where is the blended data, is the blending operator, and is the unblended data. The formulation of has been introduced in Mahdad (2012) in detail. When considered in time domain, the corresponds to blending different shot records onto one receiver record (node) according to the shot schedules of different shots. The Born modeling from seismic reflectivity to the primary reflections can be expressed as:

where denotes the subsurface reflectivity model and denotes the Born modeling operator. One way to remove the effects caused by the blending operator is first solving equation 1 and then solving equation 2, which is referred to as deblending. The general deblending framework can be summarized as (Chen et al., 2014a,2015a):

where is called the shaping operator, which is used to constrain the current model, and is the step size of the updated misfit. denotes the adjoint of . denotes the deblended data after th iteration.

Another way for dealing with the simultaneous-source data is to solve the following equation for directly, which is known as direct imaging of blended data,

where .

Equation 4 can be best solved using a least-squares (LS) based migration approach. More robust LS solvers involve adding constraints of structural coherency when inverting , either in a preconditioned LS formulation (Dai and Schuster, 2011; Chen et al., 2015c) or in a shaping-regularized LS iterative framework (Xue et al., 2014; Fomel, 2007b).

Because of the great success of deblending reported in the literature (Chen, 2015; Zu et al., 2015; Li et al., 2013; Beasley et al., 2012; Mahdad et al., 2011; Abma et al., 2010; Gan et al., 2015b) in the recent years, more and more focus is currently moving towards the direct imaging of blended data, which can be more efficient and can illuminate the surface better (Berkhout et al., 2012; Verschuur and Berkhout, 2011). It is worth mentioning that the deblending step for the massive blended data requires large computational resources (mainly for the parallel processing of a huge number of common receiver gathers) and a long processing period because of the thousands of iterations used for each common receiver gather. If the direct imaging can obtain a good result, we can obtain a big saving in both computational resources and processing period. However, a key aspect for the success of direct imaging is the macro velocity model of subsurface. Either tomography based velocity analysis or Born-approximation wave-equation based velocity inversion, requires an initial acceptable velocity model from the very noisy blended data (Figure 9a shows an example). In the next section, we will introduce a way for obtaining high-resolution and high-fidelity velocity spectrum from blended data, using the recently developed similarity-weighted semblance.

2020-04-11