Ground Penetrating Radar using Electromagnetic Models EECS 725 Introduction to Radar Systems Spring 2017

Ground Penetrating Radar Classification: nadir looking, bistatic with TX & RX antennas co-located Can be pulsed (low SNR), FMCW, or chirp Useful for identifying irregularities and subsurface material properties (electrical permittivity, conductivity, magnetic permeability) under the surface of the earth Mostly interested in VOLUME scattering not surface scattering (non specular) Doppler information not needed (target can be stationary relative to radar) Applications: remote sensing of snow and ice sheets, civil infrastructure integrity (roads and bridges) Note: zenith looking version (non GPR, but same technique) used for ionospheric sounding

GPR as an Electromagnetic Inverse Problem FORWARD PROBLEM: given total knowledge (geometry, material electrical properties) about an ground target predict the scattered field due to an incident plane wave. Can be computationally expensive but many methods exist (MoM,FEM,FDTD,GTD,etc.) INVERSE PROBLEM: given limited measurements of the scattered field due to an incident plane wave predict ground target properties (geometry, material electrical properties) Very computationally expensive. Fewer methods with lower accuracy (Born, Riccati, Layer Stripping, etc.) Mathematically ill-posed (sensitivity to measurements, non-uniqueness)

Some common assumptions Plane waves (far-field measurements) Magnetic permeability is free space 𝜇 0 Same linear polarization maintained for transmit and receive 1-dimensional variation of permittivity and/or conductivity as a function of depth - pretty good assumption for snow and ice sheet which are largely planar Depth is approximately known or can be estimated from time-domain data Large dielectric contrast dominates scattering due to layering of substantially different material

How do I set up an inversion? Geometry: 1-dimensional variation, RX&TX located outside object space Physics: Maxwell’s Equations or some close relative Measurement: Reflection coefficient, RX waveform Inversion: estimate permittivity/conductivity as function of depth

Example Inverse Problem - Circuit Analogy Geometry: unknown capacitive circuit element Measurement: Noisy voltage signal Physics: Kirchhoff's current law Inversion: Estimated capacitor size 𝐶 ≈1𝜇𝐹

Subtle Issues in GPR inversion Inhomogeneous medium – natural materials (ice, snow, mixtures) change as a function of depth with both slow, smooth variation and sharp transitions between distinctly different materials Range Resolution – how big? Unambiguous range – how far?

Layer Stripping [1] Time domain method which iteratively removes each “layer” by finding peaks in the RX time waveform and time-of-flight physics Ignores multiple scattering Takes into account energy change due to loss Computationally efficient, but not very accurate Disadvantages: requires large discontinuities or high contrast; not well suited for smooth variation

Inversion example – Layer Stripping [1]

Inverse Riccati Methods [2] Frequency domain method for 1-dimensional problems but can handle both smooth and sharp transitions Usually a linearized version, 𝑟 𝑧= 0 − ,𝑘 ≪1, of this nonlinear differential equation is “inverted” to give estimate [3] Inversion suffers from longitudinal distortion [4]

Inversion Example – Riccati [3]

References [1] S. Caorsi and M. Stasolla, "A Layer Stripping Approach for EM Reconstruction of Stratified Media," IEEE Transactions on Geoscience and Remote Sensing, vol. 52, pp. 5855-5869, 2014. [2] T. J. Cui and C. H. Liang, "Reconstruction of the permittivity profile of an inhomogeneous medium using an equivalent network method," IEEE Transactions on Antennas and Propagation, vol. 41, pp. 1719-1726, 1993. [3] M. J. Akhtar and A. S. Omar, "Reconstructing permittivity profiles using integral transforms and improved renormalization techniques," IEEE Transactions on Microwave Theory and Techniques, vol. 48, pp. 1385-1393, 2000. [4] C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave and Optical Technology Letters, vol. 31, pp. 10-13, 2001.