Transdimensional post- and pre-stack seismic inversions: preliminary results on synthetic data

In most geophysical inverse problems, the properties of interest are parametrized using a fixed number of unknowns. For example, in most cases Earth models are parametrized with basis functions whose size and shapes are fixed in advance (i.e. fixed layer thicknesses, or cells with fixed dimensions). It is well known that this size value must constitute a compromise between model resolution and model uncertainty. In other cases, the number of unknowns is set at some arbitrary value and regularization is used to encourage simple, non-extravagant models. Recently, variable or self-adaptive parametrizations have gained in popularity. However, rarely the number of unknowns is itself directly treated as an unknown. This situation leads to a transdimensional inverse problem, that is, one where the dimension of the parameter space is a variable to be solved for. In the past few years, transdimensional inversions have been successfully applied to solve inverse problems especially at a seismological scale (Bodin and Sambridge, 2009; Bodin et al. 2012; Mandolesi et al. 2018), whereas more theoretical insights into this kind of problems can be found in Malinverno et al. (2004). As the authors are aware, applications of transdimensional algorithms to exploration geophysics are still rare although some example can be found in Zhu et al. (2016) and Dadi et al. (2016). However, these examples refer to a single parameter inversion, that is only one subsurface physical parameter is treated as an unknown (i.e. the P-wave velocity values or the resistivity values). In this work we check the applicability of a transdimensional inversion algorithms (the reversible-jump Markov chain Monte Carlo; rjMCMC) to the 1D inversion of post-stack and pre-stack seismic data. The first application is still a single parameter inversion, in which only the acoustic impedance (Ip) is considered unknown. Then, we extend the formulation to a multiparameter pre-stack inversion in which the parameters to be simultaneously determined are the P-wave, S-wave velocities and density. In both cases the optimal layer number, together with their thickness, are unknowns and constitute a final output of the inversion procedure. Therefore, the implemented method can help to automatically determine a proper subsurface parameterization, namely an optimal number of layers for a given data set. This characteristic enhances the uncertainty estimation since the transdimensional sampler can prevent a biased sampling of model space related to a suboptimal number of unknown parameters fixed a-priori (Sambridge et al. 2006). In addition, note that the rjMCMC, being based on Markov chain Monte Carlo principles, also ensures a correct uncertainty appraisal even for highly non-linear forward modellings. To draw essential conclusions about the applicability of rjMCMC algorithms we here limit the attention to synthetic inversions. In both case we simulate field dataset by adding Gaussian random noise to the observed data, so as to impose a signal-to-noise ratio equal to 10.