Least-squares migration/inversion of blended data

Problems with direct migration

Within limits of the Born approximation of the acoustic wave equation, the seismic data can be modeled with a linear operator as follows:

$${\bf d} = {\bf L}{\bf m},$$ (1)

where ${\bf d}$ is the modeled data, ${\bf L}$ is the forward Born modeling operator, and ${\bf m}$ denotes the reflectivity, a perturbed quantity from the background velocity. Equation 1 models the data for the conventional acquisition geometry, i.e., without interference between different shots. For the blended acquisition geometry, however, two or more shot records are often blended together, creating one or more super-areal shot record(s). This blending process can be described by a linear transform as follows:

$$\widetilde{\bf d} = {\bf B} {\bf d},$$ (2)

where ${\bf B}$ is the so-called blending (Berkhout, 2008) or encoding (Tang, 2008a; Romero et al., 2000) operator, and $\widetilde{\bf d}$ is the set of super-areal shot records after blending. Substituting equation 1 into equation 2 leads to the modeling equation for the blended acquisition geometry:

$$\widetilde{\bf d} = {\bf BLm} = \widetilde{\bf L}{\bf m},$$ (3)

where $\widetilde{\bf L} = {\bf B L}$ is defined as the combined Born modeling operator.

There are many choices of the blending operator; which one produces the optimal imaging result might be case-dependent and is beyond the scope of this paper. In this paper, we mainly consider two different blending operators: a linear-time-delay blending operator and a random-time-delay blending operator. The first operator seems to be common and easy to implement in practice for acquiring marine data, while the second one is interesting and has been partially adopted in acquiring both land and marine data with simultaneous shooting (Hampson et al., 2008). For example, Figure 1 shows a scattering reflectivity model with a constant velocity of $2000$ m/s. Figure 2 and Figure 3 show the snapshots of the corresponding blended source wavefields. For both cases, $41$ point sources with an equal spacing of $100$ m are blended into one composite source. Figure 4 shows the modeled blended data. Given the complexity of the super-areal shot gathers shown in Figure 4, it might be very difficult or even impossible to deblend them.

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Figure 1. A reflectivity model containing many point scatterers. [ER]

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Figure 2. Source wavefield after linear-time-delay blending. [CR]

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Figure 3. Source wavefield after random-time-delay blending. [CR]

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Figure 4. Modeled blended shot gather. (a) Linear-time delay blending and (b) random-time-delay blending. [CR]

We can directly use the adjoint of the combined modeling operator, which is also widely known as the migration operator, to reconstruct the reflectivity as follows:

$$\widetilde {\bf m}_{\rm mig} = \widetilde{\bf L}' \widetilde{\bf d}_{\rm obs} = {\bf L}'{\bf B}'{\bf B} {\bf d}_{\rm obs},$$ (4)

where the superscript $^\prime$ denotes the conjugate transpose and the subscript $_{\rm obs}$ denotes observed data. Contrary to the imaging formula in conventional acquisition geometry, now we have an extra ${\bf B}'{\bf B}$ in our imaging formula, which has a direct impact on the imaging quality of blended data. If ${\bf B}'{\bf B}$ is close to unitary, i.e., ${\bf B}'{\bf B}\approx {\bf I}$ with ${\bf I}$ being the identity matrix, then direct migration of blended data would produce exactly the same results as migration of conventional data, and the blending process would produce little impact on the final image we obtain. However, in reality, ${\bf B}'{\bf B}$ is often far from unitary, because ${\bf B}$ is usually a short matrix (its number of rows is much smaller than its number of columns); thus its normal operator, ${\bf B}'{\bf B}$, is rank deficient. In other words, there are many non-negligible off-diagonal elements in ${\bf B}'{\bf B}$. As a consequence, direct migration using equation 4 would produce crosstalk artifacts. An example is demonstrated in Figure 5 and Figure 6, which illustrate the migrated images for the blended data shown in Figure 4; the images are severely degraded by the crosstalk artifacts. For comparison, Figure 7 shows the crosstalk-free image by migrating the data synthsized with the conventional acquisition geometry (when ${\bf B}={\bf I}$).

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Figure 5. Migration of linear-time-delay blended data. [CR]

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Figure 6. Migration of random-time-delay blended data. [CR]

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Figure 7. Migration of the data acquired with the conventional acquisition geometry. [CR]

Least-squares migration/inversion of blended data