Three-dimensional quantitative seismic imaging methods such as full waveform inversion, still suffer from their prohibitive computational cost. The computational burden mainly results from the computational cost of the forward problem, namely, the seismic full wavefield modeling, that must be performed for a huge number of sources at each iteration of the inversion. Building super-shots by source assembling allows to mitigate the number of simulations per iteration but adds artifacts in the imaging because of interferences between individual shots of the super-shot. These artifacts can be mitigated by using phase encoding technique which consists of applying a deterministic or random phase shift to each individual source at each iteration of the inversion. We have implemented source assembling with phase encoding in acoustic frequency-domain full- waveform inversion and have assessed the performances of the method with a 2D and 3D synthetic case study corresponding to the SEG/EAGE overthrust model.


Because 3D prestack depth imaging methods such as prestack depth migration and full waveform inversion (FWI) are computationally expensive, the simultaneous-shot technique (Capdeville et al., 2005) can provide an interesting compromise between computational efficiency and imaging accuracy. Taking advantage of the linear relationship between the seismic wave-field and the source, the simultaneous-shot technique consists of assembling individual sources to mitigate the number of seismic wave modelings performed during the imaging. The computation cost of one migration or FWI iteration is reduced proportionally to the number of shots stacked in each shot assemblage or super-shot. However, the imaging is altered by artifacts associated with the interference between the individual shots of a given super-shot, that may require additional iterations of the inversion before convergence towards an acceptable model. These artifacts can be reduced by encoding specific phase for each shot in the super-shot. This technique, called phase encoding (PE) (Morton and Ober, 1998; Jing et al., 2000; Romero et al., 2000), was originally proposed for prestack migration. Since the imaging kernels of prestack migration and frequency-domain FWI are basically the same, the technique can be used without any particular modification for FWI. We have implemented the simultaneous-shot technique with phase encoding in frequency-domain FWI. In a previous study, we assessed the performances of the method when only a limited number of shots are assembled with random PE within several super-shots (Ben-Hadj-Ali et al., 2009). In this study, we tested the method when all the sources are assem- bled to form one single super-shot. We applied the method to a realistic synthetic case study using the SEG/EAGE overthrust model and assessed the number of iterations required to obtain an acceptable velocity model.


FWI is generally recast as an iterative local optimization based on the minimization of the least-squares objective function (Pratt et al., 1998) given by,

where m is the model, do the recorded data and dkc (m) the modeled (computed) data at iteration k. The model perturbation, based upon the Born approximation and the steepest descent method, is given by

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