GNGTS 2018 - 37° Convegno Nazionale

720 GNGTS 2018 S essione 3.3 with a probability equal to the ratio between the error associated to the unfiltered model and the error given by the filtered model. To preserve the best fitting solution at each iteration (elitist strategy) the filtered version of the best current model is accepted only if it decreases the error value associated to the best model. The two-layer approach ensures that the optimization is performed without any bias towards any preferred blocky solution. A 11-sample EPS filter is applied to the predicted 1D Vp , Vs and density profiles. Obviously, the scattering of the final solution decreases as the length of the EPS filter increases. The search ranges for Vp , Vs and density are centred on the true model ± 25%. Fig. 2 compare the final result achieved by the proposed approach (Fig. 2a) and by the standard approach without the EPS regularization (Fig. 2b). It is clear that the proposed method achieves more reliable elastic parameter estimations. Note that the match between predicted and observed data is optimal in both cases: This proves the high ill-conditioning of this optimization problem. 2D cross-hole tomography. I now consider a 2D cross-hole tomography example performed on a subsurface model with horizontal and vertical dimensions equal to 140 m and parametrized in 14x14 cells. I simulate an acquisition with 25 equally spaced sources and receiver positioned at the leftmost and rightmost edges of the model, respectively. Gaussian random noise resulting in a S/N ration equal to 10 is imposed to the observed traveltimes. A numerical solution of the Eikonal equation based on the fast-marching algorithm is used as forward modelling. In this case I employ a 5-sample 2D EPS filter. To preserve sharp edges in the model solution, the matrix of the model derivatives should be sparse so that the velocity can be allowed to change abruptly on the edge. For edge preserving regularization, different definitions of the model edge may yield different edge preserving regularization methods. In this application, I borrow the concept of edge preserving smoothing operator that yields to the following objective function E ( m ): (4) where D is the 2D first order partial derivative operator, c is a positive constant that must be accurately set case-by-case, F EPS is the 2D filter operator, whereas W i is a weighting matrix; the weighting is inversely proportional to a function of the model gradient. The weighting should tend toward zero on the edge of the model and should tend to one in non-edge areas so that the smoothing is turned off on the edge but applied in non-edge areas. For accepting the filtered models, we adopt the same two-layer strategy previously discussed. The search ranges are 600 m/s wide and centred on the true model. Fig. 3 compares the results achieved by the proposed strategy and that yielded by the standard approach in which no EPS filter is introduced and where the objective function is the simple L2 norm data misfit. It is clear that the proposed method is able to attenuate the ill-conditioning of the cross-hole tomography and provides reliable predictions in which the sharp velocity contrasts are preserved. Conversely the more standard method estimates a highly scattered and unphysical final model that clearly lies in the null-space of solutions. In this case the two velocity anomalies are almost masked by the noise amplification effects produced by the ill-conditioning of the problem. Note that the optimal match between observed and predicted traveltimes achieved by both optimization approaches proves the severe ill-conditioning of the cross-hole tomography problem. Conclusions. I tested different strategies that could be used to make global optimizations applicable in high-dimensional and ill-conditioned geophysical optimization problems. The first strategy is based on a reparameterization of the subsurface model into series of Legendre polynomials. The second approach includes a 1D EPS filter into the optimization framework as model space preconditioning. The third approach uses a 2D EPS filter and an edge-preserving regularization with the aim to precondition the model space and to preserve the edge-structure