xueyang duan and mahta moghaddam
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04/22/23 1
Vector Electromagnetic Scattering from Layered Rough Surfaces with Buried Discrete Random Mediafor Subsurface and Root-Zone Soil Moisture Sensing
Xueyang Duan and Mahta MoghaddamRadiation Laboratory, Dept. of EECS, University of Michigan
IGARSS, Vancouver, Canada, July 24 – 29, 2011
Outline
• Motivation and Background• Problem Description• Formulation• Model Validation• Simulation results• Conclusion and Future Work
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Motivation and Background Ground subsurface sensing is a high-priority topic in
microwave remote sensing, where we focus on soil:• Monitoring facility constructions • Landslide warning• Locating the permafrost depth• Mapping the soil moisture profile
EiEs
Low frequency radar systems receive scattering from both subsurfaces and sublayer inhomogeneities, e.g. vegetation roots, rocks, ice particles.
A forward model of scattering from discrete random media representing the root structures and other inhomogeneities is required to study their impact on the evaluation of backscattering cross section.
Significant Contribution
• In the past investigations:– 3D single or multilayer rough surface scattering model without considering
sublayer inhomogeneities– No 3D full wave solution to random media scattering, especially with rough
surfaces
• In this work: – First time to include the sublayer inhomogeneities, especially vegetation roots,
in the subsurface scattering model– 3D full wave solution to scattering from discrete random media, especially
root-like clusters, with rough surface
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EiRegion 1
Region 2
Region 3
Surface 1
Surface 2
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Problem Description
• Root of an individual plant can be simulated as cylinders distributed with certain patterns.
• For a vegetated area with multiple plants, roots are modeled as layers of randomly arranged cylinders, whose sizes and orientations follow statistical distributions.
Simulate the roots:
• cylindrical scatterers:
• Randomly distributed within each layer
• background:• spherical scatterers:
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Problem Description (cont.)
S-matrix of random media is required
The inhomogeneities can be modeled as collections of regular-shaped or irregular-shaped scatterers:
0 0( , )
- length and radius - dielectric property- cylinder center location: - cylinder orientation: - cylinder axis:
( , )s s L a
( , )c c
ˆ ˆˆ ˆsin cos sin sin cosc c c c c cz x y z
( , , )c c cx y z
- radius- center location- dielectric property
a
( , )s s
Solution Strategy: • Transition matrix (T-matrix): spherical wave basis, for volumetric scattering• Scattering matrix (S-matrix): plane wave basis, for layered problem
( , , )s s sx y z
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Formulation
Overview of approach:
Random media of cylindrical scatterers or cylindrical clusters:• Decompose the cylindrical cluster to cascaded short sub-cylinders• Solve single short cylinder T-matrix using extended boundary condition method (EBCM)• Solve T-matrix of random or root-like cylindrical cluster using generalized iterative EBCM and incident field variation• Obtain S-matrix from T-matrix to S-matrix transformation
Random media of spherical scatterers:• Obtain single sphere T-matrix analytically• Solve multiple sphere scattering using recursive T-matrix method• Obtain S-matrix from T-matrix to S-matrix transformation
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Formulation – single scatterer T-matrix
1 1
0
0
M
N
TT
T
2 2 2 2
2 2 2 2
diag
diag
n s n n s n sM
n s n n s n s
s n s n n s n sN
s n s n n s n s
j k a kaj ka j ka k aj k aT
j k a kah ka h ka k aj k a
k a j k a kaj ka k a j ka k aj k aT
k a j k a kah ka k a h ka k aj k a
1
1 1 RgT Q Q , ,
, ,
MM MNnm lk nm lk
NM NNnm lk nm lk
Q QQ
Q Q
20 0 1
, 1 0 1 01 0
1ˆ Rg , , Rg , ,
1
m
MM k m k mnm lk l n l n
S
k kQ ds n N k r M k r M k r N k r
n n k
T-matrix of a single scatterer:• T-matrix of single spherical scatterer is from Mie-scattering coef.:
• T-matrix of single cylindrical scatterer is from extended boundary condition method:
, , and are similar as the above formulation with different spherical harmonics.
,MNnm lkQ ,
NMnm lkQ ,
NNnm lkQ
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Formulation – aggregate T-matrix (spheres)
1
1,0 1,0 0, 1 1,0 1,01 1 1 1 1 1n n n n nn n n n n nT I T T
0, 1 0, 1 1,01 1 1n n nn n n n nT
Scattering from discrete random media is solved using the recursive T-matrix method
Using Addition theorem and translational matrices:
The aggregate T-matrix is computed recursively,
if | | | |
if | | | |
Rg Rg | |
T
T ji j j j ji
i i T
ji j j j ji
T T
i i ji j j j
M r N r r rM r N r
M r N r r r
M r N r M r N r r
0 01
n
i in i ni
T
Basic idea: include one scatterer at a time into the cluster, and solve the new T-matrix including interaction between the added scatterer and the rest
Where and are translational matricesjiji
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Formulation – Generalized iterative EBCM (cylindrical clusters)
Limitations when considering cylindrical structures representing roots: EBCM: not valid for cylindrical scatterer with large length-to-diameter ratio Recursive T-matrix method: external circumscribed circles of the scatterers must be exclusive
Generalized iterative EBCM
[1] W.Z.Yan, Y.Du, D.Liu, and B.I.Wu, EM scattering from a long dielectric circular cylinder, PIERS, Vol. 85, pp. 39-67, 2008.
• Knowing the T-matrices of all sub-cylinders, scattered field from N sub-cylinders is obtained from iterative procedure:
Iteration starts assuming no field interaction among sub-cylinders
Compute scattered field and update the exciting field on every sub-cylinder
Repeat till the scattering coefficients of all sub-cylinders converge
• The long cylinder or cylindrical cluster can be treated as cascaded sub-cylinders.
Field interactions among sub-cylinders need coordinates transformation using Addition theorem
Total scattered field is the superposition of scattered fields from all sub-cylinders transformed to the main frame.
• [1]: only valid for vertically oriented long cylinder• Our work: generalized to 3D randomly arranged cylinders
iE12E
21E
12
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• T-matrix of the overall cluster:
Formulation – Generalized iterative EBCM (cylindrical clusters)
M Msca inc
totalN Nsca inc
a aT
a a
sca totalC T I total scaT C
Rotation (XZX): ith frame to the main frame Description
Rotate cylinder axis pointing to the main frame origin
Rotate the cylinder frame to be parallel to the main frame
• 3D coordinates translation is implemented using rotation with z-axis translation. Euler angles are,
0ˆˆ ˆc c ix z d
0ˆˆ ˆix d z
10
ˆˆ0, cos , =0c iz d
1
0 0 1
ˆ ˆ ˆ ˆcos 05= , = ,
ˆ ˆ ˆ ˆ2 cos 02c c
i ic c
x x x z
x x x z
Rotation (XZX): main frame to the ith frame Description
Rotate z axis pointing to the ith origin
Rotate to be the ith frame
0 00, = , =2i i
11
0 1
ˆ ˆ ˆ ˆcos 0ˆˆ0, = cos ,ˆ ˆ ˆ ˆ2 cos 0c i
x x x yz d
x x x y
Where (x’,y’,z’) denotes the coordinates after 1st rotation,
0 0 0 0 0 0ˆ ˆ ˆ ˆsin cos sin sin cosi i i i i id x y z
0 0 0 0 0 0ˆ ˆ ˆ ˆsin cos sin sin cosi i i i i id x y z
and
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Scattering matrix is obtained from multipole expansion of plane wave and numerical plane wave expansion of spherical harmonics with near-to-far field transformation. [1]
Numerical plane wave expansion of vector spherical wave => non-trivial ! when , where is an integer relaxation constant, and is the number of truncated terms in the expansion [MacPhie and Wu, 2003],
- practically computable in near-field only Near-to-far field transformation
0nj kr 0kr N N 0N N
Transformation: incident plane wave to spherical harmonics, using multipole expansion and Addition theorem
Each row ( ) of matrix P corresponds to a given direction of incidence.
1 2 i
T
NP p p p
Formulation – T matrix to S matrix transformation
ip
S-matrix can be obtained from the T-matrix as
U
S U T P
[1] X. Duan and M. Moghaddam, ‘3D Electromagnetic Scattering from Discrete Random Media Comprising Long Finite-Length Cylinders and Root-like Cylinder Clusters’, APS&URSI, Spokane, WA, July 3 – 8, 2011.
Model Validation – Experiment Setup
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Vector Network Analyzer and switching matrix based measurement system.
• freq: 2.2 GHz – 2.8 GHz• 1 transmitter and 14 receivers mounted on an octagon• ‘E’-shape wideband patch antennas• Rotation platform in the center
Measurement objects:• PEC/dielectric spheres: dia. 1”• PEC cylinders: length of 2” and 3”, diameter of 3/8, 5/8, 13/16 inches• positioned ‘randomly’
x
y
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Model Validation – Simulation
Antenna Model [1]: relating the vector fields and S-parameters measured
[1] Haynes, M., and M. Moghaddam, “Multipole and S-parameter based antenna model,” IEEE Trans. Antennas Propagat., vol. 59, no. 1, pp. 225-235, January 2011
Propagation Model: computing the transmission parameter Sji between antennas j and i in the presence of object.
RX Antenna
TX Antenna
00
00
0 0
0 0
T Tjjjs jsji sr
Tjjjs js sr
Ti is si si t
i is si si t
A B DDcS
B A DDd
D A B D aT
D B A D b
With Addition theorem,
The cylindrical cluster T-matrix is computed from cascading half-inch sub-cylinder.
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Model Validation – PEC spheres
• Measurement of 3D ‘randomly’ arranged PEC spheres
• Comparison between predicted and measured scattering parameters at 14 receivers for HH and VV.
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Model Validation – PEC cylinders
• Measurement of 3D ‘randomly’ arranged PEC cylinders (dia. 5/8 inch)
• Comparison between predicted and measured scattering parameters at 14 receivers for HH and VV.
• The amplitude comparisons show some agreements; better agreements in phase can be observed.
HH VV
Simulation results for single rough surface above random spheres
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Description Values
Freq. P-band ( = 68.97 cm)
vol. fraction
#sphere
sphere info
Media region
distribution Uniform random
Surf. info RMS = 0.5 cm, cor. Length = 5 cm
• Enhancement in all scattered field components, especially crosspol component
1478N
5 %vf
5.5 cm 0.08 , 10spha
8 8 , 6.17 0.95, 2 cmbg sepi d
Simulation results for single rough surface above random cylinders
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Description Values
Freq. P-band ( = 68.97 cm)
vol. fraction
#cylinder
cylinder info
Media region
distribution Uniform random
Surf. info RMS = 0.5 cm, cor. Length = 5 cm
200N
5 %vf
0.4 , 0.1 , 15cylL a
8 8 , 6.17 0.95, 2 cmbg sepi d
• Enhancement in all scattered field components, especially crosspol component
Tree Root model
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• Model parameters:• Depth of the root zone;• Horizontal range of the root zone;• Trunk diameter : assumed to be the upper end of cylinders
• Distribution:• Azimuth plane: uniformly distributed roots at• Elevation plane: N cylinders in every root group, with elevation angle
• Lower end is at
rd
i
rR
tr
11
1
, , if tansin
, , if tansin
ri i i t
i r
r ri i i t
i r
i d R
d
R R
d
11tan /i ri d d
Reference: Dennis S. Schrock, ‘How to Prevent construction Damage’, University of Missouri Extension, 2000.
Tree root facts: depending on the soil conditions,
implemented tree root model:
• In this model, determines the root density, which in reality depends on soil moisture.1d
• 85% tree roots are within the top 0.5 m of soil• Usually the roots extend the range of one (at least) to three times of the radius of the canopy spread or two or three times the height of the tree.
Simulation results for single rough surface above one single root
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Description Values
Freq. P-band ( = 68.97 cm)
Root area
#root
Root distr.
Subcyl. info
Surf. info RMS = 0.5 cm, cor. Length = 5 cm,
• Enhancement is small in copol components, but large in crosspol component
6, 2N N
0.5 m 0.73 , 0.5 m, 2 cmr r sepR d d
1 0.3 m 0.43 , 0.2 m 0.29td r
0.03 m, 0.06 m 0.09 , 15r rootr L
1 6.17 0.95i
Simulation results for single rough surface above multiple roots
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Description Values
freq P-band ( = 68.97 cm)
Root area
#branch
Branch distr.
Subcyl. info
Surf. info RMS = 0.5 cm, cor. Length = 5 cm,
• Enhancement is small in copol components, but large in crosspol component
(1) (1)6, 2N N
(1) (1)0.5 m, 0.5 m, 2 cm, 4 cmr r sep mgR d d d
(1) (1)1 0.3 m, 0.2 mtd r
0.03 m, 0.06 m, 15r rootr L
1 6.17 0.95i
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Conclusion and Future Work
Solution to scattering from randomly arranged spherical scatterers Solution to scattering from randomly arranged cylindrical clusters that can be used to represent root structures Experimental validation of the scattering model of discrete random media of both spherical and cylindrical scatterers Simulations results of single rough surface with buried random media or simplified roots
The presence of roots can significantly impact overall scattering cross section from the subsurface, especially in cross-pol Root geometry, density, and dielectric properties have strong impact on the scattering cross section.
In this work, we showed:
Further integration of the discrete random media model with multilayer rough surface model, soil moisture profile, and above-ground vegetation Use in root-zone soil moisture retrieval
Future work:
Acknowledgement
• Measurement setup from Mark Haynes and Steven Clarkson
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Thank you !Question?
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