Monitoring Spatio-temporal Dielectric Permittivity Variation in the Vadose Zone through Bayesian Inversion of GPR Data
Terry, Neil Carl
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A minimum-relative-entropy (MRE) based Bayesian inversion framework is applied to monitor spatio-temporal distribution of dielectric permittivity (and therefore water content) using tomographic radar data. This framework provides a tool that is used to track the spatio-temporal variation of water content and identify flow pathways in the study area, which is critical for hydrogeological site characterization and further study of contaminant transport in unsaturated/saturated zones. The radar data used in this study are collected during a synthetic, transient infiltration experiment. Among all soil mechanical properties, the saturated hydraulic conductivity field is treated as a random field with a known pattern of spatial correlation. Given a constant, non-point source flux at the surface, the TOUGH2 -EOS9 model is used to compute water saturation at grid points throughout the field at different times during infiltration. Cross-borehole tomographic ground penetrating radar data in the form of first-arrival travel times are simulated at temporal increments throughout the experiment using a curved-ray radar simulator. Gaussian random noises are added to these travel times data to represent measurement errors. The suite of GPR first-arrival travel times at each temporal increment are then used as observational data in the Bayesian inversion procedure. To estimate water saturation at each temporal snapshot, it is assumed that variation in water is mainly determined by changes in dielectric constant. The study area is discretized into fine grids and water movement is simulated using TOUGH2 simulator to obtain a realistic distribution of water content and dielectric permittivity. However, it is unreasonable to estimate the dielectric properties in each grid block because of the non-uniqueness relationship between dielectric permittivity and the resulting travel times. Therefore, the field is divided into 16 zones, each representing a spatially average dielectric permittivity value, and these zones are the targets of the parameter inversion. Minimally subjective probability distribution functions (pdfs) are assigned to each parameter using the minimum relative entropy (MRE) method. It is assumed initially that nothing is known about each parameter except for loose upper and lower bounds, which results in uniform prior pdfs. A quasi-Monte Carlo sampling technique is used to draw several pseudorandom dielectric constant parameter sets from these probability distribution functions. Then, first arrival GPR travel times are calculated for each set using the curved ray GPR simulator and are compared to the observational data. Each parameter set is given a weight based upon the misfit between the corresponding calculated model responses and observations. The output from this inversion is new probability distribution functions for each target parameter, along with corresponding statistics. This information can be used to derive new MRE pdfs and serve as prior pdfs for the future Bayesian updates when more information becomes available. These intermediate pdfs can be called memory functions of the unknowns (or parameters of interest). The inversion approach used in this study addresses many difficulties in parameter estimation, and the methods used can be tailored to any inversion problem provided a forward model exists to link parameters to data uniquely. Particularly, I demonstrate the benefits of this approach to time-lapse estimation of spatially varying parameters in the context of utilizing geophysical data for vadose zone parameter estimation. First, nonlinearity and non-uniqueness between data and model parameters can be handled by the incorporation of a numerical modeling technique. Second, uncertainty in the parameter estimates is quantified by treating those estimates as probability distributions rather than deterministic estimates. Third, multiple sources of information can be incorporated into the inversion procedure through use of the MRE method in defining the parameter prior probability distributions. Fourth, a zonation technique is utilized to reduce the ill-posedness of the inverse problem. Fifth, a robust geophysical radar simulator is used to synthesize realistic GPR data. Sixth, standard error between source-receiver pairs is pre-calculated to provide more robust estimates of the data misfit. Finally, the inversion of each successive observational dataset compliments the next by providing memory functions for the step-wise inversion.