ARL-TR-7799 ● SEP 2016

US Army Research Laboratory

A Discrete Scatterer Technique for Evaluating 

Electromagnetic Scattering from Trees

by DaHan Liao

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ARL-TR-7799 ● SEP 2016

US Army Research Laboratory

A Discrete Scatterer Technique for Evaluating 

Electromagnetic Scattering from Trees

by DaHan Liao

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution is unlimited.
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A Discrete Scatterer Technique for Evaluating Electromagnetic Scattering from 

Trees

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DaHan Liao

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14. ABSTRACT

An analytical solver is developed for characterizing the coherent scattering responses of trees. Realistic 3-D tree structures are 

first constructed using an open-source random tree generation engine. The trees are then parsed into discrete, canonical 

scatterers, such as cylinders and disks, and a multi-ray approach is applied for the calculation of the aggregate response of the 

scene, with the transmissivity of each ray determined from a cell-based representation of the computational domain. As each 

scatterer in the outlined framework is assigned a deterministic position, the spatial distribution of the trees and their canopy 

structures is fully preserved. A cell-by-cell strategy is also proposed for speeding up the calculations of the responses from small 

components such stems and leaves, which are expected to far outnumber those scatterers composing the main trunks. The 

accuracy of the analytical solver is assessed by comparing simulation results for a forest stand with solutions from a large-scale, 

full-wave solver. The primary aim of this study is to develop an accurate and computationally efficient solver to facilitate 

scattering simulations of realistic forest scenes for airborne foliage-penetration radar applications.

15. SUBJECT TERMS

analytical solution, discrete scatterer technique, electromagnetic scattering, foliage-penetration radar, tree scattering

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Contents

List of Figures iv

1. Introduction 1

2. Tree Generation 2

3. Cell-Based Discrete Scatterer Technique 3

4. Numerical Results 7

5. Conclusions 15

6. References 16

Appendix. Scattering Solution for Branch and Leaf Scatterers 19

Distribution List 23
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List of Figures

Fig. 1 Cellularized DST computational domain...............................................4

Fig. 2 Ray paths for a scatterer: a) direct response, b-c) single ground-bounce 

returns, and d) double ground-bounce return.........................................5

Fig. 3 Forest stand with 72 randomly generated trees: a) tree trunks only, Nc

= 2017, b) tree trunks + primary branches, Nc = 25397, c) tree trunks + 

primary and secondary branches, Nc = 292877, and d) tree trunks + 

primary and secondary branches + leaves, Nc = 1878880. ....................8

Fig. 4 Comparison of DST solution with FDTD solution for scene containing 

trunks only (Fig. 3a): a) radar cross sections and b) phase differences.9

Fig. 5 Comparison of DST solution with FDTD solution for scene containing 

trunks + primary branches (Fig. 3b): a) radar cross sections and b) 

phase differences..................................................................................10

Fig. 6 Comparison of DST solution with FDTD solution for scene containing 

trunks + primary and secondary branches (Fig. 3c): a) radar cross 

sections and b) phase differences.........................................................11

Fig. 7 Comparison of scatterer-by-scatterer vs. cell-by-cell DST solutions for 

scene containing trunks + primary and secondary branches (Fig. 3c): a) 

radar cross sections and b) phase differences.....................................12

Fig. 8 Radar cross sections of the scenes in Fig. 3: a) σvv, b) σhh, and c) σhv..14

Fig. 9 Phase differences of the scenes in Fig. 3: a) ξc and b) ξx. ....................15
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1. Introduction

Developing sensing capabilities for the detection of ground targets concealed in a 

forest environment necessitates a fundamental understanding of the scattering 

properties of not only the targets and the trees but also their interactions over the 

observation time interval. For a given target in motion under the tree canopy, the 

time-frequency behaviors of these scattering interactions are very much dependent 

upon the tree arrangement and the precise structure of the canopy. In this respect, 

the spatial distribution of the various tree constituents (trunks, branches, and 

foliage) potentially becomes an important determinant of the detection performance 

of the radar sensor and processing algorithm, and therefore, needs to be preserved 

in a wave scattering solver. For example, in the application of a clutter-limited 

technique such as along-track interferometry for moving target indication, the trees 

surrounding the target can introduce deleterious effects into the interferometric 

phase, leading to decorrelation and a nonlinear target phase-velocity relationship, 

as a result of tree occlusion, attenuation, and phase distortion. From the radar 

simulation perspective, such effects can only be studied with a physics-based solver 

that faithfully captures the actual positions of the target and the trees and their 

scattering phase centers over the entire coherent processing interval. 

Existing foliage-penetration radar performance analysis tools are rather simplistic 

in that they do not take into account the localized interactions of the target with its 

surrounding, nor do they fully consider the polarimetric scattering responses of the 

scene (i.e., the target and clutter responses are often rudimentarily incorporated into 

the signal model just by using radar cross section values or coefficients). Higher 

fidelity scattering analysis in the form of the discrete scatterer technique (DST) has 

been undertaken in the microwave remote sensing community within the context 

of developing algorithms for retrieving tree/forest biophysical parameters from the 

scattering returns.

1,2 In Lin and Sarabandi,1 wave theory is applied in constructing 

a 3-D coherent scattering model for forest stands consisting of fractal-generated 

trees; unfortunately, in this model, although individual trees have deterministic 

positions, the propagation or transmissivity effects of the trees are considered only 

in a statistical manner within a layered-canopy representation. A similar approach 

to that of Lin and Sarabandi1 is also employed by Thirion et al.,

2 wherein the main 

physical features of the canopy such as the trunks and crown shapes are captured, 

but the branches and leaves are specified only indirectly through probability 

distributions. 

Moreover, notwithstanding the usefulness of the analytical solvers developed in

these aforementioned studies, and others not discussed here,3–6 the application 
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focus in existing works has not been on the detection of moving targets under the 

canopy but rather only on the responses of the trees themselves. It should be noted 

that a variety of radiative transfer-based solvers also have been put forth for tree 

response characterization. While traditional radiative transfer theory has been 

applied successfully to a number of remote sensing scenarios—some involving 

complex forest canopy structures7—this method does not include the coherent 

scattering effects occurring within the tree structure and therefore cannot be used 

to compute the phase response of a scene. 

In order to account for the spatial distribution of the trees coherently and 

systematically, this work follows the framework proposed by Liu et al.,

8 in which 

a multi-ray approach is exploited to calculate the aggregate scattered fields from a 

collection of trees, with the transmissivity of each ray determined from a cell-based 

representation of the scene. A cell-by-cell, interaction-tracking procedure is also 

proposed here for speeding up the calculations of the responses from small 

components such stems and leaves, which are expected to far outnumber those 

scatterers composing the main trunks. Furthermore, tree and scene generation for 

the analytical solver is carried out in conjunction with that for a large-scale, fullwave solver, allowing for an assessment of the accuracy of the ray-based solution 

for any scene. Note that the details of the full-wave solver itself have been published 

elsewhere,

9 which demonstrated the practicality of finite-difference time-domain 

(FDTD)-based scattering simulations of a realistic forest scene for foliagepenetration radar applications. 

In brief, this work is organized as follows. In Section 2, an overview of the 

procedure for generating 3-D realistic trees for both the analytical and numerical 

solvers is presented. In Section 3, the formulations for the DST are described in 

detail. In Section 4, computational examples are given: the DST solutions are 

compared with full-wave simulation results; the accuracy of the cell-by-cell 

approach is evaluated; and a general discussion on the scattering effects of a forest 

stand is put forth. Finally, Section 5 summarizes the study.

2. Tree Generation

Realistic 3-D tree structures are created using the open-source tree generation 

engine Arbaro,10 which implements the recursive, rule-based, growth algorithm 

developed by Weber and Penn.

11 Given a random seed and an input parameter file 

defining the physical characteristics of a tree (e.g., tree and trunk shape, number of 

branch levels, branch sizes and angles, leaf shape and density), the algorithm 

constructs the hierarchical structure for the tree by following a series of recursive 

rules derived from geometrical observables. For the DST solver of this work, each 
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tree is exported from Arbaro as a multilevel Persistence of Vision Raytracer

(POV-ray) file in which the trunk/branches and leaves are represented by tube and 

disk primitives, respectively. Additional geometric transformation or processing of 

the exported tree object (i.e., decomposition of the tube primitives into cylinders of 

desired sizes, global translation, and scaling, etc.) is carried out before the entire 

model is passed on to the scattering solver. For the FDTD-based simulations, the 

mesh form of the tree structure is first exported from Arbaro as a Wavefront OBJ 

file; subsequently, voxelization of the mesh is performed using the parity count and 

ray stabbing methods.

9,12–13

3. Cell-Based Discrete Scatterer Technique

Once the trees have been parsed into canonical components or scatterers (i.e., 

dielectric cylinders and disks), each with deterministic locations and well-defined 

physical and electrical properties, the scattering solver is derived by applying the 

3-step procedure as follows. First, the tree scene—consisting Nt trees—is 

discretized into Nb cubic cells (Fig. 1), with each scatterer assigned to a cell. 

Second, the propagation paths for each scatterer are identified and tabulated: this 

necessitates deducing all the cells intercepted by the 4 rays shown in Fig. 2. Finally, 

the ray-based scattering response for each scatterer is calculated analytically, while 

taking into consideration the effects of attenuation and phase change encountered 

along each ray path, and then the total scattering response of the entire scene is 

obtained by coherently summing the returns from all the scatterers. Succinctly, in 

the multi-ray approach, for incident direction ki

ˆ and scattering direction s k

ˆ , the 

scattering matrix for the nth ( n∈{1,2,...,Nc}) scatterer can be written as 

( ) ( ) ( ) ( ) ( ) ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ , , , , ,, n nn n n

c cg gc gcg si si si si s i S kk S kk S kk S kk S kk =+ + + (1)

where

( ) ( ) ( ) ( ) ( ) , ˆ ˆ ˆˆ ˆ ˆˆ ˆ ,, ; n n o io so n n fs n n jk k r k r

c c c si s si i S kk T k S kke T k − ⋅ −⋅

= ⋅ ⋅

 

(2)

( ) ( ) ( ) ( ) ( ) ( ) ( ) , ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆ , , ; n n o ig so n n fs n n n jk k r k r

cg c cg g s i s s gi gi i i S kk T k S kk e T k k T k − ⋅ −⋅

= ⋅ ⋅ ⋅Γ ⋅

 

(3)

( ) ( ) ( ) ( ) ( ) ( ) ( ) , ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ , , ; n n o io sg n n n fs n n jk k r k r

gc g gc c s i s s gs gs i i S kk T k k T k S k ke T k − ⋅ −⋅ = ⋅Γ ⋅ ⋅ ⋅

 

(4)
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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ,, ; n n o ig sg n n n fs n n n jk k r k r

gcg g gc cg g s i s s gs gs gi gi i i S kk T k k T k S k k e T k k T k − ⋅ −⋅ = ⋅Γ ⋅ ⋅ ⋅ ⋅Γ ⋅

 

(5)

in which n

or

 is the location of the scatterer center; r r (r z)z n

o

n

o

n

g = − 2 ⋅ ˆ ˆ    ; 

k k (k z)z gi i i ˆ ˆ ˆ 2 ˆ ˆ = − ⋅ ; k k (k z)z gs s s ˆ ˆ ˆ 2 ˆ ˆ = − ⋅ ; (⋅,⋅) fs,n

S is the scatterer free-space response; 

(⋅) n

T u (u = c, g, cg, gc) is the transmissivity matrix of the propagation path as 

determined by the cells intercepted; and Γ(⋅) is the ground reflection coefficient 

matrix. Essentially, in the presence of the ground, the multi-ray technique captures 

the first 4 orders of scattering effects: Eq. 2 is the direct scattering response from 

the scatterer; Eqs. 3 and 4 correspond to the single ground-bounce returns; and

Eq. 5 is the double ground-bounce contribution.

Fig. 1 Cellularized DST computational domain
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Fig. 2 Ray paths for a scatterer: a) direct response, b-c) single ground-bounce returns, and 

d) double ground-bounce return

In Eqs. 2–5, the free-space response of the scatterer, (⋅,⋅) fs,n

S , is derived with the 

application of 2 sets of approximate formulations: for a finite-length, circular 

cylinder scatterer, the volumetric current integration method is employed wherein 

the internal cylinder fields are assumed to be the same as those of the infinite-length 

case;14,15 and for a thin-disk scatterer, the generalized Rayleigh-Gans solution is 

applied wherein the internal disk fields are related to the incident fields through the 

polarizability tensor of the scatterer.15 In the current work, the leaves are simply 

modeled as circular disks. The analytical formulations for these 2 sets of solutions

are included in the Appendix. 

The aforementioned transmissivity matrix 

( ) ( ) ( )

( ) ( )





 





⋅ ⋅

⋅ ⋅ ⋅ = n

u hh

n

u hv

n

u vh

n

u vv n

u

T T

T T

T

, ,

, ,

(6)

is calculated with Foldy’s approximation,16 which derives the distortion effects of 

a random medium on a mean field. It can be shown that 

( ) ( ) ( )

, , , ; n pp u n j T e pp vv hh u pp

−Κ ⋅⋅ ⋅ ⋅ = =

 

 (7)

and, assuming a medium with azimuthal symmetry, ( ) ( ) 0 , ⋅ = , ⋅ = n

u vh

n Tu hv T . In Eq. 7,

( ) n Nb u ⋅ ∈R  is the propagation path length vector (i.e., the bth element of (⋅) n

u 

 is 
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the path length of the ray in cell b, b∈{1,2,...,Nb }); and ( ) Nb Kpp ⋅ ∈C  is the 

aggregate forward scattering vector for all the cells, with its bth element computed 

as 

( ) ( ) ,

cell 

2 , , b fs m

pp pp

o m b

S

k v

π

κ ∈⋅ = ∑ ⋅ ⋅ (8)

where v is the cell volume. Together, Eqs. 1–8 give the most general framework for 

analyzing the scattering from a tree scene, taking into account propagation 

attenuation and phase change effects experienced by the incident and scattered 

waves for each discrete scatterer in the random medium. In view of the fact that the 

scatterers that compose the tree trunks are the largest scatterers in the scene and 

therefore would provide the dominant responses, extra consideration must be taken 

in evaluating Eq. 6 for the trunk elements. In most realistic scenarios, the individual

scatterers of a trunk are not expected to be shadowed by other scatterers belonging 

to the same trunk—in other words, a trunk cannot shadow itself. As such, in 

calculating the transmissivity matrices for the scatterers of a particular tree trunk t

( n∈trunk t, t ∈{1,2,..., Nt}), the effects of self-shadowing should be excluded by 

modifying Eq. 8 as

( ) ( ) ,

cell 

trunk 

, . b fs m

pp pp

m b

m t

κ S ∈

∉

⋅ = ⋅⋅ ∑ (9)

To increase the computational efficiency of the solver, instead of a scatterer-byscatterer brute force approach, a cell-by-cell scattering and propagation strategy can 

be applied for the treatment of the smallest branch elements and the leaves, which 

are expected to far outnumber those scatterers composing the trunks and larger 

branches. In this approach, all the constituents of a cell are assumed to have the 

same set of transmissivity matrices (for each ray path) and the responses are 

propagated together as a group rather than one by one. To this end, the 

approximation is made such that the ray paths of all the scatterers belonging to the 

same cell are identical. Here, the center of the cell is used as the common reference 

point for identifying all those paths and evaluating the transmissivity matrices using 

Eqs. 6–8. The accuracy of the previous approximation is investigated in the next 

section. 

Essentially, in this cell-by-cell strategy, computational speedup is provided by not 

having to calculate the transmissivity matrices on a scatterer-by-scatterer basis, as 

well as by not having to determine and tabulate the ray paths for every scatterer in 

a cell. Note that the same cell-based strategy can potentially be applied to the 

scatterers for the trunks and larger branches, as well. However, in general, since 
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7

those scatterers are expected to provide the more dominant responses—as 

compared to the smallest branches and the leaves, the associated errors induced by 

the underlying assumptions of the technique are also expected to be more 

significant.

4. Numerical Results

The scenes of interest are displayed in Fig. 3, which shows a forest stand composed 

of Nt = 72 randomly generated quaking aspen (Populus tremuloides) trees modeled 

with different levels of structural fidelity. The trees are constructed assuming a

4-level hierarchy—as such, there are 4 cases of consideration: the trunks only, the 

trunks with primary branches, the trunks with primary and secondary branches, and 

the trees with all their branches and leaves. As the starting point, the backscattering 

returns of the forest stand are first examined by comparing the DST results with

those from the full-wave solver. Note that since the FDTD method as employed 

here cannot efficiently model extremely thin structures such as leaves, validation 

of the DST results is only carried out for scene configurations without the foliage. 
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Fig. 3 Forest stand with 72 randomly generated trees: a) tree trunks only, Nc = 2017, b) 

tree trunks + primary branches, Nc = 25397, c) tree trunks + primary and secondary branches,

Nc = 292877, and d) tree trunks + primary and secondary branches + leaves, Nc = 1878880

The FDTD- and DST-calculated backscattering radar cross sections σ pq

(p,q = v,h) and phase differences d ξ (d = c,x) of the forest stand are illustrated in 

Figs. 4–6, as a function of the elevation incidence angle θi and structural fidelity. 

Formally, the plotted co- and cross-polarized responses are defined as follows: 

2

4 pq pq σ = π S , * ξc = arg SvvShh , and * ξ x = arg SvvShv , where pq S is the 

pq-polarized scattering matrix element. Note that in the backscattering direction, 
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hv vh S = −S . The results at each elevation incidence angle are obtained by averaging 

the responses over 36 evenly spaced azimuth incidence angles spanning 

φi = [0°, 360°] and over 301 frequencies spanning f = [200 MHz, 500 MHz] 

(P-band). The ground and the tree trunk/branches are assumed to be electrically 

homogeneous, with relative dielectric constant and conductivity (εr, σd) of 

(5.45, 20 mS/m) and (13.9, 39 mS/m), respectively. The FDTD computational 

domain has dimensions 30 m × 30 m × 9.2 m and is discretized into 2 billion voxels; 

parallelized simulations are performed after partitioning each scene into 256 

subdomains.9 On the other hand, the DST scene has dimensions 30 m × 30 m × 

10 m and is divided into Nb = 9,000 cubic cells (i.e., each cell has a side length of 

1 m).

Fig. 4 Comparison of DST solution with FDTD solution for scene containing trunks only 

(Fig. 3a): a) radar cross sections and b) phase differences
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Fig. 5 Comparison of DST solution with FDTD solution for scene containing trunks + 

primary branches (Fig. 3b): a) radar cross sections and b) phase differences
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Fig. 6 Comparison of DST solution with FDTD solution for scene containing trunks + 

primary and secondary branches (Fig. 3c): a) radar cross sections and b) phase differences

As evident from Figs. 4–6, in general, except for the cross-polarized response of 

the trunks-only scene (Fig. 4), the DST results are in good agreement with those 

from the full-wave method—in both amplitude and phase. For the trunks-only 

scene, the observed discrepancy between the analytical and numerical crosspolarized returns is in part due to the fact that the FDTD solver as implemented

here cannot capture low cross-polarization signal levels. Once the branches are 

introduced into the scene, the cross-polarized response increases; therefore, those 

cases with the branches (Figs. 5 and 6) do not pose a problem for the FDTD solver. 

Note that at the cost of increased computational time and memory, the fidelity of 

the numerical cross-polarized solution can be improved by reducing the FDTD

voxel size. Although the limitation of the full-wave analysis here precludes a 

complete validation of the DST results when the cross-polarized response is low, 

the DST co-polarized solutions are seen to match those from FDTD reasonably well 

in all cases.
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The analytical solutions presented thus far are obtained by calculating the responses 

from the scene on a scatterer-by-scatterer basis. To speed up the computations, the 

cell-by-cell interaction algorithm discussed in Section 3 is applied for the treatment 

of the secondary branches in the scene in Fig. 3c. A comparison of the results from 

this new approach with those from the original scatterer-by-scatterer technique is 

shown in Fig. 7. It is observed that the more computationally efficient approach

retains the accuracy of the brute force technique. Note that the memory usage and 

runtime for the scatterer-by-scatterer simulation are 369 MB and 33 min/frequency, 

respectively, on a Dell Precision T7500 workstation with Intel Xeon CPU of 

2.67 GHz, while those for the cell-by-cell simulation are 200 MB and 

40 s/frequency, respectively. 

Fig. 7 Comparison of scatterer-by-scatterer vs. cell-by-cell DST solutions for scene 

containing trunks + primary and secondary branches (Fig. 3c): a) radar cross sections and 

b) phase differences
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The DST-simulated backscattering responses of the forest stand for the 4 levels of 

scene structural fidelity are plotted together in Figs. 8 and 9. The foliage is modeled 

by considering the leaves as circular dielectric disks with a thickness of 0.2 mm, a 

mean radius of 2.6 cm, and (εr, σd) = (28.0, 0.37 S/m). The cell-by-cell strategy is 

implemented for the secondary branches and the leaves when the scene contains 

those components. (The memory usage and runtime for the case with all the 

branches and leaves are 1.0 GB and 56 s/frequency, respectively.) For the 

co-polarized responses, it is noted that as the elevation incidence angle increases, 

the dominant contribution tends to be provided by the tree trunks—though this is 

more apparent for hh than vv. In general, the inclusion of the primary branches is 

seen to lead to an enhanced backscattering intensity, compared to the response of 

the scene with only the trunks. However, further addition of the secondary branches 

and the leaves to the scene can weaken the return—especially at the higher 

incidence angles where the propagation loss of the canopy plays a more important 

role. Unlike the co-polarized returns, the cross-polarized response of the forest 

stand is mainly determined by the primary branches and not by the trunks. This is 

as expected, since the trunks here resemble slightly asymmetric cones that have 

only a small cross-polarized backscattering response. As in the co-polarized cases, 

the inclusion of the secondary branches and the leaves tends to decrease the crosspolarized return, relative to that for the scene with only the trunks and the primary 

branches.
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Fig. 8 Radar cross sections of the scenes in Fig. 3: a) σvv, b) σhh, and c) σhv
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15

Fig. 9 Phase differences of the scenes in Fig. 3: a) ξc and b) ξx.

5. Conclusions

An analytical solver is developed in this study for characterizing the far-field 

electromagnetic scattering responses of trees. The solution is based on a multi-ray 

discrete scatterer approach, applicable to a cellularized scene comprising realistic

3-D tree structures of which the constituents are represented by cylinders and disks. 

The accuracy of the solver is determined by comparing the backscattering results 

for a forest stand at P-band with those from large-scale, full-wave simulations. A 

cell-by-cell strategy is also proposed and implemented to increase the 

computational efficiency of the solver for evaluating the scattering contributions

from the smallest branches and the leaves. Overall, the analysis and computational 

framework put forth is intended as a tool for investigating scattering effects relevant 

to airborne foliage-penetration radar applications.
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6. References

1. Lin Y-C, Sarabandi K. A Monte Carlo coherent scattering model for forest 

canopies using fractal-generated trees. IEEE Trans Geoscience and Remote 

Sensing. 1999 Jan;37(1):440–451.

2. Thirion L, Colin E, Dahon C. Capabilities of a forest coherent scattering model 

applied to radiometry, interferometry, and polarimetry at P- and L-band. IEEE 

Trans Geoscience and Remote Sensing. 2006 Apr;44(4):849–862.

3. Zeng T, Hu C, Sun H, Chen E. A novel rapid SAR simulator based on 

equivalent scatterers for three-dimensional forest canopies. IEEE Trans 

Geoscience and Remote Sensing. 2014 Sep;52(9):5243–5255.

4. Jiang W-Q, Zhang M, Chen H, Lu Y-G. CUDA implementation in the EM 

scattering of a three-layer canopy. Prog Electromagn Res. 2011;116:457–473.

5. Williams ML, Pottier E, Ferro-Famil L, Allain S, Cloude SR, Hajnsek I,

Papathanassiou K, Moreira A. Forest coherent SAR simulation within 

PolSARpro: an educational toolbox for PolSAR and PolInSAR data 

processing. ACRS 2007. Proceedings of the Asian Conference on Remote 

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Laboratory (US); 2015. Report No.: ARL-TR-7346.

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(NY): Wiley-Interscience; 1985.
Approved for public release; distribution is unlimited.

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Approved for public release; distribution is unlimited.

19

Appendix. Scattering Solution for Branch and Leaf Scatterers
Approved for public release; distribution is unlimited.

20

Analytical formulations for calculating the electromagnetic scattering responses of 

tree branches and leaves are given. Note that the following treatment assumes the 

incident and scattered wave directions ( ˆ

i k , ˆs k ) and their associated polarization 

vectors ( h

ˆ , vˆ) are defined by

ˆ sin cos sin sin cos ; ˆ ˆ ˆ i i i ii i k x yz = +− θ φ θφ θ (A-1)

ˆ sin cos sin sin cos ; ˆ ˆ ˆ s s s ss s k x yz = ++ θ φ θφ θ (A-2)

,

,

,

ˆ ˆ ˆ ; ˆ ˆ

i s

i s

i s

z k h

z k

× = ×

(A-3)

, ,, ˆ ˆ ˆ ; is is is v hk = × (A-4)

in which the subscripts i and s identify quantities related to the incident and 

scattered waves, respectively; θi is the incidence angle in elevation (measured from 

- zˆ ); θs is the scattering angle in elevation (measured from + zˆ ); and φi,s is the 

incidence/scattering angle in azimuth (measured from + xˆ ). 

A-1 Scattering from Branches

In this study, the tree branches are approximated with circular cylinders. The freespace scattering matrix elements of a finite-length, dielectric circular cylinder with 

length L, radius a, and complex relative dielectric constant εr,t are derived as1,2

( ) ( ) ( ) ( )

2

,

1, 2, 3,

1 cos cos ˆ ˆ , ˆ sinc , , , 2 2

s i fs o rt o is jn

pq s i s n n n

n

k L k L

S k k p K K e pq vh φ φ ε θ θ +∞ −

=−∞

−   + = ⋅     Κ+ + =     ∑   

(A-5)

where

( )( ) 1

1,

,

ˆ ˆ ; 2

s j

o n

n n n

r

jk I e j x jy k

φ

ρ

β α Κ= − − +  ( )( ) 1

2,

,

ˆ ˆ ; 2

s j

o n

n n n

r

jk I e j x jy k

φ

ρ

β α

−

Κ =− − + +  3, ˆ. n nn Κ = I z γ 

(A-6)

For q = ν,

( , )

sin cos

; ni i

n

nrn

jM

J k aR ρ

θ θ

α = ( , )

sin cos

; i i

n

nrn

nW

J k aR ρ

θ θ β = ( , )

sin

; n i

n

nrn

jM

J k aR ρ

θ

γ = (A-7)

 1 Liao DH. Scattering from the finite-length, dielectric circular cylinder: Part I—derivation of an 

analytical solution. Adelphi (MD): Army Research Laboratory (US); 2015. Report No.: ARL-TR-7346.

2 Karam MA, Fung AK, Antar YMM. Electromagnetic wave scattering from some vegetation samples.

IEEE Trans Geoscience and Remote Sensing. 1988 Nov;26(6):799–808.
Approved for public release; distribution is unlimited.

21

for q = h,

( )

2

,

sin cos

; i i

n

nrn

nW

J k aR ρ

θ θ

α = ( , )

sin

; n i

n

nrn

jN

J k aR ρ

θ β − = ( , )

sin cos

; i i

n

nrn

nW

J k aR ρ

θ θ

γ = (A-8)

and 

( ) ( ) ( ( ) ( ) ( ) ( )) 2

2 2 , , 1, , , 1,

, ,

; n rn s n r sn r n s

r s

a I k aJ k a J k a k aJ k a J k a

ka ka ρ ρ ρρ ρ ρ

ρ ρ

= + + −

−

(A-9)

( ) ( ) 2 (2)

, , 22 2 cos ; 2

i ni

n n n i

ka H ka

R M N Wn

π ρ ρ = ⋅ ⋅−   θ   (A-10)

( )

( )

( )

( )

(2)

, ,

(2)

, ,, ,

' ' ;

n i nr

n

i n i rn r

H ka J ka

M

k aH k a k aJ k a

ρ ρ

ρ ρρ ρ

= − (A-11)

( )

( )

( )

( )

(2)

, ,

, (2)

, , ,,

' ' ;

n i n r

n r t

i n i rn r

H ka J ka

N

k aH k a k aJ k a

ρ ρ

ρ ρ ρρ

= −ε (A-12)

( ) ( ) 2 2

, ,

1 1 .

i r

W

ka ka ρ ρ

= − (A-13)

The previous equations assume , k k ρ io i = sinθ , , k k ρ so s = sinθ , and 

2

, , cos r o rt i k k ρ = − ε θ .

A-2 Scattering from Leaves

The scattering response of a circular disk-shaped leaf with thickness τ, radius a, and 

complex relative dielectric constant εr,t is given by

( ) ( )

( ) ( ) ( )( )( )

2

1

, ˆ ˆ , 1 ˆˆ ˆ ˆ ˆ ˆ , , , fs o c si

pq s i r t T s i N T s i

si

k v J Qa S k k a p q a a p z z q pq vh Q a

ε

π

= ⋅ − ⋅+ − ⋅ ⋅ =    

(A-14)

where

2

; 2 c

a

v

τπ = (A-15)

( , )

1

; 1 1 T

rt T

a

ε g = − +

(A-16)
Approved for public release; distribution is unlimited.

22

( , )

1

; 1 1 N

rt N

a

ε g = − +

(A-17)

( ) 













 −















 −

− ⋅ − = − 1 1 sin

2 1 1

1 2

1

2

2

2

m

m

m

m

m

gT ; (A-18)































 −

− ⋅ − − = −

m

m

m m

m

gN

1 sin

1

1 1

1

2

1

2 2

2

; (A-19)

τ

a

m

2 = ; (A-20)

( ( ))2

1 2 2 sin sin 2sin sin cos si o s i s i s i Q = k θ + θ − θ θ φ −φ . (A-21)
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23

1 DEFENSE TECHNICAL

(PDF) INFORMATION CTR

DTIC OCA

2 DIRECTOR

(PDF) US ARMY RESEARCH LAB

RDRL CIO L

IMAL HRA MAIL & RECORDS MGMT

1 GOVT PRINTG OFC

(PDF) A MALHOTRA

3 DIRECTOR

(PDF) US ARMY RESEARCH LAB

RDRL SER U

TRAIAN DOGARU

DAHAN LIAO

ANDERS SULLIVAN
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