Theory and practice of interpolation in the pyramid domain

Properties

The equation of a plane-wave in $(t,x)$ is given by

$$P(t,x)=f(t-px),$$ (7)

where $f$ is a waveform and $p$ is a constant slope. The prediction operator $B$ from one trace to another at a distance $\Delta x$ in $(\omega ,x)$ for the plane-wave in equation (7) is given by Fomel (2002):

$$B(\omega,\Delta x)=e^{i\omega p \Delta x}.$$ (8)

If we now introduce the new variable $\Delta u=\omega \Delta x$ in equation (8), then we have a new prediction operator $A$ in the $(\omega ,u)$ domain:

$$A(\omega,\Delta u)=e^{ip\Delta u}.$$ (9)

From Equation (9), we notice that the dependency in $\omega$ of the prediction operator has vanished in the pyramid domain. Sun and Ronen (1996) extend this property to more than one plane-wave. We illustrate the pyramid domain in Figure 1. Figure 1a shows two plane-waves with the same wavelet (Ricker 2 with a fundamental frequency of 20Hz). Figure 1b displays the real part of the Fourier transformed data in the $(\omega ,x)$ domain and Figure 1c the real part of the $(\omega ,u)$ pyramid domain. Notice that we limited the range of frequencies for display purposes only in Figures 1b and 1c. In 2-D, the pyramid domain maps into a triangle-shaped area. In 3-D, it will map into a pyramid-shaped volume. Consequently, each trace in $(\omega ,x)$ is transformed into a radial trace in $(\omega ,u)$.

As anticipated from the definition of the prediction operator in equation (9), the information at each frequency in Figure 1c is independent of $\omega$, which means that any scheme involving pefs in the pyramid domain will require only one filter (one 1-D filter for 2-D data, and one 2-D filter for 3-D data). In addition, we observe that the slope $p$ now acts as a wavenumber on the $u$ axis. This fact implies that the low velocity event in Figure 1a will look like a high wavenumber component on the $u$-axis whereas the high velocity event will look like a low wavenumber. An added feature is that since we only have one pef for the whole domain (in theory), the filter estimation should be relatively robust to the noise present in the data (especially for random noise).

Theory and practice of interpolation in the pyramid domain