Quantifying the influence of static-like errors in least-squares-based inversion and sequential simulation of cross-borehole ground penetrating radar data

Abstract

Unknown borehole irregularities and small-scale velocity fluctuations near transmitter and receiver antennae positions may cause relatively strong travel time effects on cross-borehole ground penetrating radar (GPR) data. Previous studies have demonstrated that such effects may severely contaminate cross-borehole GPR tomographic images of radar wave velocity if they are not properly accounted for prior to, or during, inversion. In this paper we calculate the travel time effect of cavities in the borehole walls and small-scale velocity anomalies near the antennae positions using a full waveform modeling algorithm. We define covariance matrices for static-like errors which approximately capture the overall correlation properties of these effects. In synthetic tests, we investigate to which extent the resolution of least-squares-based tomographic inversion is affected by the calculated error types under different assumptions made about the statistical properties of the data errors. We find that the effects of the correlated data errors may be significantly suppressed if static-like errors are accounted for during inversion, even though the errors are not strictly static. Furthermore, we demonstrate that when static-like errors are accounted for, model resolution does not decline significantly, even when the expectation to the standard deviation of the data errors is increased above the level of the correlated errors. We implement this approach of accounting for correlated data errors in a sequential simulation algorithm, which we use to solve the inverse tomographic problem in order to obtain multiple realizations of the fine-scale GPR velocity distribution between the boreholes. Synthetic tests show that the assumptions made about the error correlation properties are significant for obtaining reliable images of the fine-scaled velocity distribution between the boreholes, even in cases where the correct prior knowledge about model correlation properties are available. We observe that the static-like data errors may introduce artifacts in the velocity distributions near the borehole walls if they are not properly accounted for during the conditioned simulation process. We apply the sequential simulation algorithm to a real data set from Arrenæs, Denmark and demonstrate that accounting for correlated data errors has a significant effect on the interpretation of the field data. In comparison to the standard approach in which errors are considered to be uncorrelated, higher resolution images with stronger contrasts between high- and low-velocity anomalies are produced when correlation of the data errors are accounted for.