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Accounting for Correlated Data Errors during Inversion of Cross-Borehole Ground Penetrating Radar Data

Dep. of Geography and Geology, Univ. of Copenhagen, Øster Voldgade 10, DK-1350 Copenhagen K, Denmark

View larger version (26K): [in this window] [in a new window]   FIG. 1. (a) Location map of the Danish area. The black circle indicates the location of the field site in northern Zealand. (b) Layout of boreholes for ground penetrating radar (GPR) recording and electrical resistivity tomography (ERT) measurements at the field site (see also Looms et al., 2008). Tests made in the field have shown that the electrodes placed in the boreholes for the ERT measurement do not significantly affect the GPR signal recorded in the other boreholes.

View larger version (68K): [in this window] [in a new window]   FIG. 2. Ray geometry. A total of 702 data points were used. Rays dipping >45° were omitted from the study. Transmitter positions were 1 m apart, whereas receiver positions were 0.25 m apart.

View larger version (47K): [in this window] [in a new window]   FIG. 3. Data error covariance matrices and corresponding simulated data noise: (a–b) spatially correlated transmitter static errors; (c–d) spatially correlated receiver static errors; (e–f) uncorrelated errors related to mispicking of first arrivals. In (a–d), the spatial correlation length, L, is 5 m. The data error covariance matrices were set up using Eq. [3], and the different types of noise were simulated as outlined by Jacobsen (1993). The total data set consists of 702 travel times (Fig. 2). For illustration purposes, the data errors are shown for only 200 data points.

View larger version (56K): [in this window] [in a new window]   FIG. 4. Synthetic test study: (a–c), (e–g), and (i–k) inversion results; (d, h, and l) true slowness test pattern that should ideally be fully recovered by inversion. In (a–c), data error correlation was not taken into account during the inversion process. In (e–g) and (i–k), the inversion was performed assuming that the data errors were correlated; different assumptions about the correlation properties and the standard deviations of the correlated data errors were made. Only in (j) was the inversion performed using the correct data error covariance matrix of the total error budget.

View larger version (37K): [in this window] [in a new window]   FIG. 5. Results from inversion of the data set acquired between GPR Boreholes 1 and 3 at the Arrenæs field site (Fig. 1b). Borehole 1 is to the left and Borehole 3 to the right. In (a–c), the data errors were assumed to be uncorrelated. In (d–h), the data errors were assumed to be correlated; different assumptions about the correlation properties and the standard deviations of the correlated data errors were made.