GNGTS 2019 - Atti del 38° Convegno Nazionale

GNGTS 2019 S essione 3.3 725 A MARKOV CHAIN MONTE CARLO ALGORITHM FOR ELASTIC AMPLITUDE VERSUS ANGLE INVERSION WITH NON-PARAMETRIC PRIORS AND NON-LINEAR FORWARD MODELLINGS M. Aleardi 1 , A. Salusti 1,2 1 University of Pisa, Earth Sciences Department, Pisa, Italy 2 University of Florence, Earth Sciences Department, Florence, Italy Introduction. One of the main objectives of reservoir characterization is to exploit the acquired seismic and well log data to infer the distribution of elastic parameters and litho-fluid facies around the investigated area. From the mathematical point of view this process is an ill- conditioned inverse problem in which many models can fit the observed data equally well. For this reason, one goal of reservoir characterization studies is the quantification of the uncertainties affecting the recovered solution, which are expressed by the so-called posterior probability density function ( pdf; Tarantola 2005). One challenge of this inversion process concerns the simultaneous estimation of discrete (i.e. litho-fluid facies) and continuous (elastic properties) model parameters from the observed data. Another challenge is related to the complexity of the property distribution and correlation. For example, the distribution of elastic properties is often multimodal due to the presence of multiple litho-fluid facies. An analytical and computationally fast derivation of the posterior model is only possible in cases of linear forward operators, Gaussian, Gaussian-mixture, or generalized Gaussian distributed model parameters and Gaussian errors in the seismic data. However, the validity of the Gaussian or Gaussian-mixture assumption is often case dependent because they could not be adequate to reliably capture the complex relations among elastic attributes and litho-fluid facies. At the same time, the linear forward model might not be sufficiently accurate to describe the relation between seismic data and elastic parameters in cases of strong elastic contrasts at the reflecting interface and far source-receiver offsets. For this reason, it is often advisable to numerically evaluate the posterior model through a Markov Chain Monte Carlo (MCMC) algorithm. From the one hand, MCMC methods have been successfully applied to solve many geophysical problems (Sambridge and Mosegaard, 2002) as they can theoretically assess the posterior uncertainties in cases of complex (i.e, non-parametric) prior distributions and non-linear forward modellings. From the other hand, these methods convert the inversion problem into a sampling problem and for this reason they require a much larger computational effort with respect to the analytical approach. Moreover, the use of non-parametric priors often complicates the inclusion of geostatistical a-priori information (e.g. a semivariogram model) into the inversion procedure and for this reason the use of non-parametric models is not so common in geophysical inversions. Finally, classical MCMC methods, such as the Metropolis-Hastings algorithm, are known to mix slowly between the modes if the target distribution is multimodal. To partially overcome this issue, multiple MCMC chains are usually employed so that the ability to exhaustively explore the high probability regions of the model space is enhanced. We present an amplitude versus angle inversion algorithm for the joint estimation of elastic properties and litho-fluid facies from pre-stack seismic data in case of non-parametric mixture prior distributions and non-linear forward modellings. The algorithm inverts the pre-stack seismic responses along a given time interval using a 1-dimensional convolutional forward modelling based on the Zoeppritz equations. The distribution of the elastic properties at each time-sample position is assumed to be multimodal with as many modes as the number of litho-fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a MCMC algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis-Hasting algorithm with the more advanced Differential Evolution Markov Chain method. For the lack