Transcript Document

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)
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)
Quiz
• Define the ormsby wavelet (f1-f2-f3-f4) that has the same resolving
power as a 65Hz Ricker wavelet.
– 24-26-99-101
– 11-13-88-110
– 14-18-31-125
– all of the above
– roll on your own: __-__-__-__
amplitude
f2
f1
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
f3
f4
frequency
G. Partyka (Oct 06)
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)
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 twoway travel times through the bed?
•
This questions may be further divided into two related questions:
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?
1. 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)
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)
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)
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)
Rayleigh’s Criterion
sin2x
x2
b
• 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.
0.81A
b
A
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
G. Partyka (Oct 06)
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.
769-792.
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
G. Partyka (Oct 06)
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. 1176-1180.
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
G. Partyka (Oct 06)
Ricker’s Criterion
Relates to resolution of two “Ricker” Wavelets.
Y(f) =
f
f1
2
2
e1 - (f/f1)
b
Flat
d2(Kt)
A a7R
b
Rayleigh’s Criterion
dt2
= 0
Ricker’s Criterion
“Wavelet Contraction, Wavelet Expansion, and the Control of Seismic Regulation”
by Norman Ricker (1953) Geophysics, Vol. 18, No. 4, p. 769-792.
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
G. Partyka (Oct 06)
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)
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. 1176-1180.
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
G. Partyka (Oct 06)
(Kt)
Sonic Log
Sonic Log
The Widess Criterion
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)
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)
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)
Temporal Resolution (TR) of the Ricker Wavelet
amplitude
TR
Kt = [ 1 – 2(pf1
Tb
peak frequency:
temporal resolution:
wavelet breadth:
peak-to-trough:
relationship of Tb to TR:
f1
t)2
]e
-(pf1t)2
frequency
f1
TR
Tb
Tb / 2
TR
= 1 / (3.0)f1
= 1 / (1.3)f1
= 1 / (2.6)f1
= 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)
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)
Peak-to-trough separation (ms) or apparent thickness (ms)
Temporal Resolution
TR
Tb / 2
Peak-to-Trough Time Separation
60
50
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)
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)
TR
amplitude
Temporal Resolution of the Low-Pass Sinc Wavelet
Kt =
T0
Tb
fm
f4
2 f4 sin (2p f4 t)
(2p f4 t)
frequency
temporal resolution:
wavelet breadth:
peak-to-trough:
1st zero crossings:
TR
Tb
Tb / 2
T0
= 1 / (3.0)fm
= 1 / (1.4)fm
= 1 / (2.8) fm
= 1 / 2fm
= 1 / (1.5)f4
= 1 / (0.7)f4
= 1 / (1.4)f4
= 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)
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)
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)
Temporal Resolution (TR) of the Bandpass Sinc Wavelet
Kt =
f4 sinc
f1 sinc
f1
-
(2p f4 t)
amplitude
band-pass sinc
f4
2 f4 sin (2p f4 t)
2 f1 sin (2p f1 t)
(2p f1 t)
fm = (f1 + f4) / 2
TR
T0
f1
Tb
fm
f4
frequency
temporal resolution:
wavelet breadth:
peak-to-trough:
1st zero crossings:
TR
Tb
Tb / 2
T0
= 1 / (1.5)f4
= 1 / (0.7)f4
= 1 / (1.4)f4
= 1 / (2fm)
; 2 octaves (where f4 / f1 .ge. 4)
; 2 octaves (where f4 / f1 .ge. 4)
; 2 octaves (where f4 / f1 .ge. 4)
; 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)
Sinc Bandwidth and Temporal Resolution with Constant fmax
30
peak-to-trough separation (ms)
peak-to-trough separation (ms)
30
20
10
TR
0
0
10
2–64 Hz Sinc
(5 octaves)
20
spike separation (ms)
30
20
10
TR
0
0
16–64 Hz Sinc
(2 octaves)
10
20
30
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)
Temporal Resolution (TR) for All Band-Pass Sinc Wavelets
2.0
TR for all band-pass sinc wavelets
1.9
TR = 1 / C f4
Resolution constant (C)
1.8
1.7
1/2 ms dTR for 64 Hz f4
TR 30-64 sinc = TR 4-64 sinc within 1/2 ms
1.6
1.51
1.5
1.4
1
2
1
3
4
2
5
6
7
8
9
10
11
12
13
14
3
Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
15
16
4
f4 / f1
octaves
G. Partyka (Oct 06)
Temporal Resolution (TR) with Spike Pairs of Unequal Amplitude
30
1.0
peak-to-trough separation (ms)
25
1.51
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
15
10
5
TR
0
0
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)
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)