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Vector Electromagnetic Scattering from Layered Rough Surfaces with Buried Discrete Random Mediafor Subsurface and Root-Zone Soil Moisture Sensing

Xueyang Duan and Mahta Moghaddam

Radiation 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( , )ε µ

- length and radius - dielectric property- cylinder center location: - cylinder orientation: - cylinder axis:

( , )s sε µL a

( , )c cθ φ

ˆˆˆˆ sin cos sin sin cosc c c c c cz x y zθ φ θ φ θ= + +

( , , )c c cx y z

- radius- center location- dielectric property

a

( , )s sε µ

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

µµ

− − − ′ ′ ′ ′ ′ ′= ⋅ ⋅ × − × + ∫

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 Tβ α τ α β α τ−

+ + + + ++ + + + ⋅ = − ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅

( ) ( ) ( ) ( )0, 1 0, 1 1,01 1 1n n nn n n n nTτ τ β τ α β+ + ++ + + = + + ⋅ ⋅ ⋅

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

α

β

β

⋅ < = ⋅ >

= ⋅ ∀

( ) ( )0 01

n

i in i ni

Tτ β β=

= ⋅ ⋅∑

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 matricesjiαjiβ

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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 dα β γ−= = ⋅

( )( )

1

0 0 1

ˆˆˆ ˆcos 05= , = ,

ˆˆˆ ˆ2 cos 02c c

i ic c

x x x z

x x x z

πα φ β θ γπ

−

−

′′ ⋅ ⋅ >− = ′′− ⋅ ⋅ <

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

πα β θ γ φ= +

( ) ( )( )

11

0 1

ˆˆˆˆcos 0ˆˆ0, = cos ,ˆˆˆˆ2 cos 0c i

x x x yz d

x x x yα β γ

π

−−

−

′ ′′ ′′ ′ ⋅ ⋅ <= ⋅ = ′ ′′ ′′ ′− ⋅ ⋅ >

Where (x’,y’,z’) denotes the coordinates after 1st rotation,

0 0 0 0 0 0ˆ ˆˆ ˆsin cos sin sin cosi i i i i id x y zθ φ θ φ θ= + +

0 0 0 0 0 0ˆ ˆˆ ˆsin cos sin sin cosi i i i i id x y zθ φ θ φ θ= + +

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 = L

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 λ ε= = =

8 8 , 6.17 0.95, 2 cmbg sepi dλ λ λ ε× × = + =: :

λ

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 aλ λ ε= = =

8 8 , 6.17 0.95, 2 cmbg sepi dλ λ λ ε× × = + =: :

• Enhancement in all scattered field components, especially crosspol component

λ

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

iφ

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

θ φ θ θ πθ

θ φ θ θ πθ

−

−

⋅ > = −

< = −

( )11tan /i ri d dθ π −= − ⋅

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 Nφ θ= =

0.5 m 0.73 , 0.5 m, 2 cmr r sepR d dλ= = = =

1 0.3 m 0.43 , 0.2 m 0.29td rλ λ= = = =

0.03 m, 0.06 m 0.09 , 15r rootr L λ ε= = = =

λ

1 6.17 0.95iε = +

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 Nφ θ= =

(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 ε= = =

λ

1 6.17 0.95iε = +

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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?