Estimation of geometry and complex permittivity of the buried object and the medium Reza Firoozabadi, Eric L. Miller, Carey M. Rappaport and Ann W. Morgenthaler Contact: {rfirooza,elmiller,rappaport}@ece.neu.edu Boundary Definition by B-Spline: Reconstruction of dielectric properties: Dielectric Properties: Reconstruction of the geometry and electric properties of the subsurface objects from the noisy scattered field data has been a challenging problem in different environmental and biomedical research areas. In this work, we propose a method to solve this category of problems which utilizes an iterative inversion algorithm to reconstruct the shape of the object and the rough surface as well the electric parameters of the lossy object and the half-space it is buried inside. The boundaries are modeled by 2-D parametric B-spline curves determined by control points. The object and the ground are lossy and characterized by their complex permittivities. Ground is dispersive and we use single-pole conductivity model to determine its complex permittivity in each required frequency. The forward model exploited in the algorithm is SAMM (Semi-Analytic Mode Matching) which is a fast and efficient method to determine the scattered electromagnetic fields from buried objects in half space with rough interface. As an efficient numerical method, Levenberg-Marquardt algorithm is used to update the unknowns in an iterative routine coded in Matlab. The numerical experiments validate the accuracy and reliability of our inverse method. Abstract: Dispersive ground: Number of control points Basis functions of order k Control points @ ² g a v = b i ! ( 1 + z ¡ ) 2 ¢ Soil complex permittivity: Knot vector: T={t0,t1,…,tm} ² g ( ! ) = a v + i ¾ Basis Functions Final curve Single-pole conductivity model: ¾ g = b + 1 z ¡ 2 a Estimated εm= (3.5 + 0.0027i) ε0 Conclusions: Reconstruction of the buried object and the rough ground surfaces and permittivity and conductivity of soil and object was performed from observed electromagnetic field data by: Modeling the object boundary by a low-order B-spline model. Modeling the soil conductivity by single-pole conductivity model Using SAMM forward model to determine scattered field data at the location of the receivers. Using Levenberg-Marquardt iterative algorithm to solve the nonlinear least-squares optimization problem. z = e ¡ i ! ¢ T Non-dispersive object: Object complex permittivity: @ ² t r = i ² t = r + i The Forward Model: C and C’: Coordinate scattering centers, C0: Global coordinate center, I : incident plane wave, R : reflected plane wave, T : transmitted plane wave, t: cylindrical modes in object, q: cylindrical modes in ground, r: cylindrical modes in air Modes and Scattering Centers: C C’ Object Air Ground I R T t q r C0 x y z SAMM (Semi-analytical Mode Matching) forward model computes the scattered fields at receivers by enforcing boundary conditions at surfaces and computing the cylindrical rescattering modes from scattering centers. Field vector at receivers Matrix including Bessel and Hankel functions Coefficient vector: Boundary mismatch vector Problem Geometry: Unknown vector: Lossy dielectric object and lossy dispersive ground Multi-frequency incident uniform plane wave at desired angle Receivers above the interface detect the scattered fields. u = £ 1 T ; 2 g t ¤ Future Work: Simulation in 3D space. Simulation for multiple objects. Developing regularization procedures. Defining interface boundary in other compact forms. Ground Object Air Receivers Rough Interface Incident Plane Wave x y z u 1 = h p ( ) y ; 2 : N c i T x g [ ² a v b ] t r C. Boor, A Practical Guide to Splines, New York: Springer-Verlag, 1978. A. W. Morgenthaler, and C. M. Rappaport, “Scattering from dielectric objects buried beneath randomly rough ground: Validating the semi-analytic mode matching algorithm with 2-D FDFD,“ IEEE Trans. Geoscience and Remote Sensing, vol. 39, pp. 2421-2428, Nov. 2001. K. Levenberg, “A method for the solution of certain problems in least squares," Quart. Appl. Math., vol. 2, pp. 164-168,1944. C. T. Kelley, Iterative Methods for Optimization, SIAM, 1999. C. Rappaport, S. Wu, and S. Winton, “FDTD wave propagation in dispersive soil using a single pole conductivity model," IEEE Trans. Magn.,vol. 35, pp. 1542–1545, May 1999. References: Initial guess of control vector u0 Call forward solver (2-D SAMM) to Compute scattered field vector f(u) Compute cost function e (u) = || f(u) – f0 ­|| Compute new control vector u by Levenberg-Marquardt algorithm Desired accuracy achieved? Generate the surface by new control vector u No Yes Numerical Results: Example specifications Source fields: multi-frequency TM plane wave. Incidence angle: -45 degrees. SNR: 30 db. Frequency range: 300,400, … ,900 MHz. Object permittivity: εm= (2.9+0.0029i)ε0. Ground: Bosnian soil with density=1.263 g/cc, moisture level=25.3% (εav=5.03815, a1=-.925, b0=1.76106, b1=-.32102, b2=1.56193) Interface: 80 points, 11 control points. Object: 30 points, 4 control points. Publications:* R. Firoozabadi, E. L. Miller, C. M. Rappaport and A. W. Morgenthaler, “New Inverse Method for Simultaneous Reconstruction of Object Buried Beneath Rough Ground and the Ground Surface Structure Using SAMM Forward Model", IS&T/SPIE 17th Annual Symposium on Electronic Imaging: Science and Technology, pp. 382-393, San Jose, CA, 16-20 Jan. 2005. R. Firoozabadi, E. L. Miller, C. M. Rappaport and A. W. Morgenthaler, “Characterization of the Object Buried Beneath a Random Rough Ground Using a New Semi-Analytical Mode Matching Inverse Method,” IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, Washington, DC, 3-8 July 2005. R. Firoozabadi, E. L. Miller, C. M. Rappaport and A. W. Morgenthaler, “Subsurface Estimation of the Geometry and Electromagnetic Properties of Buried Anomaly and Half-space Background with Unknown Rough Boundary”, submitted to Progress In Electromagnetics Research Symposium PIERS 2006, Cambridge, 26 - 29 March 2006. R. Firoozabadi, E. L. Miller, C. M. Rappaport and, A. W. Morgenthaler, “A New Inverse Method for Subsurface Sensing of Objects Under Randomly Rough Ground Using Scattered Electromagnetic Field Data”, to be appeared at IEEE Transactions on Geoscience and Remote Sensing. Goal: Inverse Problem Formulation: The Levenberg-Marquardt iterative algorithm is an efficient method in optimization problems in form of nonlinear least-squares minimization: Hessian : Levenberg Marquardt parameter C=I2Nc Gradient : Jacobian: The Jacobian is determined by finding the elements of: , and . Finding a good estimate to the boundaries from observed data by minimizing the cost function: Boundary reconstruction Residual vector: Predicted field Unknowns vector Observed data State of the Art: * This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821). Current forward methods: Born approximation (less accurate for large targets) FDFD & FDTD (sensitive to grid discretization and computationally expensive) SDFMM (computationally expensive) Current minimization methods: Newton Method Steepest descent method