Seismic full-waveform inversion using a compressed simultaneous sources and receivers scheme
Authors: Abubakar Aria , Habashy Tarek M. , Pan Guangdong 
Published: May 2016

Abstract
We apply the source-receiver compression approach for reducing the computational time and memory usage of the seismic (acoustic) full-waveform inversions. By detecting and quantifying the extent of redundancy in the data, we assemble a reduced set of simultaneous sources and receivers that are weighted sums of the physical sources and receivers employed in the survey. Because the number of these simultaneous sources and receivers can be significantly less than those of the physical sources and receivers, the computational time and memory usage of an inversion method can be tremendously reduced. The scheme is based on decomposing the data into their principal components using a singular-value decomposition approach and the data reduction is done through the elimination of the small eigenvalues. Consequently this will suppress the effect of noise in the data. Moreover, taking advantage of the redundancy in the data, this compression scheme effectively stacks the redundant data resulting in an increased signal-to-noise ratio. We present inversion results for the Marmousi model for surface measurements. We show that this approach has the potential of significantly reducing both computational time and memory usage of the Gauss-Newton method by 1–2 orders of magnitude.

Introduction
In the last decades, developments in the numerical computation techniques have motivated research on the full-waveform inversion (FWI) approaches for geophysical applications (Pratt et al. (1998); Shin et al. (2001); Abubakar et al. (2003); Operto et al. (2007); Vigh and Starr (2008); Hu et al. (2009b); Abubakar et al. (2009)). In FWI seismic inversion, one usually deals with a large size data set. One of the major bottlenecks for inversion is dealing with such a large data set, particularly if a significant part of this data set is lacking sensitivity (redundant) to the unknown model parameters. Such redundancy is usually the result of collecting data employing a non-optimally designed survey. A large number of sources in the survey contributes to a large computational cost in running the forward simulator a number of times corresponding to the number of these sources. On the other hand, a large number of receivers contributes to the computational cost of constructing the sensitivity (Jacobian) matrix as well as in inverting the Hessian matrix in a gradient-type inversion approach.
To deal with these issues, the simultaneous-source encoded FWI approach was proposed to reduce the number of sources used in the inversion, see Krebs et al. (2009); Herrmann et al. (2009), and Ali et al. (2011). In this approach, a large number of physical sources are converted into one simultaneous source or several simultaneous sources by summing the individual physical sources using a phase-encoding technique (Morton and Ober, 1998). It was pointed out that this approach is more sensitive to noise than the standard FWI method. Krebs et al. (2009) attempted to reduce this noise problem by randomly varying the encoding scheme in each of the inversion iterations. They obtained a fifty-fold improvement in CPU-time, which corresponds to the total number of sources used for two-dimensional (2D) Marmousi case study. Herrmann et al. (2009) recast the simultaneous source technique in the framework of the compressive sensing and discuss its consequences for FWI. Ali et al. (2011) studied the design of the simultaneous source assembling (either all source shots combined into one simultaneous source or several simultaneous sources) as wells as the effects of the crosstalk noise resulting from the use of the phase-encoding technique.