# Inversion-Based Deblending in Common Midpoint Domain Using Time Domain High-Resolution Radon
## 2 Theory
### Sparse Regularization  
We utilize sparse regularization for our inversion scheme as a way to solve for the \(\ell_{1}\) norm in the model space, as a more robust solution to the Radon transform [17] and, in turn, the deblending scheme. To solve for the sparse solution in this paper, we utilize the iteratively reweighed least-squares inversion (IRLS) method [20]. IRLS is a popular way to solve for sparse and/or robust inverse solutions due to its simplicity. Effectively, IRLS reformulates the calculation of the least absolute value solution as a repetition of least-squares iterations using weights in the data and model spaces. This approach is more efficient than directly solving for the least absolute value solution as we can utilize faster least squares solvers like the conjugate gradient to reduce computational times. Utilizing IRLS also allows for mixed norm inversions; for this paper, we use an \(\ell_{2}\)-norm in the data space and an \(\ell_{1}\)-norm in the model space. As many researchers have explored, applying a sparse constraint to the Radon transform offers numerous benefits and few drawbacks for the mapping of seismic data [17]. These improvements address issues such as missing data due to aliasing and limited aperture information in mapping and reconstruction. The advantages also extend to the use of the Radon transform in deblending. The primary benefit of enforcing a sparse constraint in the model space for our method is that, as a path integral, the erratic and random noise from the random delay times maps poorly in Radon space. This poor mapping results in low-amplitude smears in Radon space, in contrast to the high-amplitude focal points seen from coherent hyperbolic arrivals. This is complemented by the fact that in the shot frames where the data belong, the signals sum and focus correctly. Consequently, the solver will converge on the sparsest solution by downweighting smeared blended noise and upweighting coherent focal points. This is also the most deblended solution, as the data are remapped into the frames where they appear most coherent, and blending noise is fully explained by the mixing of neighbouring shots, effectively deblending the data. IRLS exists as a way to indirectly solve the sparse solution of an inversion problem by framing it as multiple iterations of a least squares problem, in this case, conjugate gradient (CG), with the outer iterations updating data and model weights to have the CG algorithm solved, driving the solution to the \(\ell_{1}\)-norm. Enforcing \(\ell_{1}\)-norm in the data space allows for reduced sensitivity to outliers which would otherwise negatively impact the inversion, often referred to as robust inversion. Enforcing \(\ell_{1}\)-norm in the model space solves for the sparsest solution, mainly the model that explains the data with the smallest amount of data points; this is referred to as the sparse solution. The utilization of IRLS allows us to mix and match which norms we wish to utilize through the calculation of the weights and, thus, are able to solve for a mixed norm solution. Byadding model and data weights, we can then dictate the algorithm to converge towards the least absolute value solution (\(\ell_{1}\)) or least squares (\(\ell_{2}\)) by reformulating the original least-squares equation through the minimization of the modified equation:  
\[ \begin{split}&\text{Minimize }||\mathbf{W_{m}m}||_{2}^{2}\\ &\text{Subject to }||\mathbf{W_{r}(b-\Gamma Lm)}||_{2}^{2}.\end{split}  \tag{7}\]  
where \(\mathbf{W_{r}}\) is the data weight operator and \(\mathbf{W_{m}}\) is the model weight operator. These weights allow the user to adjust the preference to either the residual or model, where \(\mathbf{W_{m}}\) can be changed to obtain a certain fit to either a sparse \(\ell_{1}\) or smooth \(\ell_{2}\) model, and \(\mathbf{W_{r}}\) can be changed to prefer non-spontaneous events or \"weight\" all events equally in the input data. The operators are usually diagonal matrices. For example, \(\mathbf{W_{r}}\) can be constructed as follows:  
\[