عنوان مقاله [English]
We derive the discretized adjoint states full waveform inversion (FWI) in both time and frequency domains based on the Lagrange multiplier method. To achieve this, we applied adjoint state inversion on the discretized wave equation in both time domain and frequency domain. Also, in this article, we introduce reliability tests to show that inversion is performing as it should be expected. Reliability tests comprise of objective function descent test and Jacobian test. Influence of data imperfections is also being studied. We define data imperfection as any factor which causes deterioration in FWI results. Some of these factors are coherent and incoherent noises in data, source wavelet inaccuracy in phase and amplitude, and the existence of gaps in the seismic survey. We compare time and frequency domain inversion methods sensitivity to data imperfection. In all cases, we find that time domain full waveform inversion is more sensitive to imperfections in the data. In general, we find that time domain FWI result shows more deterioration than frequency domain FWI. All tests have been done using 2D full waveform inversion codes. We employ the multi-scale inversion and finite difference scheme (FDM) for discretization and the misfit function is minimized via limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method.
Keywords: FWI, Sensitivity Analysis, Discretized Adjoint State Method
The full waveform is a powerful method to obtain high-resolution seismic velocity model using the whole wavefield. Foundation of the FWI theory is a study published by Tarantola (1984), who demonstrated minimization of the misfit between recorded and modelled data might be computed without calculating the partial derivatives explicitly. In other words, Tarantola (1984) proposed applying an adjoint state method in the seismic inversion context. The adjoint state method is based on control theory that was implemented in geophysical inverse theory by Chavent (1974).
FWI was originally developed in the time domain (Tarantola 1984, Tarantola 1986, Mora 1987, Tarantola 1988), then Pratt and Worthington (1990) and Pratt (1990) introduced the frequency domain FWI (FDFWI) approach and successfully applied to 2D cross-hole data.
To mitigate the nonlinearity of FWI and negate the cycle-skipping effect, various multiscale strategies have been proposed (e.g. Bunks et al. 1995; Sirgue and Pratt 2004 and Boonyasiriwat et al. 2009). In the time domain, the multi-scale FWI method successively inverts low-passed seismic sub-bands of increasing high-frequency content, hence increasing the chances of convergence to the global minimum (Bunks et al. 1995). Furthermore, by applying Fourier transform on the dataset, the frequency content of the waveform is discretized and the multi-scale method can be easily employed in FDFWI. The inversion process is done incrementally adding from low to high-frequency components (Sirgue and Pratt 2004).