ANTI-ALIASING THEORY

Claerbout (1992) proposed a method to anti-alias Kirchhoff space-time operators by local lowpass triangle filtering. Here, I extend Claerbout's anti-aliasing method to the Kirchhoff prestack time migration operator, which is given by:

 

$$t_k \; = \; t_s + t_r \; = \; \sqrt{\left(\frac{\rho_{s}}{... ...2 } + \sqrt{\left(\frac{\rho_{r}}{v}\right)^2 + \tau^2 } \; ,$$ (1)

where t_k is the two-way Kirchhoff migration reflection traveltime, $\tau$ is the one-way vertical traveltime to the reflection point (pseudodepth) as shown in the geometry of Figure , v is the rms migration velocity which may vary over $(x,y,\tau)$ in general, and $\rho_{s}$ ($\rho_{r}$) is the distance measured along the planar recording surface between the source (receiver) and the vertical projection of the image point to the recording surface:

 

$$\rho_{\star} \; = \; \sqrt{ (x_i-x_{\star})^2 + (y_i-y_{\star})^2 } \; ,$$ (2)

assuming the recording surface is horizontal at $z=\mbox{ \it constant\/}$.

 

geometry
Figure 1 Geometry of the Kirchhoff 3-D prestack time migration operator for the anti-aliasing criterion.

The spatial derivative of (1) is needed to determine the migration operator aliasing criterion:

 

$$\frac{\partial t_k}{\partial \rho} \; = \; + \sqrt{ \left(... ..._r} \right) \; \approx \; \frac{\Delta t_k}{\Delta \rho} \;.$$ (3)

To remain unaliased, the temporal period T of any local waveform on a seismic trace at time t_k must be greater than $2 \Delta t_k$:

 

$$T \; \ge \; 2 \Delta t_k \; = \; 2 \frac{\Delta \rho}{v^2} \left( \frac{\rho_{s}}{t_s} + \frac{\rho_{r}}{t_r} \right) \;.$$ (4)

I evaluate the effective rms spatial sampling interval $\Delta \rho$ for the prestack migration operator as

 

$$\Delta \rho \; \approx \; \sqrt{ \frac{1}{2} \left( dx_s^2 + dy_s^2 + dx_r^2 + dy_r^2 \right) }$$ (5)

where I define $dx_{\star}$ and $dy_{\star}$ in the prestack case as

 

$$dx_{\star} \; = \; \frac{\vert x_i - x_{\star} \vert \Delta x }{\rho_{\star}}$$ (6)

and  

$$dy_{\star} \; = \; \frac{\vert y_i - y_{\star} \vert \Delta y }{\rho_{\star}} \; .$$ (7)

Here, $\Delta x$ and $\Delta y$ are the true inline and crossline sampling intervals of the seismic trace data. For the poststack case, (3) reduces to

 

$$\frac{\Delta t_k}{\Delta \rho} \; \rightarrow \; \frac{4}{v^2} \left( \frac{\rho_m}{t_k} \right) \;\;\; \mbox{ (poststack) }$$ (8)

where $\rho_m$ is the surface distance from the midpoint to the image point surface projection, and (5) reduces to

 

$$\Delta \rho \; \rightarrow \; \sqrt{ dx_m^2 + dy_m^2 } \;\;\; \mbox{ (poststack) }$$ (9)

where dx_m and dy_m are still validly defined by (6) and (7). Evidently,

 

$$\min\{\Delta x,\Delta y\} \; \le \; \Delta \rho \; \le \; \sqrt{(\Delta x)^2 + (\Delta y)^2} \;,$$ (10)

as expected. As an aside, the anti-aliasing criterion (4) may also be a good approximation for the case of Kirchhoff depth migration with the use of the rms migration velocity equivalent $v(x,y,\tau)$ to the depth migration interval velocity model v_int(x,y,z).

The Kirchhoff migration operator anti-aliasing criterion (4) suggests an anti-aliasing method by local lowpass filtering of the input trace waveform in the vicinity of t_k. This local anti-alias lowpass filter is a function of trace time t_k and migration operator dip given by (3), in general. In particular, the anti-aliasing filter is not necessarily a simple function of offset or aperture. For example, a constant offset trace will be subject to relatively stronger lowpass filtering at early traveltimes t_s and t_r, compared to later arrival times (as expected by the greater relative curvature of a shallow versus deep migration impulse response).

I implement the local lowpass filters as triangular smoothing. Claerbout (1992) showed that for an arbitrary N-point triangle filter, the smoothing can be implemented efficiently by using only 3 filter coefficients instead of N, if the input trace data are first subjected to a sequential process of causal followed by acausal temporal integration. This observation offers a large savings in computational effort when applying the triangle filters. I relate an N-point triangle to the aliasing period T by the relation $T = 2(N+1)\Delta t$, where $\Delta t$ is the true temporal trace sample interval. My chosen anti-aliasing criterion (4) in terms of triangle filter length N becomes:

 

$$N \; \ge \; \max \left(\; \frac{\Delta \rho}{v^2 \Delta t} ... ..._s} + \frac{\rho_{r}}{t_r} \right) - 1, \;\; 1 \; \right) \;.$$ (11)

For poststack geometries, this result reduces to:

 

$$N \; \ge \; \max \left(\; \frac{4 \rho_m \Delta \rho}{v^2 ... ...a t } - 1, \;\; 1 \; \right) \;\;\; \mbox{ (poststack) } \;.$$ (12)