GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.3 603 Bibliografia Egorov, Y. V., & Shubin, M. A. (1988). Linear partial differential equations. Foundations of the classical theory. Itogi Nauki I Tekhniki. Seriya “ Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya, ” 30, 5 – 255. Fedi M. and Florio G., 2006. SCALFUN: 3D analysis of potential field scaling function to determine independently or simultaneously Structural Index and depth to source. 76 ° SEG Annual Meeting, New Orleans, 1-6 october, 2006, pp. 963-967 Fedi M., 2007, DEXP: A fast method to determine the depth and the structural index of potential fields sources: Geophysics, 72, I1 – I11. Fedi M., Florio G. e Quarta T., 2009. Multiridge Analysis of Potential Fields: Geometrical Method and Reduced Euler Deconvolution. Geophysics, 74, n. 4, L53-L65 Florio, G., & Fedi, M. (2014). Multiridge Euler deconvolution. Geophysical Prospecting, 62(2), 333 – 351. doi:10.1111/1365-2478.12078 Fedi, M., Florio, G., & Paoletti, V. (2015). MHODE: a local-homogeneity theory for improved source-parameter estimation of potential fields. Geophysical Journal International, 202(2), 887 – 900. http://doi.org/10.1093/gji/ ggv185 Hsu, S. (2002). Imaging magnetic sources using Euler’s equation. Geophysical Prospecting, 50(1), 15 – 25. Thompson, D. (1982). EULDPH: A new technique for making computer-assisted depth estimates from magnetic data. Geophysics, 47, 31. Efficient gradient computation of a misfit function for FWI using the adjoint method B. Galuzzi, E. Stucchi Department of Earth Sciences “A. Desio”, University of Milan, Italy Introduction. The objective of Full Waveform Inversion is to find an optimal Earth model that minimizes a misfit function used to quantify the difference between the predicted and the observed data (Tarantola, 1986; ������� ��� ������� ������ �� ��� ��������� ������ ���� Virieux and Operto, 2009). If the inversion starts from a model in the attraction basin of the global minimum, the use of a gradient based method can significantly improve the rate of convergence towards the optimal solution. However, the traditional gradient computation that makes use of the finite difference approximation of the partial derivatives, evaluating one or more closely spaced points for each model parameters can be computationally very expensive. The adjoint method is a mathematical tool that allows to considerably shorten the time required for computing the gradient (Plessix, 2006; �������� ������ ���� ������ �� ������������ Fichner, 2010). This method is particularly effective when the computation of the predicted data is expensive and the number of model parameters is high. In this work we illustrate the adjoint method, where the misfit function is the L 2 -norm difference between the observed and predicted data. The predicted data are computed using the numerical solution of 2D acoustic wave equation. At the beginning we analyze the numerical implementation of the 2D acoustic seismic wave equation, and explain the modelling parameters used. Then we describe the misfit function and formulate the procedure to obtain the gradient of the function using the adjoint method. Successively we illustrate some important numerical aspects of the implementation concerning storage, stability and absorbing boundaries. Finally, we compute the gradient of a misfit function, in which the number of model parameters is high and the computation of the predicted data is very expensive. The gradient is computed using both the adjoint method and the finite difference approximation of the spatial derivatives and the results are compared. The 2D Acoustic seismic wave equation and its implementation. The generation and the propagation of the seismic waves in a geological medium, is often modeled by the acoustic 3D