# Using Volumetric Calculators For Better Accuracy

Interpreters routinely use horizon slices and stratal (or proportional) slices to interpret seismic attributes. While some attributes such as dip magnitude and dip azimuth can be computed from a picked surface, most attributes are computed volumetrically on a 3-D grid of voxels whose vertical size is defined by the time or depth sample increment.

In general, picked horizons occur at fractional sample increments, requiring subsequent interpolation.

For attributes that vary smoothly, such as the original seismic amplitude, simple linear, quadratic and sine function interpolation provide excellent results.

For attributes such as envelope and spectral magnitude components, the results are almost always acceptable, but can be less accurate as the values approach zero.

For attributes that are cyclical, such as phase, azimuth and strike, such interpolation gives erroneous results.

We present a simple workflow that allows an interpreter to more accurately extract such attributes using the volumetric calculators available in most commercial interpretation software packages.

Interpreters routinely use horizon slices and stratal (or proportional) slices to interpret seismic attributes. While some attributes such as dip magnitude and dip azimuth can be computed from a picked surface, most attributes are computed volumetrically on a 3-D grid of voxels whose vertical size is defined by the time or depth sample increment.

In general, picked horizons occur at fractional sample increments, requiring subsequent interpolation.

For attributes that vary smoothly, such as the original seismic amplitude, simple linear, quadratic and sine function interpolation provide excellent results.

For attributes such as envelope and spectral magnitude components, the results are almost always acceptable, but can be less accurate as the values approach zero.

For attributes that are cyclical, such as phase, azimuth and strike, such interpolation gives erroneous results.

We present a simple workflow that allows an interpreter to more accurately extract such attributes using the volumetric calculators available in most commercial interpretation software packages.

Many attributes of interest have a cyclical behavior, including instantaneous phase, spectral phase components, dip azimuth and strike of azimuthal anisotropy and curvature. While human interpreters perceive these attributes to be continuous, computer software does not, and injects a numerical discontinuity in phase and azimuth between -180 degrees and +180 degrees (or depending on the software package between 0 degrees and 360 degrees).

Similar discontinuities appear in strike between -90 degrees and +90 degrees.

Such discontinuities do not pose a problem if we only wish to look at the data at discrete voxels – but if the number of pixels used on the computer screen is larger than the number of voxels being displayed, the data need to be either replicated or interpolated.

The most common implementation is to interpolate the data, linearly, bilinearly or with a spline. Such interpolation between samples works very well for seismic amplitude and envelope, but fails for cyclic (to a computer, discontinuous) attributes like phase (figure 1a). Many commercial software packages use splines to interpolate along the vertical axis, which for phase results in erroneous values beyond ±180 degrees.

The simplest way to avoid such artifacts is to disable the interpolation. This can almost always be done when displaying vertical or horizontal slices through the data volume.

Examining figure 2, we note that the vertical slices through the seismic amplitude and instantaneous envelope display nicely, but the same slice through the instantaneous phase in figure 2c looks “green.” Using figure 1a as a guide, we realize that much of the green is an artifact of inaccurate interpolation between phase values that approach +/-180 degrees. Disabling interpolation and instead replicating the nearest pixel provides the lower resolution but acceptable image in figure 2d.

This artifact becomes particularly ugly when we wish to extract cyclical attributes along a picked horizon. Horizon picks rarely fall on an integer sample value such that the data need to be interpolated.

In figure 3 we show a time-structure map and amplitude extraction where the base Oligocene corresponds to a trough. The initial phase extraction is totally erroneous (figure 3c). A more accurate means of interpolating the phase is to compute the original data and its quadrature (the “real” and “imaginary” components of a complex trace), using the envelope, as shown in figure 1b.

Be careful to compute the angle using “ATAN2” (the same one as in Excel) to obtain values of phase range between -180 degrees and +180 degrees. By doing so we obtain the geologically reasonable image (phase close to ±180 degrees, appearing as magenta), consistent with our picked trough.

Interpolation of other vector components, such as dip azimuth and dip magnitude, are similar to the technique shown in figure 1b.

In contrast, interpolation of attributes that are defined by a strike require a slight variation.

For azimuthal anisotropy, the “azimuth” is really a strike and also varies between -90 degrees and +90 degrees, while the azimuthal intensity, ε, is a strictly positive number. Here, be careful to use “ATAN” rather than ATAN2 to obtain strikes between -90 degrees and +90 degrees.

This workflow works well in many, but not all software packages.

Figure 4, for example, illustrates a limitation faced in the one that we use.

We wish to extract and display volumetric dip azimuth along the same horizon shown in the previous image. Using figure 1b eliminates some of the artifacts. However, while the software allows us to disable interpolation on vertical and horizontal slices, it does not allow us to disable it on horizon slices.

The result is the appearance of blue “rings” (corresponding to north, or 0 degrees) circling anomalies that wrap around between SSW and SSE (±180 degrees) azimuths.

Presented with such an image, the only recourse is to realize that they are artifacts and not geology!