Xueyang Duan and Mahta Moghaddam

1. 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

2. Outline • Motivation and Background • Problem Description • Formulation • Model Validation • Simulation results • Conclusion and Future Work

3. 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 • Low frequency radar systems receive scattering from both subsurfaces and sublayer inhomogeneities, e.g. vegetation roots, rocks, ice particles. Es Ei 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.

4. 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

5. Problem Description • Simulate the roots: • Root of an individual plant can be simulated as cylinders distributed with certain patterns. Ei Region 1 • For a vegetated area with multiple plants, roots are modeled as layers of randomly arranged cylinders, whose sizes and orientations follow statistical distributions. Surface 1 Region 2 Surface 2 Region 3

6. Problem Description (cont.) • The inhomogeneities can be modeled • as collections of regular-shaped • or irregular-shaped scatterers: • background: • spherical scatterers: - radius - center location - dielectric property - length and radius - dielectric property - cylinder center location: - cylinder orientation: - cylinder axis: • cylindrical scatterers: • Randomly distributed within each layer • Solution Strategy: • Transition matrix (T-matrix): • spherical wave basis, for volumetric scattering S-matrix of random media is required • Scattering matrix (S-matrix): • plane wave basis, for layered problem

7. Formulation Overview of approach: • Random media of sphericalscatterers: • 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 • Random media of cylindricalscatterers 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

8. Formulation – single scatterer T-matrix • 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.

9. Formulation – aggregate T-matrix (spheres) • Scattering from discrete random media is solved using the recursive T-matrix method • 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 • Using Addition theorem and translational matrices: Where and are translational matrices • The aggregate T-matrix is computed recursively,

10. Formulation – Generalized iterative EBCM (cylindrical clusters) • Limitations when considering cylindrical structures representing roots: • Knowing the T-matrices of all sub-cylinders, scattered field from N sub-cylinders is obtained from iterative procedure: • 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 • [1]: only valid for vertically oriented long cylinder • Our work: generalized to 3D randomly arranged cylinders • The long cylinder or cylindrical cluster can be treated as cascaded sub-cylinders. • 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 2 1 • Field interactions among sub-cylinders need coordinates transformation using Addition theorem Generalized iterative EBCM • Total scattered field is the superposition of scattered fields from all sub-cylinders transformed to the main frame. [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.

11. Formulation – Generalized iterative EBCM (cylindrical clusters) • 3D coordinates translation is implemented using rotation with z-axis translation. Euler angles are, Where (x’,y’,z’) denotes the coordinates after 1st rotation, and • T-matrix of the overall cluster:

12. Formulation – T matrix to S matrix transformation • 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] • 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. • 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 • S-matrix can be obtained from the T-matrix as [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.

13. Model Validation – Experiment Setup • 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 y x • 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’

14. Model Validation – Simulation • Antenna Model [1]: relating the vector fields and S-parameters measured • Propagation Model: computing the transmission parameter Sji between • antennas j and i in the presence of object. With Addition theorem, RX Antenna TX Antenna • The cylindrical cluster T-matrix is computed from cascading half-inch sub-cylinder. [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

15. Model Validation – PEC spheres • Measurement of 3D ‘randomly’ arrangedPEC spheres • Comparison between predicted and measured scattering parameters at 14 receivers for HH and VV.

16. Model Validation – PEC cylinders • Measurement of 3D ‘randomly’ arrangedPEC 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. VV HH

17. Simulation results for single rough surface above random spheres • Enhancement in all scattered field components, especially crosspol component

18. Simulation results for single rough surface above random cylinders • Enhancement in all scattered field components, especially crosspol component

19. Tree Root model • Tree root facts: depending on the soil conditions, • 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. • implemented tree root model: • 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 Reference: Dennis S. Schrock, ‘How to Prevent construction Damage’, University of Missouri Extension, 2000. • In this model, determines the root density, which in reality depends on soil moisture.

20. Simulation results for single rough surface above one single root • Enhancement is small in copol components, but large in crosspol component

21. Simulation results for single rough surface above multiple roots • Enhancement is small in copol components, but large in crosspol component

22. Conclusion and Future Work In this work, we showed: • 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. Future work: • 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

23. Acknowledgement • Measurement setup from Mark Haynes and Steven Clarkson Thank you ! Question?