Fresnel Volume Ground Penetrating Radar Attenuation Difference Tomography and Incorporating Geostatistical Constraints in Nonlinear Inverse Problems

Publication Date


Type of Culminating Activity


Degree Title

Doctor of Philosophy in Geophysics



Major Advisor

Warren Barrash, Ph.D.


Partha S. Routh, Ph.D.


The role of geophysics in the characterization and monitoring of hydrological systems is rapidly expanding to advance the understanding of fluid flow in the Earth. The non-invasive probing of the subsurface using different geophysical methods provides a wide range of possibilities to study the static and dynamic nature of subsurface flow systems. Water resource management is both a local and global concern. Fate and transport of contaminants, environmental remediation, and contaminant monitoring require accurate and spatially extensive knowledge of the heterogeneous subsurface. Direct measurements using borehole techniques are accurate but lack sufficient spatial coverage to determine hydraulic parameters at a large scale. In this thesis I focus on two broad areas: (a) improvement of the physics in tomographic imaging using cross-hole ground-penetrating radar and (b) development of a methodology to incorporate geostatistical scale information into the geophysical images that naturally leads to uncertainty estimation in a statistical sense. Although this method is applicable do inverse problems in general, it is demonstrated in conjunction with a ground-penetrating radar velocity tomography problem.

Tomography, a commonly used method in exploration geophysics, is increasingly used in near-surface applications to determine property distributions such as the seismic or radar velocity of the material traversed by propagating seismic or electromagnetic energy. Near-surface investigations using tomographic methods pose significant scientific challenges in terms of determining the uncertainty in the geophysical parameter estimates. This difficulty results from the less-than-optimal coverage needed to image the desired scale of the heterogeneity and infinite frequency assumption that is the basis for commonly used ray theoretical radar tomograms. I examine the attenuation responses from cross-hole radar data to monitor the movement of an electrically conductive tracer as it migrates through a shallow aquifer. Accurate time-lapse tomographic images of the tracer plume provide valuable information concerning the tracer movement and distribution, leading to a better understanding of the distribution of the properties that govern transport, such as porosity and permeability. The tomographic problem is solved using a first order electromagnetic scattering formulation to account for the finite frequency effect of propagating waves. Accounting for finite frequency wave propagation naturally leads to the Fresnel volume sensitivity distribution (in contrast to the ray sensitivity distribution in the infinite frequency approximation). Incorporating the finite frequency nature of the waves in the tomographic reconstruction procedure provides better resolution in the corresponding tomograms. The Fresnel volume method provides results similar to full-waveform methods but without the prohibitive computational efforts required in full-waveform inversions.

The Fresnel volume approach is applied to a data set obtained from a time lapse tracer test. The field test was carried out in an unconfined alluvial aquifer at the Boise Hydrogeophysical Research Site (BHRS) in Boise, Idaho. The aquifer at the BHRS consists of coarse (cobble-and-sand) fluvial deposits that overlie a clay layer at approximately 20 m depth. A bromide tracer was injected into one of the 18 monitoring wells at the BHRS. The tracer movement was monitored by direct sampling in isolated sections of monitoring wells and also through radar attenuation-difference tomographic data acquisition. The tomographic data are used to construct time-lapse tomograms of the tracer plume using the Fresnel volume (or Fresnel zone in two-dimensions) approach and also using the ray-based approach for comparison. The Fresnel zone tomograms are more resolved than the ray-based tomograms and demonstrate the importance of accounting for finite frequency propagation in the tomographic reconstruction. In comparison to the ray-based tomograms, the Fresnel zone tomograms lead to a more accurate understanding of the movement and shape of the plume. The movement and shape of the tracer plume is sensitive to important aquifer parameters such as hydraulic conductivity and porosity, and better resolution of the plume characteristics can lead to a better understanding of the distribution of such properties.

Other information such as borehole measurements (e.g. neutron logs) provide detailed information about porosity, but only in the near vicinity of the borehole. Geophysical methods may potentially provide information concerning the distribution of aquifer properties between boreholes at a relatively fine scale through tomographic imaging. However, the reliability and accuracy of the geophysical parameter estimates are often not well understood and sometimes not considered in the overall analysis. Therefore one of the goals of this thesis is to better understand the uncertainty associated with the parameter estimates. Although the scope of this problem is generic and broad I focus my research by considering a crosshole radar tomography data set as an example. I develop a generic statistical procedure and guideline to address the uncertainty in parameter estimates. This method will ultimately provide models which satisfy the data and also agree with the statistical variability of the subsurface at all scales. That is, incorporating geostatistical constraints provides the flexibility to generate an ensemble of models that satisfy both the objective of fitting the data and the objective of honoring specified spatial covariance properties (as expressed through one or more semivariograms) at all scales. I demonstrate the practical application of the theoretical developments accomplished in this thesis using tomographich radar data sets acquired at the BHRS.

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