On the separation of simultaneous-source data by inversion

Method

We can represent the simultaneous-source acquisition process for $n$ sources as follows:

$$\sum^{n}_{i=1} {\bf S} {\bf d}_{i} ={\bf d},$$ (1)

where ${\bf S}$ is a shifting operator built from the relative time-delays between sources, ${\bf d}_{i}$ is the data due to source $i$ , and d is the simultaneous-source data. We can rewrite this equation in the form

$$\top{\bf H}_{i} {\bf m}\approx {\bf d},$$ (2)

where $\top$ is the summation operator, and ${\bf H}_{i}$ is an operator that models data ${\bf d}_{i}$ from model ${\bf m}$ . Note here, that all component shots of the encoded data ${\bf d}$ are modeled from a single, consistent model ${\bf m}$ . In this paper, ${\bf H}_{i}$ is a modified hyperbolic Radon operator that maps data from the the velocity-CMP space to shot-offset space, honoring the time delays at source ${i}$ relative to a reference shot. Adding a regularization operator A, we have

$$\begin{array}{cc} {\bf\top H}_{i} {\bf m} \approx {\bf d}, \\ {\bf\epsilon Am} \approx{\bf0}, \end{array}$$ (3)

where ${\bf\epsilon}$ , regularization parameter determines the regularization strength.

There are many possible choices for the regularization operator ${\bf A}$ . Taking ${\bf A}$ to be an identity matrix and minimizing the regressions in equation 3 with a hybrid solver leads to a sparse Radon inversion problem. Alternatively, we can regularize the problem with a shot-space operator ${\bf B}_{i}$ by re-writing equation 3 as follows:

$$\begin{array}{cc} {\bf\top H}_{i} {\bf m} \approx{\bf d}, \\ {\bf\epsilon B}_{i}{\bf H}_{i}{\bf m} \approx{\bf0}, \end{array}$$ (4)

which in matrix form can written as

$$\left[\begin{array}{c}{\bf\top}\\ {\bf\epsilon B}_{i}\end{array}\... ...}_{i} {\bf m} \approx \left[\begin{array}{c}{\bf d}\\ {\bf0}\end{array}\right].$$ (5)

In this paper, we define ${\bf B}$ as a system of non-stationary dip-filters. First, we compute local event dips using the plane-wave destruction method (Fomel, 2002), then we compute dip-filters using factorized directional Laplacians (Hale, 2007). Because of the random delays between simultaneous sources, for any given source, events from other sources are random in its corresponding common-offset gathers. By destroying predictable events corresponding to source $i$ , operator ${\bf B}_{i}$ ensures that only these events are allowed in the final separated data sets, whereas unpredictable events are not. Events that are not predictable by ${\bf B}_{i}$ are passed on to other sources, where they are predictable by the corresponding operator ${\bf B}_{j}$ . We call this inversion method dip-constrained sparse inversion (DCSI). In this paper, we refer to solution of equation 5, with ${\bf B}_{i}$ as an identity matrix, as unconstrained sparse inversion.

However, because the operator ${\bf B}_{i}$ is a function of the separated data, the problem becomes non-linear. To linearize this problem, we start by solving the equation 3 to get an initial estimate for ${\bf d}_{i}$ . Then, using ${\bf d}_{i}$ , we obtain an estimate of the operator ${\bf B}_{i}$ , which is used to regularize the problem starting from initial model estimate ${\bf m}$ . Results from this new step can then serve as inputs into the next inversion step. This process can be repeated as as many times as necessary.

Following the approach of Abma et al. (2010), instead of using ${\bf B}_{i}$ as a regularization operator, we can use ${\bf B}^{-1}_{i}$ as a smoothing operator by re-writing equation 4 as follows:

$$\begin{array}{cc} {\bf\top }{\bf B}^{-1}_{i} {\bf H}_{i} {\bf... ... d}, \\ {\bf\epsilon }{\bf I}{\bf m} \approx{\bf0}. \end{array}$$ (6)

In this paper, we implement ${\bf B}^{-1}_{i}$ as polynomial division (Claerbout and Fomel, 2008) with non-stationary directional Laplacians.

Equation 5 can be directly extended to multiple surveys. For example, for two surveys, we can minimize the regressions

$$\begin{array}{cc} \left[\begin{array}{c}{\top}_{1}\\ {\bf\eps... ...{1,2}{\bf m}_{2} \end{array}\right] \approx {\bf0}, \end{array}$$ (7)

where for survey $k$ , ${\bf H}_{ik}$ and ${\bf B}_{ik}$ are the modeling and shot-space regularization operators, respectively, for source $i$ ; ${\bf m}_{k}$ and ${\bf d}_{k}$ are the Radon model and simultaneous-source data; ${\bf Z}_{k}$ is a temporal regularization operator; and ${\bf S}_{k,k+1}$ is a shifting operator that aligns the models ${\bf m}_{k}$ and ${\bf m}_{k+1}$ . Note that ${\bf H}_{ik}$ incorporates both geometry and relative shot timing for survey $k$ . Because of differences in geometry and relative shot timing between surveys, operator ${\bf H}_{i1}$ is different from ${\bf H}_{i2}$ . The last regression in equation 7 minimizes the difference between models ${\bf m}_{1}$ and ${\bf m}_{2}$ . Because we are interested only in production-related differences between ${\bf m}_{1}$ and ${\bf m}_{2}$ , the difference between the two models is also very sparse. We can generalize equation 7 to an arbitrary number of surveys as follows:

$$\begin{array}{cc} \left[\begin{array}{c}{\bf\top}_{k}\\ {\bf\... ...+1}{\bf m}_{k+1} \end{array}\right] \approx {\bf0}. \end{array}$$ (8)

In this paper, we refer to the method of solving the joint-inversion problem represented by equation 8 as spatio-temporal constrained sparse inversion (STCSI).

On the separation of simultaneous-source data by inversion