Iterative deblending with multiple constraints based on shaping regularization

Deblending via shaping regularization

The blending process can be summarized as the following equation:

$$\mathbf{d}=\Gamma\mathbf{m},$$ (1)

where $\mathbf{d}$ is the blended data, $\Gamma$ is the blending operator, and $\mathbf{m}$ is the unblended data. The formulation of $\Gamma$ has been introduced in Mahdad (2012) in detail. When considered in time domain, $\Gamma$ corresponds to blending different shot records onto one receiver record according to the shot schedules of different shots. Deblending amounts to inverting equation 1 and recovering $\mathbf{m}$ from $\mathbf{d}$ .

Because of the ill-posed property of this problem, all inversion methods require some constraints.

Chen et al. (2014a) proposed a general iterative deblending framework via shaping regularization Fomel (2007,2008). The iterative deblending is expressed as:

$$\mathbf{m}_{n+1} = \mathbf{S}[\mathbf{m}_n+\mathbf{B}[\mathbf{d}-\Gamma\mathbf{m}_n]],$$ (2)

where $\mathbf{S}$ is the shaping operator, which provides some constraints on the model, and $\mathbf{B}$ is the backward operator, which approximates the inverse of $\Gamma$ . The shaping regularization framework offers us much freedom in constraining an under-determined problem by allowing different types of constraints. In this paper, the backward operator is simply chosen as $\lambda\Gamma^*$ , where $\lambda$ is a scale coefficient closely related with the blending fold, and $\Gamma^*$ stands for the adjoint operator of $\Gamma$ (or the pseudo-deblending operator). For example, $\lambda$ can be optimally chosen as $1/2$ in a two-source dithering configuration Chen et al. (2014a); Mahdad (2012). In the next two sections, I will first introduce the conventional way for choosing the $\mathbf{S}$ and then propose a novel way for designing the $\mathbf{S}$ .

Iterative deblending with multiple constraints based on shaping regularization