Bayesian wavefield separation by transform-domain sparsity promotion

The forward model

The primary-multiple separation problem is cast into a probabilistic framework where the recorded total data vector $\mathbf{b}$ ,

$$\mathbf{b} = \mathbf{s}_1 + \mathbf{s}_2 +\mathbf{n},$$ (1)

is assumed to consist of a superposition of primaries, $\mathbf{s}_1$ , multiples, $\mathbf{s}_2$ , and white Gaussian noise $\mathbf{n}$ , each component of which is $N(0,\,\sigma^2)$ --i.e., zero-mean Gaussian with standard deviation $\sigma$ . We denote by $\mathbf{b}_2$ the vector that consists of the SRME-predicted multiples. As the SRME predictions contain errors, we assume

$$\mathbf{b}_2 = \mathbf{s}_2 + \mathbf{n}_2,$$ (2)

where $\mathbf{n}_2$ , the error in the predicted multiples is also assumed white Gaussian; each component of $\mathbf{n}_2$ is $N(0,\sigma_2^2)$ . Moreover we assume that $\mathbf{n}$ and $\mathbf{n}_2$ are independent.

Following earlier work (Herrmann et al., 2007a), we write the two unknown signal components as a superposition of curvelets--i.e., $\mathbf{s_i}= \tensor{A}\mathbf{x}_i, \ i = 1, 2$ , where $\tensor{A}$ is the curvelet-synthesis matrix (Candes et al., 2006), and obtain the system of equations

$$\begin{array}[l]{ccccc} \mathbf{b}_1&=& \tensor{A}\mathbf{x}_... ...thbf{b}_2&=& \tensor{A}\mathbf{x}_2 + \mathbf{n}_2. \end{array}$$ (3)

Here, $\mathbf{n}_1=\mathbf{n}-\mathbf{n}_2$ . Moreover, the unknown curvelet coefficients for the primaries, $\mathbf{x}_1$ , and multiples, $\mathbf{x}_2$ , are related to the SRME-predicted primaries, $\mathbf{b}_1=\mathbf{b}-\mathbf{b}_2$ , and SRME-predicted multiples, $\mathbf{b}_2$ . With this formulation, we are in a position to exploit the sparsity of the curvelet coefficient vectors $\mathbf{x}_1$ and $\mathbf{x}_2$ for separating the two signal components.

Remark: Note that our assumption that $\mathbf{n}$ in Equation 1 is white Gaussian noise is consistent with the assumptions underlying the matched filter used in the SRME-multiple prediction, see (Verschuur et al., 1992). Furthermore, our formulation can be extended to the case where both noise contributions $\mathbf{n}$ and $\mathbf{n}_2$ are colored Gaussian as long as the corresponding covariance matrices are known. In the absence of an accurate probabilistic model for the prior distribution of the error in the predicted multiples, we make the simplifying assumption that this error is Gaussian. As we will show in the next section, this assumption leads to an optimization problem with a $\ell_2$ -norm penalty for the error. In this way, we merely assign a cost function that penalizes the $\ell_2$ misfit between the predicted multiples and the estimates produced by our algorithm. In addition, the fidelity of the predicted multiples can be incorporated by choosing the parameters in this optimization procedure appropriately.

Bayesian wavefield separation by transform-domain sparsity promotion