SPECTRAL ESTIMATION

The phrase ``spectral estimation'' has been used for a long time to represent correct estimation of the power spectrum of a time series. Here, I use the phrase to mean estimation of the spectrum which is supposed to result from interpolation.

Let us consider a plane wave, which is a linear event in the time-space (t-x) domain, and its spectrum, which is also a linear event in the frequency-wavenumber (f-k) domain. If we sample densely enough to avoid aliasing in the time direction and sparsely along the space direction, the spectrum of the plane wave will show spatial aliasing as is then the case in a conventional seismic survey. Figure 1a shows a spectrum containing several linear events with spatial aliasing. Figure 1b shows the spectrum of such data from $-3\pi/\Delta x$ to $3\pi/\Delta x$ along the wavenumber axis. From Figure 1b, we can see that the aliasing is caused by a portion of the original spectrum shifted by $\pm 2\pi/\Delta x$, which is still located in the region $-\pi/\Delta x$ to $\pi/\Delta x$ along the wavenumber axis. If we oversample with $\Delta x$ = $\Delta x_{orig}/2$,the spectrum which caused the aliasing effect will be shifted further, to $\pm 4\pi/\Delta x$ (Figure 1c). As a result, one of the aliased events will disappear where $-2\pi/\Delta x \leq k \leq 2\pi/\Delta x$.On the other hand, we can mimic a spectrum like Figure 1c by taking the aliased spectrum only from $-\pi/2\Delta t \leq f \leq \pi/2\Delta t$, as in Figure 1d.

 

fig1
Figure 1 (a) The spatially aliased spectrum. (b) The spatially aliased spectrum with replication. (c) The oversampled spectrum which is the spectrum after interpolation. (d) The spectrum estimated from the aliased spectrum.

Let us describe a linear event in the spectrum as a line segment

f = ak

where a = df/dk is the inverse dip of a linear event, and f and k are defined on $-\pi/\Delta t\leq f \leq \pi/\Delta t$ and $-\pi/\Delta x\leq k \leq \pi/\Delta x$, respectively. Then, the aliased event can be described by a shifted line:

$$f = a(k\pm {2\pi \over {\Delta x}}).$$ (2)

If we oversampled by 1/N times the given sampling interval in the space domain, the line will be

$$f = a(k\pm {2N\pi \over {\Delta x}})$$ (3)

where k is defined on $-N\pi/\Delta x \leq k \leq N\pi/\Delta x$.By pulling out N from the bracket on the right-hand side of equation (3) and changing variables according to f'=f/N and k'=k/N, we find that equation (3) becomes

$$f' = a(k'\pm {2\pi \over {\Delta x}})$$ (4)

where f' and k' are defined on $-\pi/N\Delta t\leq f' \leq \pi/N\Delta t$ and $-\pi/\Delta x\leq k' \leq \pi/\Delta x$, respectively. Equation (4) tells us that, by taking a portion of the given spectrum, we can obtain a spectrum which corresponds to the spectrum after interpolation.