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Published online 25 February 2008
Published in Vadose Zone J 7:249-262 (2008)
DOI: 10.2136/vzj2006.0144
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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Right arrow Ground Penetrating Radar, GPR
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Inferring the Subsurface Structural Covariance Model Using Cross-Borehole Ground Penetrating Radar Tomography

Thomas M. Hansena,*, Majken C. Loomsb and Lars Nielsenb

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


Figure 1
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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).

 

Figure 2
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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.

 

Figure 3
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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).

 

Figure 4
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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.

 

Figure 5
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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.

 

Figure 6
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FIG. 6. Geometry of tomography setup. Each line represents a ray. The color of each line indicates the mean velocity observed along the ray.

 

Figure 7
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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.

 

Figure 8
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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.

 

Figure 9
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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.

 

Figure 10
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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.

 

Figure 11
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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°).

 

Figure 12
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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).

 

Figure 13
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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).

 

Figure 14
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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.

 

Figure 15
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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.

 

Figure 16
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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.

 





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