comparing IRLS and Huber on a Geophysical problem

We now compare IRLS and the Huber solver doing velocity-stack inversion. This method was first introduced by Thorson and Claerbout (1985) and completed with an IRLS algorithm by Nichols (1994). Guitton and Symes (1999) introduced the Huber misfit function with a specific non-linear solver as an alternative to IRLS. The velocity estimation is made using inverse methods to minimize the function

$$f(\bold{m})= E(\bold{Hm}-\bold{d}),$$

where E is an error measure function, H is the modeling operator that transforms the model space $\bold{m}$ (velocity domain) into the data space $\bold{d}$ (CMP gathers). The adjoint operator (H^) is the well-known hyperbolic Radon transform operator. The forward operation is

$$d(t,x) = \sum_{s=s_{min}}^{s_{max}}w_o m(\tau=\sqrt{t^2-s^2x^2},s),$$ (7)

and the adjoint transformation becomes

$$m(\tau,s) = \sum_{x=x_{min}}^{x_{max}}w_o d(t=\sqrt{\tau^2+s^2x^2},x),$$ (8)

where w_o is a weighting function Claerbout and Black (1997). We now solve the inverse problem using both the Huber solver and IRLS on synthetic and real data. Note that in the following sections, I assume that when $\epsilon=max\vert\bold{d}\vert/100$, we nearly solve the l¹ problem for both the Huber solver and IRLS.