RE6OLUTION OF SEISMXIC REFLECTIONS
by
ROBERT W. WYLIE
B.S., The Pennsylvania State University
(1954)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIEiENS FO. THE
DEGREE OF ASTER OF
SCIENCE
at the
IL
MASSACHUS4ETTS INSTITUTE OF
TECHNOLOGY
June, 1956
Signature of Author.
Department of Geology and Geo
y&ica, Kay 15, 1956
Certified by
Thesis Supervisor
Accepted by
. .- a
-
.
-
.
-
-W,
-
--
hairman, Departmental Committie on
Graduate Students
I
ABASTRACT
Resolution of Seismic Reflections
by
Robert W. Wylie
Submitted to the Department of Geology and Geophysics on May 15,
1956 in partial fulfillment of the requirements for the degree
of Master of Science.
The problem under investigation is that of resolving seismio
echoes from two reflecting horizons in the earth where the horizons
represent the boundaries of a bed of rook which may be thinning
and hence of possible importance as a trap for petroleum. The
echoes from the boundaries of a thinning bed are recorded by standard
seismic methods as two waveforms which overlap when the bed thins
sufficiently. One needs special methods to resolve the two events
in these cases.
Assuming that these reflections can be represented as two
distinct waveforms or wavelets, the purpose of this thesis was
to investigate some approaches to the problem of determining their
times of occurrence in the case where overlapping does obscure
the second event. A brief discussion of Norman Ricker's papers
concerning seismic wavelets and seismic resolution (representing
the published work on the problem to date) is presented.
The use of the mathematical inverses of wavelets as contractor operators is investigated and the limitations of such an
approach are noted. Such operators should give perfect resolution
in a noiseless system. These studies were carried out on theoretical wavelets and on data from a group of reflection seismographs.
It was decided that in practice the noise problem is severe.
Another group of experiments is presented representing the
uses of a symmetric operator in resolution of our two events. The
specific group of symmetric operators which were studied were unfolded cosine transforms of the reciprocals of the amplitude
spectrums of the wavelets. In general the results were superior
to those obtained from inverse operators but more work is necessary.
The limitations and possible applications of the approach are noted
along with some suggested future investigations.
Since the above approaches had definite limitations a
numerical method for applying a least squares fit
-2-
was set up and
is presented. Experiments were not carried out on this method
and it is left in mathematical form as a future possibility,
Computationally the least squares fit method is relatively involved but it is felt that this may be a closer answer to the
original problem and that it should be investigated.
Thesis Supervisor:
Stephen M. Simpson, Jr.
Assistant Professor of Geophysics
Title:
4=-
TABLE OF CONTENTS
page
Title
Page
Table of
................................
1
Contents....................................
4
Acknowledgements.....................................
7
Introduction.... ...................................
8
Review of Previous Work on Contraction............... 10
Formation of an
Inverse..............................
15
Fig. 1 Ricker wavelets V(25) and V(oo) with their
18
spectral characteristics...........*..........
Fig. 2 Computed inverses to V(25) and V(coD) of
Fig. *
.
.
......
...............
.
.
.
.
.
19
Fig. 3 Inverses of V(25) and V(co) showing exponential divergence...... ....
*..............,
20
Fig. 4 Comparisons of re-inverted approximate inverses with original wavelets................. 21
Fig. 5 Stable and unstable wavelets and corresponding amplitude and phase spectra............... 23
with the inverse of the
1
Fig. 6 Comparison of
stable wavelet associated with the amplitude
characteristic of V(25)....................
Fig. 7 Smoothed inverse operators and their
spectral characteristios......................
24
25
Fig. 8 Convolutions of V(25) and V( oo) with
their approximate inverses.................... 26
Fig. 9 Artificial complexes and their resolution
by inverse operators.......................... 28
-4-
Table of Contents
(Continued)
Fig. 10 Phasing test................................
page
29
Fig. 11 Results of convolving approximate
inverses of V(25) and V( oo) with
V(o)' and V(25) respectively................
31
Fig. 12 Four records exhibiting pinchout............ 33
Fig. 13 The average wavelet from Record 7.19
and its spectral characteristics............
34
Fig. 14 The inverse of the modified wavelet of
Fig.* 3 .
. .
.
..
.
..
.
..
35
. . ..
..
Fig. 15 Convolution of combination waveforms
with first 9 terms of the inverse of
the first wavelet from Record 7.19..........
37
Fig. 16 Four traces from Record 7.19 convolved
with 9 term inverse.........................
38
A Study of the Resolution of a Symmetric Operator
Chosen to Produce Contraction of a
39
Specific Waveform...........................
Fig. 17 Reciprocal of amplitude spectrum of
smoothed wavelet from Record 7.19 and
the symmetric operator......................
40
Symmetric operator and its amplitude
characteristic for sharp-front wavelet......
41
Fig. 19 Convolutions of symmetric operator from
sharp-front wavelet of different lengths
with the sharp front wavelet................
42
Fig. 20 Convolutions of symmetric operator formed
from smoothed wavelets to different
lengths with the smoothed wavelet...........
44
Fig. 18
Fig. 21 Convolutions of smoothed wavelet with
symmetric operator from sharp-front
wavelets........ .0...
* ..........
Fig. 22 Smoothed wavelet convolved with symmetric
operator from sharp-front wavelet which
has been modified by a method of Cesaro
oooO**oo~o
oooo ooo oooo
SU.M
-5-
45
0........
-** *Ooo
~
46
46
Table of Contents
(continued)
page
Fig. 23 A portion of Record 7.19 filtered by the
symmetric operator formed from the
smoothed wavelet............****.*.........
48
Fig. 24 A portion of Record 7.19 filtered by the
symmetric operator formed from the sharp
front wavelet... .......
49
.*........*....
Fig. 25 Phase test on Fig. 18........................ 51
Fig. 26 Phase Test on Fig. 20........................ 52
Fig. 27 Phase Test with 1/2 spacing on Fig. 20...... 53
Fig. 28 Phase test with 1/2 spacing on Fig. 18...... 54
Fig. 29 Phase test with 1/4 spacing on Fig. 20....., 56
Least Squares Method of Resolving Overlapping
Seismic
57
Signals..........................
Fig. 30 Geometrical Configuration.................... 57
Fig. 31 First and second wavelets (W(t) and U(t)).... 58
Fig. 32 W(t - x) and U(t - y)........................ 58
Fig. 33 Short interval of seismic interest........... 59
Functional Relationships in Interval of Interest
59
Discrete Notation.....................*...*
Fig. 34 Geometrical Configuration.................... 61
Fig. 35 Arrival of a reflection across 2 seismograms.v..o.....................0040.0...
.62
Steepest descent Method Applied to Discrete Data0.... 66
Conditions Which may be Enforced in Order to get
a Closer Fit................................. 68
Conclusions and Suggestions for Future Study......... 69
.00000000
References.....................000.00..00.
-6-
71
Acknowledgements
of
completion
In
it
thesis
this
pleasure
great
me
gives
to express my gratitude to the many people who's encouragement, suggestions, and efforts helped to make this study
possible.
Especially I should like to thank Professor
S. N. Simpson
who's ideas and guidance make up much of the
train of thought behind this work.
I appreciate also the
assistance of Mr. Harold Posen who worked with me on this
problem and who's results are incorporated into this thesis.
grateful to Mr. D. R. Grine and Mr.Sven Treitel
Likewise, Ia
for the use of their WWI computer programs which made
possible the numerical work.
I am grateful to the Geophysical Analysis Group project
which
under
a
deal
great
of
this
and
done
was
work
to
which
I am indebted for the considerable amount of computer
time on WWI which the experiments represent.
The majority of the raw data was supplied by Dr. Dunlap
of
the
Atlantic
Refining
Company
and
to
him
and
to
Dr.
of Phillips Petroleum Company, who's suggestion started
this investigation, go my sincere thanks.
Many other suggestions aided the investigation and
for these I am indebted to all the members of the GAG
project and to the project's sponsors.
0-7-.
Piety
Introduct$on
One of the major problems of interest in exploration
seismology is that of establishing the boundaries of a bed
which is thinning and may or may not be pinching out.
Exploration-wise one or both of these possibilities,
pinch-outs or thinning, may be of economic interest in
different areas, but in any case the boundaries need to be
defined.
Much of this mapping has been carried out by
intuition and extrapolation from areas in which the boundaries
have been established with the maximum resolution of the
instruments at hand.
Even if a definite answer as to the
exact boundaries is unobtainable thenvis, nevertheless,
much room for refinement of seismic resolution in these
areas.
By increasing this resolution, extrapolation and
intuition need .be used to a much lesser extent and uncertainties
will be decreased greatly.
The goal of this paper has not been to find a method
which will give the exact boundaries but has been to
investigate methods of refining the data produced by present
seismic instruments in order to obtain greater resolution
in pinch-out or thinning areas.
Here we are dealing
with a specific problem, present over small intervals of
seismic records, and have therefore attempted to investigate
specific tools which will aid in the solution of this
special problem.
It is felt that much of the work will
be of interest, however, to persons working with more
-8-
general questions of seismic resolution.
All of the tools investigated require a much closer
insoection of the data coupled with varying degrees of
more difficult comoutation than would be encountered
in an ordinary interpretation process.
Increasing
resolution, however, is quite often worth a great deal of
effort.
In this investigation the experiments have been
carried out numerically on WWI.
It is worth mentioning
however that, where a numerical method
is known to work,
large quantities of data are to be studied, and high
accurracies are not to be required, electronic analog
equipment may often be economically built and put into use.
The investigation is far from complete and
I have
therefore included descriptions of all the studies so
that if the problem is carried on the research can at
least start a few steps closer to a final answer.
Some of
the investigations which proved fruitless are mentioned
in a sentence or two where they logically came up in the
investigation in order that these blind alleys could be
avoided.
The experiments concerning methods having
possible adaptation in some areas are presented in full.
-9-
REVIEW OF PREVIOUS WORK ON CONTRACTION
Our interest in this paper is sighted at improved
resolution of two or more signals where their arrival
times are such that the first signal is superimposed on the
second.
We are therefore interested in the character or
shape of these signals as they appear ox a seismic record
so that we might distinguish the arrival of trailing
signals which may be obscured by the original reflection
wavelet.
The signal or wavelet shapes as observed in
practice are fairly common knowledge but little work has
been published on them.
Some work has been done in determin-
ing them theoretically.
In 1953 Norman Ricker published two papers in Geophysids(6)
(7) concerning the wavelet theory of seismogram structure
and wavelet contraction, wavelet expansion, and the control
of seismic resolution respectively.
In the first of these
articles Ricker attempted to describe the action of an
imperfect elastic medium on a seismic disturbance using
two basic assumptions: first that the initial disturbance
at the origin is given by a doublet, and secondly that the
disturbance is transmitted according to Stoke's differential
wave equation for a visco-elastic medium.
-010-.
3f Ct2&
e
37J
c'
= elastic disDlacement
where
= viscosity of the medium
=
density of the medium
is set = 0
wave velocity when
= time
With these basic assumptions Ricker proceeds to develop
theoretical velocity and displacement type seismic wavelets for varying distance relationships.
However, many
questions have been raised concerning this work.
interpretation of the paper, Bowman
In an
(3) points out four
general points of controversy due to some of the mathematics
and to the initial and some additional assumptions incorporated in the study.
The questions were:
(1). "Are the mathematical parts of the theory on
a sound basis?"
(2). *Does
experiment bear out the theory?"
(3). 'How does the non-realization of the assumed original
pulse shape in
-practice affect the applicability,"
of the theory?"
(4).
"What is the physical meaning of the assumed
wave equation?'
In general the answers to the above questions as partially
offered by Bowman and others seem to support the mathematics but indicate a reasonable amount of doubt as to
the actual occunrame of Ricker type wavelets in field
studies.
Strong objection has been made due to the
difference in Ricker's model of the earth from the actual
and to the lack of field evidence supporting his theoretical
damping conclusions.
Ricker's experiments and theory were based on an
earth model which was visco-elastic.
As far as this
earth model is concerned Ricker's theory has been shown to
be correct.
Ricker's experiments were carried out on the
Pierre Shale and in this area there was a very good agreement of theory and observed data.
However the properties
of Pierre Shale differ from most rocks in that this shale
is almost truly visco-elastic.
Jeffreys
(4) pointed out
the non-visco elastic state of most rocks and the fact
thau a more complicated wave equation was in order.
Van melle (0)and Treitel (9) have shown that the wave
equation used by Ricker is also non-physical from the energy
relationships.
In his second paper in 1953 Ricker incorporated the
study of his wavelets into a paper concerning seismic
resolution.
Using these wavelets as models Ricker considers
a seismogram, free of distortions, as being made up of
large numbers of fundamental wavelets with different time
origins superimposed on one another.
The analysis of a
seismogram he then considers as the breaking down of the
record into its wavelet components.
To do this, Ricker
proposes a wavelet contractor as an electronic filter,
which when coupled with proper redording techniques would
produce a distortionless seismogram.
The individual
wavelets on such a seismogram would be contracted to a
-12-
lesser breadth without altering the relative arrival times
of the wavelet centers.
Ricker set down two conditions which
were to be met within the band of relavent frequencies.
First: The chase characteristic must be a linear
function of the frequency f with intercept on the phase
axis of zero or an intergral multiple of r.
For the wavelet
to emerge erect, the phase intercept should be zero or
an even multiple of 2r.
Second: The amplitude response characteristic must be
of the form
A= A0ekf2
where f is the frequency and A
and k are constants.
In-as-much as the characteristics of this wavelet
contractor have been derived from the wavelet theory which
has been seriously questioned a great deal of reserve must
be used in evaluating the usefulness of the method and the
equipment.
It was shown by Robinson(S) that Ricker's
wavelet contractor could be duplicated by a mathematical
linear oDerator.
Using Ricker's conditions and assumptions
Robinson achieved the same results.
On actual seismograms
the maximum contraction achieved by Ricker's contractor
was approximately 0.8 of the original breadth.
Within this range Ricker's contractor, even with its
non-physical basis, seems to have applicability and sufficiency
for seismic studies.
It was hoped, however, that we might
find a better method of seismic resolution using a different
approach which did not incorporate some of the weakness
-13-
of Ricker's assumptions.
accomplish this.
This thesis study was launched to
II
Formation of an Inverse
In transforming the seismic recordings through his
contractor Ricker- 's operations required that the shape of the
wave forms remain constant.
It was felt that if this condition were dropped better contraction and/or resolution
could be affected with no loss of information pertaining to
the onset times of the reflected energies. This was the
general problem and the study proceeded historically in almost the same pattern as this thesis.
The operation which we are searching for is one which will
give a better indication of the time of onset of seismic reflections. As in most studies we have assumed that the seismic
traces as time functions may be represented as a noise function
plus signal functions whose time origins are directly related
to the geometries and the velocities involved. Finding these
onset times is the broad problem. Discovering them in areas
where they are not visible on the seismic record due to interference from other signals and noise without destroying the
information from other signals is the specific problem. The
most practical uses of such a method is of course that of
finding the onsets of two overlapping reflections such as
would represent a pinch-out or thinning of a bed bounded by
two reflecting horizons. It was with these goals in mind
that we proceeded.
Considering the reflection from any
horizon to be a finite time series and having relaxed the
condition for constancy of wavelet shape, it was then quite
logical to attempt to find an operator which would produce
a spike or near spike at the onset of the waveform and which
would produce negligible output before and after this onset.
These characteristics suggest the mathematical inverse. This
inverse when convolved as a series term by term with the
original series will produce a series which will be an impulse
followed by zero terms. In finding an inverse for a time
series one merely performs a polynomial division into unity
obtaining a polynomial whose coefficients are the desired
inverse.
a0 + alz + a2 z2 + a3 z3.. 0
where a0 , a1, a2 ... ar are the ordinates of the
wavelet.
Unfortunately, as was found by experiment, most inverses
of waveforms found on seismic records are unstable. That is
to say, the higher order terms in the true inverse polynomial
become increasingly divergent. These divergent series will of
course theoretically still give a perfect spike when convolved
with the original wave form if one retains the infinite series
of the inverse whose terms diverge also to infinity. Again
there is a logical compromise. As suggested by Dr. Piety (5 )
we need only use the first group of smaller terms in our inverse to produce a spike in convolution which will be valid
to the number of terms used from the inverse. This restriction
may or may not be serious in application depending upon the instability of the inverse and the interval length over which we
wish to operate.
Experiments were then set up to test the applicability
of our proposed "chopped-off" inverse. Two waveforms were
chosen for the initial study from Ricker's theoretical wavelet
tables. Thei- were the Ricker wavelet "taken at infinity" and
the Ricker wavelet "taken at twenty-five" which we shall henceforth respectively denote as V(ao) and V(25). These wavelets
were modified so that there initial values represented a discrete jump in amplitude, the original waveforms as derived by
Ricker having extended in time from minus to plus infinity.
This modification was necessary in order that any stability
might be achieved by our inverse and it does render the wavelets
more physical. -- In our notation we will designate 1/V ( )
as the inverse,-- Using these modifications (Fig. 1) the two
wavelets were divided into unity and 180 terms of 1/V(25) and
of 1/V(co) were calculated in order that we might observe the
nature of the instability. (Fig. 2 and 3).
The stability of the inverse of any symmetric wavelet
and hence the true stability of any symmetric wavelet has been
shown to be impossible (MIT GAG Report 9, Section 4 ( I)) and
hence the instability of V(oo) was known. In general, if any
of the roots of the z plane transfer characteristic lie within the unit circle the wavelet has an unstable inverse. This
was shown to be the case for V(25) also by observing the nature
of the computed inverse (Fig. 2 and 3). We note that in Fig.
3 the inverses apear to be time series added on to exponential
functions. As we are only interested in the first portion of
our inverse this exponential function has been determined and
subtracted from the raw inverse in various ways in attempts to
create an inverse which would be valid over a wider range of
convolution.
The logarithmic plot of the inverses (Fig. 3) shows the
exponential dependence very clearly. The functions aebx and
a(ex-1) were determined by fitting the asymptotes of 1/V(25)
and l/V(oD) to straight lines on this logarithmic scale. In
subtracting these exponential functions, the term -1 has little
effect on the large terms of the inverses but it does tend to
render the first values , which are the most important in this
study, more similar to the exact inverse. In. Fig. 4 we show
the effect of retaining consistency in the first terms. Here
we have reinverted the modified inverse of V(25) and V(o) and
have found
and
1
1
V(2.5) - a(ebx-l)
-17-
1
V(00)
-a(e
bx~l)
V(25)
0.8r
V(25)
30
0.6Tu
20
+
o.4n
10
+ 0.2Tt
4)
M
Time
Frequency
in cycles per second
in units of h
(h assumed = 25T m4S.)
-0.4
-0.6ni
-0.8I
-Ty
Ty
V(o )
0.8v
V(o )
0.4Tu
0
o *
0.2n
Ca
0
CS
Frequency in
cycles ner second
.S.
*0.2TcO
-0. 6r
-0.8r
wig 1
Ricker Wavelets
V(25) and V(ee)
with their spectral characteristics.
V(255
t
20
10
30
40
5
Time
in units of h
V00)
10
Time
in units of
305
.
PIg.
Xi IComputed inverses to V(25) and V(..o) of 1pig,
1
0
20
40
60
80
100
120
Inverses of V(25) and V(m) showing exponential
..20"'
140
160
divergence
180
200
OhLIllAL *xVELET V(25)
OhIGINAL WAVE LET V(oc)
-
:5
dH
a
C
time in units of h
h assumed = 2.5 m-s
tipa
= 0.39
a(e
a = 0.4
15,,b1
b=
1
V 25)-
time
l)
1
-
.85
a(e)
-
1
a ebX- 1)
-
- -o
00
0
00
F~T~
.time
tie
time
Fig.
Comparisons of re-inverted approximate inverses
with original wavelets
a(eb)
to be substantially similar to V(25) and V( o) whereas
I
I
va)
V(oo)
differ from the original wavelets completely.
The question also came up as to whether the inverse was
not just an exponential plus the inverse of the stable wavelet
whose amplitude spectrum was the same as that of the original
wavelet (Fig. 1). Using a factorization orogram developed by
other members of the GAG project this wavelet and its inverse
were formed (Fig. 5) from report lOb MIT GAG project. As we
can see in Fig. 6 this guess proved to be false.
As a result of these experiments the modified inverses
V(6)
ak
Y&)
were used in further studies. These are shown with their
amplitude and frequency spectrums in Fig. 7. It can be
noticed in Fig. 3 that V(25) seemed more stable than V( cc).
This was interpreted to indicate the probability that the
anomalous roots of the characteristic polynomial of V( m)
were further inside the unit circle than those of V(25).
The amplitude and phase characteristics of the modified inverse of V(25) at high frequencies are close to what we would
expect for a true inverse (amolitudes related reciprocally,
phases being the negative of each other) but the phase condition breaks down at low frequencies. The spectral characteristics for V( m) and its modified inverse do not fit these conditions as well. This phenomena would seem to indicate that
convolving the modified inverses with the original wavelet will
produce a spike plus low frequency. The results of this convolution are shown in Fig. 8 and the preceding analysis is
shown to have been correct.
Using these convolved results we then proceeded to set
up a theoretical noiseless seismic situation with V(25) in
-22-
Seconds
-
STABLE AND UNSTABLE WAVELETS
Fig.
7IT
30
25
20
15
Spectrum of Ricker Wavelet
-Spectrum of Stable Wavelet
Phase
Amplitude
r
-
Phase
10
5
0
0
-20
0
50
100
Cycles
150
CORRESPONDING AMPLITUDE AND PHASE SPECTBA
-23-
200
Fig.5
Inverse of factored wavelet
Ampl.
Time in units of h
Inverse of V(25)
Ampl.
Time in units of h
Fig.
6
COMPARISON OF
WITH THE INVEhSE OF THE STABLE
WAVELET ASSOCIATED WITH THE AMPLITUDE CHARACTERISTIC OF V(25)
-24-
15
Inverse of
V(25)
rinus a[eh-13
a-0.39
b-0.05
I
Ampitud
1
I
2
/\
Ampl.
0
150
~~yI
C.
hn
Phase
C
5
-C.2nT
-0.4n
Time in
(~0
units of h
-0.6n
I
I
/
I
/
\,/*--.~
-0.8n
-
-n
0.8m
10
/
I
15
Inverse of V(ad minus
i
pitude
a[ebx-1)
4
a-0.
b-0.085514
0.6r
I
0.4n
5
A
I
I
-13
10
20
5
2
'30
---
IO
---
5
5Freuercy in dps.
i
'.
i5o
I'
/V
0.2nr
- -.
....
.
/
-0.2r
assuming h-25mss
Time in units of h
/
/
-0.4n
/
Fg*
7
Smoothed inverse operators
and their spectral
characteristict.
I
-
I
-
-6"
-0.8en
- a(e
V(25) Convolved with [1)
-1)
Relative
Amolitude
Time
in units of h
V(oo) Convolved with [
)
- a(ebx
Relative
Amolitude
Fig. 8
Convolutions of V(25) and V(oo) with their
approximate inverses
order to show its possible use under ideal conditions. These
results are shown in Fig. 9. In this test we assumed that a
Ricker wavelet occurred on a noiseless section of a seismic
trace and that after a time lag another positive or negative
V(25) of different amplitude occurred. The resulting combination hid the second event in both cases. Using the modified
inverse as an operator a convolution was made and definite
resolution was oroduced.
This was of course an ideal case where the first and
second wavelets were similar, in phase, and where there was
no noise. As our data is taken at discrete intervals in
digitalizing any seismic information we were first interested
in the effects of phasing. The occurrence of any reflection
over a seismic record will have different onset times on
various traces. We must of necessity continue our digitalization at a constant spacing over the interval of interest
on the seismic record or traces. It is obvious, therefore,
that we very seldom hit the first term of our wavelet with
a data point. It is this first term which our inverse has
been developed to spike and hence there is need to study the
sensitivity of the approach to a shifting. Such an experiment was carried out on V(25) for shifts of 1/4, 1/2, 3/4
and 1
and the results are shown in Fig. 10. There appears
to have been a shift in the convoluted results and some slight
changes in amplitudes. The characteristics are still quite
good however. The data roints of V(25) appear to be close
enough together and l/V(25) insensitive enough to small
changes that we can conclude that for this operator and wavelet phasing is of only minor importance. In a later section
of this paper we shall see how the spread of the data points
can make the phasing a major problem.
The problem of noise or variation in wavelet shape
-27-
Component
Wavelets
V(25)
Wavelet
domolex
Resolution
by inverse
Operators
Fig. 9
Artificial wavelet complexes and their resolution by inverse operators
V(251
ORIGINAL
WAVELET
e
e= b minus first term
CONVOLVED
WAVELETS
Ampli
Time in units of h
READING a
Amoli
Time in units of h
RFADING b
Ampli
Time in units of h
READING c
Amolil
Time in units of h
READING d
Ampli
READING e
Phasing test. Inverse operator of wavelet from
readings (c convolved with wavelets from readings in
different phase relations
Fig. 10
-.29-
appears to be much more critical in the case of our inverse
operators. If there were only the original wavelet for which
we had established an inverse followed by another waveform in
our interval of interest we would still be in good straits.
The first waveform would produce a spike followed by a relatively flat period until the onset of the second waveform
which in convolution would at worst produce a transient. This
ideal case is in practice unobtainable as we invariably have
some noise which will distort the shape of the first waveform.
This will vary also from trace to trace. In order to discover the effects of this phenomenon a convolution was made
with the modified inverse of V(25) and V( co) and the modified
inverse of V( oc) and V(25). These results are shown in Fig.
11. The modified inverse of V(25) appears to be relatively insensitive to changes in shape compared to the modified inverse
of V(oo). Even the former points out definite limitations of
this process in situations where we must contend with noise.
The operation must be sensitive enought to indicate the
onset of a set waveform but not sensitive to the degree that
it will blow up when it is convolved with a wavelet other than
its inverse. Thus if we were concerned with wavelets of the
same form as V(25) we would appear to have solved our general
problem. Ricker points out, however, that ordinary seismic
filters do not yield wavelets of the same form as V(25). They
are more likely of the form shown in Fig. 13. It should be
pointed out also that this dissimilarity of waveforms may not
be totally electronic and. as stated previously the occurrence
of Ricker's wavelets in actual practice is questionable. Going back to Fig. 1 and noticing the form of V(25) there is
reason to expect difficulties in applying the inverse operator
approach to contracting waveforms usually observed on seismic
records. Most waveforms in practice do not have their maximum
100
I
V(25) convolved with
-25
bxa(ebx
-A~' f~
-
Ampl. _
-
§
/
Time
in units of h
-50
-75
750
V(o ) convolved with [E
)
- a(ebx-Jy)
500
250
Ampl.
0
-250
Time in units
of h
-500
-750
Fig. U
Results of convolving approximate inverses of
V(25) and V(c4 with V(oo) and V(25) respectively.
-31-
amplitudes within the first few terms. Usually they build up
to maximum values in later terms, a characteristic which will
tend to increase the instability. One would expect the inverses of these waveforms to diverge quite rapidly.
In order to study the overlapping of two waveforms from
an actual seismogram a suite of records was obtained from the
Atlantic Refining Company which exhibited several pinch-out
structures. Of these we chose four records for our study (Fig.
12). The reflection picks at 0.98 and 1.07 seconds on record
7.19 were seen to get closer and closer together as we moved
over horizontally to record 7.16 where they seemingly combined
into one wavelet complex at 0.98 seconds. Analysis was started
on the two distinct waveforms which were present on record 7.19.
The first waveform aopearing around 0.98 seconds on record
7.19 was averaged over this record by lining up the maximum
positive peaks on every other trace, digitalizing each over a
fairly wide range, and averaging these time series term by term
(Fig. 13). Thus we have assumed that the noise is uncorrelated
on the average with the signal. The averaged wavelet was then
modified so' that it would have a definite zero origin and would
tail-off to zero. It is noticed on Fig. 13 that the averaged
wavelet from 7.19 has its largest positive amplitude not in the
first swing as did V(25) but rather in the second positive form
later in time. This is much more in line with the results of
other field observations.
An inverse was calculated for this modified wavelet (Fig, 14)
and as predicted it was widely divergent after only the first few
terms. The nature of the terms in this inverse also show no
signs of possessing any generalized smoothing function such as
a(ebx-l) which was so useful in rendering the inverses of the
Ricker wavelets stable over long intervals of time.
-32-
0
0
0
CQ
H
b;
9
D UP
AVEBAkGE WAVELET
MiODIFIED FOhM
/
25
/
20
/0.
15
0.81
/7
0
/\
o.-r
-0.8.
Fi.13
The average wavelet from 7.19 and its spectral characteristics
-
-- powl
aws!
-
-
--
-
Time
in milliseci
Fig. 14
THE INVERSE OF THE PiODIFIED kAVELET OF
-35-
Fig. 13
The first nine terms seem all that we should really employ in
forming an inverse operator. This operator, although of very
limited use, is nevertheless good enough to separate the first
wavelet in the noiseless case from another waveform. In order
to check this the wavelet complex occurring around 1.07 seconds
on 7.19 was averaged in the same manner as was the original wavelet. This complex was added on to the first wavelet after a
time lag of four spacings (10 milliseconds) and this new complex was then convolved with the first nine terms of our inverse. In this noiseless case the onset of the two events was
quite clear (see Fig. 15). This limiting condition of a noiseless system in which we are only able to discern between events
occurring within 20 milliseconds is quite restricting. As we
notice when we convolve our nine term operator with some of
the actual traces from 7.19 the noise in the actual cases tends
to obscure any spiking (Fig. 16). Our inverse approach is applicable however in discerning the onset of a second event shortly after the known onset of a known waveform in a relatively
noiseless system. These crnditions are quite restrictive and
we are therefore left to search for another app-roach.
We were next led to attempting to fit a Ricker wavelet to
our wavelet in the hope that this would be suitable means of
creating an inverse operator. Due to the configuration of our
waveform which we previously discussed the closest fit was to
V(oo). Convolution of the inverse obtained from our modified
V(oo) and the first modified wavelet from 7.19 produced diverging terms with no recognizable spike. This method was then
dropped.
-36-
First Wavelet From 7,19
Ampl.
Second Wavelet From.7.,19
Amu,
V
Combination Wave-Form
Ampl
V
Convolution of Combination
Wav-Form with First 9 Terms
of the Inverse of the First
Wavelet from 7.19
Fig.
-37-
is
1.1ial
rPnJIZ saonds
ii
£
ii;
1k)
Trace 1 convolved with 9 term tWverse
~A!IAAP~A
e 7 convolved with 9 term inverse
Trace 12 convolved with 9
inverse
18 convolved with 9 term inverse
L
t
I
0-38-
I
A STUDY OF THE RESOLUTION OF A SYMMETRIC
OPERATOR CHOSEN TO PRODUCE CONTRACTION OF
A SPECIFIC WAVEFORM
In the ideal case in which we are concerned with a stable
wavelet (i.e. having a stable inverse) it is seen that our convolution process will produce a single spike which has a white
light amplitude spectrum. It was hoped therefore, that we
might be able to produce contraction, although surely not a
single spike, if we convolved experimental wavelets with the
damped series which are the cosine transforms (unfolded to produce a symmetric operator) of the reciprocals of the amplitude
spectrums for the original waveforms. This convolution would
have as its output a series which would have a white light
spectrum and which it was hoped would be a close approximation
to a spike.
The reciprocal of the amplitude spectrum was, therefore,
computed for both the sharp-front wavelet (i.e. initial value
was quite large - as used in finding an inverse) modified from
the averaged wavelet from Record 7.19 and a modification of the
same wavelet smoothed so as to have gradually increasing amplitudes before its first positive maximum (compare wavelets on
Figs. 19 and 20).
The cosine transforms for both of these reciprocal amplitude spectrums were computed and are also shown on
In order to investigate the contraction properFigs. 17 and 18.
ties various lengths of these symmetric operators were then convolved with the smoothed and sharp-front wavelet. In Fig. 19
the symmetric operator formed from the sharp-front wavelet to
49, 25 and 15 terms was convolved with the sharp-front wavelet.
These results show very good spiking for all three lengths of
the operator with similar forms and amplitudes in each case.
The dependence on operator length was evident in Fig. 20 where we
convolved 49, 25 and 15 terms respectively of the symmetric operator(formed from the smoothed wavelet and the smoothed wavelet
itself. Here we notice that the 25 term and 49 term operators
-39-
-
-
-
t
- !
{
A. J
.
.
-T
.
.-
,
i
4
,
l
.
4
-
_4
--
-t
--
4
-
-
oit
-4I
T7
tin~ue in
-
maI
'Sea4n
tjt
4
I
A
--
4 t
**--
-4o-
'6
- -
r -
''9
I
i
*
Ampl.
0
10
N/
20
-
Time in millisecondS
90
7G
60
50
40
30
20
10
Ampl.
Fig. 18 Symmetric operator and its amplitude characteristics obtained from
cosine transform of reciprocal of amplitude characteristic of
Sharp-Front Wavelet
Sharp-Frontl Modifticattar. of
Averaged Wavelet from Record 7.19
Sharp-Front Wavelet Convolved
with 49 Terms of the Symmetric
Operator which is the Cosine
Transform of the Reciprocal
of the Amplitude Spectrum of
the Sharp-Front Wavelet
+
Sharp-Front Wavelet Convolved
with 25 Terms of the Symmetric
Operator which is the Cosine
Transform of the Reciprocal
of the Amplitude Spectrum of
the Sharp-Front Wavelet
Sharp-Front Wavelet Convolved
with 15 Terms of the Symmetric
Operator which is the Cosine
Transform of the Reciprodal
of the Amplitude Spectrum of
the Sharp-Front Wavelet
/
-- +.10
time 'mliiseconds
Fig. 19
p
A
+
--
~
-
~
ef Waeet t
i
mUhiBeeo~s
Smoothed Wavelet Convved
with 49 Teres of Syisetr4o
Operator Formed frox
Smoothed, Wavelet
Sinoothed Vavelet ConVeIfd-
with 25 Terms ot sy atri4
Operater Formed tromn
Smoothed Wavelet
p e
A
WVV\ IV1
vl*'*,
Smoothed Wavelet C.avo1ved
with 15 TOems of SM-" trte
operater Formed From
Smoothed Wavget
,+91
UUI'
Fig* 20
produce relatively low leading and trailing amplitudes in comparison with the results of the 15 term convolution. The leading
and trailing amplitudes for the convolutions in these two experiments were small compared with the amplitude found with the
25 and 15 term convolutions shown in Fig. 21 where the symmetric
operator developed from the sharp-front wavelet was convolved with
the smoothed wavelet. In this experiment the 49 term operator produced a usable spike despite the variation in waveform
This
seems to indicate that this method is at least less sensitive
to small variations in wavelet shape than our inverse approach.
It was hoped that we might be able to decrease the leading
and trailing amplitudes by smoothing our syMmetric operator. A
method of Cesaro sums was used on our operator formed from the
sharp-front wavelet giving zero initial and terminal values and
weighting each of the ordinate values linearly from these zero
extremes to the maximum at the point of symmetry. The convolution
of this revised symmetric operator (taken to 49, 25 and 15 terms)
with the smoothed wavelet resulted in a drastic reduction in contraction as is shown in Fig. 22. It was concluded that the
amplitudes of leading and trailing terms of the operator could
not be diminished by this process without a marked loss in contraction and this variation was dropped.
It was noticed that the symmetric operator formed from the
sharp-front wavelet to 49 terms produced a spike when convolved
with the smoothed wavelet and that the symmetric operator formed
from the smoothed wavelet to 25 and 49 terms produced similar results when convolved with the smoothed wavelet. Now inasmuch as
a smoothed wavelet seems to be more physical than one with a
sharp-front - and since the convolutions from the latter's operator seem to be somewhat better - it was decided that the 25
term operator from the smoothed wavelet and the 49 term operator
from the sharp-front wavelet would be used in further experiments
concerning this method.
-44-
-T---
Ample
Original wavelet
10ms.
Convolved with 49 term
operator
Ampl.
Convolved with 25 term
operator
Ampl
Convolved with
15 term
operator
Ampl
Fig. 21
Convolutions of Symmetric Operator Formed from
Smoothed Wavelet with Sharp-Front Wavelet
-45-
t
II
T
14K
iK F
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-46-
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In an attempt to find how our contractor operators would
behave on an actual seismic record we convolved 25 terms of the
symmetric operator formed from the smoothed wavelet and 49 terms
of the symmetric operator formed from the sharp-front wavelet with
alternate traces from Record 7.19 over the interval from 0.91
seconds to 1.16 seconds (refer to Fig. 12). These results are
shown in Figs. 23 and 24 respectively, and are both encouraging
and discouraging. The results are noticed to be very similar
and in this experiment there seems little advantage of one operator over the other. It is seen that the first event or reflection is evident when looking at the filtered record section
as a whole, but that the second event, expected at about 1.07
seconds, is obscured. This may be attributed to the fact that
our second waveform on Record 7.19 is a complex and is probably
much different from our first wavelet (see Fig. 15). In practice
we might then find the cosine transform of the reciprocal of the
amplitude spectrum for the second waveform and use it as an operator over the same interval in order to pick out the second
event. Carrying out these two independent operations on the
complex on Record 7.16 we might then expect to find the onset
times of the two events.
Another note of interest in connection with Figs. 23 and
24 is that the spiking appears to die out on certain traces
where there was an identifiable waveform on the original records.
This suggests again the possibility of phasing and noise effects.
In order to study the effects of phasing, ordinate values
of the smoothed wavelet were interpolated for each 1/8 th
spacing (original time interval = 2.5 milliseconds, here we interpolated an ordinate for every Le milliseconds interval).
The convolutions were then performed of 25 terms of the symmetric
operator formed from the smoothed wavelet and 49 terms of the
symmetric operator formed from the sharp-front wavelet with the
smoothed wavelet with ordiante values corresponding to a shift
of 1/4, 1/2, and 3/4 data spacings (by a similar arrangement
to that shown in Fig. 10). These results are shown respectively
-47-
Record time
tn seconds'
Irace
1.10
.00
.9o
.
/lA
A
1AA AflA\
I/v
.
i9.
Ia
At A
III
1
Ip
V
uIV
V
It
I
........
.
I.
I
.V
{
.......
..... .i...
..
..
.....
....
[r~e J.1
A I~I
_
V
._..LW....
1
____
___
___
__
J
IIAF
'_
___
___
_
_
___
_
_
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_
_
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_
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...
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LLW
Trace 12.
I..
1-7.
A
Trace 16. 1
AA
,!
A
r-
i
I
'V
\ /L
AAAA A
Record time
in seconds
1.20
1.00 sec.
0.90 sec.
Fig. 24
1.10
sec.
1.20 seo.
A portion of record 7.19 filtered by symmetric operator
or1aW
A
M 5 'arp-
waveekA
W-/
Here we notice that the spacing and relative
breadth of our waveform are very critical. As expected, the worst
resolution was produced with the half shift where the convolution
in both cases produced two diminished spikes instead of the single
event which was produced in the convolutions where there was no
shift. It would be very difficult to tie down the onset of the
firwt event let alone pick out the onset of the second overlapping event on any single trace if the onset of the first
signal satisfied this worst case. With the present phasedependent convolutions we must depend on the ensemble of convolutions
across the record for resolution, weighting more strongly the results of the in-phase traces.
It was hoped that we might eliminate the phasing effects
by decreasing our data spacing and therefore another phase test
was set up. The values of our symmetric operators, the 25 term
operator formed from the smoothed wavelet and the 49 term operator formed from the sharp-front wavelet, were interpolated for
half spacing. These new operators were then convolved with the
smoothed wavelet with 1/2 spacing shifted 1/8, 1/4, and 3/8
of the original 2.5 millisecond spacing. The results are shown
in Figs. 27 and 28 and, as the results are similar for all members
of the two sets, we can conclude that finer spacing will solve
the phasing problem. However, it is also noticed that there has
been a decided decrease in the contraction.
This may quite
probably be due to the inaccuracies of the interpolation used
to form the symmetric operators for the half spacing and hence
to the difference of the interpolated values from actual values
which could be calculated exactly. It is felt that the 1/2
spaced operators will be more sensitive to changes in wave
shape but that with a more accurate computation of the symmetric
operator the resolution should be almost as good for any shift
with 1 /2 spacing as it is for the exact fit with regular spacing.
Mathematically it can be shown that the above relationship holds
in Figs. 25 and 26.
-50-
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IN
Smoothed Wavelet from Record 7.19
time in milliseconds
Smoothed Wavelet Convolved with Symmetric
Operator from Sharp-Front Wavelet
No Shift
Convolution with shift of 1/ 4 spacing
V2. 5 ms.,
+
v.7Ji
0
r-mr-19
04
Convolution with shift of 1/2 spacing
2 .2ms
24
Convolution with shift of 3/4 spacing
(2. ma
4 -3)
owl--
709
-4
06- a
v
1
1
PHASE TEST ON CONVOLUTION OF SMOOTHED WAVELET WITH SYMMETRIC OPERATOR
FROM SHARP-FRONT WAVELET (49 terms)
Fig. 26
------
fmooth1ed Wa velet from 7.19
v
i
1
i
-r-
1 timei in
ft
a
*
a
hi
r- "--T-
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2op
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Cdnvolitio8
with
fro
moothed *veleit
6f
(d
1alf S a in
7.19
every 1..25 ms )
poiitI take
ta
convolited with symmetric operator formed from
ite Sharp- Vr6nt! WaveLet Wiith Valuesl Inte poJlaqted
tolale Spakbing I
N.Shi..
LK....
N
hit
0
V
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pa ihhia
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shirt in original
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ivy
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oi
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e t Data
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.
2 Spacing with 2.mA
Wavelet Data
~
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3 shift ii
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IT H-/2 SPACI14G
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27
Smoothed Wavelet from 7.19
~ii'
rL-A-
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.
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milliseconds
p
I
p
Convolutions of Smoothed Wavelet from 7.19
(with 1/2 spacing) with Symmetric Operator
Formed from Smoothed Wavelet(With 1/2 spacing)
With and Without Shifting
4
No Shift
vw
u4
r---T
L-
r-
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1/2 Spacing with 1/8 shift in data points
1/2 Spaoing with 1/4 shift in data pointw
~2I12-
~h
It:
4
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,3pacin
i
l
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Ll,
Fig. 2s
and is only in the nature of our data handling that we have
failed to produce the desired contraction. For completeness
we also convolved the snoothed wavelet and the symmetric operator
from the sharp-front wavelet to 1/4 spacing (Fig. 29). These
results show a larger, decrease in the conLraction than did the
experiments at 1/2 spacing and further indicate that our interpolations were not yielding accurate ordinate values.
Further investigation is proposed inasmuch as there is a
good possibility that, with closer calculation of the cosine
transform of the reciprocal of the amplitude spectrum for the
half spaced wavelet d ata, we can produce an operator which will
have little or no phase dependence and which will contract the
wavelets. If, due to the nature of our data, we are unable to
obtain an operator which has the above properties and is also
fairly insensitive to small changes in wavelet shape, our
method may still
be of some value.
Looking at an ensemble of
seismic traces operated on by one of the symmetric operators
such as in Fig. 23 or 24 the contraction obtained with using
data of the original type (spacing = 2.5 milli3econds) may be
enough to pick out the first event. Subsequent operation with
an operator designed to contract a second event may likewise indicate the time of occurrence of that event. Adding the results
found in both operations and taking into account the overall effects we may be able to obtain the desired information as to
the boundaries of our thinning bed.
P, V-A
IE6
V, %ID,IC-_
V
NN
11I
1
V
time
-+10
VI
+milliaeconds
V
Partial Convolution of Smoothed Wavelet and
Symmetric Operator from the Sharp-Front Wavelet
Both Taken at 1/4 Spacing
(Data points every 0.625 millisecond7
Pige P9
LEAST S4UARES 1ETHOD OF RESOLVING
OVEhL"PPING SElSMIC SIGNALS
The above approaches to solving our general problem seem
to have limitations and therefore a method of least squares
fitting is also proposed. This method is computationally much
more involved than any plan set up thus far in this paper. It
is therefore suggested that the method, mathematically set up
below, be programed for a digital computer and tested. If the
resolving power is good enough it is then suggested that,
economics having been weighed, special purpose analog equipment
be built to do the same work. This analog system would of course
run large amounts of data much more economically than a digital
machine to the accuracies required.
A Least Squares Fitting Method of Resolving 0verlaooing Seismic
Signals
In applying a method of least squares we first will describe the procedure using functional notation and then will
develop the discrete form which we will have to deal with in
an analysis by digital means.
A
B
-01-
aGovveirical
Fig. 30
-57-
t ov'
We will in general be interested in extending the resolution of reflections from two horizons from an area A
where these two events are distinguishable on seismic records to an area B where the reflectors are close enough together that their waveforms overlap and where one or both of
their onsets can not be distinguished. (see Fig. 30.) We
further assume that in area A the two events are reoresented
by the onset of two distinct waveforms W(t) and U(t) resDectively. These two events may be determined by an averaging process over records in area A, (See Fig. 31) requiring
that similar filters are used, it is assumed that the average
wavelet representing the reflection of a horizon in area A
will remain constant in amplitude and duration for the same
reflector in area B. This is of course a first aoproximation
which might be relaxed later if more information were available.
Our two averaged waveforms W(t) and U(t) can be expressed
Fig. 31
as distinct time functions and their existence at set times
after an arbitrary origin may be represented as W(t - x) and
U(t - y)
Fig. 32
-58-
Short Interval of Seismic Interest
As we are only interested in a short interval of the records in area B representing that period of time over which
reflections from the two horizons are possible, we may set up
a new time origin and limit encomoassing this interval (Fig. 33).
Fig. 33
Interval of Interest on Seismic Record
This restriction to a short seismic interval will allow us to
deal with such oarameters as velocities, normal move-out, and
associated the geometries as constants. This will simplify
our computation greatly. Likewise we will later realize
simplification if we can restrict ourselves to small angles
of dip.
Functional Relationshios in Interval of Interest
Over this short interval we can then express the contribution of the first wavelet W(t) on the Ith trace as
W(t - T1 -
e (a)).
Where T
is its time of onset on the
middle trace from the time origin 0 and e (a) is the moveout time to the 2th trace for an angle of dip of the first
reflector (a). Here we see that e is a known function of
a-for the Ith trace. Likewise we may reoresent the comnonent
of the 2th trace due to the second wavelet as U(t - T2 - d1(b))
where T2 is again the onset time on the middle trace and
-59-
d I(b) is the move-out time associated with the Ith trace
and the angle of dip b. of the second bed.
The Ith trace may now be represented from 0 to P as the
time series F (t) = N'(t)+ W(t - T
- e (a)) + u(t - T2
Here N (t) represents the noise function from 0
Le+ta
-
d (b))
to P
4Otfme
fl(t) = W(t -
where Tl,
a,
T2 ,
T
eI (a)) + u(t - T 2 - d ()
-
@ are arbitrary functions.
Then if the
noise function N1(t) is uncorrelated with
W(t
-
-
T
e (a))'4#u(t - T 2 - d (b))
on the average we can set up an error function
E2 (,1'
aE @ =
2
1
o
(F (t) - f (t))2 dt
and expect to find a minimum as
T,
l*.T 1
T2 ---
T2
a
a
simultaneously
pN
-
b
In practice there is of course no unique minimum and
there will probably be several over the interval. If we restrict ourselves first to a certain range of a's and @'s and
to possibilities that T2 > T1 our field is somewhat simplified.
Making a good guess as to Tl, T2 , a, and P from information
available in area A will be our best aid to solution. Having
found the error function for the values of our first guess
we next the gradient of this error function and use it to
determine a new set of variables. With this new set of variables a new error function is calculated and the procedure
-6o-
is cycled until the variations reach a lower designated
limit. The function E2 is seen to be dependent on four
independent variables and thus in finding the gradient
E2 we have the direction of steepest descent in four dimensions.
Discrete Notation
Using the same geometries we now proceed to setting up
the problem in a discrete form. We may consider the Ith
trace from 0 to P to be represented by
F1 (t):
Y
O*ih*P
h = spacing
t.vt.O
We consider our geophone-shotpoint configuration to be an in
line split-spread set up with 2M recorders each spaced a
distance s apart all occurring on a horizontal plane (i.e. our
data has been assumed corrected to a plane surface as in Fig.
34).5%o
7
;7
7~
7 7 7'
7, 7
'17
/
I
St4Vaee
Ig
iS=
jeophoanv 1A*ebV 4
Fig. "34 Geometrical Configuration
-61-
2D
=
oswhere V is the average velocity
T will now be
considered for the interval 0 to P. The move-out time
el(a) is somewhat more complicated. Considering the numbering of jugs from the down dip side to the up dip side separately (as shown in Section 4. Report 10b, MIT GAG)(2).
L
P)
+e.
(M-) V
Ms
-(M
sA6be-mvuts
and
cos
I
up di/O
*-
a
1
Ve 4
e,eiI
Ow
di
?e$pee1fi/e/
From these expressions we can find the exact move-out
times for any angle of dip and any trace number. For a
first approximation, in order that we might simplify the
computation, we assumed that our move-out times were linear
functions of the trace numbers. That is to say, that the
onset times of a given event on all the traces on one side
of a split spread were assumed to lie on a straight line
rather than along a curve (Fig. 35)
,4|~
/,;.,,
of A
"e-Aciaa/ Cabrve
'e00
Fig. 35
wunrsD( 4pphe tt ,
oftArrtPve 1
oe/d l
Arrival of a Reflection Across a Seismogram
ce
This assumption leads to relatively small errors (decreasing with depth) when compared with anomolous times of
arrival due to small perturbations (MIT GAG Report lOb,
Section 4 ( 2 )).
r
( 4 ))
'
O
_ES+
Csof
5m)
For the end traces
COO)
(A,+
:A
(-M)
Vcos
Ot
+ (a
the middle trace
an assumption which improves with depth.
The move-out times of our interval of interest
mt.,
I
M-o) k*Iot,(AM
%
*L
C(o')=
0
2D, ( M-1)
0()
Cos L/ 'SIV%
"pg
,(M
(
- )v eY at
Now the move-out time per trace on the down dip side of the
record will be
(M-I
(M
4(o
)(M)-..
4D-a
-63-
)V Cos
C
,(
-
It is seen that these functions can be found for any angle
if we assume a value for V and D. The step-out on the trace
is now (m - 2) - (a)L or (2 - m)-(a)u depending on which
side we are calculating. These move-out times when general
will not produce integers and these quotients must be rounded
off in order that the points of the digitalized traces will
coincide. Otherwise a method of recalculating the amplitude
values on the traces for each non-integer shift
(m - I) +_ (a) L
.r(I - M) e (Wu
O"
h
h
would be necessary.
Likewise we find the move-out time per trace for oIr
)
second wavelet
Ms(M; Ve~dr)
(M+d)
c~
These move-out times may also be represented in discrete
units of h. We shall consider these shift to be represented
on the 2th trace as
e I(CL)
h
d
u
nd
h
where these functions will be rounded off to the nearest
integers,
We now transfer our functional representations of our
wavelet s
1( )otatf)n
an
Into discrete notation.
-64-
Wc{) Wt
4.
t;
A
.
(t) =Ai
40v tz c k
LA
ta
W
)
f
(S"re
~t:L~1
,st4 3z)
Assume that W(t) is Q, units of h long and that U(t)
Q units of h in duration.
is
?4e(t
I
k
Then
; Ov.
W4 (I.*-
.A
0~-LOA
4-
4d
<
Likewise
QA (t- )
=
L4
(f -1.-
.4.1's
4d )
7
ando(A
?
d
(j)
=-o
44
L
-. 65-
We may now also set up f (t) in discrete form.
ee 4
(Or
7t
.1
-
r
f+
L
+
Having formed f (t) we now proceed to find the error function.
(Ft.
+
where subscripts u and L again denote up and down dip.
Steepest descent method aoplied to discrete data
Estimating Tl, T 2 , a and P from what we observed in
area A and using these values in the above error function
a value of E2 1(T,
T2 , a and A) is calculated.
To improve
this first estimate the gradient of the error function is
taken at these values and we proceed along the gradient to
our next estimate. This process is repeated until tolerances
are met. In discrete notation we will proceed to this end by
-66-
means of variations.
First we vary T 1 , by a small amount and
calculate E2(T 1 + A , T2 , x, a) holding the other functions
at the original estimated values. Likewise we do the same
thing for T2 , a, and P increasing each of these by the same
increment while holding the other variables constant.
a) Subtracting the four new error functions from the original,
.,
E'(
p)
T,
,Af,,--
-K
b) Finding the square root of the sum of the squares of these
remainders,
c) Find the ratios
x
wV
V
an increment of change H is chosen
$x
d) Multiply this incremental change (H) by
and change T by
this amount. Similarly multiply Z times the increment
times the increment and add these
times the increment and
estimates.
values algebraically to I the original
II
e) With these new values T 1, T2, at and p' recompute the
error function and call it
in a similar
f ) Compute E - 'E2 and proceed to form "
manner from the values of 'E , Tj, Ir, aL P! and cycle
is within tolerances.
E the process until
The values of T1 N + 1
T2N +
1
N + 1
N + 1 are then
the estimates of Tl, T2 , a, and b respectively. The value
of W e
at which the iterative method is stopped
would be determined previously.
-67-
I
Conditions which may be enforced in order to get a closer fit
If we no longer relax the assumption that any event
occurs in general at non-integral values of spacing onthe
trace and furthermore falls on a curve rather than a straight
line we may come closer to the exact values of Tr,
T2
, a,
P.
We
To utilize this information all data must be readjusted.
assumed that there is no curvature across our record and that
there was no change in shape or amplitude of a wavelet from
area A to area B. It is seen that these assumptions may be
dropped with an increase of course in computational labor. It
is felt however that the method proposed above will yield
values of T1 , T2, a, P which are as reliable as our seismic
records permit for small angles of a and P and for non-shallow
reflections.
-68-
CONCLUSION6 AND SUGGE6TIONS FOR FUTUhE STUDY
From the experiments concerning the inverse we may conclude that this approach will be of most use in separating two
events when the onset and shape of the first are known and where
the difference in onset times from that of the first wavelet to
that of the second waveform does not exceed the length of the
time interval during which the terms of the inverse of the first
wavelet are not greatly divergent. We must further specify a
low noise level., There may also be specific instances in which a
form of an inverse may be used as an operator over an interval
longer than itself with good results but this would depend on
the occurrence of fairly singular wavelet shapes.
From the experiments concerning the use of a symmetric
operator, formed from the cosine transform of the reciprocal of
the amplitude spectrum of a wavelet, our conclusions are not
quite as definite. The possibilities for applications of this
approach, however, appear to be more numerous and more promising. With the limited experimental background we can only say
that if enough data is available we can visually correlate the
events spiked by the symmetric operator and neglect nill traces,
considering them to be the result of phasing. It appears also
that the phasing problem can be solved. Thus in the case where
we have a section of a seismic record with low noise level and a
complex of two similar wavelets we would expect that convolution
with a symmetric operator determined for these wavelets should
produce two spikes considerably larger in amplitude than the
other out-put. Likewise for two different wavelets in a low
noise case e expect that it will be possible to use two different operators and obtain the two spikes and hence relative
times of arrival on successive convolutions.
It is suggested that the above experiments be continued
-69-
and that the least squares approach, developed qualitatively in
this thesis, be set up for a digital computer and be evaluated
in several experiments. From these results it is hoped that
there will be conclusive evidence as to the areas of applicability
and validity of each of these approaches.
-70-
REFEhENCES
1. MIT Geophysical Analysis Group, Simpson, S. M. Jr., Directors
1955, Stability and physical realizability: MIT G.A.G.
Report 9.
2.
, 1956, Move-oiat averaging
automation: MITG.A.G. Report 10b, Section 4.
3. Bowman, Robert, 1954, An Interpretation of the published work
of Norman Ricker: Special paper for MIT G.A.G.
4. Jeffreys, Harold, 1931, Damping in bodily seismic waves:
Monthly Notices of the Royal Astronomical Society, Geophysics Supplement, p. 318-323.
5.
Piety, R. G., 1955, Personal communication.
6. Ricker, Norman, 1953,
The form and laws of propagation of
seismic wavelets: Geophysics, 18, p. 10-40.
7.
,
1953,
Wavelet contraction, wavelet expansion,
and the control of seismic resolution; Geophysics, 18, p.
769-792.
8. Robinson, Enders A., 1953, Evaluation of linear operators in
seismogram analysis: apecial paper for MIT G.A.G.
9. Treitel, Sven, 1956, Personal communication,
10. Van Melle, F. A., 1954, Note on "The primary seismic disturbance in shale": Bulletin of the Seismological Society
of America,
44,
p. 123-125.
-71-