Estimation of the velocity model uncertainties and of the associated uncertainties on microseismic event locations by a Bayesian approach

The reservoirs and mines are often monitored by detecting and analyzing the induced seismicity. This microseismicity is interpreted to derive the location and orientation of the fractures and also to derive several physical parameters. Very precise location of the induced seismicity with its associated uncertainties is thus of primary importance. The largest contribution to the microseismic location errors is due to the lack of knowledge of the wave-propagation medium, the velocity model has thus to be preliminary inverted. We here present a tomography algorithm that estimates the true uncertainties on the resulting velocity model. Including these results, we develop an approach that allows to obtain accurate event locations and their associated uncertainties due to the velocity model uncertainties. The inversion of the velocity model is often a totally non-linear inverse problem. To solve it, we choose a Bayesian approach as this formulation allows for a complete understanding of all possible solutions. In a Bayesian context, a large number of samples of statistically near-independent models from the a posteriori probability distribution are sought. Such solutions are consistent with both the data and the prior information, as they fit the data within error bars, and adhere to soft prior constraints given by a prior probability distribution. We apply the Bayesian approach to the tomography of calibration shots for a typical 3D geometry hydraulic fracture context. We develop a Monte-Carlo Markov chain (MCMC) algorithm which generates samples of velocity model distributed according to the posterior distribution. This leads us to express the best and the mean velocity model as well as the associated standard deviation. This allows identifying the regions of the velocity field where the velocities are well constrained and thus reliable and also the regions where the velocities are poorly constrained. This strategy results in the estimation of the true uncertainty on the velocity model, which is much more accurate than the common roughly estimated percentage of uncertainty on the velocity model. We then develop the Bayesian formalism for the event location by including the posterior distribution of the velocity model previously obtained. For the location problem, we use a full grid search algorithm as the whole posterior probability distribution can be explored at a reasonable computational cost. We compute the probability density of the hypocenter location, taking into account the velocity model uncertainties. The resulting probability map shows that the commonly used probability density associated to the picking uncertainties must not be used to represent the probability density associated to the velocity model uncertainties. Traveltime picking and velocity model uncertainties should thus not be mixed.