Definitions of operators

The HRT maps the data (t, x) into a velocity space ($\tau$,v) that clearly exhibits the moveout inherent in the data and, therefore, forms a convenient basis for velocity analysis. Thorson and Claerbout (1985) were the first to define the forward and adjoint operators of the HRT, formulating it as an inverse problem, where the velocity domain is the unknown space. In their approach, the forward operator H stretches the model space (velocity domain) into the data space (CMP gathers) using a hyperbola superposition principle, whereas the adjoint operator ${\bf H}^ \dag$, the HRT, squeezes the data summing over hyperbolas (related to the velocity stack as defined by Taner and Koehler (1969)). The forward operation is

$$d(t,x) = \sum_{s=s_{min}}^{s_{max}}w_o m(\tau=\sqrt{t^2-s^2x^2},s),$$ (1)

and the adjoint transformation becomes

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

where x is the offset, s the slowness, $\tau$ the two-times zero offset travel time, and w_o a weighting function that compensates to some extent for geometrical spreading and other effects Claerbout and Black (1997).