Introduction

The Radial Trace Transform (RTT), is a simple coordinate transform of normal (t,x) domain seismic gathers; a horizontal deformation, accomplished by the following linear mapping

$$\begin{eqnarray} t &\rightarrow& t \\ x &\rightarrow& v = \frac{x}{t} \nonumber\end{eqnarray}$$ (1)

The radial coordinate is termed ``v'' because the RTT sorts the data by apparent velocity. Neglecting dispersion effects, ground roll maps to zero temporal frequency in the RT domain. The RTT is not a new concept. Nearly twenty years ago, this coordinate transform was an active subject of research for use in multiple suppression Taner (1980), migration Ottolini (1982), and even for the subject of this paper, ground roll removal Claerbout (1983). Henley's 1999 recent paper reminded the world of the RTT's usefulness in attacking ground roll. Sava and Fomel (2000) use the RTT to compute angle gathers, exploiting the fact that slant-stack in the (t,x) domain is equivalent to computing the RTT in the Fourier domain.

Figure 1 shows 30 radial traces (thick lines) overlain on a rectangular (t,x) mesh. Points along radial traces rarely fall exactly on (t,x) data locations, making the RTT an exercise in interpolation. In matrix notation, let us denote the RTT of data $\bf d$ as follows:  

$$\bf Rd = r$$ (2)

$\bf R$ is generally non-square, and even if square, is usually noninvertible. Often, however, such interpolation operators are nearly unitary: $\bold R^T \bf R \approx I$. We define the interpolation error as follows  

$$\bf e = \left[\bold I - {\bf R}^T \bf R \right] d.$$ (3)

 

figure1 Figure 1 30 radial traces overlaying a (t,x) grid.