# Trace interpolation

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

A typical 3-D survey has a trace spacing in the crossline direction that normally is coarser than the trace spacing in the inline direction, in some cases, by as much as four times. This coarse spacing can cause spatial aliasing in the crossline direction. The problem of spatial aliasing was discussed extensively in the 2-D Fourier transform and further aspects of migration in practice within the context of 2-D Fourier transform and 2-D migration, respectively.

Recall that the coarser the trace spacing and the steeper the event dip of interest, the lower the threshold frequency at which spatial aliasing begins to take effect. In this section, we shall review trace interpolation as a means to circumvent the adverse effect of spatial aliasing in the crossline direction in 3-D migration. Trace interpolation does not create data, it merely unwraps the f − k spectrum of the input data so that aliased frequencies are mapped to the correct quadrant in the f − k domain. Finally, data do not necessarily need to be trace-interpolated in the crossline direction down to the trace spacing of the inline direction. Instead, signal bandwidth and subsurface dip can be taken into consideration to compute the optimum trace spacing to avoid spatial aliasing (equation 1-7).

During the first decade of 3-D seismic exploration, various methods have been developed for trace interpolation [1] [2]. Presently, trace interpolation usually is done using one-step complex spatial prediction filters [3] [4] in the frequency-space domain [5]. Consider a 2-D CMP-stacked data P(x, t) to be trace interpolated such that the trace spacing of the interpolated data is half of the trace spacing of the original data. Assume that P(x, t) is made up of a discrete set of dipping linear events and that the waveform for a given event is invariant from trace to trace. Also assume that the data to be interpolated do not contain any random noise. Under these assumptions, Pk(ω), the discrete Fourier transform of P(x, t) in the time direction, may be represented by the following combination of amplitude and phase spectra in the frequency-space domain [5]:

 ${\displaystyle P_{k}(\omega )=\sum _{j=1}^{N}A_{j}(\omega )\exp(-i\omega k\Delta \tau _{j}),k=0,1,2,\ldots ,M,}$ (11)

where Aj(ω) is the amplitude spectrum of the wavelet associated with the jth dipping event, Δτj is the time shift along the jth dipping event from trace to trace, k is the trace index, M is the number of traces, and N is the number of dipping events.

A one-step complex prediction filter is designed as a trace interpolation operator for each frequency component in the frequency-space domain and applied to the input data associated with the frequency component with twice the frequency [5]. The following are the steps involved in trace interpolation using one-step prediction filters. Details of the design and application of prediction filters for trace interpolation are provided in Section G.5.

1. Start with a 3-D volume stacked data P(x, y, t) that is to be migrated, and assume that the data volume is adequately sampled in the inline direction, but needs to be interpolated in the crossline direction before migration. Apply Fourier transform in the time direction, P(x, y, ω).
2. Sort the data into crosslines.
3. Then sort each crossline complex matrix of data P(y, ω) into complex arrays P(y) for each frequency component ω.
4. Design a one-step prediction filter from the data array P(y) of frequency ω/2 and apply it to P(y) of frequency ω to obtain the interpolated array Q(y) (Section G.5).
5. Interlace the original data array P(y) with the interpolated data array Q(y) to obtain the output array R(y) with twice the number of elements as in the input array P(y).
6. Repeat steps (d) and (e) for all frequencies. Then, combine the complex arrays R(y) and sort them into their crossline complex matrix R(y, ω).
7. Apply inverse Fourier transform to obtain the interpolated crossline section R(y, t).
8. Repeat steps (b) through (g) for all the crossline sections.

Figure 7.2-21 shows a CMP-stacked section with trace spacing of 10 m and the same stack with every other trace dropped so as to make the trace spacing 20 m, 40 m, and 80 m. Note the increasingly less distinctive character of the steep flanks of the diffractions and steeply dipping reflections at coarser trace spacing. Migrations of the four stacked sections in Figure 7.2-21 are shown in Figure 7.2-22. Note that migration of the stacked section with sufficiently small trace spacing provides a clear image, albeit in time, of both gently dipping and steeply dipping reflectors. With coarser trace spacing, however, the image quality of the steeply dipping events at the upper half of the stacked section begins to deteriorate rapidly. Spatial aliasing at coarse trace spacing has caused the aliased frequency components of the dipping events to move in the wrong direction. As a result, the migrated section in Figure 7.2-22d with very coarse trace spacing (80 m) is very noisy, especially in the upper half, and void of coherent reflections that are present in the unmigrated stacked section (Figure 7.2-21d).

Figure 7.2-23a shows the stacked section in Figure 7.2-21b with trace spacing of 20 m after interpolation to trace spacing of 10 m. Compare this interpolated section with the stacked section in Figure 7.2-21a and subtract the two sections to obtain the difference section shown in Figure 7.2-24a. The difference section contains alternating traces with zero sample values since trace interpolation retains the original traces in the input section with 20-m trace spacing (Figure 7.2-21b) while creating extra traces in between the original traces to produce a section with 10-m trace spacing (Figure 7.2-23a). Note that the traces with live samples in the difference section carry negligibly small energy associated with steeply dipping events within the upper half of the section.

Figure 7.2-23b shows the stacked section in Figure 7.2-21c with a trace spacing of 40 m after interpolation to trace spacing of 20 m. Compare this interpolated section with the stacked section in Figure 7.2-21b and subtract the two sections to obtain the difference section shown in Figure 7.2-24b. Again, the difference section contains alternating traces with zero samples interlaced with traces that contain nonzero samples corresponding to errors in trace interpolation.

Finally, Figure 7.2-23c shows the stacked section in Figure 7.2-21d with trace spacing of 80 m after interpolation to a trace spacing of 40 m. Compare this interpolated section with the stacked section in Figure 7.2-21c and subtract the two sections to obtain the difference section shown in Figure 7.2-24c. This difference section indicates that interpolation from a coarse trace spacing to a finer trace spacing may not be sufficiently accurate, and as a result, may not reproduce the signal coherency that is present in the original stacked section (Figure 7.2-21c). Hence, there is a limit as to how coarse the input trace spacing can be for trace interpolation to provide acceptable results. The degree of accuracy in trace interpolation has a direct impact on the fidelity of the image obtained from the migration of the interploated data. Compare the migrated sections in Figures 7.2-25a,b,c with those in Figures 7.2-22a,b,c. Note that the image quality is comparable when interpolation is performed using input data with a reasonably coarse trace spacing — from a 20-m to 10-m interval or from a 40-m to 20-m interval. The image quality however is not faithfully preserved when interpolation is performed using input data with very coarse trace spacing — from an 80-m to 40-m interval.

Cascaded interpolation to increasingly finer trace spacing causes accumulation of interpolation errors. Figure 7.2-26a shows the stacked section with a 20-m trace spacing after trace interpolation applied to the already interpolated stacked section in Figure 7.2-23c with a 40-m trace spacing. Compare this section with that in Figure 7.2-21b and subtract the two sections to obtain the difference section shown in Figure 7.2-27a. Note the significant energy content of the difference section indicating that cascaded interpolation produces unfavorable results. A third cascade in the interpolation yields the stacked section in Figure 7.2-26b with a 10-m trace spacing from the already interpolated section in Figure 7.2-26a with a 20-m trace spacing. Compare this section with that in Figure 7.2-21a and subtract the two sections to obtain the difference section shown in Figure 7.2-27b. The energy content of the difference section indicates further deterioration of the signal quality with an increased number of cascades in interpolation.

## References

1. Rothman et al., 1981, Rothman, D., Larner, K. L., and Chambers, R., 1981, Trace interpolation and design of 3-D surveys: Presented at the 39th Ann. Mtg. Eur. Assoc. Expl. Geophys.
2. Ronen, 1987, Ronen, J., 1987, Wave-equation trace interpolation: Geophysics, 52, 973–984.
3. Canales, 1984, Canales, L., 1984, Random noise reduction: 54th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 525–527.
4. Gulunay, 1986, Gulunay, N., 1986, F − X decon and complex Wiener prediction filter: 56th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 279–281.
5. Spitz, 1991, Spitz, S., 1991, Seismic trace interpolation in the f − x domain: Geophysics, 56, 785–794.