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Inferring the Subsurface Structural Covariance Model Using Cross-Borehole Ground Penetrating Radar Tomography

^a Univ. of Copenhagen, Niels Bohr Institute, Juliane Maries Vej 28, DK-2100 Copenhagen Ea, Denmark
^b Univ. of Copenhagen, Geological Institute, Oester Voldgade 10, DK-1350 Copenhagen K, Denmark

View larger version (8K): [in this window] [in a new window]   FIG. 1. Prior probability density function (pdf) given by geologist no. 1 (light yellow) and geologist no. 2 (light gray). Posterior pdf using the prior information given by geologist no. 1 (dark yellow) and geologist no. 2 (dark gray).

View larger version (100K): [in this window] [in a new window]   FIG. 2. Three unconditional realizations using a range of 10, 20, and 40 m in a model of size 100 by 100 m, using a grid cell size of 1 m.

View larger version (11K): [in this window] [in a new window]   FIG. 3. Ergodic variation of the mean computed from 100 unconditional realizations using a range of 10, 20, and 40 m. The red curve is the best-fitting normal distribution to the observed data. The vertical line indicates the a priori choice of mean velocity (0.13 m/ns).

View larger version (17K): [in this window] [in a new window]   FIG. 4. Ergodic variation of the semivariance computed from 100 unconditional realizations using a range of [10,20,40], with experimental semivariogram (black lines) and theoretical semivariogram (blue line). Mean and 95% confidence intervals for the variance of the experimental semivariogram are the red solid and dashed lines, respectively.

View larger version (21K): [in this window] [in a new window]   FIG. 5. Likelihood of 100 samples of a Gaussian random function with a specific range of 10, 20, and 40 m being a realization of a Gaussian random function with range from 5 to 75 m.

View larger version (78K): [in this window] [in a new window]   FIG. 6. Geometry of tomography setup. Each line represents a ray. The color of each line indicates the mean velocity observed along the ray.

View larger version (21K): [in this window] [in a new window]   FIG. 7. The 5% (light gray), 50% (gray), and 95% (black) highest probability density regions using (a) 10, (b) 30, (c) 70, and (d) 702 rays. The red circle denotes the choice of horizontal and vertical correlation lengths (hmax, hmin) = (6 m, 1.5 m) used to generate the reference model.

View larger version (13K): [in this window] [in a new window]   FIG. 8. One-dimensional marginal distributions of (left) horizontal correlation length (hmax) and (right) vertical correlation length (hmin), using data observations for 10, 30, 70, and 702 rays.

View larger version (10K): [in this window] [in a new window]   FIG. 9. One-dimensional marginal distributions of (left) horizontal correlation length (hmax), (middle) vertical correlation length (hmin), and (right) dip (β), using data observations for all 702 rays. The dashed line indicates the value used to generate the reference random field.

View larger version (42K): [in this window] [in a new window]   FIG. 10. Ground penetrating radar reflection profile recorded close and parallel to the cross-borehole profile considered. The green area indicates the approximate location of the model parameter space considered.

View larger version (71K): [in this window] [in a new window]   FIG. 11. Sensitivity kernel using a (a) linear and (b) nonlinear (linearized) approach, in the case considering a covariance model with horizontal and vertical correlation lengths and dip (hmax, hmin, β) = (10 m, 3 m, 6.5°).

View larger version (75K): [in this window] [in a new window]   FIG. 12. Eight realizations from the prior probability density function (pdf) using horizontal and vertical correlation lengths of (a) (hmax, hmin) = (2 m, 1 m), (b) (hmax, hmin) = (10 m, 3 m), and (c) (hmax, hmin) = (18 m, 6 m).

View larger version (76K): [in this window] [in a new window]   FIG. 13. Eight realizations from the posterior probability density function (pdf) using horizontal and vertical correlation lengths of (a) (hmax, hmin) = (2 m, 1 m), (b) (hmax, hmin) = (10 m, 3 m), and (c) (hmax, hmin) = (18 m, 6 m).

View larger version (14K): [in this window] [in a new window]   FIG. 14. The 20% (white), 40%, 60%, and 80% (black) highest probability density (HPD) intervals for the likelihood function using (a) 702 rays and a linear sensitivity kernel, (b) 702 rays and nonlinear sensitivity kernels, and (c) 70 rays and nonlinear sensitivity kernels, as a function of the horizontal (hmax) and vertical (hmin) correlation lengths. The HPD plots are based on approximately 300 sample points.

View larger version (17K): [in this window] [in a new window]   FIG. 15. One-dimensional (1D) marginal probability density function distributions of the likelihood of the horizontal (hmax) and vertical (hmin) correlation lengths using (a) 702 rays and a linear sensitivity kernel, (b) 702 rays and nonlinear sensitivity kernels, and (c) 70 rays and nonlinear sensitivity kernels.

View larger version (13K): [in this window] [in a new window]   FIG. 16. One-dimensional (1D) marginal probability density function (marg. pdf) distributions of the likelihood of the horizontal (hmax) and vertical (hmin) correlation lengths and the dip using 702 rays, based on 175 samples found using the Metropolis–Hastings algorithm.