Interpolation of seismic data is an important application in exploration seismic signal processing. The need for interpolation on incomplete data may be due to acquisition limitations, economic constraints or regularizing of merged data with a variety of shooting parameters from different vintages.

Furthermore, the deliverable of interpolation is regularized prestack data, which will improve many state-of-the-art inversion processes and migration imaging.

Better and more accurate interpolation algorithms have come out over the past 10 years. Some examples are prediction filters and rank reduction method, both of which operate in the frequency-space domain, and the minimum weighted norm interpolation (MWNI) method in the mixed frequency-space and frequency-wavenumber domain.

MWNI is the most popular method among them because it closely maintains the original input trace characteristics, including its signal-to-noise, better than other methods.

## Challenges to Most Interpolation Schemes

There are five data scenarios that challenge most interpolation methods:

#### Data with random missing traces.

This is considered the least evil kind of data scenario for interpolation because it boils down to a de-noising exercise for interpolators.

#### Data with a gap of missing traces.

Depending on the size of the gap, this data can be very difficult to interpolate.

#### Data with aliasing in space.

This happens when structural data is not adequately sampled and recorded in space.

#### Data with curving events or diffractions.

This violates many interpolators that hang on the plane wave or bandlimited sparsity assumptions.

#### Data with regularly missing traces.

This may be considered as up-sampling of spatial data. An example of this is the popular megabin acquisition. It is the most challenging of interpolations, especially when the data is structural and steep dips are aliased. We will demonstrate how our new data interpolation method, which we refer to as “6-D interpolation,” stands up against these challenges.

## Why 6-D Interpolation?

The reason “5-D interpolation” is given its name, in spite of interpolator choice, is that interpolation is applied in 3-D prestack seismic data, which can be defined in 5-D space: inline x, crossline y, offset x, offset y and time (or frequency).

In some applications, offset x and offset y are redefined as radial offset and azimuth. The industry 5-D MWNI starts data fitting from the stable low frequency slices, one slice at a time, and recursively layer-by-layer works its way up to higher frequencies. This unconstrained fitting can lead to inaccurate results under challenging scenarios such as meager data support, aliased dips or up-sampling of regularly missing data.

Recently, we proposed the 6-D interpolation method, which has an additional dimension along multiangular directions to be added to the 5-D MWNI in order to guide the a priori model in the frequency-wavenumber domain. The angular weights are derived from a scanning of different dips of the input data in the frequency-wavenumber domain along many radial directions pointing away from the origin. Angular weights connect data information across all frequency-wavenumbers globally, which is crucial to de-aliasing of data, but is completely missing in the conventional 5-D MWNI.

A good analogy for 5-D interpolation is the brick-laying process of building a fire pit from low to high level without a global plumb line reference. Even with a good foundation level to start with, this could eventually lead to a slanting brick structure at the top levels. But with a plumb line reference, which in our case is the angular weight function, or the sixth dimension, it guides the 5-D MWNI engine resulting in a stable 6-D interpolation method. The objective is to preserve structural integrity of the data after interpolation.

Our first example illustrates the proof of concept.

We show this example to test and compare the recovery performance of the new 6-D interpolation with the conventional 5-D interpolation, both operated by an MWNI engine, under the earlier mentioned most challenging scenarios: spatial aliasing of incomplete curving diffractions and up-sampling of regularly missing data (deliberately decimated in this experiment) of three times. A complete structural 3-D real dataset is considered the “hidden” control reference when two out of three of its crossline prestack gathers are zeroed and treated as the input to both 5-D and 6-D interpolators.

Figure 1(a) exhibits an inline from the CMP stack of complete prestack data gathers and is considered as the “hidden” reference data.

The data is 60-fold. Conflicting diffraction curves with different amplitudes are present in both left and right sides of the figure.

In figure 1(b) we show the CMP gathers that have been regularly decimated 3:1 in the crossline direction. Consequently, only 33 percent of the original data are used and treated as input to the 5-D and 6-D interpolation tests. The steeply dipping diffraction curves become aliased inside two yellow highlighted circles.

Figure 1(c) shows the data recovery results of the conventional 5-D interpolation by MWNI, performed in small overlapping data blocks so as to preserve local structural details. The yellow highlighting circles depict the poor recovery of the aliased steeply dipping events. Poor quality recovery of aliased steeply dipping data is shown in the yellow circles.

Figure 1(d) shows the data recovered by the proposed 6-D interpolation method using the extra dimension angular weights on MWNI. Identical data blocking parameters are used as in (c). The new 6-D interpolation recovery is acceptable when compared to the reference data shown in (a). When the data complexity is less challenging outside the highlighted circles, and the data support is sufficient, the 5-D and 6-D interpolations converge to similar results.

Our next example is from a real super merge of different data volumes.

Figure 2 shows that 6-D interpolation is used to address all five aforementioned data challenges (regular and random missing data, block gaps, up-sampling and data aliasing) on a super merge of 17 different 3-D’s (i.e. nine megabin and eight orthogonal surveys) with widely different acquisition parameters. Note that 6-D interpolation in figure 2(c) and (d) preserves structural integrity – long and short wavelength features; the AVO response is also well honored, although not displayed here.

## Where Does That Lead?

The 6-D interpolation, in contrast to the conventional 5-D interpolation (MWNI), has an additional angular weight function dimension that connects dipping data information across all frequency-wavenumbers, which in effect can delineate aliased data in highly structural and deficient in data support situations. This idea can lead to improvements in other frequency-space interpolators as well. Furthermore, the 6-D interpolation leads to an important application: a natural CMP grid size can be subdivided by half in both directions, and 6-D interpolation can be used for up-sampling to achieve effective resolution of the data without actual field acquisition for the finer grid size. This could result in substantial savings in acquisition costs of up to four times. It’s comparable to a 1080p high-definition television producing an effective 2160p ultra-high resolution at a small cost.

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