Axes and allies

The standard seismic acquisition grid is presented in panel (a) of Figure 1. Ease of drafting and understanding conventionally has lead geophysicists to draw these axes 90^o to each other. However, there is no reason to do so. Further, imposing the intuition that these axes are mathematically orthogonal leads to difficulties in interpreting Fourier sampling criteria that we aim to investigate here. These axes inhabit the same physical space along a 2D seismic acquisition line. Plotting them orthogonally casts an inappropriate feeling of a second physical dimension.

 

nybox Figure 1 (a) The standard representation of the seismic acquisition grid. (b) Viable alternative to conventional orthogonal drawings. (c) Fourier transform of (a) with Nyquist sampling limits included. (d) Fourier transform of (b) with Nyquist sampling limits included.

At the limit of this argument, we contend that it is much easier to de-couple completely the origins of these axis and plot them parallel to each other. Having performed a Fourier transform across space of both the source and receiver axes, we present them in Figure 2.

 

waveline
Figure 2 Lengths of the source, receiver and image axes in the Fourier domain. The circle with an x indicates the multiplication of the data axes to produce the image surface location axis during the imaging condition. Aliased replications from the source axis are represented by the bold lines pointing toward the origin. (a) If the image spaced is sampled as finely as the receiver axis, alias contamination will enter into the image. (b) Aliased contamination from the subsampled source axes is avoided when the two fields are compared during the imaging condition.

Viewing these three distinct axes separately aids in the interpretation of this entire argument. Unfortunately, there is an historic tendency when analyzing the acquisition grid coordinates to include midpoint-offset, mh, axes as diagonal axes to those shown in Figure 1. We will avoid the use of midpoint while casting this presentation largely in the terms of shot-profile migration, as well as explain later the development of our x-axis during the imaging condition. Further, when they are superposed, an incorrect stretching is implied. We will briefly consider the mid-point axis, in order to highlight the danger of this practice.

The mapping transformation of energy from one coordinate frame to the other has been historically defined as:

$$\begin{eqnarray} m=\frac{s+r}{2}\; \; \\ h=\frac{s-r}{2} \;.\end{eqnarray}$$ (1) (2)

The Jacobian of this system is thus,

$$\begin{eqnarray} J^2= \left( \begin{array} {cc} \frac{\partial m}{\partial r} & ... ...& -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \\ \end{array}\right),\end{eqnarray}$$ (3)

which makes the determinant $1/\sqrt{2}$. Drawing a midpoint axis along the 45^o line of the sr-axis is confusing when someone attempts to find value of a particular midpoint location on the plane. Some measurement of distance must be employed as a zero-offset location does not lie on one of the original axes. The next problem we then face is which of the multitude of distance measures we should select: l_?. The historic choice has been the square-root of the sum of the squares: l₂. If we make that choice, we then apply the determinant presented above to cancel the $\sqrt(2)$ factor associated with the norm that so naturally lends itself to pieces of paper. We could have chosen any norm. Each of them would return a different number, and none of them have any more or less value for locating a seismic image on the surface of the earth.

The cross-correlation imaging condition with subsurface offset Rickett and Sava (2001) 

$$I(x,h,z) =\sum_s \sum_\omega R(x=r-h,z,\omega) S^*(x=s+h,z,\omega),$$ (4)

combines the two independently propagated wavefields, Source and Receiver ,and generates the surface location axis x with the relationship within the arguments of the two wavefields. Note that these relations are equations for a line where both the source coordinate, s, and receiver coordinate, r, coexist. Here we see the distinction between the surface location coordinate x and midpoint. The x-axis fallows directly from the mathematics of migration. With the multiplication (across space) associated with the correlation (along time) of equation (4), the cartoons in Figure 2 should be recalled.

The Appendix develops in rigorous detail the wavenumber limits acceptable in the image to eliminate completely alias contamination. The analysis of the problem centers around the effects of the migration process on the data grid, without needing to consider the values of the data on each grid node. We thus draw an analogy to the body of work available from crystallography, where structure can be analyzed mathematically without need to know what atom resides at any particular location. Thankfully, the regular Cartesian grid on which we normally acquire and process seismic data is a simple rectilinear crystal, though of several more dimensions than seen under a microscope.

The reference crystal we will consider will be the archetypal seismic grid where sources and receivers occupy all locations and share the same spacing increment. The suspicious or simply inquisitive reader can now turn to the appendix to work through the details of the following result. The maximum allowable wavenumbers, $B_x_{\xi}$, to avoid artifacts due to migration operator aliasing is  

$$B_x_{\xi}=\frac{1}{{\rm lcf}(2a_1\Delta r_{\xi},2a_2\Delta s... ...rac{N_\r}{a_1},\frac{N_s_{\xi}}{a_2})={\rm min}(B_\r,B_s_{\xi})$$ (5)

where $N_\r$ and $N_s_{\xi}$ are the Nyquist frequencies defined by fundamental sampling intervals $\Delta r_{\xi}= \Delta s_{\xi}$, a's are subsampling factors, and ${\rm lcf}$ stands for least common factor (which will change to $\mbox{min}$ if the subsampling factors are integers. The subscript $\xi$ denotes the sampling associated with the model space and is included to maintain parallelism with the appendix.

We consider three approaches to remove the aliasing problems associated with the acquisition and subsampling situations mentioned above during shot-profile migration. First, wavenumbers from the source and receiver wavefields at each depth level are band-limited to prevent the entry of aliased duplications into the image during the imaging condition. This does not require eliminating these components from the propagating wavefields, as we can save appropriate portions of the wavefields in temporary buffers for imaging condition evaluation. Second, a band-limited source function, with a wavenumber spectrum limited to the cutoff frequencies imposed by the resampled shot axis, is propagated throughout the migration process. This effectively zeros energy in the aliased band during the convolution in the imaging condition. No additional computational overhead is required for the latter alternative, though anti-aliasing by band-limited imaging requires two additional Fourier transforms for a split-step Fourier migration strategy. It should be noted, however, that both of these approaches will remove energy across both k_x and k_h axes.

A third alternative is to restrict the wavenumbers of the subsurface offset axis k_h during imaging. Casting the imaging condition in terms of it Fourier dual can allow similar mitigation options. Because k_s-k_r=k_h, we can select (k_s,k_r) combinations during the imaging condition that do not exceed our prescribed bandwidth. The multiplication of the source and receiver wavefields shown above takes the form of a convolution in the Fourier domain, which can be utilized to insert our anti-aliasing criteria. Lastly, decimating the receiver wavefield to match the shot increment, will be discussed in more detail with reference to shot-geophone style migration.