Introduction

Multisource migration (Romero et al., 2000; Morton and Ober, 1998), lsm, and waveform inversion (Tang, 2009; Dai and Schuster, 2009; Virieux and Operto, 2009; Krebs et al., 2009) of phase-encoded supergathers were developed to significantly reduce the cost of migration and inversion. The key idea is to blend $N$ encoded shot gathers into an $N$ -shot supergather, and iteratively migrate encoded supergathers or, in the case of LSM or FWI, encoded supergather residuals. A representative formula for iteratively estimating the model parameter $s_i$ in the ith cell is given by the steepest descent formula

$$s_i^{k+1} = s_i^{k} - \alpha \frac{\partial \epsilon}{ \partial s_i},$$ (3.1)

where $s_i$ can represent either the reflectivity or the slowness in the ith cell, $\alpha$ is the step length, and $\epsilon$ is the misfit function^3.1 that is encoded after each iteration with a different encoding function. The benefit of this approach is that wave equation migration of each supergather costs about the same as the migration of a standard shot gather. If the number of iterations is fewer than $N$ , then the computational cost of phase-encoded multisource imaging can be much less than separately migrating each of the $N$ shot gathers (Schuster et al., 2011).

The problem with the above approach is that it is efficiently suited for land data where the receiver spread is fixed for each shot, but not for marine data with a receiver array that moves with each shot. As an illustration, Figure 3.1(a1) shows two shot gathers to be blended, where one shot is at the red source and the other is at the dark blue source; this 2-shot supergather will be denoted as ${\bf d}^{obs.}$ . Typical of marine surveys, the receiver array is at a different offset for either source so that only certain receivers are $\it selectively$ listening for the red shot but not for the dark blue shot at the uncommon receiver positions. In comparison, the predicted 2-shot supergather ${\bf d}^{pred.}$ generated by a finite-difference^3.2solution of the wave equation does not $\it discriminate$ and generates traces at every receiver, as shown in Figure 3.1(a2). Hence, there will be discrepancies between the predicted and observed traces at the uncommon receiver positions (indicated by the dashed ovals in Figure 3.1). I denote this problem in multisource FWI as the aperture mismatch problem, where the observed supergather is for a blended marine survey while the predicted supergather is for a blended land survey.

The aperture mismatch will lead to a non-zero misfit function $\epsilon=\frac{1}{2} \vert\vert{\bf d}^{pred.}-{\bf d}^{obs.}\vert\vert^2$ even if the exact velocity model is used for prediction. The remedy to this mismatch is to use an encoding function in the multisource finite-difference modeling that only activates specified receivers for any one shot. This orthogonal encoding function strategy was developed by Huang and Schuster (2012) for wave equation migration, and will now be tested for fwi.

The first part of the chapter provides the theory for multisource FWI with frequency selection, and is followed by results from tests on synthetic and field data. Speedups ranging from $4\times$ to $8\times$ compared to conventional FWI are obtained. The last part presents a summary.