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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - Nov 2020)
Ground-penetrating radar (GPR) has the potential for high-resolution imaging of near-surface material properties, including electrical conductivity and permittivity, which can be used for geological interpretation of the near subsurface. This thesis presents ray-based traveltime inversion and frequency-domain full-waveform inversion (FWI) techniques for application to borehole GPR surveys.Ray-based traveltime inversion is attractive for its speed, reliability, and ability to work in 3D, but the ray approximation involved limits recoverable detail to greater than one wavelength. The traveltime method presented here uses an efficient and easily programmed fast-sweeping eikonal solver to compute traveltimes. The inversion method also incorporates the unknown time offset between signal transmission and start of recording at the receiver as a model parameter that is recovered simultaneously with the material slowness.The resolution of FWI approaches the diffraction limit of one half wavelength, but at a substantial computational cost. The FWI inversion scheme presented here works in 2D and is unique in its simultaneous recovery of the source wavelet, conductivity, and permittivity. Its frequency-domain formulation allows for efficient factorization of the forward modeling operator and its subsequent application to multiple right-hand sides in order to quickly construct the forward model Jacobian. Efficient calculation of the Jacobian allows the use of the Gauss-Newton technique rather than the gradient descent method that is common for other GPR FWI inversions.Measured data must be converted from 3D to 2D before use with this 2D FWI technique. I present a graphical derivation of the perpendicular ray Jacobian, which is an essential part of 3D to 2D transformation. The graphical derivation provides the reader with an intuitive understanding of the Jacobian that is difficult to obtain from traditional mathematical treatments. I also illustrate that 3D to 2D transfer functions previously derived for the acoustic case are applicable to borehole GPR.
The diffraction limit defines the maximum resolution of an imaging system that collects and focuses waves. This limited resolution arises from the finite length of the waves used to create the image. Therefore, the only way to increase the resolution is to use higher frequencies with shorter wavelengths. For situations in which increasing the frequency is not possible or not desirable, super-resolution imaging techniques can be applied to overcome the diffraction limit. Super-resolution is possible with the inclusion of evanescent waves, which exhibit unlimited spatial frequencies.Evanescent waves decay exponentially away from their surface of origin so they are difficult to recover. One way to recover evanescent wave information is to scatter the wave from a small object. This scattering converts part of the evanescent wave into radiation that can propagate into the far-field where it can be detected. In order to characterize this conversion, the two-dimensional scattering of evanescent fields from a single cylinder and from multiple cylinders is investigated. The scattering models are derived using an analytical approach where the electromagnetic fields are broken down into cylindrical waves so that the boundary conditions on the cylinders can be applied directly. The incident field can be formulated from a vector plane-wave spectrum, which allows for an arbitrary combination of radiative and evanescent waves. Multiple cylinders of various sizes can be used to approximate the scattering from many two-dimensional objects. For simulating the imaging of objects buried underneath a surface, or near a planar interface, the model is separated into two dielectric half-spaces.An example of a super-resolution application for these models is the simulation of apertureless near-field scanning optical microscopy (ANSOM). In ANSOM, a probe is placed in the extreme near-field of an object in order to scatter the evanescent fields that are formed by the illumination of the object. Images created by ANSOM are fundamentally different from traditional images and are difficult to interpret. The simulations provide insight into how the images are formed and what information they contain.
The gradient method, Levenberg-Marquardt (LM) method, L₂ cooled roughness (CRL2)method and L₁ cooled roughness (CRL1) method are applied to the problem of recovering the relative permittivity structure of a dielectric object. The CRL1 method is a novel technique for the recovery of the relative permittivity structure of a dielectric object introduced in this work. The frequencies used in this work range from 0.80Hz to 1.2 GHz. The size of the permittivity structure is approximately 1 wavelength, which is approximately 30cm at 10Hz. The gradient method and LM method were unable to recover the relative permittivity structure unless the starting model is very close to the target. Both methods require a starting model that is close to the target model for them to be successful. The CRL2 method was able to recover a blurry approximation to the target relative permittivity structure. The blurriness is due to the L₂ norm. The CRL1 method is able to recover “blocky” structure. In the absence of noise, the CRL1 method was able to recover structure that was approximately one third wavelength in size. The recovery of structure at a fraction of a wavelength is highly sensitive to noise. Even at 0.0 1% noise, the CRL1 algorithm had difficulty recovering the exact structure.
Master's Student Supervision (2010 - 2018)
In the context of climate change, ground water monitoring has become an important task for which Ground Penetrating Radar (GPR) has been ideally suited. However, the limited depth of investigation has prevented GPR’s use in situations with deep water tables. In order to meet these new depth objectives, a novel Radar has been designed. This new radar, marries modern digital coding techniques and the ever improving field of digital electronics with that of a prototypical GPR. The sequences chosen for investigation include Golay-codes and M-sequences. This new GPR uses off the shelf digital equipment to meet these demands and does so in a more cost effective manner than conventional GPR. The design and implementation of this radar is covered. Simulations of theoretical performance are included for both code types and include factors for both white noise and digitizer quantization. Preliminary results demonstrate that the use of digital codes allow for greater dynamic range above and beyond that afforded by an impulse radar. Specifically, we show that when used with pre-existing dynamic range Golay-codes can add an additional 50 dB of dynamic range. Contrarily, we show that M-sequences can provide a similar dynamic range but this is total and not in addition to receiver sensitivity. In both cases, however, we achieve total dynamic ranges greater than that of an impulse radar. According to the simulation, the increase in dynamic range from the sequences, combined with a lower frequency of radar (25 MHz), allow us to achieve previously unseen depths of investigation (180 m). This depth is under a presumed attenuation of 1 dB m −¹ . As an additional benefit of using these codes, we can exploit the use of commercial FPGAs for code generation and processing. This substantially reduces the cost and opens up the radar for the intended application of remote monitoring. This lower frequency has the adverse effect of lowering theradar’s resolution. Moreover, the use of long codes increases the device’s acquisition time. However, these limitations do not unduly impact its intended use.