Estimating sparse radon domains

Given a CMP gather ${\bf d}$ and a radon transform operator ${\bf L}$, we want to minimize the least-squares objective function

$$f({\bf m}) = \Vert{\bf Lm - d}\Vert^2,$$ (3)

where ${\bf m}$ is the unknown radon domain. The main idea of this paper is to decompose ${\bf m}$ into its positive and negative parts by imposing simple bounds on ${\bf m}$ with the L-BFGS-B algorithm. Therefore, the two problems  

$$\mbox{ min } f({\bf m_{(-)}})\mbox{ subject to }{\bf m_{(-)}}\in\;\;]-\infty,0[ \;\;,$$ (4)

and  

$$\mbox{ min } f({\bf m_{(+)}})\mbox{ subject to }{\bf m_{(+)}}\in\;\;[0,+\infty[\;\;,$$ (5)

need to be solved. Note that we could decompose ${\bf m}$ into more subdomains as well. It is important to solve both problems of finding ${\bf m_{(-)}}$ and ${\bf m_{(+)}}$ independently, and not simultaneously as it can be done with linear programming techniques Claerbout and Muir (1973). Here, the main idea is to decrease the null space and its effects by constraining the model, similar to what is accomplished with the Cauchy regularization. Once the two models ${\bf m_{(+)}}$ and ${\bf m_{(-)}}$ are estimated with the L-BFGS-B algorithm, the sparse model is obtained by computing ${\bf m_{sparse}}={\bf m_{(-)}+m_{(+)}}$. In the following section, I illustrate this technique with a synthetic and real data example using the hyperbolic radon transform.