Full Waveform Inversion with Optimal Basis Functions

Abstract

Based on the approach suggested by Tarantola, and Gauthier et al., we show that the alternate use of the step (linear) function basis and the block function (quasi-function) basis can give accurate full waveform inversion results for the layered acoustic systems, starting from a uniform background. Our method is robust against additive white noise (up to 20% of the signal) and can resolve layers that are comparable to or smaller than a wavelength in thickness. The physical reason for the success of our approach is illustrated through a simple example. DOI: 10.1103/PhysRevLett.90.104301 PACS numbers: 43.90.+v, 91.30.-f Wave inversion means the recovery of the coefficients/ parameters of the wave equation from its solution(s). It is one of the most important problems in physical sciences. However, except for some cases of linear inversions (e.g., in 1D inversion) [1][2][3][4], most nonlinear inversions still present considerable difficulties. In seismic inversions, involving the imaging of Earth's deep subsurface structures, the traveltime inversion is the most tractable [5]. A more ambitious inversion approach is that of full waveform inversion, in which a common strategy is to retrieve the model parameters by minimizing a misfit function. While simple in concept, the success of full waveform inversion has been rather limited because of the extensive computational requirement and difficulty in realizing target convergence. Some time ago, Tarantola [6] proposed a scheme of full waveform inversion for acoustic systems, which was subsequently implemented by Gauthier et al. [7]. If only reflection data were used, the method can map the high spatial frequency components of the model, e.g., the interfaces, but totally fails in recovering the low spatial frequency information, e.g., the layer velocities. This so-called low frequency lacuna problem is rather well known in other seismic imaging approaches as well. Later literature on full wave inversion also met similar convergence problems [8][9][10][11][12]. The use of global minimizing processes [13][14][15][16] has shown much better convergence characteristics, but at the expense of computational efficiency. Symes [17] proposed a smooth and convex misfit function by using model parameters which are linear in the inversion calculation. This method has some successful applications [18,19]. Another proposal is to improve the initial model by using the hybrid inversion method, which minimizes a weighted combination of first-arrival traveltime and seismogram misfit functions [20,21].In this Letter, we consider the full waveform inversion for a 2D layered acoustic system, with point sources and receivers, perhaps the simplest test case for nonlinear inversion and a first-order approximation to the structure of Earth's subsurface. By using the optimal basis functions alternately in the inversion process, we not only overcame the low frequency lacuna problem encountered before, but also obtained robust and accurate results. The physical reason for this success is elucidat...