Statistical measures of instantaneous spectra Kui Zhang* and Kurt J. Marfurt 2008 AASPI Consortium annual meeting Not Gaussian!

Motivation  Spectral decomposition is a powerful analysis tool in tuning thickness identification, direct hydrocarbon detection, and geological facies delineation.  Spectral decomposition has limitations: One single frequency can not accurately represent the whole spectrum. The full 4D decomposition consumes considerable disk space and it is difficult to visualization efficiently.

Motivation The mode is a good first-order approximation of the spectrum: peak magnitude, peak frequency, and Peak phase We have developed several statistical attributes including: bandwidth, range-trimmed mean, spectral slope, and roughness Moment-based measures are useful for nearly-Gaussian spectra. The spectra of balanced data are NOT Gaussian.

1.00 Frequency index magnitude percentile p low 0.50 p high J low J 50 J high 0.00 J low J high slope roughness Schematic illustration of spectral attributes 0.00 High Frequency index Theory

 Bandwidth (as above figure)  Range trimmed mean  Spectral slope  Spectral Roughness The slope that best fits the spectrum between J low and J high A measure of how well the spectrum is fit by a linear variation between f low and f high. Theory

We formulate the misfit between our parameterized model and the data m j as E: Let’s parameterize the spectrum to have the form (1) where, m j is the magnitude at frequency f j, m 0 is the intercept, s is the slope, and (2) m 0 and s can be obtained by solving the following equations: (3) (4) (5)

(8) (6) (7b) Combining (4) and (5): Where, (7a) We define the roughness as:

Seismic (time slice t=1.05s) neg pos 0 A A’ Negative Positive Amplitude 0 N

0 Maximum Peak magnitude A A’ Peak magnitude N 1 km

Range-trimmed mean A A’ 0 Maximum R-T mean R-T mean magnitude gives higher resolution than peak magnitude N 1 km

Slope A A’ Negative Positive Slope 0 N 1 km

Roughness A A’ 0 Maximum Roughness N 1 km

Time 1.3 AA’ Slope 3Slope 2 Slope 1 > <

Seismic horizon

Peak magnitude (Phantom horizon slice 40ms above the picked horizon) 0 Maximum Peak magnitude N 1 km

0 Maximum R-T mean Range trimmed mean (Phantom horizon slice 40ms above the picked horizon) N 1 km

Range trimmed mean (Phantom horizon slice from 990ms-1040ms) 0 Maximum R-T mean N 1 km

Negative Positive Slope 0 (Phantom horizon slice 40ms above the picked horizon) N

1 km 0 Maximum Roughness (Phantom horizon slice 40ms above the picked horizon) N 1 km

Seismic amplitude (time slice t=0.88s) Slope 1 km negative positive negative positive 0 Amplitude Slope 0

RoughnessRange trimmed mean 1 km Oil well 0 maximum 1 km 0 maximum RoughnessR-T mean

Most negative curvature Slope 1 km negative positive Slope 0 negative positive Curvature 0

Conclusions Spectral slope, range-trimmed mean magnitude, and spectral roughness computed from the balanced spectra improve the interpretability of channels, Karst-modified fractures, and other features of geological interest. Spectral slope, range-trimmed mean magnitude, and spectral roughness computed from the balanced spectra improve the interpretability of channels, Karst-modified fractures, and other features of geological interest. These measures may be sensitive to anomalous attenuation. These measures may be sensitive to anomalous attenuation. Further modeling and comparison to logs is necessary to evaluate the sensitivity to upward-fining, upward-coarsening, and other subtle stratigraphic trends. Further modeling and comparison to logs is necessary to evaluate the sensitivity to upward-fining, upward-coarsening, and other subtle stratigraphic trends. Measures from spectral phase will be investigated. Measures from spectral phase will be investigated.

Acknowledgement We thank all sponsors of AASPI consortium for their support. We thank all sponsors of AASPI consortium for their support. We also thank Burlington Resources and Mull Petroleum Co. for the use of their data in education and research. We also thank Burlington Resources and Mull Petroleum Co. for the use of their data in education and research. 24