induced current. There are correlations between the incident and scattered waves in the medium. To take into account the above effects, we start with general formulations including stochastic Green s functions. We make use of the parabolic equation approximation for the stochastic Green s functions, which should be applicable to many practical problems in microwave and optical scattering in the atmosphere and the ocean, as well as optical scattering in biological media [4]. The formulation is based on the use of the circular complex Gaussian assumption. It includes two effects: backscattering enhancement [7, 8] and shower curtain effects [9]. Both phenomena have been discussed recently, but have not been included in most RCS studies. The backscattering enhancement effect is interesting as RCS can become higher than the conventional RCS, while the shower curtain effects give different RCS depending on whether the random medium is closer to or further from the transmitter.. Formulation of the problem Let us consider a Dirichlet object in a random medium. The scattered field at r is given by [] ψ s ( r) = G( r, r ) ψ( r ) ds, () s n where G is the stochastic Green s function. In general, the random surface field ψ( r )/n can be expressed using the random transition (scattering) operator T, ψ( r ) = T( r, r s ) ψ in( r s ) ds, () n n s where ψ in ( r s ) is the incident field at r s as shown in figure. Now, we assume that the object size and its surface radius of curvature are greater than the wavelength and the Kirchhoff approximation is applicable. Then, we get the well-known Kirchhoff surface field T( r, r s ) = δ( r r s ), ψ( r ) = ψ in( r ) (3). n n The scattered field is then given by ψ s ( r) = G( r, r ) ψ i( r ) ds. (4) n

Multiple scattering effects on the RCS of objects in a random medium 5 ˆ n r G G Random medium r s r Object nˆ Figure. Stochastic Green s functions G and G. The scattered power at r is given by ψ s ( r) =4 ds ds G ψ i n G ψi, (5) n where G = G ( r, r ), ψ i = ψ i ( r ), G = G ( r, r ), ψ i = ψ i ( r ) as shown in figure. This contains the fourth-order moment, and using the circular complex Gaussian assumption, we can express the fourth-order moment in terms of second-order moments [5]. (see the appendix) G ψ i G ψi = G G n n ψ i ψi n n G ψ i G n ψi n + ψ i G n G ψi n. (6) It is important to note that equation (6) includes the correlation between the incident ψ i and the scattered field G, and this gives the backscattering enhancement to be discussed in a later section. In contrast, it is often assumed that there is no correlation between the incident and the scattered field and with this assumption, we get G ψ i G ψi = G G n n ψ i ψi. (7) n n We will show later the difference between our formulation in equation (6) and the formulation in equation (7). The apparent RCS of the object is then given by RCS = 4πL ψ s, (8) ψ o where ψ o is the free space incident field at the object, ψ s is the scattered field at the observation point and L is the distance between the object and the observation point. We now examine equations (5) and (6). The incident wave ψ i is the stochastic Green s function and we write ψ i = ψ( r ) = G ( r, r ), (9) ψ i = ψ( r ) = G ( r, r ), where we used the reciprocity: G ( r, r ) = G ( r, r). Under the parabolic equation approximation [4, ], Green s function is given by G = (slowly varying function of z and ρ)exp(ikz). () Therefore, n = ikŝ ˆn = ik(ẑ ˆn ), ()

5 A Ishimaru et al where ŝ ẑ is the direction of wave propagation. Using equations (9) and (), we get ψ s =4k (ds ds [ G G +4 G G G f G f + G f G f ](ẑ ˆn )(ẑ ˆn )), () where G = G + G f, G is the coherent component and G f is the incoherent component. Note that if we ignore the correlation between the incident and the scattered field as in equation (7), we have ψ s =4k (ds ds [ G G + G G G f G f + G f G f ](ẑ ˆn )(ẑ ˆn )). (3) The difference between equations () and (3) is that equation () includes the enhancement while equation (3) does not. We now need to evaluate equation () to obtain the RCS. 3. Stochastic Green s functions Let us now evaluate equation (). Under the parabolic approximation, we get G = ( ikρ 4πL exp +ikz τ ) o, z G = ( ikρ 4πL exp +ikz τ ) (4) o, z where τ o is the optical depth. We assume that the transmitter is in the far field of the object, and therefore kρ/ z and kρ/ z are negligibly small. Next we consider the incoherent mutual coherence function G f G f =Ɣ f (z, ρ ; z, ρ ). (5) The mutual coherence function Ɣ due to a point source has been obtained previously including the inhomogeneous random medium [4, 9, ] Ɣ = Ɣ(z, ρ ; z, ρ ) = G(z, ρ )G (z, ρ ) ( = (4πX) exp ik ρ ) d ρ c X H +ik(z z ), (6) where X = X(z) = z dz n(z ),H = z a dz + z b dz p(s)s ds[ J o(knsρ)]. The phase function p(s) is normalized so that p(s)s ds =, (7) ρ = X(z ) ρd X(z),n(z) is the refractive index, a is the absorption coefficient, b is the scattering coefficient and s = sin(θ/), θ = scattering angle. This is the mutual coherence function for a random medium consisting of a random distribution of particles. First, we note that the term k ρ d ρ c /X in the exponent is negligible when the transmitter is in the far field of the object. We then write the coherent Ɣ c and incoherent Ɣ i parts of Ɣ as Ɣ = Ɣ c + Ɣ i, Ɣ c = (4πX) exp( τ o +ik(z z )), (8) Ɣ i = (4πX) [exp( H) exp( τ o)]exp(ik(z z )),

Multiple scattering effects on the RCS of objects in a random medium 53 where H is given in equation (6) and H(ρ ) = τ o = d (a + b) dz = optical depth and d is the thickness of the random medium. Let us use the Henyey Greenstein (HG) phase function for the scattering medium. We have ( g) p(s) =, g = anisotropy factor, (9) (+g g cos θ) 3/ which satisfies the normalization p(s) d =. () 4π 4π It is more convenient to use s = sin(θ/). Then, we write p(s) = (+g) ( g) [ ( ) ], s ( g) 3/ o =, g s + s o () p(s)s ds =. Even though this is the HG phase function used for <s<(or<θ<π), it can be approximated by p(s) = [ s o ( ) ] 3/ s + s o p(s)s ds =, when g is close to one. Making use of this, and using p(s)j o (knsρ)s ds = exp( kns o ρ) (3) we get H(ρ) = z a dz + z () b dz [ exp( kns o ρ)]. (4) For large optical depth, the incoherent mutual coherence function Ɣ i in equation (8) is well approximated by expanding H(ρ) in equation (4) about ρ = and keeping the first term [ z z ] Ɣ i = (4πX) exp a dz bkns o ρ dz. (5) A better approximation which reduces to the proper limit as the optical scattering depth τ s and τ s, and as ρ and ρ is given by [] where Ɣ i = ρ o = ( (4πX) exp( τ a)[ exp( τ s )]exp ρ ) d ρ o +ik(z z ), (6) z b(z )kn(z )s o X(z ) X(z) dz [ exp( τ s )],

54 A Ishimaru et al τ a = d a dz is the optical absorption depth and τ s = d b dz is the optical scattering depth. The coherence length ρ o at the object is important to characterize RCS. If the uniform random medium with n(z) = is located from d to d + d, we get = ks [ ] o (d + d ) d τ s [ exp( τ s )]. (7) ρ o Ld If the phase function is given by the Gaussian function: p(s) = 4α p exp( α p s ) (8) with normalization p(s)s ds =, the incoherent mutual coherence function Ɣ i is given by Ɣ i = ( ) X exp( τ a)[ exp( τ s )]exp ρ d +ik(z ρo z ). (9) If the medium is uniform from d to d + d, we get = τ sk [ ] (d + d ) 3 d 3 α p L d [ exp( τ s )]. (3) ρ o 4. RCS of a large Dirichlet object Making use of the stochastic Green s functions and the mutual coherence functions, we can now evaluate the apparent RCS given in equation (8). Noting equation (), RCS is given by RCS = 4πL ψ s = I ψ o + I + I 3, (3) where I,I,I 3 are the three terms in equation (). We note that in equation (), we have ds (ẑ ˆn ) = dˆρ, ds (ẑ ˆn ) = dˆρ. (3) Therefore, we have where I = 4π λ exp( τ o)f, I = 4π λ (4exp( τ o))[exp( τ a )( exp( τ s ))]F, I 3 = 4π λ [exp( τ a)( exp( τ s ))] F 3, F = dˆρ dˆρ exp(ik(z z )), ( F = dˆρ dˆρ exp ik(z z ) ρ ) d ρ o ( ) = dˆρ dˆρ exp ik(z z ) ρ d ρo ( F 3 = dˆρ dˆρ exp ik(z z ) ρ ) d ρ o ( ) = dˆρ dˆρ exp ik(z z ) ρ d ρo (33) for HG medium, for Gaussian medium, for HG medium, for Gaussian medium,

Multiple scattering effects on the RCS of objects in a random medium 55 ρ Illuminated surface z(ρ) Shadow surface z Figure 3. Illuminated surface given by z = z( ρ). Tr Random medium nˆ ρ d d z θ z ρ z L Figure 4. Inclined square conducting plate illuminated by the source. The random medium with thickness d is located between the transmitter and the object. The plate size is a a, and ρ = x ˆx + y ŷ and ρ = x ˆx + y ŷ. (x,y) Shadow Illuminated z z Figure 5. Spherical object. and ρ d = ρ ρ, ρ = x ˆx + y ŷ, ρ = x ˆx + y ŷ. The illuminated surface of the object is given by z = z ( ˆρ ) and z = z ( ˆρ ) as shown in figure 3. The coherence length ρ o is given by equation (7) for a HG medium and by equation (3) for Gaussian medium. For a large conducting plate shown in figure 4, wehavez(ρ) = x tan θ, a cos θ a a cos θ a d ρ d ρ = dx dy dx dy, (34) a cos θ a a cos θ and z z = (x x ) tan θ,ρ d = (x x ) + (y y ). The integral in (34) can be calculated numerically. For a sphere with radius a shown in figure 5, weuse a π a π d ρ d ρ = ρ dρ dφ ρ dρ dφ, (35) and z = a a x y,z = a a x y, ρ = (x x ) + (y y ), x = ρ cos φ,y = ρ sin φ,x = ρ cos φ,y = ρ sin φ. a

56 A Ishimaru et al Case A: d /L= d /L=.5 Case B: d /L=.5 d /L=.5 Tr Tr Target size is a Figure 6. Geometry of RCS calculations. L Coherence length ( ρ o ) in λ 5 5 5 (A) Case A Case B Coherence length ( ρ o ) in λ 45 4 35 3 5 5 (B) Case A Case B 4 6 8 Optical depth (τ ) o 5 4 6 8 Optical depth (τ ) o Figure 7. Coherence length for (A) Henyey Greenstein phase function with albedo (W o ) =.9 and g =.85 and (B) Gaussian phase function with W o =.9 andα p which gives the same half-power beamwidth. 5. Coherence length and shower curtain effects Equations (7) and (3) give general expressions of coherence length ρ o for HG and Gaussian media. It is seen that ρ o in terms of the wavelength depends on the optical scattering depth τ s, the medium characteristics (s o and α p ) and the ratio of the distances (d /L and d /L). In figures 6 and 7, we show the geometry and the coherence length in wavelength for case A and case B. Note that if the medium is close to the transmitter (case A), the coherence length is greater than case B, showing the shower curtain effect. For a HG medium, we use W o =.9 and g =.85 where W o is the albedo defined by W o = b. For a Gaussian medium, we use a+b W o =.9 and α p which gives the same half-power beamwidth as HG with g =.85. 6. RCS of a square plate Let us examine the RCS of the square plate shown in figure 4. The plate size a is 3λ. In figure 8,weshowI,I,I 3 and I com = I +I +I 3 at θ =. Note that I is the conventional RCS

Multiple scattering effects on the RCS of objects in a random medium 57 (A) (B) I fs I fs 3 3 RCS (db) 4 RCS (db) 4 5 I 5 I I I I 6 3 I com I fs 7 3 4 5 Optical Depth (τ ) o I 6 3 I com I fs 7 3 4 5 Optical Depth (τ ) o Figure 8. Components of RCS. Phase function is Henyey Greenstein with albedo (W o ) =.9 and g =.85 in all case. (A) Case A geometry, (B) case B geometry. fs denotes free space. I fs denotes the free space RCS. 5 5 RCS(dB) 3 35 OD=. 4 OD= OD=5 OD= 45.3.4.5.6.7.8.9 g Figure 9. Effect of asymmetry factor g. in a scattering medium and I com is increasingly larger than I as the optical depth increases. This indicates that the multiple scattering contributes significantly to the total RCS. The effects of asymmetry factor g and the albedo W o are shown in figures 9 and, respectively. It shows that as g increases, the scattering pattern becomes more peaked in the forward direction and as W o increases, the scattering increases, resulting in increased RCS. We also compare the RCS from the geometry of case A and case B, which are shown in figure. Note that if the medium is close to the transmitter (case A), the RCS is higher than that of case B where the random medium is away from the transmitter. This result illustrates the shower curtain effect. Figure shows the shower curtain effects as the random medium moves from d = tod = L/. Figure 3 compares RCS, which includes backscattering enhancement from equation () with RCS without enhancement from equation (3). Note that the enhancement approaches 3 db for large optical depth. Figure 4 shows the angular dependence of RCS at an optical depth of and 5. It is shown that RCS, as a function of angle, is very much affected by

58 A Ishimaru et al 3 RCS(dB) 4 5 6 OD=. 7 OD= OD=5 8 OD=.3.4.5.6.7.8.9 Wo Figure. Effect of albedo W o. RCS (db) 8 4 6 8 3 I fs 3 Case A 34 Case B 4 6 8 Optical Depth (τ ) o Figure. The shower curtain effect with Henyey Greenstein phase function for geometry in case A and case B geometry. I fs denotes the free space RCS. Normalized RCS(dB) 3 OD=. 4 OD= OD=5 OD= 5...3.4.5 d /L Figure. The shower curtain effect as a function of d /L. the multiple scattering. This is more pronounced for large optical depth where RCS is quite different from the conventional RCS (I ).

Multiple scattering effects on the RCS of objects in a random medium 59 5 RCS (db) 5 3 I fs 35 Case A WE Case B WE Case A Case B 4 4 6 8 Optical Depth (τ ) o Figure 3. RCS comparison for the cases with and without backscattering enhancement. WE denotes without enhancement. I fs denotes the free space RCS. (A ) (B ) I I I I 3 4 I 3 I com 3 4 I 3 I com RCS(dB) 5 6 RCS(dB) 5 6 7 7 8 8 9 9 3 4 5 6 Angles (degrees) 3 4 5 6 Angles (degrees) (A ) (B ) I I 4 6 I I 3 I com 4 6 I I 3 I com RCS(dB) 8 RCS(dB) 8 4 3 4 5 6 Angles (degrees) 4 3 4 5 6 Angles (degrees) Figure 4. Angular dependence of RCS with Henyey Greenstein phase function. (A-) Case A with an optical depth (OD) of, (A-) case A with an optical depth (OD) of 5, (B-) case B with an optical depth (OD) of, and (B-) case B with an optical depth (OD) of 5.

5 A Ishimaru et al 7. Conclusions A theory of RCS in a multiple scattering environment is given by equations (8) and (). This is applied to a flat conducting plate. The conventional RCS is given by I and the multiple scattering RCS is given by I + I + I 3 in equation (33). The shower curtain effect, the backscattering enhancement and the angular dependence are shown to highlight the difference between the conventional RCS and the multiple scattering RCS. Acknowledgments This work is supported by the National Science Foundation (grant ECS-998849) and the Office of Naval Research (grant N4-4--74). Appendix. Circular complex Gaussian random variable [, 3, 5] Consider a complex random function u and its fluctuation v u = u + v. (A.) Two circular complex Gaussian random variables u and u satisfy the following conditions: v v =, but v v (A.) and the odd moments of v are zero. Thus, the following relations hold v v v3 v 4 = v v3 v v4 + v v4 v v3 u u u 3 u 4 = u u 3 u u 4 + u u 4 u u 3 u u u 3 u 4 = u u 3 u u 4 + u u 3 v v4 + u u 4 v v3 + v v3 v v4 + u u 4 v v3 + u u 3 v v4 + v v4 v v3 u u = u u u u = u u +4 u u v v + v v u u = u u + v v = u u. We have applied this assumption to the wave propagation in a random medium. We found that the results reduce to the correct limits in the weak fluctuation as well as the strong fluctuation, and that the phenomena such as the backscattering enhancement result from the use of this assumption. Note that the circular complex Gaussian variables are similar to but different from the real Gaussian random variables [3]. References [] Kim H and Johnson J T Radar images of rough surface scattering: comparison of numerical and analytical models IEEE Trans. Antennas Propag. 5 94 [] Hesany V, Plant W J and Keller W C The normalized radar cross section of the sea at incidence IEEE Trans. Geosci. Remote. Sens. 38 64 7 [3] Bissonnette L R, Roy G, Poutier L, Cober S G and Isaac G A Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements Appl. Opt. 4 637 4 [4] Ishimaru A 997 Wave Propagation and Scattering in Random Media (New York: IEEE) [5] Rockway J D Integral equation formulation for object scattering above a rough surface PhD Dissertation Dept. of Electrical Engineering, University of Washington, Seattle, WA [6] Ocla H E and Tateiba M The effect of H-polarization on backscattering enhancement for partially convex targets of large sizes in continuous random media Waves Random Media 3 5 36

Multiple scattering effects on the RCS of objects in a random medium 5 [7] Ishimaru A 99 Backscattering enhancement: from radar cross sections to electron and light localizations to rough surface scattering IEEE Antennas Propag. Mag. 33 7 [8] Kuga Y and Ishimaru A 984 Retroreflectance from a dense distribution of spherical particles A 83 5 [9] Kuga Y and Ishimaru A 986 Modulation transfer function of layered inhomogeneous random media using the small-angle approximation Appl. Opt. 5 438 5 [] Ishimaru A 99Electromagnetic Wave Propagation, Radiation, and Scattering (Englewood Cliffs, NJ: Prentice Hall) [] Ishimaru A 978 Limitation on image resolution imposed by a random medium Appl. Opt. 7 348 5 [] Goodman J W 985 Statistical Optics (New York: Wiley) [3] Middleton D 96 Introduction to Statistical Communication Theory (New York: McGraw-Hill)