Radial smoothing

We have to choose $\bold S$.Inserting a smoother signifies the assertion that the dips in seismic data should change in a gradual way. Choosing an isotropic smoother means we expect the dips to vary similarly in all directions. However, we know that the dip spectrum of the data probably changes more quickly in some directions than in others. We want to smooth most heavily along directions where the dip is nearly constant.

In a constant-velocity, flat-layered earth, events fall along hyperbolas like

$$t^2 = \tau^2 + \frac{x^2}{v^2},$$

where x is offset, v is stacking velocity, $\tau$ is zero-offset time. The time dip of an event is dt/dx. If velocity is constant, differentiating gives

$$\frac{dt}{dx} = \frac{x}{v^2 t},$$

which means that the dip does not change along radial lines, where x/t is constant. In a real earth, we suppose that dips will change, but slowly. Real earth velocity may change quickly in depth, but hyperbola trajectories are functions of RMS velocity, which is smooth.

We want a smoother with an impulse response which is highly elongated in the radial direction. To get a big impulse response cheaply, I apply the inverse of a directional derivative, pointed in the radial direction. To directly apply the inverse, the roughener has to be causal, which means that the inverse will only smooth in one direction. We want $\bold S$ to have an impulse response which is smoothed both in towards zero radius and out towards large radius, so we make it the cascade of the causal smoother and its anticausal adjoint.