114 A new concept of karst development based on hydrogeology and geophysics The geostatistical processing applied to each delay-time map (Bourges et al., 2012) involved two main steps: • detection of anomalous delay-time lines responsible for high spatial variability; • kriging and extraction of long-wavelength structures, to reveal underlying geological features. An initial variographic analysis was performed on both delay-time maps. The variograms were modeled using the following nested structures: • in-line shot delay-time map: A cubic structure with a range of 50 m, consistent with the size of geological heterogeneities, and a spherical structure with a range of 140 m. The cross-line variogram revealed an additional source of variability attributed to acquisition artifacts, modeled by a spherical structure with a range of 10 m; • cross-line shot delay-time map: A cubic structure with a range of 60 m, and an anisotropic spherical structure with ranges of 120 m in the in-line direction and 110 m in the cross-line direction. As with the in-line data, an additional variability source — attributed to acquisition artifacts — was modeled by a spherical structure with a range of 15 m. Using the previously defined variogram models, both delay-time maps were interpolated. During this process, small-scale variability — associated with acquisition artifacts in the cross-line direction — was filtered out using factorial kriging. The interpolated maps were then compared to the original delay-time profiles. For the in-line shot survey, one anomalous acquisition line responsible for additional variability was identified, while two such lines were highlighted in the cross-line shot survey. After removing the anomalous delay-time lines, omnidirectional variograms were computed, focusing on mid-scale variability (up to 50–70 m). For both delay-time maps, the updated variograms consist of: • a cubic structure with a range of 49 m (in-line shots) and 60 m (cross-line shots), and • a large-scale structure with a range of 120 m. Finally, the two large-scale delay-time maps — representing long-wavelength geological features — were interpolated using factorial kriging. Given the high correlation between the two resulting maps (correlation coefficient ≈ 0.8), an average long-wavelength structure was computed. The picked first-arrival times from all shots (both in-line and cross-line), the weathering zone (WZ) depth map derived from the averaged delay-time map (long-wavelength structure), and the velocity model obtained using the Plus–Minus method serve as input data for the inversion procedure — tomography — which is suitable for determining the subsurface velocity distribution (Mari and Mendes, 2012). The velocity distribution obtained through the joint Plus-Minus tomographic inversion can be effectively used to compute 3D static corrections. For this purpose:
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