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

2. 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

3. 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)

4. 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

5. 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

6. 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

7. 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?

8. 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

9. Inversion example – Layer Stripping [1]

10. 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]

11. Inversion Example – Riccati [3]

12. 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 , 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 , 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 , 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 , 2001.