Stochastic borehole radar velocity and attenuation tomographies using cokriging and cosimulation

A new GPR constrained velocity and attenuation tomography algorithm using ray approximation is presented. This algorithm is based on slowness and attenuation covariance modeling using experimental travel time and amplitude covariances, respectively. The computed covariances, the measured travel times and additional slowness and attenuation values allow cokriging and conditional simulation. Starting with a homogeneous velocity field, an iterative solution is computed updating the raypaths in applying Snell–Descartes' law on the cokriged velocity field after each iteration. Once the raypaths are known, attenuation tomography is performed using cokriging and conditional cosimulation of the amplitudes and any available attenuation information. Advantages of the stochastic tomography algorithm are first demonstrated using a synthetic velocity model. Then, stochastic tomography of data collected on a mine waste rock pile is presented. Stochastic tomography using the synthetic model allows providing several statistically equivalent velocity fields. These realizations yield an estimate of the uncertainty associated with the estimation process. Also, it is shown that application of constraints along boreholes can dramatically reduce the uncertainty of the estimates. Information along boreholes allows better estimation of the slowness covariance model and its parameters along the vertical direction. Using inverted slowness and attenuation fields of the real data set, conductivity and the dielectric constant are computed. Cluster analysis of these electrical parameters characterizes the different geological units. Also, threshold probability maps are generated to define the probed materials. Both synthetic and real data analysis demonstrate that the new method leads to accurate estimates of the probed material physical properties and provides the user with an estimation of the uncertainty inherent to the tomographic process.