Using orthogonal Legendre polynomials to parameterize global geophysical optimizations: Applications to seismic-petrophysical inversion and 1D elastic full-waveform inversion

We use Legendre polynomials to reparametrize geophysical inversions solved through a particle swarm optimization (PSO). The subsurface model is expanded into series of Legendre polynomials that are used as basis functions. In this framework the unknown parameters become the series of expansion coefficients associated to each polynomial. The aim of this peculiar parameterization is three-fold: Efficiently decreasing the number of unknowns, inherently imposing a 1D spatial correlation to the recovered subsurface model and searching for maximally decoupled parameters. The proposed approach is applied to two highly non-linear geophysical optimization problems: Seismic-petrophysical inversion, and 1D elastic full-waveform inversion. In this work, with the aim to maintain the discussion at a simple level we limit the attention to synthetic seismic data. This strategy allows us to draw general conclusions about the suitability of this peculiar parameterization for solving geophysical problems. The results demonstrate that the proposed approach ensures fast convergence rates together with accurate and stable final model predictions. In particular, the proposed parameterization reveals to be effective in reducing the ill-conditioning of the optimization problem and in circumventing the so-called curse-of-dimensionality issue. We also demonstrate that the implemented algorithm greatly outperforms the outcomes of the more standard approach to global inversion in which each subsurface parameter is considered as an independent unknown.