Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization

Numerical blending

We assume that the seismic record is blended using two independent sources, which correspond to two shooting vessels in the ocean bottom nodes (OBN) acquisition (Mitchell et al., 2010). Here, one source means a collection of shots from one shooting vessel. The two sources shoot pseudosynchronously, which means that each shot in one source has a random time dithering compared with the corresponding shot in another source. By precisely picking the shooting time of each shot of one source at the long seismic record on the node, we can get one common receiver gather. Using the second source, we get another gather. Thus, the forward problem can be formulated as follows:

$$\mathbf{d}=\mathbf{d}_1+\mathbf{Td}_2.$$ (1)

Here, $\mathbf{d}_1$ and $\mathbf{d}_2$ denote the seismic records to be separated out, $\mathbf{d}$ is the blended data, and $\mathbf{T}$ denotes the dithering operator. Note that we perform the processing in the common-receiver domain, because each blended record in this domain is coherent for one source and incoherent for another (Hampson et al., 2008). In this case, $\mathbf{d}$ is usually referred to as the pseudodeblended data (Mahdad et al., 2011), which means picking traces from the original blended field record according to the shooting time of one source to form a common receiver gather with one source coherent and the other one incoherent. We propose to augment equation 1 with another equation through applying the inverse dithering operator $\mathbf{T}^{-1}$ :

$$\mathbf{T}^{-1}\mathbf{d}=\mathbf{T}^{-1}\mathbf{d}_1+\mathbf{d}_2.$$ (2)

By combining equations 1 and 2, we formulate an augmented estimation problem:

$$\mathbf{Fm}=\mathbf{\tilde{d}},$$ (3)

where

$$\mathbf{\tilde{d}}= \left[\begin{array}{cc} \mathbf{d}\\ \mathbf... ...bf{m}= \left[\begin{array}{cc} \mathbf{d}_1\\ \mathbf{d}_2 \end{array}\right],$$ (4)

and $\mathbf{I}$ is the identity operator.

We choose to solve equation 3 rather than equation 1 because equation 3 is a more convenient form to use in the following derivations.

Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization