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FIRST PRINCIPLES

Let us review Ronen's pyramid transform (Hung et al., 2005). Fourier transform data from $ (t,x)$ to $ (\omega,x)$. Ronen then shrinks and stretches the space axis to transform from $ x$ to $ u=\omega x$. All the information on a trace (constant $ x_0$), in the $ (t,x)$ plane, lands on the line $ u=\omega x_0$ in the the $ (\omega ,u=\omega x)$-plane. This is a line through the origin. Ordinary traces have become radial traces.

Look ahead to Figure 1 and you will see an example of the basic property of Ronen's space, the property we will prove next, namely: Whereas dips in $ (t,x)$ space are curves in $ (\omega,x)$ space, they return to straight lines in $ (\omega, u)$ space. Dips in physical $ (t,x)$ space become frequencies on Ronen's $ u$-axis, the same $ u$-space frequency for all $ \omega $. To track the three events in Figure 1 into Ronen's space, it's helpful to notice that each has a different $ \omega $ spectrum.

synth3
synth3
Figure 1.
LEFT: Input data $ (t,x)$ with three events, each with different frequency content (and spatial extent). MIDDLE: Real part of ($ \omega ,x$)-space of the panel on the left. RIGHT: Same in $ (\omega ,u=\omega x)$-space, Ronen's pyramid space in which we again see straight lines as we did in $ (t,x)$-space. Observe three frequencies on the $ u$-axis, each corresponding to a different slope in the $ (t,x)$-plane. To help identify which is which, notice each of the three events in the $ (t,x)$-plane has a different $ \omega $ spectrum. [ER]
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To establish the basic property of pyramid space we consider a dipping signal in $ (t,x)$-space, say $ g(t,x)=g(t-px)$. Transform time $ t$ to frequency $ \omega $. Now we have a plane of complex valued information say $ G(\omega,x)$. What is it in Ronen's $ (\omega, u)$ space? Say $ g(t,x=0)$ fourier transforms to $ G(\omega)$. Then $ g(t-t_0)$ transforms to $ G(\omega)e^{i\omega t_0}$. Take the time shift $ t_0$ to be distance $ x$ times slope $ p$. Then $ g(t-xp)$ transforms to $ G(\omega)e^{i\omega xp}= G(\omega)e^{ip u}$. Add some extra plane waves.

$\displaystyle G(\omega,u) \quad = \quad G_1(\omega) e ^{ip_1 u} + G_2(\omega) e ^{ip_2 u} + \cdots$ (1)

Examine any $ \omega $. You see sinusoids on the $ u$-axis. The frequencies of these sinusoids are $ (p_1,p_2,\cdots)$. We see those same frequencies for each $ \omega $. Thus the PEF on the $ u$-axis informs us of a spectrum on the $ p$-axis. To a seismologist, $ p$ is a dip axis though it would be more correct to call it an axis of stepout.

What happens in 3-D? Instead of $ (x,y)$-axes we have $ (u,v)=(\omega x,\omega y)$ axes. It is not immediately clear what applications will bring us to do in the $ (u,v)$-plane. Some applications may call for a 2-D PEF there, while others call for two 1-D PEFs. What are we expecting to describe in $ (p_x,p_y)$-space? Globally, there may be a bounding range in a slowness circle corresponding to a slowest material velocity. Locally things will look more like plane waves.


next up previous [pdf]

Next: APPLICATION: AN ALGORITHM FOR Up: Claerbout and Guitton: Pyramid Previous: INTRODUCTION

2009-04-13