This paper presents a computer simulation of backscattering enhancement from randomly distributed spherical water scatterers at 30 GHz, corresponding to laboratory measurements. Because of the finiteness in the scattering volume size, we adopt a very simple direct simulation method, together with a sampling of the scattering paths for triple scattering and higher to minimize computer time. The spherical wavefront and the directivity functions of transmitting and receiving antennas are taken into account. The simulation results agree favorably with measurements. Some of the knowledge on the detailed characteristics of enhanced backscattering is also obtained.
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 This paper reports on a computer simulation of the enhanced backscattering corresponding to the laboratory measurements made by Ihara et al. . The enhanced backscattering has been intensively studied in various branches of physics, such as in optics and condensed matter physics, as well as in radar remote sensing [Barabanenkov et al., 1991; Ishimaru, 1991]. In the theoretical treatment of enhancement in electromagnetic wave scattering from random discrete scatterers, a second-order theory in conjunction with a plane wave incidence is commonly used [Kuga et al., 1985; Mandt et al., 1990]. However, actual radar waves have a spherical wavefront and a finite beam width, hence the theory was very recently extended to the case of finite beam width, and several interesting results have been obtained that includes the effect of foot print size on the magnitude of enhancement [Kobayashi et al., 2005]. On the other hand, the computer simulations of the problem are mostly based on Monte Carlo technique. This technique has been successfully applied to show vector character of light in enhanced backscattering from a plane-parallel medium containing Rayleigh scatterers [van Albada and Lagendijk, 1987], as well as in the analysis of enhanced backscattering from random rough surfaces [e.g., Johnson et al., 1996].
 To examine the multiple-scattering effects in detail, laboratory measurements of radar signatures of large water scatterers were conducted in the millimeter wave band under controlled scattering environment [Tazaki et al., 2000]. Recently, the effect of enhancement has been confirmed under the same scattering environment by using a mirror image technique [Ihara et al., 2004].
 The computer simulation model in this paper is constructed to follow exactly the scattering environment used in the experiment. Although our scattering volume is made up of very many scatterers, we adopt a simple direct simulation method: In view of the finiteness in scattering volume size, we directly evaluate the multiply scattered waves of nth order from all the scatterers by constructing n particle scattering channels within the scattering volume. The spherical wavefront and the directivity functions of transmitting and receiving antennas are taken into account.
 The objective of this paper is twofold: One is to see how the simulation results are consistent with the measured results, and the other is to look into detailed characteristics of enhanced backscattering.
2. Simulation Model and Problem Formulation
 When a transmitted wave travels through n successive scattering centers (n ≥ 2) in a scattering medium from scatterer 1 to scatterer n and enters into a receiver, another wave also exists in the medium that travels in the opposite direction from scatterer n to scatterer 1 (time-reversed path) exactly on the same channel, as shown by the dashed lines in Figure 1.
 If the transmitting and receiving antennas are in the same position, these two waves will be added in phase resulting in enhanced backscattering. In triple scattering, however, there is the case that scatterer 3 is scatterer 1 itself as “Case 2” in Figure 1. In this case, the time-reversed path will not exist. In quadruple and higher-order scattering, the construction of the scattering path becomes much more complicated. Note that, in the analysis of meteorological radar data, the analysis is usually based on single scattering.
Figure 2 shows the geometry and the symbols used in the following equations. While the transmitting antenna (t) with its beam center along Z axis is fixed at the origin of a Cartesian coordinate system (X, Y, Z), the receiving antenna (r) is able to move along the X axis on the Z = 0 plane, so that its beam center always points to the center of the rectangular scattering volume. The scattering volume is made of two cubic Styrofoam blocks, each containing 868 spherical scatterers, and is extending from Z = z0 toward the positive Z direction. (The total number of scatterers is thus 1736.)
 The interaction of the waves is written as follows. Let the electric fields of horizontally and vertically polarized wave, Eh and Ev, be expressed by a two-dimensional vector E
 We then define the four-dimensional vector D whose elements are the elements of coherency matrix
where ⊗ denotes the Kronecker product.
 Letting E(1,n) and E(n,1) be the received electric fields after the successive scattering from scatterer 1 to scatterer n, and scatterer n to scatterer 1, respectively, D may be written as
where the terms in the upper and lower parentheses in the last equation correspond to the ladder and cyclical terms in the diagrammatical theory.
 Letting E0 be the effective electric field vector of unit amplitude at the input port of transmitting antenna, E(1,n) and E(n,1) may be written by the following equation
 The position vectors of both scatterers and antennas easily give the length and angle parameters in equation (5). T(t,n:1,r) is given by replacing 1 by n, and n by 1. In equation (5), k is the propagation constant in free space, ke is the effective propagation constant in scattering volume given by [e.g., Bringi and Chandrasekar, 2001]
where k0′ is the propagation constant in background medium (Styrofoam blocks), f is the forward-scattering amplitude of a water sphere embedded in background medium, and nd is the number density of water spheres. Moreover, C is a constant factor given by
where λ is the wavelength, η is the characteristic impedance in free space, Pt is the transmitting power, Gt0 and Gr0 are the gains of transmitting and receiving antennas in the direction of beam center, respectively, Fe(i,j) is the field directivity function of an antenna with a half-power beam width θb, when looking at a direction making an angle θ(i,j) with its beam center
and for n = 1, we set
S(1,n) and S(n,1) are the product of n scattering matrices of the scatterers in a specific path and its time-reversed path, respectively,
where the overbars denote that the matrices are those in time-reversed path.
 It is well known that, for any number of successive scatterings, the reciprocity relation exists for the matrices S(1,n) and S(n,1) [Mishchenko, 1992]
where Q = diag( −1,1). In this paper, we use the local right-handed coordinate system always looking at the wave propagation direction, also known as the forward scattering alignment (FSA) convention. By the use of these relations in equation (3), we have
The terms E(n,1) ⊗ E(n,1)* and E(n,1) ⊗ E(1,n)* in equation (3), corresponding to the second terms in the upper and lower brackets in equation (12), are not evaluated in the numerical computation, because these terms are automatically evaluated in the various combination of scatterers: Note that the numbers 1, 2, … n attached to the scatterers are not intrinsic to the scatterers, but mean that the scattering has occurred first, second, etc., on these scatterers in a scattering chain.
 Excluding the case that the transmitting and receiving antennas are in the same position, the scattering matrices of the scatterer 1 and scatterer n that directly face to the antennas do not exactly satisfy the reciprocity relation, because the scattering angle is slightly different when looking at either transmitting or receiving antenna (see Figure 2). We, however, assume that the relation of equation (11) is always satisfied since the distance between these two antennas is normally very small.
3. Computational Procedure of Numerical Simulations
 The major parameters used in the numerical simulations are given in Table 1. These parameters are the same as those employed in our measurements. Although the scatterers used in the measurements were water-injected thin polystyrene spheres of radius 12.5 mm, we assume that, as in our preceding work [Tazaki et al., 2000], their scattering properties are the same as those of water spheres of radius 12.5 mm without polystyrene shell.
Table 1. Major Parameters of Numerical Simulation
Size of the scattering volume (X, Y, Z), m
0.9 × 0.9 × 0.9
Size in radius, mm
1736 (868 × 2)
Distance from transmitter to the front face of scattering volume, m
 In order to calculate the scattering matrix of spherical scatterers, we have to know the direction of incident and scattered waves for each scatterer in a scattering chain. For this purpose, we introduce a local coordinate system (x′, y′, z′) with its origin at the center of each scatterer, and z′ being always directed upward (i.e., in the direction of Y axis in Figure 2). The axis x′ is adjusted for each scatterer so that the propagation vector of the incident wave is upon the x′z′ plane. By using the spherical coordinate system (r,θ,ϕ) on the basis of the coordinate (x′, y′, z′), we can define the incident and scattering angles of a scatterer i from the position vectors of the scatterers in front and in the rear: Denoting the position vectors of these three scatterers by ri−1, ri, ri + 1, respectively, the angles that the vectors ri − ri−1 and ri+1, − ri make with respect to z′ axis are the incident and scattering angles θi and θs. The azimuth angle of a scattered wave ϕs is given by the angle between x′ axis and the projection of ri+1 − ri onto the x′ y′ plane. By definition of the coordinate system, the azimuth angle of an incident wave = 0. In this manner, we can successively define the incident and scattering angles for each scatterer in a scattering chain. In spherical scatterers, it is easy to convert the scattering matrix defined on the reference plane of scattering that contains propagation vectors of both incident and scattered waves into the scattering matrix defined in the above spherical coordinate [e.g., Ishimaru and Cheung, 1980]. Thus the scattering matrix data were calculated in advance in an angular range from 0 to π with very small angular interval on the reference plane of scattering, and were stored in a file for use in further calculations.
 Random numbers are then generated to give three-dimensionally randomized positions of the scatterers x(i), y(i), z(i) for i = 1 to 1736. The minimum intercentral distance between scatterers is kept above 6 times the wavelength, since a preliminary examination shows that the far-field approximation can be used above that distance. In double scattering for example, all the combinations of two numbers i, j in the range of 1 to 1736 are selected under the condition i ≠ j. For each combination of these numbers, different scatterer positions, x(i), y(i), z(i); x(j), y(j), z(j), can be assigned. We evaluate the double scattering for each combination of scatterer positions in equation (12) with n = 2, on reference to the scattering matrix data, and the results are added for all the combinations of scatterer positions. We then restart the random number generator, and execute the computations again for different positioning of scatterers. We repeat this process 32 times and the values are averaged. This average is the required power value for double scattering. The number of averaging (32 times) corresponds to the maximum attainable number in our measurements [Tazaki et al., 2000]. In terms of D in equation (3), the required average Ap may be written as follows
where np is the total number of scattering path.
 As the order of scattering increases, the number of scattering path increases remarkably. This directly affects the computation time. For our scatterer number of 1736, the computation time exceeds acceptable level for triple scattering and higher. For those cases, we sampled the scattering path to reduce the computation time. The sampling will not significantly affect the overall accuracy, because the relative importance of higher-order scattering diminishes. By the same reasoning, as well as its extremely small proportion to the total number of scattering paths, we have ignored the effect of the specific path in which some scatterer enters into the scattering path more than once in quadruple scattering and higher. In triple scattering, however, calculations were made for both total and sampled paths, taking account of the effect of specific paths. In Table 2, we show the number of total and specific paths, their ratios, and the number of sampled paths for each order of scattering.
Table 2. Total Number of Scattering Paths, Number of the Specific Paths That Contain Same Scatterer More Than Once, Their Ratios, and Number of Sampled Pathsa
Order of Scattering
Total Number of Scattering Paths
Number of Specific Paths
Ratio of the Specific to Total Paths
Number of Sampled Paths
The sampled paths in triple scattering contain 46,872 specific paths. In quadruple scattering and higher, the specific paths are ignored.
5.225750 × 109
5.763 × 10−4
1.012435 × 107
9.066677 × 1012
1.567122 × 1010
1.728 × 10−3
8.185320 × 106
1.573068 × 1016
5.434259 × 1013
3.454 × 10−3
1.179360 × 107
2.729273 × 1019
1.569897 × 1017
5.752 × 10−3
1.953504 × 107
4.735290 × 1022
4.080560 × 1020
8.617 × 10−3
1.729728 × 107
 The accuracy of sampling may be estimated as follows. If we write any element of D in nth-order scattering by xi, its mean value with respect to total scattering path (population) p and that of sampled path s after 32 times random number generation, are given respectively, by
where ns is the number of sampled path for each random number generation. The quantity to be obtained is now Ap. The mean value p is in the following confidence interval around sample mean s, under some confidence level
where t(∞) is the value derived from Student's t distribution under the above confidence level with an infinitely large sample number, and s′ is evaluated by calculating the sample variance s′2 from a given data set. Hence the confidence interval of Ap is given by multiplying the above equation by np
As an example, the ladder term confidence interval of both copolarization and cross polarization was calculated for each order of scattering and for the scattering angle of 0 degree. The incident wave polarization is vertical. Table 3 summarizes the confidence interval of Ap normalized by the estimated value (np/ns)As corresponding to specified confidence levels (%). Note that the confidence interval shown in Table 3 is calculated excluding the effect of specific paths. In triple scattering, we compared the results obtained for both total and sampled paths. The comparison between the total and the sampled paths is made for the following four terms: copolar ladder, cross-polar ladder, copolar cyclical, and cross-polar cyclical terms. As a result, it is found that the values of three terms for total scattering paths are within the confidence interval for the confidence level of 99.9%, except the cross-polar ladder term for the total paths, that is −4.2% out of the interval compared to the estimated value (±1.95% in Table 3). Table 3 also shows that as the order of scattering increases, the error of simulation increases. However, simultaneously, the magnitude decreases as the order of scattering increases. Thus the effect of increasing error seems to be compensated. For that reason, we assume that the estimated value (np/ns)As is a valid approximation for the present simulation.
Table 3. Ladder-Term Confidence Interval of Required Average Normalized by the Estimated Average Corresponding to Specified Confidence Levela
Order of Scattering
Confidence Level, %
Confidence Interval, %
The confidence interval is calculated excluding the effect of specific paths. Incident wave polarization, vertical; scattering angle, 0°.
 We then repeat the whole procedure for higher-order scattering, until the magnitude of the values calculated for some scattering order becomes very small and negligible compared to the magnitude of those in lower scattering orders, because the final received power is the sum of the values for each order of scattering. Since the results for different order of scattering might not be correlated, we add the results independently. Further, it must be noted that in the above calculation, we have disregarded the product of the fields pertaining to different scattering channel, such as E(1,n) ⊗ E(1′,n′)* or E(1,n) ⊗ E(n′,1′)* because of their random nature.
 We further repeat the computations for necessary scattering angles and for both horizontally and vertically polarized incident waves.
4. Simulation Results
 In the following, we will show the results for vertically polarized incident wave (electric field vector in the direction of Y axis), because the measurements were performed for this polarization. Figure 3 shows the relation between copolarized and cross-polarized received powers and scattering angles, calculated for both single and double scatterings. The cross-polar component of single scattering is not shown here, since it is fundamentally nonexistent in exact backscattering (scattering angle of 0 degree) and is also very small in an angular range less than 3 degrees. Since the curves must be symmetric about an angle of 0 degree, calculations were made only for positive angles, and the values calculated in the positive angular range were also used in the negative range. The symbols l and c in Figure 3 signify ladder and cyclical terms respectively.
 The received power Pr of single-scattering contribution, in exactly the backscattering direction, is given by
where (Ap)4 is the copolar element of single-scattering contribution of Ap. Inserting the calculated value of Ap in equation (18), we have −60.6 dBm as a received power. On the other hand, a conventional radar equation gives −60.5 dBm, resulting in almost complete agreement with the simulated value. In the double scattering, we notice that the enhancement certainly appears as a sharp peak about 0 degree. This effect is more clearly seen in the cross-polar channel. Since the transmitting and receiving antennas have the same directivity function, ladder and cyclical terms of both copolar and cross-polar channels have the same value at an angle of 0 degree. Figure 4 shows the result of calculations for triple scattering, made both for total and sampled paths. It is evident from Figure 4 that the sampling of scattering paths does not affect an overall estimate of scattered powers significantly.
 Note that, in the copolar channel, the calculated value of cyclical term for an angle of 0 degree is slightly smaller than that of ladder term (see Figure 4a). This is due to the fact that one scatterer enters into some specific path twice as shown in Figure 1, and the cyclical term is regarded as nonexistent. It can be shown theoretically that the copolar component of E(1,n) ⊗ E(1,n)* in equation (3) has the same form as the copolar component of E(1,n) ⊗ E(n,1)*, when the scattering angle is 0 degree and under the condition that both transmitting and receiving antennas have the same directivity functions. Therefore, if we ignore the case that some scatterer enters into a scattering path more than once, cyclical term will have the same value as that of the ladder term in the copolar channel at an angle of 0 degree, regardless of the order of scattering. The ladder term is generally unequal to the cyclical term in cross polarization as shown in Figure 4b. The cross-polar components of E(1,n) ⊗ E(1,n)* and E(1,n) ⊗ E(n,1)* indicate that, in order that the cyclical term might have the same value as that of the ladder term, the off-diagonal elements of S(1,n) should have the same magnitude but opposite sign. It is easy to show that double scattering of spheres can satisfy this condition. However, triple- and higher-order scattering cannot satisfy this condition [van Albada and Lagendijk, 1987]. Although not shown here, calculations up to septuple scattering indicate that the enhancement curve in copolar channel for each order of scattering becomes sharper as the order of scattering increases [van Albada and Lagendijk, 1987].
Figure 5 shows the power values of ladder and cyclical terms of both copolar and cross-polar channels at an angle of 0 degree as a function of the order of scattering. Although not plotted in Figure 5, the cyclical term curve in copolar channel is identical to the ladder term curve if we ignore the effect of specific scattering paths noted above. Figure 5 indicates that the power values calculated by adding up to septuple scattering must be sufficient for the estimation of total scattered power. The effect of absorption may be contributing for the sharp decrease of the scattered power as the order of scattering increases, since the albedo of the scatterers is 0.724, contrary to the optical wave case for the Rayleigh scatterers with albedo = 1, where the evaluation should extend up to an order of 1000 [van Albada and Lagendijk, 1987]. Figure 5 also shows that, as the order of scattering increases, the relative ladder term contribution in cross-polar channel increases and it approaches the value of copolar ladder term, resulting in unpolarized state. However, it is interesting to note that the ratio of copolar to cross-polar cyclical terms remains almost unchanged even in septuple scattering. Figure 6 illustrates the total scattered powers obtained by summing up to septuple scattering. Although not shown here, the results for horizontally polarized incident wave case show that, in copolar channel, the enhancement curve of vertical polarization is very slightly sharper than that of horizontal polarization. The difference of H and V plane patterns of a sphere reflects on the shape of the curves of cyclical term, as noted in [van Albada and Lagendijk, 1987]. No difference could be observed in the curves of cross-polar channel. It is suggested that the angular range of enhancement is of the order of a wavelength divided by the mean free path of scattering [Kuga et al., 1985]. The numerical value estimated by the parameters of the present scattering environment is approximately 0.8 degrees, in fairly good agreement with the simulated curve. (The mean free path in our scattering volume is 0.722 m.) Through Figures 3–6, it is noted that the averaging over 32 different scatterer configurations is enough to give reliable results. The statement in this section may not be specific to the present scattering environment, but general since the characteristics of enhancement largely depends on the nature of scattering matrix.
 When the incident wave is a beam wave with spherical wavefront as in the present simulation, as well as in all other radar rainfall measurements, the multiply scattered power may depend on the footprint size of a radar beam. In fact, Kobayashi et al.  showed that, in their analytical theory including up to second-order scattering with both ladder and cyclical terms, the multiple-scattering contribution actually depends on the footprint size. Battaglia et al.  derived similar results in their Monte Carlo simulation although the cyclical term was disregarded in their simulation.
 To examine this point, we calculated the multiply scattered power in exactly backscattering direction for several different antenna positions (Z0 in Figure 2), thus changing the effective footprint radius within the scattering volume. The distance Z0 for which all the simulation results so far mentioned is 3.15 m. The distances Z0, the corresponding 3 dB footprint radii at the center of scattering volume, and the radii normalized by the mean free path of scattering, used in our simulation, are shown in Table 4. Figure 7 shows the total multiply scattered powers of copolar and cross-polar channels as functions of normalized footprint radius. The total multiply scattered powers in Figure 7 are those obtained by summing up from double- to septuple-scattering contributions for both ladder and cyclical terms and normalized by the first-order received power. The dependence of multiply scattered power on footprint radius is evident, and may be detectable in future laboratory measurements although the calculated value of first-order contribution should be subtracted from the measured received power to create multiple-scattering contribution alone.
Table 4. Dependence of the Footprint Radius on Antenna Position
3 dB Footprint Radius, m
Normalized Footprint Radius
5. Comparison With Measurements
Figure 8 compares the simulated values with measurements [Ihara et al., 2004]. Incident wave polarization is vertical. The simulated copolarized and cross-polarized received powers are the same as those labeled by “single + multiple (l + c)” and “multiple (l + c)” in Figure 6, respectively. The calculated peak value of copolarized received power is −57.67 dBm (1.708 × 10−6 in mW), while the calibrated measured value was −56.0 dBm. In view of unavoidable various experimental errors, the agreement seems to be satisfactory. To obtain better fit, however, the measured copolarized curve was shifted vertically, and to some extent horizontally: A very small horizontal shift was needed to compensate for an error in angles. The cross-polarized values were also shifted in the same way. Because we used the same value in the vertical shift, the relative difference between the copolarized and cross-polarized values therefore remains unchanged. Note that the measured relative magnitude of copolarized and cross-polarized received powers is in good agreement with the calculated one. The sharpness of the cross-polarized curves agrees fairly well, although the agreement is poor in copolarized curves.
 We have made a computer simulation of backscattering enhancement, corresponding to the measurements [Ihara et al., 2004], by adopting a simple direct simulation method together with a sampling of the scattering paths for triple scattering and higher. The results agreed fairly well with measurements. Some of the knowledge on the detailed characteristics of enhanced backscattering has also been obtained. Future simulation work may include the calculations for nonspherical water drops, and an extension to real rain situation. Further measurements must be needed to confirm agreements with simulation especially in the copolar channel, and to see if the effect of footprint size on the magnitude of enhancement could be observable in our scattering environment.
 The authors would like to thank Satoru Kobayashi at Applied Materials, Inc. (Santa Clara, California) for helpful discussions on fundamental process of scattering.