Introduction
Seismic Resolution of Zero-Phase Wavelets
Designing Optimum Zero-Phase Wavelets
USP Job Scripts
October 24, 2006
Introduction
Seismic Resolution of Zero-Phase Wavelets
Designing Optimum Zero-Phase Wavelets
USP Job Scripts
Introduction In 1977, Bob Kallweit and Larry Wood presented their now famous work on temporal resolution and the effects of filter design on side-lobe tuning. This material was presented internally at a 1977 technology meeting at Amoco in Houston, Texas. Kallweit's and Wood's concepts hold up as well today as they did in 1977. Their 1977 material is available in pdf form, via a FreeUSP web document.
The purpose of this exercise is to revisit this Kallweit and Wood material, and in the process capture key observations and conclusions, and recreate displays using todays graphics technology (to see if the display options that are readily available today, but not available in 1997, reveal any additional insight).
Here is a link to a summary PowerPoint slide-set of this material.
Seismic Resolution of Zero-Phase Wavelets Here is a PowerPoint summary of this portion of the material.
Kallweit and Wood start with a review of previous work on temporal resolution, and in the process summarize the work of Rayleigh, Ricker and Widess. They then go-on to show that:
They propose a definition of seismic resolution, in context of thin bed resolvability that ties together both Ricker's and Widess' criteria and relates both of them to parameters that can be measured on the incident wavelet itself.
They determine that the low-pass sinc wavelet can be analyzed in terms of its mid frequency in order to establish a similarity to the analysis of the Ricker wavelet.
Even though a low-pass sinc wavelet is not realizable in actual practice (because it has frequencies extending to zero Hertz), it is instructive to study this wavelet because it allows temporal resolution to be examined in terms of the maximum frequency. The learnings from the analysis of the low pass sinc wavelet can then be used in the discussion of the bandpass sinc wavelet. The low frequency limit of the band-pass sinc wavelet has negligible effect on the temporal resolution of the wavelet for band-pass ratios of 2-octaves and greater.
The resulting ability to relate temporal resolution to the highest, and only the highest frequency of a wavelet leads to some very useful and quite accurate approximations.
Designing Optimum Zero-Phase WaveletsHere Kallweit and Wood capture and illustrate how changing the shape of zero-phase wavelets effects both (1) side-lobe interference and (2) our ability to resolve geology (layering and reflectivity).
Here is a PowerPoint summary of the material.
Computer and graphics technology was substantially different in 1977, so in this case, it was worthwhile recreating the experiments and capturing snapshots of the results via current day workstation tools. The scripts used to recreate the experiments are archived in the Scripts directory.
The key message from Kallweit and Wood is: Wavelets designed with a vertical or near-vertical high end slope exhibit high frequency side-lobes that can cause significant distortions in reflection amplitudes and associated event character. They propose an alternate wavelet called the "Texas Double" in recognition of the primary characteristic being a 2-octave slope on the high frequency side.
In time domain, Texas Double wavelets exhibit negligible high frequency side-lobe tuning effects, and maximum peak-to-sidelobe amplitude ratios. In frequency domain, Texas Double wavelets have a vertical or near-vertical low-end slope, and a 2-octave linear slope on the high-end (with amplitude measured using a linear rather than decibel scale).
They then proceed to show the effects of varying the low and high side filter slopes on a wedge model:
Following a walk through the wedge model experiments, Kallweit and Wood propose a standard equi-resolution comparison. They explain that one of the difficulties involved in trying to compare traces containing different zero-phase wavelets designed over identical bandpasses is the question of what to compare and measure each trace against. Too many trace-to-trace unknowns yield rather unsatisfactory comparisons. A standard comparison is needed, and the standard trace proposed by Kallweit and Wood is one where the convolving wavelet has the same temporal resolution as the sinc wavelet over a given bandpass but has no side-lobes whatsoever.
They go on to explain that over a given low-pass, temporal resolution of the Texas Double wavelet equals 80% of the temporal resolution of the sinc wavelet. The question then becomes: Can the benefits associated with attenuating high-frequency side-lobes outweigh the 20% loss in temporal resolution? The well-log based comparisons included next in their report, suggest that they can. To determine differences associated with side-lobes as opposed to those associated with temporal resolution, Kallweit and Wood filter a well-log (layering and reflectivity) and compare the application of the Texas Double filter with that of an equi-resolution sinc filter. Any remaining observed differences are then due to side-lobe tuning or temporal resolution.
In this well-log example, the 2-octave and 3-octave bandpass wavelets:
3-octave comparison: shallow, mid, deep
2-octave comparison: shallow, mid, deep
Kallweit and Wood then illustrate the sensitivity of the side-lobe tuning effects of the sinc and Texas Double wavelets to small changes in high frequency components. The layered log and corresponding reflectivity were filtered holding the low side constant for each filter and varying the high side in 1 Hz increments. Since the filters change in a linear and gradual manner, the hope is that the traces would do likewise. Unfortunately, significant trace-to-trace variations are apparent.
Two sets of Texas Double filters are also applied, and compared with the sinc wavelet results. One Texas Double set exhibits the same temporal resolution as the bandpass sinc set. The other Texas Double set mirrors the f1 and f4 filter positions of the sinc wavelets. The Texas Double design reduces tuning effects to a negligible level, and allows trace-to-trace variations to be gradual and consistent.
Kallweit and Wood finish off with a discussion about using the Texas Double in practice. They discuss a processing flow in which the data is first whitened, then filtered by a 2-octave slope Ormsby filter. They state that the Texas Double design criteria should not be a goal of data acquisition: that it is of utmost importance that the signal-to-noise ratio of the high-frequency components be as large as possible, and therefore filtering process such as the Texas Double should occur in data processing and not in data acquisition.
USP Job Scripts
