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**Seismic Resolution of Zero-Phase Wavelets R. S. Kallweit and L. C**

Seismic Resolution of Zero-Phase Wavelets R. S. Kallweit and L. C. Wood Amoco Houston Division DGTS January 12, 1977 This PowerPoint version of the material, was compiled by Greg Partyka (October 2006) G. Partyka (Oct 06)

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**Seismic section whitened between 5Hz and 60Hz **

Quiz Given: Seismic section whitened between 5Hz and 60Hz 11ms measured two-way interval time 18,000ft/sec interval velocity of bed Find the thickness of a carbonate encased in shale: 99ft 49.5ft 18ft any one of the above none of the above Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**roll on your own: __-__-__-__**

Quiz Define the ormsby wavelet (f1-f2-f3-f4) that has the same resolving power as a 65Hz Ricker wavelet. all of the above roll on your own: __-__-__-__ f2 f3 amplitude f1 f4 frequency Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**The time interval between the wavelet’s primary lobe inflection points **

Summary Temporal Resolution: The time interval between the wavelet’s primary lobe inflection points The minimum two-way time through a thinning bed as measured on a seismic trace. Different types of zero-phase wavelets may be compared in terms of temporal resolution. The ability of zero-phase wavelets to resolve thin beds can be separated from variations in side-lobe tuning effects. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**What are the Limits of Temporal Resolution?**

As interval times through a thinning bed becomes less and less, how accurately do the measured times represent the actual, vertical two-way travel times through the bed? This questions may be further divided into two related questions: 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? 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 G. Partyka (Oct 06)

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Why Zero-Phase? The use of wavelets that are not zero-phase greatly complicates the analysis of seismic resolution. The use of zero-phase wavelets simplifies resolution because 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 G. Partyka (Oct 06)

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**A Historical Perspective**

Since past work on the subject of seismic resolution is considered by many investigators to be definitive, new concepts and results will be examined carefully and compared with those in the literature. We begin our study in the field of optics by examining the Rayleigh criterion of resolution, We consider next a resolution criterion developed by Ricker (1953), and finally We consider the criterion established by Widess in 1957 and published again in Geophysics in 1973. In each case, the theoretical limits of resolution will be related to parameters that can be measured on the wavelet that is convolved with the reflectivity sequence. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Rayleigh’s work relates to resolution of two diffraction patterns. **

Rayleigh’s Criterion Optical diffraction patterns caused by light transmitting through a narrow slit may seem far removed from seismic resolution. It establishes a criterion of resolution which is often cited by many investigators in regard to seismic wavelets. Rayleigh’s work relates to resolution of two diffraction patterns. Two wavelets are resolved when their separation is greater-than or equal-to the peak-to-trough time of the convolving wavelet. The text book “Fundamentals of Optics” by Jenkins and White (1957) is a good reference. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**sin2x x2 b A b Rayleigh’s Criterion**

Rayleigh chose to keep the mathematical relationships involved simple. When applied to wavelets other than sin2x / x2, the “dimple-to-dimple” amplitude ratios may vary. A 0.81A b Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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Ricker’s Criterion A bed, represented by two upright polarity spikes convolved with a wavelet, reaches the limit of resolvability when the bed becomes so thin as to cause a flat spot to appear in place of the two maxima. This occurs a a spike separation interval that can be derived equating to zero the second derivative of the convolving wavelet. This observation was made first by Ricker in his classic paper “Wavelet Contraction, Wavelet Expansion, and the Control of Seismic Regulation” by Norman Ricker (1953) Geophysics, Vol. 18, No. 4, p Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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Ricker’s Criterion Ricker also recognized that as a bed, represented by two spikes of equal amplitudes but opposite polarities, becomes thinner and thinner, the complex waveform produced by convolving the spike pair with a wavelet looks more and more like the time derivative of the convolving wavelet. It was left to Widess to expand on this concept further in his paper: “How Thin is a Thin Bed” by M. B. Widess (1973) Geophysics, Vol. 38, No. 6, P Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Relates to resolution of two “Ricker” Wavelets.**

Ricker’s Criterion Relates to resolution of two “Ricker” Wavelets. 2 f Y(f) = e1 - (f/f1)2 f1 b Flat d2(Kt) A a7R = b dt2 Rayleigh’s Criterion Ricker’s Criterion “Wavelet Contraction, Wavelet Expansion, and the Control of Seismic Regulation” by Norman Ricker (1953) Geophysics, Vol. 18, No. 4, p Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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The Widess Criterion Widess considered a thin bed as one where the complex waveform across it does not differ significantly from the derivative of the convolving wavelet itself. This definition is useful for thin bed “detectability” studies, but causes problems when it comes to thin bed “resolvability” considerations. At the bed thickness Widess first considers a bed to become a “thin” bed, i.e., when the bed thickness is about l/8; the apparent thickness is actually l/4.6 which is the peak-to-trough time of the derivative of a Ricker wavelet. J. Farr in his paper: “How High is High Resolution (1976) SEG Preprint, states that a bed as thin as l/40 may be detectable. It should be understood, however, that the apparent thickness remains at l/4.6. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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The Widess Criterion The complex waveform across a thin bed approaches the time derivative of the incident wavelet as the bed thins to zero thickness. Widess states, “A thin bed is one whose thickness is less than about l/8 where l is the predominant wavelength…” Comment: The minimum time directly measurable through a “thin” bed may be calculated from d2(Kt) / dt2 = 0 For a Ricker wavelet then, a “thin” bed is one which has a thickness less than l/4.6 “How Thin is a Thin Bed” by M. B. Widess (1973) Geophysics, Vol. 38, No. 6, P Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**t = “Predominant Period”**

The Widess Criterion (Kt) Sonic Log Sonic Log t Thick Bed Thin Bed t = “Predominant Period” 1 / t = “Predominant Frequency”. Not to be confused with “Peak frequency”. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Kallweit and Wood Proposition**

We now 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. In order to separate the concept of “resolvability” from the related concept of “detectability”, the term Temporal Resolution (TR) will be used to denote resolvability. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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Temporal Resolution The term “temporal resolution of a zero-phase wavelet” may be defined as the time interval between the wavelet’s primary lobe inflection points. This time may be derived from the equation: d2(Kt)/dt2 = 0 This is the minimum two-way time through a thinning bed as measured directly on a seismic trace. A wavelet’s inflection points are found by setting equal to zero the second derivative of the wavelet itself. “Peak frequency” (f1)is not to be confused with the term “predominant frequency” which is defined in the literature as the reciprocal of the wavelet’s breadth (Tb). Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution (TR) of the Ricker Wavelet**

Kt = [ 1 – 2(pf1 t)2 ] e -(pf1t)2 amplitude TR Tb f1 frequency peak frequency: f1 temporal resolution: TR = 1 / (3.0)f1 wavelet breadth: Tb = 1 / (1.3)f1 peak-to-trough: Tb / 2 = 1 / (2.6)f1 relationship of Tb to TR: TR = 0.43Tb = 0.86Tb / 2 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution in Relation to a Sinc Wavelet**

Let us now discuss temporal resolution in relation to the band-pass sinc wavelet. This wavelet represents the output of an amplitude whitening process such as programs DAFD and WELCON. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution 8 Hz – 64 Hz Sinc Wavelet**

TR Tb / 2 Peak-to-Trough Time Separation 60 50 Peak-to-trough separation (ms) or apparent thickness (ms) Peak Amplitude of Wavelet 40 30 20 8 Hz – 64 Hz Sinc Wavelet 10 10 20 30 40 50 60 Model spike separation (ms) or actual thickness (ms) Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution of the Low-Pass Sinc Wavelet**

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. A low-pass sinc wavelet is not realizable in actual practice because it has frequencies extending to zero Hertz; nevertheless, it is instructive to study this wavelet. Temporal resolution is established in terms of the maximum frequency, and results can be used in the discussion of the bandpass sinc wavelet. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution of the Low-Pass Sinc Wavelet**

2 f4 sin (2p f4 t) amplitude Kt = (2p f4 t) TR T0 fm f4 frequency Tb temporal resolution: TR = 1 / (3.0)fm = 1 / (1.5)f4 wavelet breadth: Tb = 1 / (1.4)fm = 1 / (0.7)f4 peak-to-trough: Tb / 2 = 1 / (2.8) fm = 1 / (1.4)f4 1st zero crossings: T0 = 1 / 2fm = 1 / f4 relationship of Tb to TR TR = 0.47Tb = 0.93Tb / 2 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution (TR) of the Bandpass Sinc Wavelet**

The band-pass sinc wavelet is the difference between two (f1 and f4) sinc functions. the f1 sinc function 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. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Non-Binary Complexity**

What is the effect on temporal resolution when the amplitude of the second spike of a set of alternate polarity spike pairs is varied. temporal resolution decreases slightly and in fact approaches the peak-to-trough time of the convolving wavelet. This behaviour is due to the constructive interference of the main trough of the positive spike wavelet on the center lobe of the negative spike wavelet. Note also that the “pseudo-thinning” pocket just prior to temporal resolution, is enlarged. Application of these temporal resolution concepts is discussed in the attached report on “Designing Optimum Zero-Phase Wavelets”. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution (TR) of the Bandpass Sinc Wavelet**

2 f4 sin (2p f4 t) 2 f1 sin (2p f1 t) Kt = - f4 sinc (2p f4 t) (2p f1 t) f1 sinc band-pass sinc amplitude fm = (f1 + f4) / 2 f4 TR T0 f1 f1 fm f4 frequency Tb temporal resolution: TR = 1 / (1.5)f4 ; 2 octaves (where f4 / f1 .ge. 4) wavelet breadth: Tb = 1 / (0.7)f4 ; 2 octaves (where f4 / f1 .ge. 4) peak-to-trough: Tb / 2 = 1 / (1.4)f4 ; 2 octaves (where f4 / f1 .ge. 4) 1st zero crossings: T0 = 1 / (2fm) ; all octaves relationship of Tb to TR: TR = 0.47Tb = 0.93Tb / 2 ; for sincs .ge. 2 octaves TR of sinc and Ricker wavelets are equal when f1 (peak frequency Ricker) = f4 (sinc) / 2 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Sinc Bandwidth and Temporal Resolution with Constant fmax**

30 30 20 20 peak-to-trough separation (ms) peak-to-trough separation (ms) 10 10 TR 2–64 Hz Sinc (5 octaves) TR 16–64 Hz Sinc (2 octaves) 10 20 30 10 20 30 spike separation (ms) spike separation (ms) Conclusion: Temporal Resolution is the same for all sinc wavelets of 2 octaves and greater bandwidths having the same fmax TR = 1 / (1.5)fmax Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution (TR) for All Band-Pass Sinc Wavelets**

2.0 TR for all band-pass sinc wavelets 1.9 TR = 1 / C f4 1.8 1.7 1/2 ms dTR for 64 Hz f4 TR sinc = TR 4-64 sinc within 1/2 ms Resolution constant (C) 1.6 1.51 1.5 1.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 f4 / f1 1 2 3 4 octaves Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Temporal Resolution (TR) with Spike Pairs of Unequal Amplitude**

30 1.0 25 1.0 -1.0 1.0 -0.8 20 -0.6 16-to-64 Hz 2-octave sinc wavelet convolved with alternate polarity spike pairs of unequal amplitude peak-to-trough separation (ms) 15 10 1.51 5 TR 5 10 15 20 25 30 spike separation (ms) Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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**Application of these temporal resolution concepts. **

Next? Application of these temporal resolution concepts. “Designing Optimum Zero-Phase Wavelets”. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka (Oct 06)

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