# Inversion-Based Deblending in Common Midpoint Domain Using Time Domain High-Resolution Radon
## 2 Theory  
### Blending Operator  
The blending operator allows us to transition between the blended domain, where sources are overlapped, referred to as a super shot, to the unblended domain, where sources belong to their respective shot records with firing times adjusted to zero for each shot. This can be represented as the blending operator \(\mathbf{\Gamma}\),which contains the overlap and delay time information. We can then represent our unblended data cube as \(\mathbf{D}\) and our blended data as \(\mathbf{b}\) with the following relationship [6; 13]:  
\[\mathbf{b}=\mathbf{\Gamma}\mathbf{D}. \tag{1}\]  
Applying the adjoint of the blending operator to the blended data, we obtain the pseudo-deblended data, in the unblended space, with each shot occupying its own data frame as opposed to multiple shots per data frame. This can be expressed by the following [6; 13]:  
\[\mathbf{\tilde{D}}=\mathbf{\Gamma^{H}}\mathbf{b}. \tag{2}\]  
Here, \(\mathbf{\tilde{D}}\) represents the pseudo-deblended data in standard shot frames. The blending matrix's role is to amalgamate shots with random delay times, effectively superimposing individual shot record data. This also means that pseudo-deblending acts to redistribute the data to their respective frames. This forms an overcomplete representation of the data with redundant information stored in each pseudo-blended source panel. It is important to note that the adjoint operator does not eliminate source interference from the data; it merely copies it around.  
The successful deblending of data using the random firing technique hinges on the inclusion of random delay times. These delays are critical in converting source interferences into what resembles random noise when remapping the data via the adjoint blending operator. It is essential that these delay times are sufficiently randomized to the point that no patterns emerge in the remapped data domains, as coherent patterns in the delay times reduce the effectiveness of our coherency constraint. Furthermore, the delays should be sufficiently long to ensure the separability of events, adhering to the minimum delay length determined by a function of the lowest frequency and blending fold, as detailed in [PERSON] [14]. Concurrently, they must be short enough that we gain an advantage in reduced acquisition times.  
Random delays introduced into the blending schedule, and consequently the blending operator, are crucial for simplifying deblending. When the adjoint blending operator (pseudo-deblending) is applied to the blended dataset, the random delays become pivotal. Blending can be conceptualized as an operation on standard acquisition data that introduces source interference.  
The influence of the addition on the random delay times can be seen when data are sorted into one of the other geometric domains after pseudo-deblending. After the application of the adjoint blending operator, coherent events can be seen in addition to the incoherent ones seen before the transform is applied; these coherent events are what we aim to extract, as they represent the part of the data that have their delay times properly corrected for, with the spontaneous data due to interfering shots that need to be removed. Through the use of an inversion method and a coherency constraint such as the high-resolution Radon transform, we can effectively remove the interfering signal and deblend our data.