Some strategies to make global methods suitable to solve high-dimensional and ill-conditioned geophysical optimization problems

Geophysical optimisation problems are often non-linear, multi-dimensional, and characterised by objective functions with complex topologies (i.e. multiple local minima). Global methods are often used to solve these problems, but they are affected by the curse of dimensionality problem, that is their ability to explore the model space exponentially decreases as the dimensions of the model space increase. In addition, their limited exploitation capabilities make global search algorithms unable to converge in ill-conditioned optimization problems or in cases with highly correlated model parameters. In this work, I test three different strategies that could be used to partially attenuate the previous issues. The first strategy uses Legendre polynomials to reparametrize the subsurface model. More in detail, 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 finally searching for maximally decoupled parameters. This approach is applied to 1D seismic-petrophysical inversion in which the objective function to minimize is a weighted sum of data misfit and a-priori model information. The second strategy combines the global algorithm with a 1D edge-preserving smoothing (EPS) filter to solve the non-linear amplitude versus angle (AVA) inversion. In this case the simple L2 norm misfit between observed and predicted seismic gather is used as the objective function to minimize. The third example concerns a 2D cross-hole tomography. Due to the severe ill-conditioning and the non-linearity of this inverse problem, the 2D EPS filter is used in conjunction with model constraints in the objective function. In particular, following Zhang and Zhang (2012) the objective function is a weighted sum of L2 norm data misfit and an edge-preserving regularization that impose sparseness constraints into the first order derivatives of model parameters. Note that EPS filters are extensively applied to reduce the noise of geophysical subsurface images while preserving structural and stratigraphic discontinuities and/or edges (i.e. for sharpening seismic stack images for interpretation; AlBinHassan et al. 2006). In the context of global search methods, EPS filters drive the optimization in a suitably preconditioned model domain instead of relying completely on the random perturbation. This will decrease the ill-conditioning of the inversion because the modified model space is designed to be smaller than the complete suite of solutions. In all the following tests the firefly algorithm (FA) is used as the optimization tool. This is a quite new global search method inspired by the swarm intelligence that was proposed by Yang (2008). Over the last decade, this optimization strategy has been extensively applied in engineering applications but found very limited applications to geophysical optimization problems. In all cases the predictions yielded by the proposed strategies are compared with those provided by the more standard approach in which only the L2 norm misfit function and where no preconditioning strategies are applied in the optimization framework. In all the following tests, I focus the attention to synthetic data optimizations with the aim to maintain the discussion at a simple level and to draw general conclusions.