Seismic Resolution of Zero-Phase Wavelets Designing Optimum Zero-Phase Wavelets R. S. Kallweit and L. C. Wood Amoco Houston Division DGTS January 12, 1977.

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Seismic Resolution of Zero-Phase Wavelets Designing Optimum Zero-Phase Wavelets R. S. Kallweit and L. C. Wood Amoco Houston Division DGTS January 12, 1977 G. Partyka Oct 2006

Questions Can different types of zero-phase wavelets be compared in terms of temporal resolution. What wavelet shape is well suited for comparing against? Can we separate the ability of zero-phase wavelets to resolve thin beds from variations in side-lobe tuning effects? What is an optimum wavelet shape? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

What are the Limits of Temporal Resolution? 1.How thin can a bed become and still be resolvable? In other words, when is the measured interval time essentially the same as the true interval time? 2.What are the errors between the true interval times and the measured interval times through thick beds? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 Review: Rayleigh Ricker Widess G. Partyka Oct 2006

Why Focus on Zero-Phase? Non zero-phase wavelets greatly complicate resolution. Zero-phase wavelets simplify resolution: –traces containing zero-phase wavelets will have seismic interfaces located in general at the centers of the peaks and troughs of the trace (neglecting tuning effects and noise). Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 Thick Bed Well Log Thin Bed Well Log G. Partyka Oct 2006

What is an Optimum Wavelet Shape to Compare Against? Low Pass Sinc Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 frequency amplitude TimeFrequency time G. Partyka Oct 2006

Temporal Resolution of the Low-Pass Sinc Wavelet TbTb TRTR fmfm frequency amplitude T0T0 f4f4 Temporal resolution is established in terms of the maximum frequency temporal resolution:T R = 1 / (3.0)f m = 1 / (1.5)f 4 wavelet breadth:T b = 1 / (1.4)f m = 1 / (0.7)f 4 peak-to-trough:T b / 2 = 1 / (2.8) f m = 1 / (1.4)f 4 1 st zero crossings:T 0 = 1 / 2f m = 1 / f 4 relationship of T b to T R T R = 0.47T b = 0.93T b / 2 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

What effect does a low cut have on resolution? Negligible for wavelets with bandpasses of 2-octaves or greater Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 frequency amplitude TimeFrequency time ….but, must have signal in bandpass. G. Partyka Oct 2006

Temporal Resolution (T R ) of the Bandpass Sinc Wavelet TbTb TRTR fmfm frequency amplitude T0T0 f4f4 f1f1 f4f4 f 4 sinc f 1 sinc band-pass sinc f m = (f 1 + f 4 ) / 2 f1f1 The band-pass sinc wavelet is the difference between two (f 1 and f 4 ) sinc functions. Effects of the f 1 sinc are negligible for wavelets with band-pass ratios 2-octaves and greater. temporal resolution:T R = 1 / (1.5)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) wavelet breadth:T b = 1 / (0.7)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) peak-to-trough:T b / 2= 1 / (1.4)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) 1 st zero crossings:T 0 = 1 / (2f m ); all octaves relationship of T b to T R :T R = 0.47T b = 0.93T b / 2 ; for sincs.ge. 2 octaves Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Sinc Bandwidth and Temporal Resolution with Constant f max peak-to-trough separation (ms) spike separation (ms) 2–64 Hz Sinc (5 octaves) TRTR peak-to-trough separation (ms) spike separation (ms) 16–64 Hz Sinc (2 octaves) TRTR Temporal Resolution is the same for all sinc wavelets with 2 octaves or greater bandwidths having the same f max T R = 1 / (1.5)f max Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Non-Binary Complexity peak-to-trough separation (ms) spike separation (ms) TRTR 16-to-64 Hz (2-octave) sinc wavelet convolved with alternate polarity spike pairs of unequal amplitude Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 What is the effect on temporal resolution when the amplitude of the second spike of a set of alternate polarity spike pairs is varied? G. Partyka Oct 2006

Questions Can different types of zero-phase wavelets be compared in terms of temporal resolution. What wavelet shape is well suited for comparing against? Can we separate the ability of zero-phase wavelets to resolve thin beds from variations in side-lobe tuning effects? What is an optimum wavelet shape? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Wavelet Shape and Sidelobe Interference Wavelets designed with a vertical or near-vertical high end slope exhibit high frequency sidelobes that can cause significant distortions in reflection amplitudes and associated event character. An alternate wavelet is proposed called the Texas Double in recognition of the primary characteristic being a 2-octave slope on the high frequency side. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Texas Double Wavelets Time Domain: –negligible high frequency sidelobe tuning effects. –maximum peak-to-sidelobe amplitude ratios. Frequency Domain: –vertical or near-vertical low-end slope. –2-octave linear slope on the high-end. Amplitudes are measured using a linear rather than decibel scale. –end frequencies correspond to the highest and lowest recoverable signal frequency components of the recorded data. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Development of High Frequency Side-Lobes REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006

Development of High Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

Development of High Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

Development of High Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

Development of High Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

Development of High Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

Development of Low Frequency Side-Lobes REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006

Development of Low Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

Development of Low Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

Development of Low Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

Development of Low Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

Development of Low Frequency Side-Lobes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

High Frequency Held Constant (Klauder Wavelets) REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) G. Partyka Oct 2006

High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

Decreasing the Low Frequency Slope REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) G. Partyka Oct 2006

Decreasing the Low Frequency Slope frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octaves G. Partyka Oct 2006 Wavelet

Decreasing the Low Frequency Slope frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave slope G. Partyka Oct 2006 Wavelet

Decreasing the Low Frequency Slope frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave slope G. Partyka Oct 2006 Wavelet

Decreasing the High and Low Frequency Slopes REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) G. Partyka Oct 2006

Decreasing the High and Low Frequency Slopes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) octave sinc G. Partyka Oct 2006 Wavelet

Decreasing the High and Low Frequency Slopes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) G. Partyka Oct 2006 Wavelet

Decreasing the High and Low Frequency Slopes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) G. Partyka Oct 2006 Wavelet

Decreasing the High and Low Frequency Slopes frequency 100 amplitude REFLECTIVITYIMPEDANCE Travel Time (ms) Temporal Thickness (ms) Temporal Thickness (ms) Texas Double G. Partyka Oct 2006 Wavelet

Proposed Standard Equi-Resolution Comparison 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. It is rather unsatisfactory to compare the traces against one another since there are too many unknowns. A standard comparison is needed. The standard trace proposed is one where the convolving wavelet has the same temporal resolution as the sinc wavelet over a given bandpass but has no sidelobes whatsoever. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Low-Pass Sinc vs Low-Pass Texas Double peak-to-trough separation (ms) spike separation (ms) 0– Hz Sinc TRTR peak-to-trough separation (ms) spike separation (ms) TRTR Conclusion: Over a given low-pass, temporal resolution of the Texas Double wavelet equals 80% of the temporal resolution of the sinc wavelet. T R = 1 / 1.5f 4 0– Hz Texas Double T R = 1 / 1.2f 4 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Equivalent Temporal Resolution: Ormsby to Low-Pass Sinc fsfs frequency amplitude f4f4 f3f f s /f f 3 / f 4 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Is it worth giving up the 20% loss in Temporal Resolution? Can the benefits associated with attenuating high- frequency sidelobes outweigh the 20% loss in temporal resolution? Well-log based comparisons, suggest that they can. …as long as that 20% is not critical to the required imaging. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Well-Log Comparison raw rc Reflectivity raw Raw InputDesired Standard High Frequency Tuning Effects Only Negligible EffectSinc WaveletTexas Double G. Partyka Oct 2006

Well-Log Comparison - 3 Octaves raw rc raw layering LayeringReflectivity G. Partyka Oct 2006

Well-Log Comparison - 2 Octaves raw rc raw layering LayeringReflectivity G. Partyka Oct 2006

Gradually Increasing Frequency Content to Examine Tuning Effects When filters change in a linear and gradual manner, we would hope that the traces would do likewise. Unfortunately, sidelobe interference gets in the way. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

Well-Log Example - Layering Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering Raw Layering

Well-Log Example - Layering 10 Hz High-Cut Slope Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering Raw Layering

Well-Log Example - Layering Texas Double Wavelets – high frequency side varies 52 to 112 Hz; low frequency held constant Raw Layering Raw Layering

Well-Log Example - Layering Texas Double Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering Raw Layering

Well-Log Example - Layering Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering Raw Layering

Well-Log Example - Reflectivity Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC Raw RC

Well-Log Example - Reflectivity 10 Hz High-Cut Slope Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC

Well-Log Example - Reflectivity Texas Double Wavelets – high frequency side varies 52 to 112 Hz; low frequency held constant Raw RC

Well-Log Example - Reflectivity Texas Double Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC

Well-Log Example - Reflectivity Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC Raw RC

Texas Double in Practice One may implement the Texas Double on real data by first running an amplitude whitening program followed by a 2-octave high-side slope Ormsby filter. The Texas Double design criteria should not be a goal of data acquisition. 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. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006