Implementation of the linear radon transform

In this section, we show the details of the linear radon transform and how it can be cast as an inverse problem. The forward transformation maps the radon domain $m(\tau,s)$ into the data space d(t,x) (recorded data) as follows:  

$$d(t,x) = \sum_{s=s_{min}}^{s_{max}} m(\tau=t-sx,s),$$ (2)

and the adjoint transformation

$$m(\tau,s) = \sum_{x=x_{min}}^{x_{max}} d(t=\tau+sx,x),$$ (3)

where t is the time, x the station location (x_min and x_max being the offset range), s the slowness (s_min and s_max being the range of slownesses investigated), and $\tau$ the travel time at x_min (the first trace is the origin of the summation path).

Equation (2) can be rewritten in a more compact way by introducing the forward linear radon transform operator L, the model space vector m (which contains all the $m(\tau,s)$ points) and the data vector d (which contains all the d(t,x) points):  

$${\bf d =Lm}.$$ (4)

Therefore, the goal is to minimize the difference between the input data d and the modeled data via the linear radon transform operator as follows:  

$$\bf{0=r_d=Lm-d},$$ (5)

where ${\bf r_d}$ is called the data residual. As explained before, the data are irregularly spaced and traces may be missing. A mask ${\bf M}$ is introduced in equation (5) such that only the recorded data are considered in the residual:  

$$\bf{0=r_d=M(Lm-d)},$$ (6)

where ${\bf M}$ is a diagonal operator that equals one where data are known and zero where they are unknown (at the missing traces). Finally, we estimate the radon domain by minimizing the objective function  

$$f({\bf m})=\Vert{\bf r_d}\Vert^2,$$ (7)

which gives a least-squares estimate of the model parameters. Note that with the linear radon transform, the model space ${\bf m}$ can be estimated without inversion by introducing the so-called rho filter Yilmaz et al. (1987), usually estimated in the Fourier domain. With missing traces, the rho filter is not appropriate anymore and inversion is required. In the next section, we describe how a sparse radon domain can be estimated with inversion.