STATISTICAL APPROACHES TO
CERTAIN PROBLEMS IN GEOPHYSICS
by
STEPHEN MILTON- SIMPSON, JR.
B.S., YALE UNIVERSITY
(1950)
SUBMITTED IN PARTIAL FULFILLMI'NT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(1953)
Signature
of Author
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-.-
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Department of Geology and Geophysics
August 14, 1953
Certified by .
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Thesis Supervisor
Chairman,
Departmental Committee on Graduate Students
ABSTRACT
STATISTICAL APPROACHES TO CERTAIN PROBLEM IN }EOPHYSICS
by
Stephen Milton Simpson, Jr.
Submitted to the Department of Geology and Geophysics
on August 14, 1953, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Several specific problems in seismic and gravitational data interpretation are considered from the
statistical viewpoint. Least squares techniques are applied
to the two types of interpretation, and, for seismic records,
other approaches are discussed.
The fitting of an nth order polynomial in x and y
to ravity data by the method of least squares is investigated as a method for approximating regional gravitational
anomalies. The normal equations for the general case are
derived and simplification considered. It is shown that,
with a symmetrical rectangular distribution of gravity
readings, each set of these equations breaks up into smaller
subsets. The resulting simplification brings fairly high
order polynomials into the range of practical computation.
For a particular gridwork the polynomial coefficients may be
expressed explicitly as linear combinations of the right hand
members of the normal equations. Once this is done, the
least squares fitting of any data taken over such a'idwork
may be effected relatively easily. The explicit expressions
for the coefficients are derived for a square gridwork of
121 points and for polynomials of order 2, 3, and 4. A set
of actual gravity readings is analysed in this fashion. The
gravity residuals are determined and contoured. The comparison of these contours with each other (for various order polynomials) and with contours derived by a standard, much more
involved process, is favorable. This consisteney, despite
certain detrimental features of the data used indioates that
the method may deserts to find practical application as a
routine, first step, Wavity reduction procedure. The
problem is pursued with regard to different gridworks, and
a table is derived which contains, in effect, the normal
equations for representative grids up to a size containing
2601 points, and for polynomials through order four.
As an approach to the understanding of linear
operators, as they apply to the analysis of seismic records,
a simple form of linear operator is studied. For this form
of operator, the so-called "cosine operator, oertain
properties are derived in the general case, and interpreted
geometrically. These include relationships between the exact
form of cosine operator chosen, the correlation properties of
the series which the cperator is to predict, the individual
errors of prediction, ead the si.ms of squared errors of
The results are applied to two classes of time
prediction.
series in connection with spectrum analysis, and, for one
class, filter characteristics are computed for a specific
cosine operator.
An iterative method for determining least squares
fits of linear operators to multiple time series is discussed
geometrically. rn argument is presented, based on the
geometry of the two term operator, to show that, in the case
of near singularity where many solutions will almost satisfy
the least squares criterion, the exact solution is necessary
for the purposes under consideration.
Several interpretive procedures are devised for
from seismic records. The first deals
information
finding
with discriminating a unknown velocity in a two velocity
system. An adaption is made for detecting reflections, and
practical example are given of the two uses. The second
employs a form of testing phase, between seismic traces and
their predictions by linear operators, to determine reflection
times, and is illustrated with an example. The third combines
the concept of ensemble averages with linear operators to
determine step-out times of reflections.
The last procedure
suggests a special experimental arrangement, coupled with a
certain type of correlation analysis, for detecting refleotions.
Included as appendixes are desariptions of four
programs written by the author for the Whirlwind I Digital
Computer.
These permit high speed computation of: two
dimensional polynomial residuals; linear operator prediction
errors and their running averages; least squares linear
operator coefficients (by an iterative method); and autocorrelations, cross-correlations and "traveling" auto- or
cross-correlations.
Thesis Supervisor:
Title:
Patrick M. Hurley
Professor of Geology
TABLE OF CONTENTS
.....................................
Acknowledgments
.
....
Introduction
LEAST SQUARES RESIDUAL GRAVITY
PART I
.
..
..
....
........
.........
=
n
=
Case
=
Applications
I-?
I-8
1-11
I-15
I-16
......-
*..............a--..
..
,.................
............
-..
....
....
...
...............
e...
m....
.....
Setting up the Normal Equations
PART II
I-6
m..................
.................--........
a=
.......
3
I-"5
m--...
.......
................................
Discussion
Example
1
2
1-2
.....
..
Simplified Solutions
Case n
Case n
Case n
s
.....
Introduction ..............................
Theory
il
a *..*
a . -
.....
.*..
*
1-17
m
1-20
SEISMIC RECORD ANALYSIS BY LINEAR OPERATORS
Introduction
11-1
11-2
11-4
.....
........
...........
a......
Statistical Methods *
Single Frequenoy Cosine Operators
Geometry of Cosine Operators
Conclusons
Example
.....
..
.........
11-6
.
.-
-
-
.
1I-8
11-9
11-10
S1-11
-
Cyolical Nature of Cosine Operators
Cosine Operator Predicting an Autoregressive Series
Cosine Operators on Series with Gaussian
Spectrum Distribution .....................-.....
Computational Example
-.--
- -
am............
A Method of Finding Linear Operators
for Least Squares Fitting Procedures
Accuracy
PART III
.
..
aa.a..
..
..
..
..
...
..
..
..
..
..
11-14
II-15
11-18
11-22
SOME INTERPRETIVE PROCEDURES
- -...
Introduction.
Velocity Separation .................................
Tests of the Method
........
a.a.aam..................
Conclusions
Phase Test ...
Ensemble Average
111-9
..................
.....................
III-10
111-13
aa.am.a..m...mea.................
................................
Travelling Correlations
III-1
111-2
111-6
II-15
.............................
Conclusions and Suggestions for Further Work
References
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
APPENDIX E
Polynomial I Description
Prediction XV Description
Description
Iteration I
kAuto Cross-Correlation I
Biographical Note
and Use Of
and Use Of
and Use Of
Description and Use Of
ACKNOWLEDGEMENTS
The author wishes to express his gratitude to
Professor P.M. Hurley, thesis supervisor, for his encouragement of and confidence in the work of this thesis.
Among his many co-workers, the author must acknowledge the help of the following persons:
Enders A. Robinson
for his stimulation, advice, and training in the fundamentals and practice of time series analysis, and in the
programming for digital computers; Theodore R. Madden for
his interest and assistance, espeoially in developing the
concept of velocity separation and suggesting the problem
of least squares residual anomalies; Markwick Smith for
many helpful discussions and comments on the various
subjects treated here; Howard Briscoe for assistance with
programming difficulties; Whirlwind tape room personnel
Margaret Mackey and Katherine Campbell, and machine operator
Robert Gilday, for their invaluable cooperation.
Finally, the author is deeply indebted to his
wifq Pauli, on whom fell the task of preparing the mannscript with its very involved mathematical notations.
INTRODUCTION
It is well known that experimental data taken in
Geophysical studies surpasses in accuracy the interpretation
that must be made on the data.
problems are very complex.
The reason is that the
For one thing,
it can be shown,
in the treatment of certain types of problems, that no
unique solution exists.
An example is the infinity of
possible mass distribution corresponding to a given gravity
profile.
In other problems the physical situation dealt
with is so inhomogeneous and anisotropic that exact solution
is impossible.
It would be hopeless to attempt to explain
rigorously the presence of any particular oscillation on a
seismogram.
Data such as this, subject to a certain amount of
randomness, and on which "best" estimates must be made, is
well suited to svatistical evaluation.
The numerical data
taken in gravity surveys does undergo evaluation of this
type.
The least squares approach, however, is not being
utilized on a large scale.
This is probably due to practical
limitations, and it is one of the problems of this paper to
see if these limitations may be minimized.
On the other hand, the raw data of seismology occurs
in analogue or ourve form.
Standard procedures of inter-
pretation consist mainly of rules of thumb, learned by long
experience, and still largely dependent on the qualification
of the individual interpreter.
There is a need to put these
procedures on a more rigorous basis,
Tbis basis may be found
in the concepts of time series as developed in economics,
meteorology, and other fields.
Huch work must be done to
determine the best means of applying these concepts to
seismic data, since, in certain ways, both the data and the
desired information are unique.
Another purpose of this
paper, then, is to propose several special methods of
applioation, and to develop certain theory necessary for a
better understanding of time series concepts as they apply
to seismology.
Statistical methods, in general, require computation,
A "program" written for a
and often on a large scale.
digital computer is a tool which will do this work automatically.
The author has written several programs for the
4hirlwind I Digital Computer to perform computations related
to the above discussed problems, and includes these programs
as appendixes, with the feeling that other investigators may
find them useful.
iii
PART I
LEAST SQUARES RESIDUAL GRAVITY
ILtroduction
Variations in the attraction of gravity over the
surface of the earth are due to many causes, but these
often fall into two general categories.
Phenomena such as
the thickening or thinning of the crust cause relatively
slow, smooth and widespread gravity fluctuations.
these regional effects.
We call
On the other hand, such things as
ore body emplacements, caverns, and local density heterogeneities cause more rapid irregular changes, and these are
termed local effects.
The actual gravity values measured over an area
usually represent a combination of regional and local effects.
The separation of these effects is of primary importance in
interpretation, and many mathematical methods have been
devised to eliminate guesswork in the problem.
Essentially,
most of the methods represent an averaging process which
gives at each point and approximate value of the regional
effect alone.
The local effect is then found simply by
subtraction from the measured values.
Many of these methods possess two undesirable
aspects.
First of all the averaging must be done at each
point individually.
Secondly, the averaging includes only
gravity values in the vicinity of the point considered.
It
is hard to say just how serious these drawbacks are, but it
seems worth-while to investigate a method which does not
I-1
In a least squares approximation all values
encounter them.
are averaged simultaneously.
Moreover, the resulting
approximation is not merely a set of discreet points but
a continuous surface of values over the area, a property
which is sometimes of value.
The purpose of Part I is to consider in some
detail how the method of least squares may be applied to
this problem, and how a simplified method of procedure may
be set up for practical application.
Part I represents an extension of the work done
by W.B. Agoes t
.
Agocs approximates the regional anomaly
by a plane surface derived from least squares criteria.
He
shows that, in an artifical example, the residual anomaly
is better derived from least squares procedures than by the
use of the "arithmetic mean regional' procedure.
For
higher order polynomials than a plane surface the algebra
rapidly becomes more involved.
Theory
It is easiest to illustrate the method for an
idealized geologic example in two dimensions.
Fig. 1.1
shows a wave in the bottom of the crust and a single ore
body emplacement.
The points on the graph would then be
the measured values of gravity across the area.
Fitting
a fairly low order polynomial to these values by least
squares gives us the curve AB which best fits all the points.
4
Ref. 1
1-2
This curve will approxitiate the regional effect more
closely than the local effect, and it is apparent that
the fit will be closest at some distance from the ore
body.
Thus the dashed line of Fig. 1.1, representing the
difference between the polynomial and the observed values,
gives a good indication of the location of the anomalous
mass.
In the two-dimensional problem the approximating
polynomial is a surface, and interpretation is made from
contours of the residuals.
Let us approximate the regional gravity by a
polynomial of order n in x and y.
n
G(xy)
n-i
E Z
=
1
1.1
7
i=0 j =0
Thus for n=2
G(xy)
000 + 0 10 x + c2 0 X2 + o
Xy + 0 1y + 00 2 Y2
The c's are unknown coefficients to be determined in
accordance with the condition that the sum of the squares
of the residuals is to be minimized.
Then the residuals are
measured values of gravity.
R(xy) =
R2 (xy)
and
Let g(xy) be the
G(xy)
g(xy)
G(xy)Y 2g(xy)
=
Hence
~
ER~
xyy
~
E
n n-k
n-i
z
(0 4 x y Z E
£x)
k=0 A=0
i=O J=O
n
-2 Eg(xy)
xy
n
n-i
EZ c
i=O J=0
I-3
x yJ
+
c
2Y")
E g(xy)
xy
or
2
Z E
R(xy)
=
XY
n
Z [Z
xy 1=0
n-i
n
E
k=0
Z
j=O
n
-2
E Cg(xy) E
xy
i=O
n-k
Z
1=0
n-i
E
k+i Y4+j g
ejk
a
o
xy]
j=0
2
g (xy)
+
XY
Differentiating this expression with respect to a,,
and
setting each derivative equal to zero for minimization, we
obtain
(n + 1)(n + 2)/
linear equations for the same
number of unknown coefficients
n n-k
k+i I+y
Z
Z: 0k-1 Z x
y
k=O 4=0
xy
where
j
xyjj
j2
xy
0, 1, ...... (n-i)
I = 0, 1,....n
There are really three variables, or sets of
variables, in equations 1.2 -
the order of the polynomial n,
the set of points xy, and the set of gravity values at
these points.
The first two of these variables determine
the coefficient matrix of the ckJ's.
Once these two are
chosen, a unique inverse matrix exists, which, if found,
may be used to compute the oki's for all sets of gravity
values taken over the same xy pattern.
This alone would
be a major simplification if the method were to be used on
a production basis.
But we shall also see that, using a
simple reasonable restriction, both the problem of finding
the inverse and the form of the inverse itself will be
greatly simplified.
G~aoFis.1.
rTT
-_____
T
/
F'~. 1.2~
M
-G{2
i
L
N
p
-
-.
'1')
Sitm21ified Solutions
In many cases gravity readings are taken over a
square, or at least rectangular, network.
When this is so
we may take the axes so that the rectangle is symmetrical
about them, and number our ordinates and absiscae in integers, as in Fig. 1.2 .
It is then easy to see that over
such a network summations of the form z y
whenever i or j is odd.
will vanish
Thus many of the coefficients of
the ck6's in equations 1.2 will drop out.
This leads to
considerable simplification, with the bigger systems
breaking up into several smaller ones.
Furthermore, if
we take a definite network we may solve the equations
explicitly for the eki's in terms of the summations
Eg(xy)x iy j.
xy
To demonstrate how this is done we shall solve
the equations for n = 1, 2, 3, and 4, over a square network
of 121 points.
The systems are positive definite and
symmetric, well adapted to solution by the matrix method
of P.D. Crout.
The non-vanishing summations over this network
are
Exy
= M=
Ex4 = Ey4 =
FIX8
4 Ref. 2
Ex2Y2
1210
Zx2 =Zy2
Ex6 = Ey6
121
=
y
Exy24 = EX4Y2
21538
451330
_EZy 8 = 10185538
I-5
12100
215380
EX26 - Ex6y2 =
4513300
Ex4y 4
3833764
h
CasTn
nm
The normal equations are
000M
+
c10ox
+
+
0 10Ex 2 +
o0 0 Ey +
01 0 ZEy +
0 00
001EY
Eg(xy)
00
zg(xy)x
c0 1 Z
y
1
F-g(XY)y
Reducing immediately to
000
010
X
14
01
Ex2
y2
giving
G(xy)
L.Eg(xy)
121
-Zg(xy)x
+
+
1210
-y-Eg(xy)y
1210
or, to six places
G(xy)
8.26448
10~4 10Eg(xy) + xEg(xy)x + yzg(xy)y]
Case n = 2
The normal equations are
00
001
002
0i1
020
ZY 2
Zy
Ey2
=
3
Zy3
Exy
ZX7
3
22
Exy 2
Exy
Exy 2
EX2
EX2 y2
EXY
2
2
Eg (xy)
Ex2
S2 2
F~g(xy)y
ZX3 y
Eg ( y )xy
2
zg (xy) y
=
x3 y
=
Eg (xy) X
g(xy)X2
which reduce to
e01 = 1210 Eg12
, =
Izg(xy)xy
12100
and three equations for c00* 002, and e20
I-6
010
-.Jjg(xy)x
1210
121000 +
1210020 = zg(xy)
1210o02 +
(Xy)y 2
1210000 +
21538002
12100020 Sg
1210o00 +
12100002
21538e2O = g (ry) X2
The solutions are
00
-l
278zg(xy)-10(Eg(xy)x2
+
Eg(xy)y 2 )3
002 = .;ij'g(xy)y2 ~ 1Eg(xy)3
s20
2
9438'
Thus
G(xy)
=
94)38
[278Eg(xy) - 10(Eg( xy)x 2 + Eg(xy)y 2 )]
+ yLxg(xy)y]
1210
lO g(xy)))
(2g(xy)y
+ Yy
+ xycj.jeg(xy)xy3
12100 -
+ x[. J.g(xy)x]
1210
+ X2
1
2
-
107g(Xy) I
Or to six plaoes
[.0294554Eg(xy) - 1.05955 10-3
G(xy) =
+
y[8.26448 10Z4Eg(xy)y]
+
y2 1.05955 10"4(Zg(xy)y2
+
xy[8.26448 10-5zg(xy)xy]
+
4
x[8.26448 10 ~Eg(xy)x]
+
x2c1.05955 10~ 4(Eg(xy)x 2
1-7
-
lOgxy)
-
iOEg(XY))3
2 + Zg(xy)y
23
The normal equations are
00
01
002
03
11
3
17172
12
02n *10
20
2
2
3
3
22
2
3
3
4
23
2
22
32
2
4
775
24
3
3
3
1)
M
y
y2
2)
y
y2
73
3)
Y2
y3
y4
4)
y3
74y
5)
2y
72
y3
4
2y2 1273
xY3
,
XY4
5
2 3
24 33
24
3 2
33
4Y2
3
2
22
3
12
13
6)
72
7)
2 2
x
9)
X2
10) x3
4
22
xy
8)
74 12
5
z
XY2
'30
3Y2
233
4
zyx3y
22
32
42
4
5
2
2
x81
2
22
2y3
3
3 2
4
3
4
5
2
x3y
x3y 2
3 Y3
4
x42
5I
X4
X5
x6
X3
Summations are assumed for all these quantities and g is
written for g(xy).
The equations reduce considerably.
Equation 5 gives us
011
121(4
1, 3, and 9, combine to give three equations for
o0
002, and o20, which have the same solutions as for
the case n = 2.
2$ 4, and 7, and 6, 8, and 10, combine to
give two independent systems which have identical coefficients.
Thus 2, 4, 7, are
1210001 +
21538003 +
12100c
= gy
21538001 + 4513300 0 3 + 215380021 = 03
12100001 + 215380003 + 215380021
L4-8
gz 7
"With solutions
001 ,
1 - 4450EigY
679536
021 =
94380
S0
0 30
I
-
2y
178gy3 - 72Egx2
-
lEgy)
f4450Egx -O178Egx3
1 --10Egx3 -178Egx]
=
6?9536
10x lEgx]
-X3
012 =27
and. from the case n = 2
000
-10 (Egx2 + Eg]
278Fg
.gr
. joyg~i
9
.
gy2 -4;19[g]
.r
002
00
020 =
C
gx
2 - lO~g)
We also have
12
12100
'-9
72Egxy2
To six places
+ £.029 4 554 ESg - 1;05955 10' 3 (Egx2 + Egy2 ))
+ y[6.54859 10~3 gy - 2.61943 10~41gy 3 - 1.05955 10~gx2y
+ y 1.05955 10* 4 (Egyz2
lozg)]
+ 731.4?159 10 5Sgy3 - 2.61943 10~4Egy]
+ xy[8.26445 10-5Egxy)
+ xy2 1l.05955 10- 5 gxy2
+ x 2 y[1.05955 10-5%gx 2y
-
1.05955 10~4Ex3
1.05955 10'4Egy3
+ x[6.54859 10"3 Sgz - 2.61943 10~4Egx3 - 1.05955 10~ 3ygxy2 j
+ X2 [1.05955 i0~4 (Egx2 . losg)3
+ x3 [1.47159 10-5 Egx 3
-
2.61943 104gxj
1-10
Case
4c
The normal equations are
00 '01
)2 03 04 11 12 13 ' l '2
9
1) M y j
2) Yj
x xy xj X9 ly
4
3)
4) y
5)
yx
|
6) xy xj x
9)
10)
xS
dg
15)
x
ZIf Z40HX 2
P
i
ff xy
yy4iVy 4
iy 94ly
l4
0
for 001s a03# 021
=
x
gx
gx
=
gZ
g
kZ Ij
= g
5, 10, 13, 15, reduce to give a
system of six equations for co
c4. 2, 4, 9, and 7,
y
x
k4 y
y
Equations 1, 3,
gxy
gxi
2y Pxx5
fy
x2
x2 2 y x 9
yY
xP x9
g
?F x# 19 IVIy#V
gxy
2V
x9 x y 7 Py X y4gly
F FxF
21V
Vy yFAF
F=g
x4 X/
0
xy xy x
0
g
E xv xfv 0y21 x1
Z Ifl
14) x
g.
Zf 4f x x lk x 4#
y I : Z9 4 x xV Zy b fy ly y
Zy o Z9 Z Z3
13)
13
x xfi x
xyk
12) x x4y xj xyB
i2
gy
xy
11)
4x
j
0
X
X j xyf X'23x yXy y r
X; 9x x xxy x xi x i
y x )x f li x2?9 2y49
7) xl xy x Xy x i9 xi
8) x9
31 10 00
12, 14,
02$ 004s 022O
zo. and
give two sets of equations
and c10, 0300 012, respectively, which
are equivalent to the corresponding equations for the
case n = 3.
I-11
The new equations to be solved are
000
002
N
y2
4
y2
y4
y6
04
022
4
2 2
K Y
020
040
2
4
=SEg
2 4
22
4Y2
= Sgy
8
2 6
2 4
4 4
Zgy4
y
Y6
2
x 22
24
2 6
4 4
4 2
6 2
22
= Zgx2y2
x2
22
2 4
4 2
4
6
2
=Sgx4
x4
x4y2
44
6 2
6
4
= Sgx
and
x2 2
2 4
4 2
= Sgxy
x2 4
2 6
S4 4
= Sgxy 3
x 4y 4
6 2
= Eg3y
x4y2
The last set has solutions
011 =
[689.84Egxy - 17.8(Sgxy3 + Sgx 3 y))
013 =
LZgzY 3 . 17.8gxy)
031
Cg
3
y -
1-12
17.8Zgxy]
The solution of the first set to six places is
400 = 4.53280 10-
2
g + 1.58932 10~4 (Egy4 + EgX)
1.35839 10~44gx2y 2 - 6.399122 10- 3 (Zgy2 +
002 = 1.62141 l3E7gy 2
-
gX2
2.81527 10-3Eg
1.35839 10-5 (gx 2 Y2 . lO~gx 2 )
-5.51847 10-5%gy 4
Egy4 - 5.51847 10-5gy2 + 1.58932 10- 5 Zg
O04 = 2.20739 10
022 = 1.35839 10 6 (Egx2y2 _ l(Zgx 2 + zgy 2 ) + 1oOEg)
020
=
1.62141 10-"3 zgx2 - 2.81527 10-3Zg
-5.51847 10-5gx 2
040 = 2.20739 10
6
-
1.35839 10 -5 (Egx 2
2
2
gx 4 - 5.51847 10-5 gx 2 + 1.58932 10-5zg
To simplify writing G(xy) we introduce the abbreviations
A = Eg
H = Egx3 y
B = Egx
I = Egxy 2
2 2
C = Egx 2
K = Egxy 3
D = Egx 3
L = Egy
2
F = Egxy
M = Egy
G = Egx2y
N = Egy3
P = Egy4
I-13
Hence
x)
C4.5328o 10 2 A
+ 1.58932 104(P + E)
+1.35839 10~*J - 6.39122 10+ y[ 6 .54859 103L -
3
2.61943 10~4N
+ y2[1.62141 10-3M-
(M + C)]
1.05955 10~3]
-
2.81527 10-3A
-
5.51847 10-5p
-1.35839 10- 5 (J - 1003)
+ Y3[1.47159 10-5N - 2.61943 10*4L]
+ y[2.20739 10-6P -
5.51847 10-5M + 1.58932 10-5A)
+ zy[1.01516 10-3F - 2.61943 10-5(K + H)]
+ xy2 [105955 10-5I - 1.05955 10~ 4 B]
+ xy 3 [1.47159 10 6 K + x 2Y[1.05955
2.61943 10-5F3
10'5G - 1.05955 10~4B]
+ x 2 Y2 [1.35839 10- 6 (J - 10(C + M) + 1oA)]
+ x 3 y[l.47159 10-6H - 2.61943 10-5]
2.61943 10~ 4 D - 1.05955 10-3I]
+ x[6.54859 10-3B -
+ x2 [1.62141 10-3C - 2.81527 10- 3 A
-
5.51847 10-5C
-
1.58932 10-*5A
-1.35839 10" 5 (J - 10R)]
+ x3[1.47159 10 5D - 2.61943 10"34B
+ x4[2.20739 10~ 6 E
-
5.51847 10-5C
Pisussioni
An interesting property which has developed in
these four oases makes the extension to higher order
approximations somewhat simplor4. If n is odd then all the
coefficients c
whose subscripts add to an even number
are the same as the corresponding coefficients for the
(n -
)rst ease.
If n is even the coefficients with
subscripts adding to an odd number are the same as for
the preceding case.
This property can be shomn directly
from equation 1.2.
Thus for the case n
coefficients (0
.5we expect nine of the
c02s
40' '40$ 020' 011, 013' 031) to
be the same as for the case n = 4,, and we need only write
the twelve remaining equations for i + j odd.
A polynomial of order equal to the number of
points taken will exactly fit the data.
However it is
practically impossible to use polynomials even approaching
such a high order for reasonably-sized gridworks, and this
danger seems slight.
choice of n.
There is still a real problem in the
If the regional effect is in reality a fairly
low order effect, polynomials of high n will begin to
approximate the local anomalies too closely.
Other systems,
however, run into the same problem, and this point would
be best settled by experience with the data.
Another important practical consideration is the
amount of work to be done, i.e., the determination of the
1-15
summations Egxi 73, of the cu' s and the solution of G(xy)
at each point.
We devote the next section to this problem.
To illustrate the work necessary we discuss a
convenient scheme for application to the case n = 3.
The
use of a computing machine with cumulative multiplication is
desirable.
Assume that the grid has been determined and the
gravity values written at each intersection as shown.
This
is done on tracing paper as shown in Fig. 1.3.
The numbers a
and
P
above each vertical line and
to the left of each horizontal line represent the sums of
g(xy) along those lines.
Then as we may easily compute the
sums Zg, Egx, Zgx2
zgy, Egy 2 2gy3 , from the relations
Zg
= EQ
Zg
Ei(i)
Zgx 3
Zgx3
=
Zgy
=
E
(i)3
pji(i)
gy2 =
(i) 2
gy~3=ZPiti) 3
2
gx2 = Eai(i) 2
Each of these computations involves one machine
operation of eleven cumulative multiplications.
For the remaining summations Egxy,
E
gx y 2 it
is convenient to have a similar grid which can be placed
under the original one. This second grid has the values of
2
2
xy, x y, xy , at each point as shown in Fig. 1.4 and can
be used for each application.
1-16
r
Pig. 1.3
-N
Pig. 1.-
The values of g(xy) then appear in the vacant
upper left hand corner of each point, making the multiplications apparent.
Each of these three summations then in-
volves a cumulative addition of 100 multiplications.
The o
's are then found as ten cumulative additions
of two of three multiplications each.
G(xy) is now completely determined with the writing
down of less than 50 numbers and it remains to solve this
equation for each point.
This involves ten cumulative
multiplications at each of the 121 points with a final
subtraction to determine the residuals.
A second tracing
paper grid laid over both of the others would simplify this
and the residuals could be written down in a form ready to
be contoured.
A nice feature of this scheme is the absence of
any tabulation of data.
It may be extended fairly simply
to higher degrees.
Examtle
As a test of this method residuals were computed
on gravity readings supplied by a mining company.
This
data was not in a convenient form for use since the readings
were taken in mine tunnels and not over a grid.
To get them
in grid form, the readings were first contoured as shown in
Fig. 1.5 and then values extrapolated to the grid.
involves several inaccuracies.
This
First of all, where readings
are sparse, the extrapolated values are bound to contain
considerable error.
Secondly, the computations weight all
values equally so that the accurately extrapolated points,
in areas of dense readings, suffer from the inaccuracies in
the less dense areas.
Three sets of residuals were computed, one for
second, third, and fourth order polynomials.
This was done
to test the effect of polynomial degree on the residuals.
The polynomials were fitted directly to the raw gravity,
without making the usual topographic corrections. The
justification for neglecting to do this is seen in Fig. 1.11.
This figure shows values of regional minus topographic
corrections computed by the mining company, and contours of
these values.
The contours demonstrate that this correction
is a low order effect (in this particular situation) and
can obviously be easily absorbed into a polynomial as low
as degree two.
Figs. 1.6, 1.7, and 1.8 then show residuals for
second, third, and fourth order polynomials respectively,
and were computed as described previously.
Once the reading
had been contoured and extrapolated, it took about a day
to compute each set of residuals.
The computed polynomials
appear in the upper right-hand corner of Figs. 1.6, 1.7,
and 1.8.
Fig. 1.10 shows the contours of the polynomials
themselves, and shows that the similarity of the residuals
is due to the similarity of the polynomials used.
1-18
In Fig. 1.9 is contoured the residual gravity as
computed by the mining company which supplied the data.
Their computational procedure required several months to
produce this diagram, which, in important respects, is
quite similar to the contours of Figs. 1.6, 1.7, and 1.8.
Part of the difference is due to the fact that the least
squares residuals are forced to oscillate arouni a mean of
zero, so that many negative contours appear.
Other differences
may well be attributable to the inaccuracies of contouring as
mentioned above.
It seems clear however, that the similarity
is sufficiently great to justify the use of the least squares
procedure, at least for a first evaluation of gravity data.
This seems particularly true in view of the relative speed
with which this procedure may be carried out.
These results were encouraging enough so that a
program was written for the WWI Digital Computer to perform
the majority of the computations automatically.
This program
finds the residuals for a polynomial up to the sixth order
over an arbitrary grid shape, once the polynomial is known.
A description of this program appears in Appendix A.
In the next section, we take up the problem of
setting up the normal equations for various sizes and shapes
of grids.
1-19
~RAW GRAVITY-1300.0
CONTOURED
AREA
ANALYSED
Fig.
1.5
RESIDUAL GRAVITY WITH
SECOND ORDER POLYNOM(AL
.5
+.w
-. V32r
1,
26
-4-.
%1e4
Fig. 1.6
3At
At)
+,o al,
FMUSbDUAL GMVifl
Wirt
TW11W OWPA POLYN4OMIAL
Fig, 1.7
+o%446"
1.14'
*
-.
rL
031
-. 24W
41
+ -OU3
01
'-f. Don
1RESIDUAL GRAVITY WITH
rAj I
+
FOURTH. ORDER POLYNOMIAL
-S
2..'I%
2,v
Fig. 1.8
of4.
di
+ DAI
0*14,
RESIDUAL
GRAVITY
COMPANY DATA
>4
Fig. 1.9
CONTOURS OF Cry-)
Z-2, ORFER POLYNOMIAL
43
f
z
4
ORDER
- -
-POLYNOMIAL
o1
ORrta
soiA
30
25
7
4s0
-
J~.
10
Nl
v-
1
JFig. 1.10
4
'REGIONAL CORRECTION MINUS
TOPOGRAPHIC CORRCCT1ON
COMPANY DATA
*pqb
/
~
4O~
$0s
/
/
4'
Fig. 1.11
Settin
U) the Normal Eauations
We are concerned here with the problem of setting
up the normal equations for various grids and polynomial
If we limit ourselves to polynomials of degree 4
degrees.
or less, we are then interested in finding the following
quantities:
4
Ex2
E6
E8
224
Ex2 2
ZY6
E4
8
4 2
2 6
24x 0 0
1.3
where the summations are to be taken over the particular
grid we are dealing with.
If the grid has the dimensions 2N by 2M as shown
in Fig. 1.2, we may set up a fairly simple procedure for
finding these summations.
First we note that Exk over the grid is
equal to
the Exk on a single horizontal line, times the number of
Thus
lines.
Zx
= (2 +1)
ik
but since in our case k is always even
Exk
N
2(2M + 1)(E ik)
iml
Likewise
Zyk
k
kN
2(2N + 1)Z i
i=l
1-20
1.4
For the cross terms Exkyl we have
xM
grid
N
J=-M
i=-N
N
MI
[EikJ CzJI
-N
-m
ItN jk
grid
iml
M j4
j=l
Hence the sums 1.3 are easily derivable from
equations 1.4 and 1.5 if we tabulate the quantities
E 1 .
Table I gives values of i from which the sums
Lk
Z I
are derived, and Table II tabulates these sums for
i=l
L up to 25 and k = 2, 4, 6, 8. The latter Table allows us
to compute the sums 1.3 for any grids measuring up to 50 by 50.
a grid this size would encompass 2601 gravity readings which
seems ample for most applications.
Table III contains the sums 1.3 computed for six
representative grids 10 by 10, 10 by 20, 20 by 20, 30 by 30,
40 by 40, and 50 by 50.
1-21
T8?5860TW~
5z9o6C
5z
9eZZLCC
688KCMItT
C?
i~o66IZCCT
9z
T9C659??8tC
0000000095Z?
1170C 95C 691
1885trOZI
9t~0966T0TT
6z5
61
TZCoCT
0z
00009T
9Z6101
T"15Z5L69
96?/Z96ir6?'ir
001
5Z906CtT
1T"
9TZI19T
9C56z5Z
9C559
00001
9T
471
79C
6089zg8i
14717
969TS 6 6t
T95QTZT
CT
69z~
95Z
0000001
69
9601
961
T959
6
T08479/5
9T961t9T
959941
96ZT
?906C
9C559
9
179
96047
T95;9
6,
91Z
47
c
95
1
9
T
;z
9
9 4171z= 4 zo;
* 4
~ft
xRV-T
5z198966
0ot6TO/
5tqC5TZ
00&t?
0o0o9/zT
t 77TtT
5CS095TVi
Zoo65Ce8
9990
O0ijTl
ooTf
LtiTIT6
OT955479TZ
9990Ct0+,64ii
61
999Z95.
91
6 Z6604y60T
6*to7ozZC69T
60TZ
9096Ir*T966
T9ZT
6 l.~9 TT
TC9Z09tZ9T
56zz6o6T
Z 99T65C0TC
IT
9MIT
019Z0T
9T
ot6T/ZOZTS
i1 Izo6ozeC
0111T
99661yZC
CCCTCZZ9T
6T
5ot 1 9Z6
CT
~t19691rt
966
96CZ06Z
IT
ccI
'ltT
068ti
M19
9 9CM
2
T6
ii zxva
55,o
tr
zI
9=31
52:90
4z
l
1
=
T=T
lxoyo
X2
Zx4
=
-
=
Ex6
N=25 ?1=25
lo
N=10 K=10
N=15 M=15
N=20 M-=20
121
231
441
961
1681
2601
1210
2310
16170
76880
235340
563550
21538
41118
1063986
11055344
59258612
219671790
451330
861630
83093010
1889941040
17749376420
10i885895150
UN-5 N-5
N=
0
0
y2
Zy
-
8
Zxy
-
E2x4
6
Ex22x62
Ex4y
Zxay
-
7044715986 351321903344 5784339914612 51429792671790
10185538
19445118
1210
8470
16170
76880
235340
563550
21538
557326
1063986
11055344
59258612
21967179u
451330
43524910
83093010
1889941040
17749376420
101885895150
10185538
3690089326
12100
84700
ce
M
7044715986 351321903344 5784339914612 51429792671790
592900
6150400
32947600
122102500
884427520
8296205680
47595554500
215380
5573260
4513300
435249100
215380
1507660
4513300
31593100
3046743700 151195283200 2484912698800 22075277282500
3833?64
99204028
2567043556 127180677376 2088984590224 18552747144100
39012820
3046743700 151195283200 2484912698800 22075277282500
39012020
884427520
8296205680
H
I-I
"4
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C
47595554500
co,
{
PART II
SEISMIC RECORD ANALYSIS BY LINEAR OPERATORS
In the study of reflection seismio records, taken
in the exploration for oil, it is becoming increasingly
difficult to pick reflection times by the standard procedures.
The reason for this is that, as the simpler geologic areas
are being fully exploited, exploration is being forced into
the more complicated areas.
seismic records taken in these
structurally complex areas contain much in the way of
unwanted information and much not-understood information.
Energy reflected from the strata of interest is largely masked
by this "noise*,
At least two different approaches to
unscrambling these records are being developed at present.
The first of these approaches is largely instrumentational.
Its principle is: take more and more information
(more traces on each record, etc.), filter it in different
ways and mix it up in a variety of combinations to see if a
procedure for averaging out the unwanted information can
be arrived at.
This approach has led oil companies to
the use of 24-trace records, each trace representing the
responses from up to thirty geophones.
The success of these
methods is not publicly available, but the oil industry is
expressing great interest in the approach described below, so
probably they are not completely satisfactory.
The second of these approaches is basioly analytic.
Rather than taking more information, we attempt to sharpen
up the interpretive procedure on the inforkation we have.
II-1
The search for such procedure has been largely carried out
at MIT in the Mathematics Department, and subsequently in the
Mathematics and Geology Departments.
The tools in this
analysis have been the statistics of time series.
Statistical Methods
After some experimentation it was found at MIT
that the use of the "linear operator" seemed most promising
in the determination of reflection times.
used are described in Rcf's. 5 and 6.
The exact methods
The linear operator
permits a measure of the change in dynamics as we proceed
down a seismic record.
As these dymanics are amplitude,
frequency, and phase relationships, it was hoped that the
dynamical change at a reflection could be discriminated
even when the 9hanges due to aRnlitude were small.
This was
hoped for since the usual interpretive procedures depend
heavily on "amplitude reflections".
The results were very
encouraging and stimulated increased research.
One direction this study has taken is the empirical
one.
We know the linear operator gives us added information.
But, since there is considerable freedom in the choice of
the exact mathematical forr of the operator we use, we try
many different forms and see which ones give us the Mjst
information.
This is a trial and error procedure and involves
an immense amount of computation.
For this reason a program
was written for the WWI Digital Computer which would compute
automatically the measure of dynamio change, for a great
11-2
variety of forms of linear operators, at very high speed. A
copy of this program and a description of its functions is
contained in Appendix B. Case studies designed to test the
effects of individual parameters of the linear operator are
being run with this program, but the results are as yet
incomplete.
Along with this empirical approach an attempt is
being made to study the linear operator from theoretical
Although the form of the operator which is being
grounds.
tested at present is relatively complicated, it is instructive to consider a simpler form, the so-called "cosine
operator".
This operator is a mathematical expression which
generates a pure cosine wave of given frequency.
We can
determine quite simply the effects of this type of operator
on various time series including those found on seismic records.
We hope to gain insight into the physical function of such
operators as well as correspondence between them and simple
filters.
We shall also consider two other more practical
problems connected with the statistical analysis of seismograms by the use of linear operators.
One concerns
certain iterative methods for approaching the values of the
linear operator coefficients for least squares fitting.
The
other is a related problem, the necessity for accuracy in
finding these values.
11-3
Single
requency Cosine Overators
A cosine operator is
a prediction mechanism which
exactly predicts equally spaced points on a cosine wave.
has the general form
t
±+2=o + ax+
where
It
+ bxi
2 oos 2nhf
a
2.1
2u
b = hI
time between observation
o
(1-a-b)i = 2(l-u)i
=
t = mean of series
f = frequency of oosine wave
= predicted value of
+
Suppose we use this operator to predict an arbitrary series x. Then the error of prediction xi+2
will be
i+ 2 =
=
- C+2
[2(l-u)i + 2ux+1 ~
1+2
-
2(1-u)i -
i+2 - x) + (x
2.2
2xi+1 + xi
-
) - 2u(xi+l
For simplicity let us deal with a series X
measured around its equilibrium mean x, i.e. a
= x
-
x
then 2.2 becomes
Ei+2
= 1+2 + X - 2uXi+1
Now if we sum the squares of these errors over
an interval of the series we get
Ref. 6
11-4
2.3
E
+2
+1 + 2Xi+2X
2+4u
-
Xi+1
2.4
If the series is stationary and the interval sufficiently
great we may write this in terms of the auto-correlations.
Let the series be normalized so that E
EE2
SE1+2 =
where R, = iA
0 + RO +u
R
22
+2u
lag auto-correlation and RO
= 1 then
u
~
o
2.5
I
or
E
2(1 + R2 - 4uR + 2u2
2
2.6
This expression has a minimum value when
-.(-4R
u
)
X 2=
R1
2.7
or
cos 2nhf = R
f=
[I-Coos"
R
+ 2n-n]
2.8
Hence we have the least squares fit for a cosine operator
predicting an aribtrary stationary series.
We find that
f is determined only by the first lag of the auto-correlation
function of the series, and that f is only determined modulo
1/.
2.l
This last fact is apparent if we refer back to equation
where we see that cosine operators have identical forms
for angular frequencies d ffering by 1/h.
11-5
Hence
If
2'
2 + 2
Min E E2+2 , 2(l + R2 - 2RJ)
.
2.9
we want a perfect least squares fit we have
+
(
2.10
+R)
with the restriction cosine 21Thf = R
One way to meet this condition is to let the interval
h shrink toward zero so that R -+ 1 and R 2-+ 1.
This is
equivalent to saying that any small segment of the original
series approaches a straight line, in the case where the
function is continous and its first derivative exists.
The Geometry of Cosine O0erators
A.
Error Sum as Function of W
Consider the sum of squared errors as a function
of u.
We have
E E 2
= 2(1 + R2 -
l + 2u2)
2.6
This is a parabola in u as shown in Fig. 2.1.
in
u = oos 2Trhf must lie
the range -l
i u
-
1-
We have shown that u = R , is the condition for a minimum
fit, and since -1
<
Ri <. +1, Z E2 will always have its
minimum in this range.
This means that, for any series, we can always
get a minimum fit with some cosine operator of frequency f,
where f must be in the range
11-6
+I
Fig.
2.2.I
Fig.
2.13
/
c
0
< f
4
t
.L_
2h v
(u
I
)
2nhf = U
( U =27Thf = n
2.11
We require E E2 to be non-negative.
the discriminant of 2.9 must be
16 R
-8(1+R2)
0
This means
or
2.12
0
Therefore the curve camnot aross the u axis, but
can be tangent to it at one point, when the equality sign
This is the condition for a perfect fit.
holds above.
B.
Error Sum as Funtion of f
The error sum as a function of frequency f is not
truly parabolic but has the general shape of a parabola.
It
is periodic in f with a period 1/h. It
appears as shown
in Fig. 2.2 .
C.
Individual Errors as Functions of u and f
Equation 2.3 gives us an expression for the
individual errors
Ei+2 = 1+2 + X
- 2uXi+1
2.3
If we fix attention on a single individual error
(i constant) and let u vary we see that E
since u = cos 2nhf
.
Thus E1
a mean given by the sum of the
is sinusoidal
varies sinusoidally about
1;- and the (-2)
value
of the series, with an amplitude of twice the (1-1)E value
of the series.
The period is
11-7
1/h.
There is no phase shift
between these curves for different i values.
Thus all
individual errors must increase or decrease simultaneously
with f.
Fig. 2.3 shows individual errors as functions of
f.
is
This figure explains why the error sum of Fig. 2.2
reflected across the line f = 1/2h.
This line in Fig. 2.3
is the axis of symmetry for the individual errors, so that
it must also be the symmetry axis for the error sum.
Conclusions
From the above, we can draw certain conclusions.
1.
If one limits himself to the general class of cosine
operators, there is a maximum error obtainable for the
particular data, using any frequency whatever.
there is such a thing as a worst fi
That is,
for cosine operators.
2. Since Z E2 is parabolic, determining 3 values of E E2
is sufficient to determine the complete shape of the error
curve for all other frequencies.
3. Moreover, since the individual errors are sinusoidual
in f, determining the individual errors for 3 values of
f determines the errors for all f.
Looking at the problem another way, much of the
information obtainable from any data series by a study of
11-8
this type is contained in the first and second lags of the
auto-oorrelation function of the series, for these two
quantities determine the shape and position of the curve E E2
The prediction program described in Appendix B
provided a means for testing the conclusions reached about
cosine operators.
Individual errors and sums of squared
errors were computed for cosine operators of frequencies
25, 30, ...-
75 ops.
The data for which these were computed
were readings taken from a typical seismic trace at intervals
of 2 ms.
The sums of squared errors are plotted in Fig. 2.4
over two intervals of 240 readings each.
Both curves exhibit
very good parabolic shapes.
The average minimum for the two
curves occurs for u = .85 -
This should equal the first lag
auto-correlation over the two intervals, which was computed
by the correlation program (Appendix D) to be -853 .
Fig. 2.4
shows several individual errors plotted
as functions of the frequency of the cosine operator used.
They appear to be sections of sinusoids as expected.
These curves, computed on an arbitrary time series,
seem to be in remarkable agreement with the theory.
11-9
Fig.
100.
2.LL
Fig.
EXAMPLE OF THE PARABOLIC
DISTRIBUTION OF ERROR SUMS
FOR COSINE OPERATORS AS
FUNCTIONS OF Cot- 21,-f)
900
800
70
60
50
'?00
40
30
60o
20
10
0
500
-10
2
400 -
-x)
-20
-30
300
-40
-50
200
-60
.70
;R,= .B9t
100
O.5
h6
01-7
Ct
0
CCLo.2irvA
t
2.5
EXAMPLES OF THE SINUSOIDAL
DISTRIBUTION OF INDIVIDUAL
ERRORS FOR COSINE OPERATORS
AS FUNCTIONS OF 'd
The CycOal Nature of Cosine Onerators
We have mentioned that cosine operators differing
in frequency by n/h have identical forms.
It is interesting
to see what this means physically.
Suppose we are trying to represent a cosine wave
of 1 cycle/ second with a spacing of h = 1/4 second.
The
points we would plot might appear as in Fig. 2.6 .
Now consider a cosine wave of frequency 1 + 1/h
5 cycles/see.
If we try to plot this frequency with a
spacing of 1/4 see. we find that it can be exactly represented by the points we plotted for the one oyole wave.
is illustrated in Fig. 2.7 .
This
We would find the same would
be true for frequencies of 1 + n/h4= 1, 5, 9, 13, 17 .-Thus, it is the fact that we cannot uniquely represent
frequencies differing by n/h that explains the identity of
form for cosine operators whose frequencies differ by this
amount.
This is also the explanation for the so-called
"condensed" power spectra met with in computational
procedures .
The computed power at a frequency f must
represent the sum of the powers at frequencies f, f,+ 1/b,
f + 2/h
....
Therefore power spectra can only have the
range 0 to 1/h cycles.
In practice h must be chosen so
that 1/h is greater than the greatest frequency from which
significant contribution is expected.
Ref. 3
II-10
2.6
Fig.
1 CYCLE/SEC.
COSINE WAVE
PLOTTING INTERVAL h: I SEC.
/ Sec.
\9
/IR/
'r
/
-
/
at
I-Se
At
A
Fig.
COSINE WAVES
C"
I
I
A
~
~'
~\:
\
I
I
I.
I
I
I
I
I
I
I
2.7
1 AND 5 CYCLES/SEC.
I( I,I
r,
j
i
Iv
\ ;
'I
'I
Cosine O2erator Fredicting an Autoreressive Series
In order to examine the filter characteristics
of cosine operators it is convenient to consider their
effect on autoregressive-type series.
The autoregressive
series has a known Cauchy-type distribution of its spectrumAt
It will be interesting to examine what the spectrum of the
error function will be when we predict such a series with a
cosine operator.
Referring back to equation 2.3 we have the error
function for cosine operators.
E1+2 = X+2 + X
- 2uXi+1
2.3
To get the spectrum of this function we first find the
auto-correlation R .
Ro
R
E (X1 + 2 + X
=
-
2uX,+1 , )1$ 1+2-
EX+2X+ 2 -T + EX1 +2X1iT
+
SX,
+
Xi+1-TXi+ 2 +
-
Mi+1-
+ F-XX+2-T
- 2u(EX1+1Xi+2-T +
Xi+1-TXI)
+ X
1i+1i-T
+ 4u2,X+1 i+1-T
If the X series is properly normalized we may
write this in terms of the correlations r. of the X series.
rT + rT+
RT
+
2
rT - 2u(r
+ rT-2 + 4u2
+ rT+l + r,+1 + r
Ref. 7
Il-11
)
RT =
rT-2 + r
(-4
+ rT(2+4u2)
+ r +1 (-4u) + rT+2
2.13
Since the series is taken to be autoregressive
r
OScos2nf 0 Th
e-th
cosaT e-bT
$
0
Substituting equation 2.14 into 2.13 we have
cos[a(T-2)]e-b( -2)
RT
4ucos[a(T-1)]e-b(T-1)
+ (2+4u2 )oaTe-bT
-
4ucos[a(T+1)]e-b(T+1)
+ cos[a(T+2)]e-b(T+2)
Using trigonometrio identities
RT
e2b[ cosatoos2a + sinaTsin2a)e"b
4ueb(cosaToosa + sinaTsinaje-bT
+
(2+4u 2 ) (cosaT )ebT
4ue'b[cosaToosa - sinaTsinale-bT
+
e-2broosaToos2a - sinaTsin2aJe-bT
or
R
2
cosaTe-bT I,2boos2a - 4uebcosa + 2 + 4u
T
-4ue-boosa + e-2bos2a]
Ref;
5
11-12
2.14
+
sinaTe-bTe2bsin2a - 4ueb sina
+ 4ue-b sina -
e-2bsin2a]
Hence
+ B sinaTe-bT
0A
cosate-b
e-bT(A cosat + B sinIaT)
2.15
where
A
=
cos2a(e2b+e-2b)
sin2a(e 2 b
B
4uoosa(eb+e-b) + 2 + 4u2
2 b)
e
-
-
4usina(eb
+
e-bT (A2+B2 1
R
eb)
(A2+B )
e-bT(A2 if
2
2.16
B
srn
(A2+B2,1A
)saosp + sinavsi4)
where P = tan
1
B
Thus
=
ebT (A2+B2 )iAoos(aT+P)
2.17
R. is now in a form similar to the rT for the
original series.
The spectrum of this type is known to be
a Cauchy distribution.
The specific shape will be controlled
by values of b, A, B, and
P.
Rather than continuing with this example we shall
proceed to another type of series.
is somewhat non-typioal.
The autoregressive series
Its spectrum, the Cauchy distribution,
is very broad, in fact there is no mean value of frequency
for this spectrum.
11-13
Cosine Oqerators on Series with gaussian Suectrum Distribution
A much more stringent series than the autoregressive type is a series with a power spectrum composed of two
Gaussian curves.
The spectrum has the form $
-(w+a) 2
e (
$()F
+ e
.I]-2
2.o18
where +a and -a are the respective means of the two
Gaussian curves, and T is their standard deviation in
radians.
With such a series the normalized auto-correlation
function may be written as
-< 2
2
rT-e
os a
If we predict such a series with a cosine
operator, we generate an error series whose auto-correlation
function is, as before
R
= rTT-2 + r
(-4u) + rT(2+ 4 u 2) + rT+ (-4u)
2*13
+ rt+2
or substituting
RT
T-2)2
e
cos[a(T-2)]
2
+ e
2
2
cos[a(T-1)] (-4u)
+C2(2 2
+ e
2
-r
+ e
f
2
(T+2)
2
2
cos[aT] (2+4u2
oos[a(T+l)] (-4u)
cos[a(T+2)]
Ref. 5
11-14
2.20
This can be reduced, as before, to the form
2
S
2
2
e
A(T)oos aT + B(T)sin aT)
2.21
where
A(T) =cos 2a [e
2
+ e
2
22
(-2+1)T+1)
+ e
2
-4u cos a [e
2
+ (2 + 4u2
2.22
B(T)
2
sin 2a [e
2
+ e
-2
-a-
-_(-2_+1)
- 4u sin a [e
(2r+1)
+ e
2
2
If we are interested in the power spectrum of
this series we want
0;
$(w)
2
/ R(T) cos wT dT
2.23
0
Probably this integral cannot be expressed in closed
form, and we shall have to resort to a computed example.
Comoutational Exemple
Here we illustrate the filter characteristics of
cosine operators in a particular case.
We choose a series
with a Gaussian spectrum peaked at 50 cycles and with a
standard deviation of 22.36 cycles.
such a series is shown in Fig. 2.8
from equation 2.18
.
The power spectrum of
,
and was computed
In general shape this is not unlike
power spectra dealt with on seismie traces.
11-15
Fig. 2.9 shows
the normalized auto-oorrelation function for this type of
series, as derived from equation 2.19 . The series has
essentially no correlation for lags greater than about .03
sec.
To examine the effectiveness of cosine operators
as frequencyfilteiing mechanisms, a cosine operator of
frequency 50 ops was taken.
chosen to be 2.5 ms.
The spacing interval was
The auto-correlation function of the
error series generated by this operator is shown in Fig. 2.9 ,
and is computed from equation 2.13 .
In this case the
function is unnormalized so that the zeroth lag autocorrelation is proportional to the total power contained
in the power spectrum of the error series.
Thus we see
that less thtan 20 per cent of the power contained in the
original Gaussian series remains in the error series.
than 80 per cent has been "filtered" out.
More
However, since
some of this is due merely to curve continuity, the shape
of the spectrum of errors is more important than the total
Dower.
The unnormalized spectrum of the error series is
shown in Fig. 2.8 , and, as might be expected, is
definitely bimodal.
This curve clearly indicates that the
operator is acting as a filter peaked at 50 ops, at which
frequency all power has been removed.
Lower frequencies are
also well reduced but the higher ones are not so much affected.
In fact, the power at 100 cycles is slightly greater than
11-16
in the original series.
This is not a computational error.
As discussed below it seems to be a necessary characteristic.
A more convenient way of showing the filter
characteristics is to plot the quantity
Power removed at w
Initial power at w
This graph is shown in Fig. 2.10
.
It shows how
frequencies lower than 50 cycles are much preferred to those
greater.
It is possible that this curve would not represent
the filter characteristics of a 50 cyole oosine operator
used on another type of series.
There is some reason, however,
to suspect that it does, and that, in fact, the curve of
Fig. 2.10 continues downward considerably below the axis
(thus representing amplification rather than filtration).
If we were to use a series containing mostly frequencies
between 100 and 200 cycles, the 50 cycle operator would
yield very high errors of prediction.
The sum of squared
errors would be far from the minimum of Fig. 2.1.
Hence
the power in the error series would probably be greater than
that in the original series.
This could only come about
by an amplification of certain frequencies, which would
naturally occur for frequencies greatly different from
50 cycles.
In this example 200 oycles is chosen as an upper
limit, because with a spacing of 2.5ms unique curves only
exist from 0 cycles to 1/2h or 200 cycles.
11-17
Fig. 2.8
POWER SPECTRA
For series with Gaussian spectrum
with a = 212/cps a = 50 cps (normalised).
----- For error series generated by a
cosine operator of frequency 50 cps
on above series (unnormalized).
11.5
x(10-3)
1.4
1.3
1.2
1.1
1.0
.9
.8
.7
.6
.5.40
.3.2-
10
20
30
'40
50
60
V0
70
80
C,4 -)p
90
100
110
120
130
14~0
Fig. 2.9
AUTOCORRELATIONS (R(ih)
For series with Gaussian spectrum
withQr = 2etops a = 50 ops (normalized).
1.0
----For error series generated by a
cosine operator of frequency 50 Ops
on above series (unnormalized).
09
'A,
2.6
e,0
|1
.7
.6
.4
3
-
L
.2
/
~//
,.5 L
4'
~
%%%~*_
_
li p
/
12
2.10
Fig.
FILTER CHARACTERISTICS
FOR 50 CYCLE COSINE OPERATOR
WITH h = 2.5 me.
o/o Power removed
/
100
/, 0
'7
I
.1
--
~
--
/0
0
IF
1~~~~-4-
30 o
c-o
To
4
--->
70
o
90 \ too
A Nethod of FiUndi
Linear O2erators for Least Sauares
Fitting Procedures
Described in this section is an iterative method
of approaching the values of coefficients for a least
squares fitting of linear operators to multiple time series.
The problem arose in connection with the determination of
linear operators to use in picking reflections from selsmograms. t
The method is extremely inefficient and is
really only possible with the aid of very high speed computing machines, but it gives interesting insight into the
behaviour of matrices, which helps in constructing other
techniques.
We are trying to fit a linear operator of the form
xi+k -e + aOxi + a,,,
+
boy,
..........
..
+ aM16i-M
+
bMNyiM
2.24
+
iOz .
+ 0Nzi-M
+ duI.......... + dMui-M
to an interval of the sequences x., yi, Z., and ui so that
IE(zk ~ ik2
is a minimum.
2.25
The plan is to guess initial values of the constants
a, a., b5 , c., and d. and compute 2.24 .
Then adjust the
constants so that I is continually reduced.
Ref.
5
11-18
The initial
values chosen are a =i
do = 0.
(mean of
series) a
b
a
These values are the values which the constants
would assume under least squares fitting procedures if the
xi series were truly random and had no predictability.
these values of the constants I = I(i,0,0, .. )
With
becomes the
sample variance about the sample mean.
The computational procedure is:
1. Find I(i,0,0,
... )
2. Find I(x+ &a., o,o,..)
3.
If 2 <1, continue adding &a until
I(i+naa, 0,0, ...)> I(i+(n-1)
a, 0,0, ... ).
If 2> 1, subtract &a and continue to
subtract until I(i-n&a, 0,0, ...
) >
I(i-(n-l)aa, 0,0, ... )
4.
Using i+(n-l)A a, 0,0, ...
,
as the new starting
point, find I("+(n-l)a,A a, 0,0, **
and repeat the steps under 3.
5.
Work successively in this fashion with each
6.
of the variables a, a0 , a1 ...dM*
Start the process over again with the
variable a.
7.
Continue recycling until the desired
accuracy is reached.
It is interesting to consider the geometry of
this process.
If we substitute equation 2.24 into 2.25,
we find that I is parabolic in each of the coefficients
a, a., b., os, d.
For 'simplicity consider the case where
we have only two coefficients a and b.
Then I is a two
dimensional paraboloid in a and b whose minimum we wish to
find.
I is positive or zero for all a, b and has one
minimum.
Contours of I = o are ellipses in the a b plane
of constant major to minor axis ratios, and are centered at
the minimum.
Figs. 2.11, 2.12, and 2.13 illustrate three
situations that might arise.
In Fig. 2.11 the contours are
circular which is the case when the matrix of the normal
equations associated with the minimum fit is well-behaved.
Fig. 2.12 is the more usual situation where the contours
are definitely elliptial.
Fig. 2.13 shows a very badly-
behaved situation corresponding to near singularity of the
associated matrix.
The solid line shows how the iterative method
described above would converge toward the minimum point
in the three situations.
The dashed curve shows how
another iterative method, the steepest descent method,
would converge in these situations.
The steepest descent
method runs into trouble in the near singular case because
with finite increments it cannot land on the long axis of
the ellipse.
It is forced to wobble back and forth, much
as a small ball would wobble rolling in such a trough.
The method described above would also encounter trouble if
the increment were not fine enough, for if
11-20
it
got near the
Fig. 2 .11
Fig. 2 ,12
Fig. 2.13
trough the next increment would carry it across the trough
to a greater value of I.
These figures illustrate a fundamental problem met
with in iterative methods.
The fine increments necessary
in treating the near singular case are very inefficient when
used on well-behaved data, whereas the larger increments
applicable in Fig. 2.11 could never find the minimum of
Fig. 2.13 .
A program was written for the WWI Digital Computer
which would do this one-variable-at-a-time type of iteration.
It is described in Appendix C. The computations it carries
out take fifteen or twenty minutes of machine time, but
they represent nearly a year of hand computation.
The
program can print out each successive value of I as it is
computed. Fig. 2.14 shows a plot of these values as the
program converges towards the solution of a particular
problem.
This diagram shows how I is parabolic in each
coefficient.
We also note that all the parabolas have
approximately the same shape.
This indicates that if there
is a predominant long ellipse axis as in Fig. 2.13
,
it
cannot be close to parallel to any of the axes a, a., ds,
for if it were, the parabolio section in the corresponding
direction would be quite flat.
One surprising feature of
this diagram is the failure of the parabolas to tend to
flatten as I is diminished.
11-21
o1
,
a ,
d
0L
J
,
d ,
0
-kJ,.
0i,
Where
Yg-
=
- a4
,
2
-2-d
M-NTiL
VALVE OF
ot 6; edi
+-,
-+
t 4z
/.,
(
a3
+d3
t
+
-3
a,
.
i0oj.
ni4
U"
\JkJ\J\J\JY
"~
cZ
ci.
j'ItkiVV<
,JV\'v
o4
*
'~
lj/ JvfJJJiJjJ'JJ
I
\A~kJ~\J\JVY
f'~.
This is a convenient point to consider the
problem of the importance of obtaining the exact solution.
If we look at Fig. 2.13 , we see that values of a and b at
the point A will reduce I almost as well as values at the
true minimum 0. Individual errors (xf.-xg ) will likewise
be practically identical.
The effect of the displacement
OA will not be felt until the values at A are used to
predict outside the interval where the minimum fit is taken.
Suppose the series is
and the minimum fit is taken in the interval I of this
series.
What happens when we predict the interval II with
coefficients chosen in I?
Consider Fig. 2.15
.
The dark solid line represents
the long axis of the ellipses for interval I and the light
solid lines, the contours for this interval. The true
minimum of these contours is at 0.
Likewise, we can draw a
similar contour picture for the interval II. If we assume
the dynamics are but slightly different in the two intervals,
the second contours will be slightly rotated with respect to
the first, and, there will be a small displacement of the
mininwm.
The heavy and light dashed lines in Fig. 2.15
11-22
comra.
\
/
/
/
//
/
//
/
~
~1 ,~/" /
/
/
/1
//
/
//
//
/
/
/
/
/
/
/
/
/
/
I.
I,,,,
/
~
//
/1
/
"/
/
/
/
/
'I
/
Cori/roUR~ FOP
ll4W~I. L
Fig* 2.o15
/
/
//
I
/
'I
//
/
represent these contours for interval II.
Since we are
considering the near singular case, the deviation of the
sum of squared errors for the second interval from its value
on the heavy dashed line will vary as the square of the
distance from a point in the ab plane to the heavy dashed
line.
Now suppose in finding our minimum point for
interval I we had landed at the point A which satisfies
the least squares criterion almost as well as the true
point 0. The deviation of the sum of squared errors when
point A is used to predict interval II will be proportional
to (AD)2 which would be about sixteen times greater than
if the point 0 were used, since OE - 1/4 AD.
On the
other hand, if we had landed at the point B for the first
interval we would get a sum of squared errors smaller than
if the point 0 were chosen. Again, if the point F were
taken, the sum of squared errors for interval 11 would not
be appreciably different than for the point 0.
These effects have been noted in computed data.
The indication is that the true minimum point 0 must be
chosen if we are to take the sum of squared errors as a
valid comparison of the changing dynamics in various intervals
of a series by this method.
11-23
PART III
SOME INTERPRETIVE PROCEDURES
In this part we present several ideas which may
be applicable to answering certain questions involving
seismogram analysis.
the others none.
Two of the ideas have had some testing,
With one exception these ideas relate
specifically to reflection seismic records, and arious
possibilities in picking reflections therefrom.
The questions
are:
1.
In a two velocity system, e.g., shear and compress-
ional waves, can we set up a method for separating
these velocities and can we apply it to reflection
determination?
2. In the use of linear operators for seismogram
analysis, is there another measure of prediction error,
other than the "error curve", which will show reflections?
3. Can we obtain information on the step-out times of
reflections, by the use of linear operators and the
concept of ensemble averages?
4. Can a special seismometer set-up be used in conjunction with correlation analysis to pick reflections?
III-1
Velocity Senaration
The determination of velocities for compressional
waves in the earth at shallow depths is relatively simple
due to (1) the ease in generating such waves, and (2) the
fact that the first arrivals are the compressional waves.
Shear waves are more difficult to generate with sifficient
amplitude to separate from the earlier arriving types.
Although with the proper equipment this can be done by
visual inspection of the seismogram,
t
it seemed of
interest to consider if a statistical test could be devised
to help in this problem.
The approach was to set up a simple model approxi-
mating the physical situation.
Assuke we have two wave forms A and B traveling
horizontally at velocities VA and VB, where VA
Vb' past
three geophones F, G, and H, equally spaced with separation
d.
The wave shapes do not change with time.
Traces F,
0, and H then represent composites of A and B with different
time lags.
Assuming VA is known, the problem is to find "B
and, if possible, the wave forms A and B.
Divide the time scale into units such that the
no. of units per sec. is
L.
Since V
A and d are known we may
line up F, G, and H so t bat very nearly
t Ref. 8
III-2
FNW AN+
N
HN1i
BN
3.1
AN + B%-j
A
3.2
AN +
3.3
-e23
where the time lag between traces is approximated by j units
so that
.....
j
L
* .
-dd
+
VA
VB
or
IVA'
B
3VA+IA3.
This is illustrated in Fig. 3.1 From equations 3.1, 3.2, and 3.3 we can get
AN
AN~j
AN- AN-2j
GN -
N-j
3.5
KHN ?N-2j
3.6
3.5 and 3.6 are recursion formulas giving AN-kj and
AN-2kj respectively (k = 1, 2, ...) once AN *.e*
.j+1
are known. Now if j has its correct value then it is easy
to show that regardl.ss of how we choose the initial A's
both formulas give the same value for AN,-2kj # If 3 is
slightly wrong then the two series will differ slightly.
The difference will increase as j strays further from its
true value. We may now set up a procedure for finding this
value.
Assume values for j and for
A.,
AN.,
..
+
use equations 3.5 and 3.6 to calculate the two series (to
111-3
(Nr*4
NJNAA'v\&'k
~J\AIVv
AKtxT 1,
vic~.
HN
--Atq + 13N--ZJ
-3.'
Rsr5oVte R~EFLECTON
73.z
PPATT ERN
a certain length), find the mean square difference between
the series, and plot this difference as a function of J.
In the ideal ease, this function will go to zero for the
correct value of J. In practice we can expect this
difference function to have a minimum at the correct value.
Thug in theory at least, j is determinable.
Equation 3.4 may be used to find VB .
Although the exact
wave shapes are indeterminate in the general case, there
may be obtained some information about them.
first j values of AN are taken to be zero.
Assume the
If j is correct,
then the series from equation 3.5 represents the true AN
with the first j values subtracted suooessively.
Thus the
series from equation 3.5 might be expeoted to have the
same frequency characteristics as the true AN'
In certain oases the assumption that AN **
0 will be fairly accurate.
should be determinable.
-
In these cases the wave forms
Examples would arise in the
separation of shear and compressional waves where it is
known that the shear waves arrive late, and in refleotion
picking.
Another possibility in this problem would be the
use of pure cross-oorrelation between two traces.
We
should expect to get a peak in the correlation at a lag
corresponding to the velocity VB and the particular geophone separation.
However, if the wave form B were of small
111-4
amplitude, the shape of the cross-correlation curve would
effectively be dominated by that of the auto-correlation of
wave form A, and the selection of the peak would be somewhat
arbitrary.
On the other hand, the mean square difference
between equations 3.5 and 3.6 should still show a true
minimum at the correct lag.
We can adapt this idea to the selection of reflections on seismic records.
Here we make the simplified
assumptions that the refleotion consists of a wave train
with zero amplitude between reflections as in Fig. 3.2 .
This is assumed to occur on two traces in the same form and
at the same time (i.e., there is no step out time of the
In
reflection which is assumed to be coming in vertically).
this case equation 3.5 alone is applicable and we need only
two traces.
AN
A -j w N ~ N-
3.5
3 is taken from the step-out time of the initial
breaks on the seismogram.
We then select some interval j
units in length in which AN is zero (a non-reflection
interval), and use equation 3.5 to predict the remainder
of the reflected wave.
For interpretation it is convenient
to plot the running variance of the predicted reflection.
Now the assumptions will certainly not be upheld
exactly on any real seismic record.
A certain amount of
random energy will be in phase between any two traces and
would be picked out by this method as part of the predicted
1II-5
reflection.
To alleviate this situation we can use three
traces and predict the reflected wave from the three
Adding the three predicted
possible pairings of these traces.
waves would tend to accentuate components in phase between
all three, and to minimize the random, in-phase components
between any two traces.
For three traces FN' N, and HN
we can express this sum as
N +"N+
3N+K *
+
2HN+K
-
j
+ AN+ K-*j + ON+ K
N -
N+K 3
G
where j corresponds to the step-out between FN and GN
K the step-out between F.
and
*
Testp of the Nethod
l. Selection of Shear Velocity
An initial test was constructed which showed that,
when the assumptions were exactly upheld, the minimum of
the plot of the squared differences between equations 3.5
and 3.6 was quite sharp.
On this basis three adjacent traces of a seismogram were converted to numerical form and the method applied
to these real series.
The seismogram was taken at Revere
Beach, Mass., in unconsolidated sediments, by Peter Southwick.
Special generating apparatus was used so that the shear
arrivals were quite prominent.
t Ref . 8
III-6
This record is now lost, but
Fig. 33 shows a very similar seismogram taken with the
same apparatus.
The first line of check marks on this
seismogram indicates the first arrivals, and the second line
of check marks was picked as the arrivals of the shear waves.
This second line permitted a direct com)utation of the shear
velocity.
The readings for the three traces were lined up
in accordance the firnt line of time breaks, and equations
3.5 and 3.6 were computed for a variety of values of J.
In
each case the first j values of A. were assumed to be zero.
The sum of squared differences between these two series were
computed for each J, and normalized by the number of terms
in the series for each j. A plot of these quantities appears
in Fig. 3.4 .
This figure shows two distinct minima (at 3
and j a 16.0) rather than just one.
13.3
Upon examination it
turned out that the value J = 13.3 corresponded to a shear
velocity which would have been computed by direct interpretation of the first two traces chosen.
The second minimum
corresponded to a eloci ty which would have been determined
directly from the second and third traces chosen.
The value
of velocity computed by the entire second line of check marks
of Fig. 3.3 lay between these two values.
Fig. 3.5 shows a running average of the points in
Fig. 3.4 (by overlapping groups of three) which exhibits
a flat minimum between j = 13.3 and j = 16.0
III-?
.
The
LEXINWTON
Shear te 5t
-
med
snd
Geophoncs in /ine
*MEMO
~
U
__
'r-
__
Io~-.
--
.
,
Fig. 3A
to
5
S.
15*
2.0
25
20
Flt .3.5
tot-
20
oorresponding velocity was quite close to that computed
from the second line of check marks.
2. Predioting a Reflected Wave
To test whether or not a reflection could be
predicted by these methods, a seismogram showing a prominent reflection was chosen (MIT Record No. 1 t).
In this
record linear operators had been computed and error curves
derived.
These error curves showed marked peaks at the
reflection so the ourves were taken as a basis of comparison.
From two traces on this record equation 3.5 was
computed.
j was selected from the initial step-out between
the traces and the non-reflection interval ohosen to ocour
after the refleotion.
The variance of the predicted wave
(in overlapping groups of ten) is plotted in Fig. 3.6 .
The dotted and dashed curves of this figure show error curves
for linear operators with different prediction distances k.
The variance curve does not reach a peak in the reflection
as rapidly as do the error curves, but it does compare
favorably with them in general shape during and after the
reflection. Before the reflection the discrepancy is more
noticeable.
This may very well be attributable to the f act
that operator interval was chosen just before the 2fleotion.
In the operator interval, the least squares fitting procedure
forces the error curves to be as low as possible.
tfRef. 4
111-8
--
Variance of predioted reflected wave
Error curve for k = 1
..,... Error curve for k = 3
----- Operator interval for error curves
Basic interval for predicted reflection
f-
AEF LECT IO
,
9
.*
'
-
.. W
.. ,
SECONDS
Conolusions
The two tests discussed show that the expected
effects are noted.
However, the data used are reasonably
ideal, in the sense that ordinary methods of interpretation
are adequate.
Whether or not the statistical teobniques are
better can only be determined by many further trials.
Situations difficult to treat by the ordinary methods will
also fail to uphold the simple assumptions of the theory
presented here.
On the other hand, only the simplest forms
of the theory were used in the mcamples.
Refinements, such
as the use of three or more traces for reflootion picking
may give more valid results.
1II-9
The "error curve' , as used by the Geophysical
Analysis Group for picking reflections, is a running average
of the squared differences between a predicted and an actual
seismic trace.
Fig. 3.? shows an actual trace (the solid
curves), and three predictions of this trace, from linear
operators with different values of prediction distance.
From this diagram we see that the error curve is a running
measure of the vertical differences between the predicted
and a etual traces.
At the reflection (shaded) these differences are
seen to become large, and hence the error curve rises to a
peak in this interval.
The reason the differences become
large is not because there is a big discrepancy between the
average amptitudes of the predicted and actual traces.
From
tie diagram it appears that the reason is that there is a
horizontal displacement of the oscillations of one trace
with respect to the other.
In other words, there is a phase
shift between the predicted and actual traces during the
reflection, which disappears shortly after the reflection.
It seems then that a test of phase relationships
might well show the reflections as well as the error curve
does.
A fairly rigorous way of testing this phase shift
would be the following:
III-10
I.T. K cORD NO. I
SHEET
MAGtIQA PETROVEUM CO
PROSPECT : PACE
PROIU
. Ffl -ft
SPREAD- N2SO-750
RECOOD: T
tNARGE: S
DEPTH: 90'
DATE - WAR, I5,1950
A.V . TRACE, C.V. PHONES
8
SEISMOGRAM TRACE N750
TRACE
47SO i FROM N750 I
##EDiCTED
OPERATOR
.ooffmo
COWPANY MARKEb REFLECT10IS
AS5SCISSA-TIME t SECONDS FROM
ORDINATES -
tNITS
SUPERVISOR
DATE
TiME
IN
ROBINSON
AUGUST
SECONDS
SHOT
OF TRACE AMPI 'UDE
1951
N250O----
1. Select highly overlapping intervals of the record.
2. Compute the cross-spectrum between the predioted
and actual traces in each interval, thus obtaining the
phase relationships.
3. Plot the phase angle of the dominant frequency as
a function of the interval chosen.
Practically, this is an involved procedure.
We
can use a simple but crude method to get approximately the
same results.
Since phase shift is expressed by horizontal
displacement we can measure this displacement directly from
graphs such as in Fig. 3.7
.
This requires that we be able
to follow corresponding waves in the two traces, which is
subject to personal interpretation.
The displacement was measured for the upper set
of curves in Fig. 3.7
.
From equally spaced points (in time)
on the solid ourve the horizontal displacements to the
dashed curve were measured.
Displacements to the right were
considered positive, those to the left negative.
Where
such measurements could not logically be made (for example
on the peak ocourring at about .96 sec.) values were taken
midway between the last value that 2cQd logically be made
and the next such value.
Once this series of displacements
was determined, its individual terms were summed in groups
of twenty overlapping by ten, in order to smooth the data.
These sums are plotted in Fig. 3.8 .
III-11
PtHi5E TEST
30
20
Sec.
-10
+
This curve indicates a rapid change of phase
oours at the reflection, the phase rising to a peak in the
middle of the reflection, and falling off more gradually
thereafter.
It seems surprising that the curve is almost
entirely positive.
If this effect is characteristio,
perhaps we should consider as significant only those portions
of the curve above a certain mean (about 25 or 30 units in
Fig. 3.7).
From the original record it appears that there
may be another reflection at about 1.23 seconds, which could
conceivably cause the rise at the end of the curve.
This is a purely empirical curve.
only holds for the particular ease treated.
Perhaps it
One would
suspect that the arrival of reflected energy would be
accompanied by a rapid change in phase relationships.
How-
ever, it does not seem reasonable that these changes should
be in one direction since the times of arrival of reflected
energy are random.
Possibly we should deal with the
original series only, and compute the rate of Qhanu
of
phase angle (between two overlapping intervals) as a function
of interval.
111-12
Snsemble Average
The step-out time of a reflection is a property
of several seismic traces rather than just a single trace.
The error ourve for linear operators, as defined elsewhere
in this paper, is a property of a single trace - a time
average of a single error time series.o
To get information
on the step-out time we must consider operators chosen for
different traces.
In this connection it is convenient to
use an "ensemble" average. This is an average across the
"ensemble" of error time series generated by the various
operators chosen.
Let us suppose that we have taken a series. of
operators on a record which consists of traces from equally
spaced seismometers.
the
4
Suppose there are T traces, and on
th trace (4 = 1, 2, ...T) we have chosen N. operators.
For the kth operator on this trace (k = 1, ... N)
there is
an associated error time series which we define as ei
.
Then, for example, we may construct a single error time
series
C
to be associated with the
4 th
trace by the
expression
zw (e
E4a E
I
]kA
3,7
We may then average these error time series over
the various traces.
Between traces we observe the effect of
step-out. Hence we construct the error time series
with an arbitrary lag or lead a
111-13
(a)
S
=
T
(A)
S E iamOtl,
A=l
2,
..
3.8
with the expectation that a peak on this error time series,
corresponding to a certain reflection, should be highest and
narrowest for that value of a most closely corresponding to
the true step-out of the given reflection.
No attempt has been made yet to compute 3.8
.
It
would be a fairly simple task to program this equation for
the WWI Digital Computer as a follow-up of the Prediction XV
program described in Appendix B.
111-14
Travelling- CgrreAtions
As an approach to the problem of using a special
geophone layout for reflection picking, consider the
following arrangement.
Two geophones G1 and G2 are placed
in the ground, one vertically under the other at a distance
d. Assuming the ground homogeneous and non-dispersive
around the geophones, the responses of G and G2 may be
considered to be due to superpositions of many plane waves
travelling with a velocity V from many different directions.
In the absence of big refleotions, the major contribution
to the responses will come from waves having directions not
far from the horizontal.
Now consider the cross-oorrelation of the two
responses at G and G2
In particular consider the value
of the function for a time lag equal to d /V. It appears
that this value will be strongly influenced by the amount
of vertical wave contribution present in the responses, since
d/V is the time of direct travel from G2 to G.
The
cross-correlation at the lag 4/V should rise rapidly at a
reflection and drop off afterward.
In practice we would have to compute this correlation
over highly overlapping time intervals of the response
functions in order to obtain the correlation as a function
of time.
The correlation program described in Appendix D
is adaptable to this type of analysis.
So far however, no
seismograms with the above geophone arrangement have been
available.
II1-15
CONCLUSIONS AND SUGGESTIONS $OR FUTURE WORK
It is difficult to make evaluations of validity on
methods which have undergone little testing.
Nevertheless
we may draw certain conclusions from the work presented here.
Polynomial gravity approximation, as presented here,
seems of sufficient simplicity and validity to justify a
considerable amount of further study.
If further trials
show more promise it would be well worth-while to find the
inverses of the matrices of Table III.
In any event,
polynomial approximations of this type have applications in
many other fields, and the simplifications brought forward
here may be of real value in these other applications.
The properties of cosine operators are of mathematical interest, but it is hoped that studies of this sort
will lead to more practical results.
In particular, further
pursuit of the filter characteristics of linear operators
will lead to a better understanding of the extent of realizability of equivalent electronic filters, and or to
simplification in the determination of such operators.
The author is more hesitant about recommending the
various procedures
iscussed in Part III.
Seismograms
exhibit extreme variability in their characteristics and,
whereas the examples given here are encouraging, the procedures may fail on other types of records.
However, the
problems they attempt to settle are of great practical ooncern and all promising techniques should be either proved
or disproved.
Phase is a crucial variable in these problems,
and probably considerable effort should be spent studying
thi s parameter.
As for the Appendixes, the author feels that the:
programs described therein have genuine value.
Anyone Con-
cerned with research depending largely on computation
appreciates the fact that obtaining errorless results is a
major problem.
Programs such as these effectively eliminate
this type of problem, and are available for the use of
persons interested in the sort of computations they perform.
REFERENCES
1.
Agoos
"Least Squares Residual Anomaly Deter-
W.B.
mination,' &eophysics XVI, 4, October, 1951
2. Crout, P:D., "A Short Method for Evaluating Determinants
and Solving Systems of Linear Equations with Real or
Complex Coeffioients," Trans. Am. Inst. of Elect. Eng.,
Vol. 6o, 1941
3. Tukey, J:W., "The Sampling Theory of Power Spectrum
Estimates, Woods Hole Symp. Autooorr. Applio, to Phys.
Prob,, pp. 46 - 67 D.N.R. NAVEXOS - P - 735, 1949
4.
Wadsworth, G.P and Robinson, E.A., Directors, M.I.T.
G.A.O. Report ilo. 1, 'Results of an Auto-Correlation
and Cross-Correlation Analysis of Seismio Records,
Presented at M.I.T. Industrial Liason Conference of
August 6, 1952,' October 1, 1952
5. Wadsworth, G.P., and Robinson, E.A., Directors M.I.T.
G.A.O. Report No. 2, 'A Prospectus on the Appleation of
Linear Operators to Seismology," November, 1952
6. Wadsworth, Robinson Hurley, *Detection of Reflections
on Seismic Records, Geophysics XVIII, 3, July, 1953
7.
Kendall, Maurice G., The Advanced Theo ry of Statis tics
8.
Southwick
i. I
.
Charles Griffin and Compmay Limited, London, 1947
Peter, YVgity of Shear Warp hrough
M.I.T. Ph.D. Thesis, 1952
Unonsoliaedaterils
APPENDIX A
APPENDIX A
POLYNOMIAL 1 (2492 m 1) - Desoritign and Use 0$f
This program was written for the WW1 Digital
Computer to eliminate the task of computing the residuals
from a least squares fitting of an n
order polynomial to
data taken over an arbitrary-sized rectangle. A copy of
the program appears at the end of this Appendix.
Polynomial I does the following.
I. It solves the equation
g(xy) = 000 + Clx + C2 0
2
+
.....
+ c 0 y + Cxy
+ *...
+ 00, (n-1)yn"'
+ Cl, (n-l)Xt
+ C
+
cn0
xn~l
Ony
where the Ck !s are given, n, the order of
the polynomial is given, and the values of
x and y are to be taken over a rectangular
grid measuring 2N by 2M, and with axes centered
as in Fig. 1.2.
II.
It then forms the differences g(xy) for
all values of g(xy) on the grid.
the residuals.
These are
III.
It prints out these residuals in the same
network fashion that the grid was chosen.
USe of Polnal
I
There are certain conventions which must be
observed in the use of this program.
The constants
defining the nature of the polynomial and grid must
appear as follows:
Eegister 440
+n
order of polynomial (less than 7)
(Octal) 441
+N
greatest value of x
442
+1
greatest value of y
The coefficients Ck of the polynomial must be scale
factored in a special way because they decrease in magnitude rapidly as J+k increases.
10 (+k)-2,
The scale factor is
which in most instances will guarantee that
all are less than unity in absolute value, but not
greatly so.
They must appear in the machine as follows.
Register.. 443
C
"x 10-2
(Octal)
444
Clo
x
445
C20
446
461
C
10
462
022
x
1
463
C32
030
x
10
464
C42
447
040
x
102
465
450
C50
x 103
466
C13
451
060
x
104
467
C23
452
001
x
10
470
033
453
011
x
1
471
C04
x
102
454
021
x
10
472
Cl4
x
103
1
10
x
102
3
x
1O
0310
x
102
1(3
455
C31
456
C41
457
C51
460
102
473
024
x 104
x
103
474
c05
x 103
x
10
475
x15104
x
476
x02
C6
0
x
104
The data g(xy) which is presumed to be taken
over the gridwork, is scale factored by 10'2
in the machine as follows:
Register
540
g(-NN)
x
10-2
541
g(-N+1,N)
x
10
542
g(~N+2,M)
x
10
.
g(N,1)
x
10
.
g(-N,M-1)
x
102
.
g(-N+1,-L)
x
10" 2
.
g(N,?-1)
x
10- 2
.
g(--N,M-2)
x
10-2
.
g(-N,-M)
x
10-2
.
g(-N+1,-N)
x
10.
-2
2
and appears
S
g(N,--M)
x
10-2
.
g(-N+1,-M)
x
10
.
g(N,-M)
x
102
Now suppose the information n, N, 1, and the
.4kIs are prepared on a tape with the tape number X, and
the date g(xy) is prepared on a tape with the tape number
Y.
Then the instructions for the operation of this program
would be
Erase storage
Read in 2492 m 1
Read in X
Read in Y
Start at 127 (Octal)
The residuals are printed out by the direct
printer in about three minutes or so depending on the
size of the grid.
They appear as four-digit numbers where
the decimal point is understood to occur after the second
digit.
As an example of the output we include a sample
of three sets of residuals.
These were derived for the
three sets of coefficients used elsewhere in this paper.
The sample illustrates the convenience of this form of
answer for contouring purposes.
In fact, with only slight
modification (inserting two extra carriage returns between
lines) these numbers would appear on a grid with square
unit cell, and could be contoured directly, on the result
sheet.
A Technical Feature in Polynomial I
We describe here a technical feature in this
program which might be of use to other programmers.
The
problem is that we are multiplying numbers rapidly decreasing in magnitude with I+k (the Ck' s ) by numbers
rapidly increasing in magnitude with J+k (x1 y ) while the
product is of a relatively constant order of magnitude,
which must be in a form which we can add to other such
products.
What we want is the product CJkxiyk to be scale
factored finally by lo .
To preserve accuracy during the
computation of the product we do the following:
1. Form x
2-15 and then
2-15+9
2-215+u ay yk
215 ad yk
scale factor to x
by use of the scale factor order.
2. Form C k0(1+k)-2x42-15+uy2-15+P
(1)
C x k2-30+u+% O(+k)-2
To get this product to the form CJk
we must multiply by
k1 0-2
It appears that we merely need to store the
negative powers of 10, multiply the expression (1) by
10-(i+k) and then shift left 30-(a+p).
However the negative
powers of 10 cannot be stored with any accuracy for high
A+k so we write 1 0 4+k) 2 30-(a+p)
in the form
2-( J+k )log2 10+30-(a+p)
2 -3.32193(AI+k)+30-(a+p)
S2-32193(,t+k)
[2-3($+k)+30-(a+p)3
(2.32193)(+k) [2-3(+k)+30-("+P)]
(.
800
)4+k [2-3(I+k)+30-(a+p)]
We can store the powers of (.800) with ample
accuracy.
Thus we multiply by the appropriate power of
(.800) and follow this by a shift left or right according
to the exponent of 2.
in as +.9999.)
(The zeroth power of (.800) is put
101b
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I
I
I
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I
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.
I
I
T
I
I
I
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ON
I-
40
qO
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4
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a 0
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tosAg
loop-10
APPENDIX B
APPENDIX B
PREDICTION XV (2539 m 2)
-
Desoriation gnd Use Of
This program was written to provide computational
facility for predicting a series xi(Y±,Z,. or u.) with a
linear operator of the general form
i+k i+k, zi+k or ui+k)
a + a0 i a xi..
+
bo01......+
+
0
z
.
ax-,
bAvi-M
..
+ doni.
.+
.
0
Mz i-N
dui-M
where the prediction distance k and the nutber of lags N
are arbitrary but have the restrictions that N 7 and
N+k
19.
The four series which this program handles
contain $00 members each so that i ranges from o through
499.
This first prediction computed is for i+k = 20 and
the last one for i+k
=
499.
After doing this computation
the program forms the running. average of squared errors
between the predicted and actual series
E
i=3-5
(xI-xi)2
j
=
25, 35, 45, 55,
-.. 495
which is called the "Error Curve".
There is considerable choice of output.
The
alternatives are any combination of or none of the following:
1. Print-out of the errors and sums of squares
2. Print-out of just the sums of squares of
errors.
3.
Photographs of oscilloscope displays of
the sums of errors squares
An additional choice is the use of magnetic tape delayed
output for 1 and 2 above, which is about fifteen times
faster than direct print-out.
This program handles up to eight operators at a
time in the above fashion.
When the magnetio tape output
is used, the error curves can be removed from the machine at
the rate of one every ten seconds whereas the individual
errors and error curves require fifty seconds for each
operator.
Each curve would represent about a week of hand
computation. Once the computations are completed the individual
errors (xi-xi) for all operators are left in magnetic drum
storage so other programs can use them for different types
of averaging processes rather than just the Error Curve as
described above.
On the next three pages are illustrated the
various output forms.
The first page is a reproduction of
the individual errors and error curves for two operators.
The results for each operator appear as a block of numbers
10 by 48 and a right-hand column of 48 numbers.
The block
represents the 480 individual errors whereas the right-hand
column is the Error Curve, each member representing the sums
of the squares of the 10 individual errors in the corresponding
row to the left.
The number appearing over the upper left
corner of each block is a number assigned to the particular
operator for identification purposes, and is printed by the
program. The number printed over the center of each blook
was inserted later.
The next page shows the output form for four
operators when just the Error Curve is desired. The +0000
identification number indicates that the operator was chosen
as theiarianoe operator which has the forn x
of series).
(means
The Error Curve for this type operator becomes
the sample variance curve and provides a basis for testing
the statistical significance of other operators predicting
the x
series.
The third page is a photograph taken automatically
by the program of an oscilloscope display of one-half of
an Error Curve. A vertical and horizontal axis are also
displayed.
+1300
+0027
+0000
-0020
+0058
+0006
+0010
+0057
-00±9
+0088
-0059
-0021
+0000
+0016
+0000
-0005
-0049
-0013
+0047
+0002
-0048
-0029
-0002
+ooo9
+0017
+0037
+0028
+0050
+0021
-0028
+0005
+0043
-0010
-0021
+0023
-0009
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
+0052
+0007
+0000
+0053
+0019
+0005
+0030
+000±
+0079
+0043
-001±
-0003
+0008
+0010
-oo6
-0073
+0002
+0032
+0025
-0092
-0021
-0035
-ooo
+0010
+0017
+0021
+0036
+0060
-ooo8
-oo40
+0028
-0002
-0003
+0036
+0012
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
+000
+0001
-0009
+0009
+0062
+0013
+0043
-ooo8
+0054
+0093
-0005
+0046
+0031
+0054
+oo14
-0022
+0018
+0040
+0051
-0036
-0025
-0005
-0060
-0043
-0025
-0001
+0017
-0002
+0013
-oo69
-ooo4
-0002
-0009
+0002
-0234
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0036
-0012
-0014
-0025
+0013
+0011
+0012
-0064
+0041
-ooo6
+0095
-0001
+0005
+0024
-0016
-0050
-0013
+0020
+0067
-oo48
-0010
+0010
-0053
-0031
-0058
+0008
-0007
+0003
+0035
-0059
-ooo4
-0021
-0011
-0002
-0242
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0009
-ooo6
-0007
+0032
-0021
-0016
-0067
-0021
+0005
+0034
+0153
-0001
+0014
+0019
+0010
-0042
-0027
+0025
+0079
-0060
+0039
-0022
-0025
-0003
+0001
+0014
-0060
+0000
+0029
-0038
-0014
+0008
-0047
+0014
+0186
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
12.4 Second Half
+0027 +0016 +0024
+0014 -0001 -nn41
-0030 -0031 -000
+0014 +0003 -0016
-0015 +0003 -0002
-0007 +0008 +0031
-0032 -0019 -0028
-oo43 -0019 +0043
-oo4o -0046 -0036
-0025 -0078 -ooo9
+0056 +0009 -0048
+0021 -0003 +0001
-0009 -0022 +0001
+0025 -0021 -0027
+0030 -0014 -0002
+0019 +0045 +0042
-0018 -00±7 +0008
+0002 -oo48 -0022
+0067 +0037 +0035
+0013 -0000 +0007
+0018 +0033 +oo44
+0025 +0o40 +0054
-0049 -0032 +0042
-0057 -0039 -0053
-0011 -0010 -0012
-0014 -0004 +0021
-0052 -o41 -0086
-0052 -0052 -oo41
+0034 +0062 +0082
-0040 -0025 -0007
-0028 -0036 -0027
+0026 -0001 +0005
-0039 -0051 -0072
+0003 -0031 -0010
-0140 -0191 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
-0075 -0075 -0075
+0032
-oooo
+0054
-0066
+0034
+0040
+0009
+0020
-0034
-0042
-0022
+0027
-0003
+0013
-0054
+0035
-00±9
-0024
-oo47
+0019
-oo43
+0086
+oo46
-0009
-oo4o
-0002
-0023
-0066
+0067
-ooo4
-0025
+0022
-0001
-0001
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
+0024
-n0i
+0073
-0024
+0050
+0055
+0003
+0053
-0049
-0091
+0075
+0048
+0026
+0000
-oo41
+0028
+0019
-0028
-0058
-0013
-0051
+0031
+0039
+0070
-0005
+0017
-0010
-oo6
+0029
+0025
-0017
-0013
+0010
-0017
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
-0075
+0007
+0001
+0009
+0012
+0007
+0005
+0011
+0011
+0026
+0031
+0043
+0005
+0001
+0005
+0004
+0017
+0001
+ooo8
+0025
+0017
+0010
+0014
+0015
+0015
+0006
+0001
+0019
+0018
+0018
+0013
+0005
+0001
+0011
+0002
+0220
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
+0054
-0023
-0001
-0021
+0134
-0041
+0022
+0069
+0014
+0071
-0054
-0068
-0084
+0033
-0027
+0065
-0126
-0020
+0014
+0002
-0±10
-0037
-0038
+oo8o
+0086
+0081
+0012
+0010
+0077
-0023
+0007
+0015
-0004
-0043
+0032
-0015
-0255
-0180
-0±80
-0180
-0±80
-0180
-0180
-0±80
-0±80
-o18o
-0180
-o±8o
-0±80
+0016
-0036
-0031
+0130
+0060
-0006
+0035
+0012
+0057
+0058
-0018
+0018
+0001
-0019
+0040
-0126
-0051
+0007
-0007
-0146
-0052
-0050
-ooo6
+0013
+0135
+0037
+0022
+0090
+0013
-0065
+0012
+0009
-0036
+0017
+0010
-0235
-0±80
-0180
-0180
-0180
-0±80
-0180
-0180
-0180
-0±80
-0180
-0180
-0180
+0030
-0052
-0003
+0033
+0069
-0012
+0024
+0004
+0046
+0112
+0029
+0004
+0002
+0019
+0001
-0056
-0027
+0052
+0018
-0082
-0037
-0071
-0083
-0079
+0052
+0034
+0042
+0042
+0046
-0078
+0009
-0010
-0009
+0027
-0219
-0180
-0180
-0±80
-0±80
-0±80
-0±80
-0180
-0180
-0180
-0±80
-0180
-0±80
-0±80
+0003
+0013
+0000
-0028
+0051
-0004
-0024
+0030
+0007
+0128
+0105
-0015
-0023
+0037
-0043
-0036
-0047
+0047
+0039
-0063
-0029
-0057
-0077
-0083
+0002
+0012
+0018
+0000
+0073
-0088
-0027
-0016
+0001
+0013
-0222
-0180
-0±80
-0180
-0180
-0180
-0±80
-0180
-0±80
-0180
-0180
-0±80
-0±80
-0±80
-0000
+0010
-0035
-0064
+0016
+0018
-oo81
+0014
-0023
+0120
+0190
+ooo4
-0012
+0050
-0035
+0025
-0056
+0027
+0057
-0008
-0000
-0003
-0110
-0102
-0090
+0003
-oo41
-0060
+0032
-0066
-0023
+ooo6
+0006
-oo4o
-0215
-0±80
-0±80
-0±80
-0180
-0180
-0±80
-0180
-0±80
-0180
-0180
-0180
-0±80
-0±80
12.4
-0013
+0033
-0032
-0077
-0047
+0007
-0096
-0024
-0061
-0023
+0209
-0018
+0001
-0011
-0019
+0057
-0049
+0005
+0067
+0029
+0039
-0027
-0095
-0039
-0±08
s0020
-0084
-0035
+0021
-0029
-0003
+oo14
+0013
-ooo8
-0213
-0±80
-0±80
-0±80
-0180
-0±80
-0±80
-0±80
-0180
-0180
-0±80
-0±80
-0±80
-0±80
Second
-0029
+0015
-0055
-0034
-0058
+000
-0068
-0003
-0083
-0149
+0132
+0021
+0041
-0068
+0008
+0098
-0023
-0000
+0074
+0049
+0078
-0006
-0017
-0016
-0057
+0008
-0120
-0074
+0027
-0030
-0010
+0014
-o045
-0003
-0219
-0±80
-0±80
-0±80
-0±80
-0180
-0±80
-0±80
-0±80
-0±80
-0±80
-0180
-0180
-0180
half
-0013
+0010
-0044
-0014
-0030
+0020
-oo46
+ooo6
-0052
-0158
+0000
-0007
+0015
-0052
+0040
+0100
-0005
-0025
+0029
+0032
+0068
+0085
+0025
-0063
-0061
-0003
-0100
-0075
+0030
-0016
-0011
+0015
-0090
-0025
-0209
-0±80
-0180
-0180
-0±80
-0±80
-0±80
-0±80
-0±80
-0180
-0180
-0±80
-0180
-0±80
+0011
+0007
-0003
-0032
+ooo8
+0035
-oo41
+ooo4
-0052
-0123
-0096
+0017
-0023
-0023
-0030
+0060
+0011
-0061
-0050
+0003
+0021
+0087
+0061
-0003
-0048
+0006
-0028
-0045
+0055
-0001
-0018
+0000
-0054
-0018
-0201
-o18o
-0180
-0180
-0±80
-0±80
-0±80
-0±80
-0±80
-0±80
-0±80
-0±80
-0±80
-0180
-0006
-0007
+0075
-0053
+0024
+0073
-0023
+0051±
-0063
-0086
-0139
+0079
+0003
+oo41
-0073
+0049
+0022
-0017
-0041
+0014
-0020
+oo42
+oo8o
+0018
-0020
+0015
-0002
-0071
+0068
+0019
-0011
-0026
+0007
-0031
-0175
-01±0
-0±80
-0180
-0±80
-0±80
-0±80
-0±80
-0±80
-0180
-0±80
-0180
-0180
-0180
+0001
+ooo4
+0013
+0050
+0018
+0007
+0030
+0003
+0029
+0119
+0142
+0014
+0003
+0013
+0016
+0065
+0011
+0010
+0019
+oo48
+0018
+0028
+0050
+0036
+0055
+0002
+0035
+0037
+0018
+0023
+0001
+0000
+0015
+ooo4
+0352
+0379
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0323
+0000
+04-54 40375 +0437 +0366 +06G5 40253 +05-1
+0339 40569 +0354 +0465 +0418 40611 +0357
+0343 40466 *0112n
3 +04PP-L03
+0396+04b3 +0423 +0313 +0593 +0216 +0395 +0399
40435 -+23r--+0644&*0530-+0342 +0346 40422 +0288 +0674 +0390
+0512 +0332+0355+0775 *0256
+0399 40399 +0399 +0399 +0399
40000
+0616 +1241
+075 4072b
+940536+1091 +0550
-40743
+07b4 +06$1 tob64 -90
+0871 +0437 +1340 +0646 +0771
+15
+0532
+0900 +0900 +0900 +0900 +0900
+0509
+0834
+0977
+0848
+1162
+0761
+0667
+1040
+0761 +0725 +0995
+1013 40594 +1030
+099 0520 +0853
+0384 0900oo0900
+0000
+0373 +0337 4509 +0437 +0484 -0316 40519 40374 +0431 +0507 +0351
40395+:0539R
+0337
+0391- 40394
+0359 +0406 +0510 +0307 +0372 40408 +0413 f0437 +0326 +0395 +0427
+03b8 +0506 +0279 40451 +G424 +0362 +0399 +0399 +0399 +0399 40399
-
40485
:+0328
+0326
+0399-
40000
+0800 +0713 40818 +0886 40927 +0833 +0680 40949 +0806 40643 40703 40846
+020i406903
06914
-+0667
+0653 +0696 +0077 +0663 +0758 +0905 +0747 +0802 +0603 +0802 +0952 +0696
+0770 +0613 +0656 +0755- 0855 -+0677 4+07b4 +07b4 +0-764 +0784 +078[+OV4
Use of Prediction Xt
It is necessary to prepare a tape containing
the operators and a tape containing the traces x., y.,
zi, u1 . These are prepared as described in the following
two pages.
Assume these are given tape numbers X and Y
respectively.
Then the operating instruction would be,:
Erase storage, put Sil switch off
Read in 2539 m 2
Read in 2539 P -
(Control Tape)
Read in X
Place Y in Photoelectric Reader
Start at 145
The control tapes control the output and serve
the following functions.**
2539 - PO
2539
-
2539
2539
-
P1
Print errors and sums of squares
and scope display sums of squares
Print sums of squares and scope
P2
display sums of squares
Scope display sums of squares
P3
Print errors and sums of squares
2539 -P4
Print sums of squares
2539 -P5
Print nothing, display nothing
DIRECT
PRINTOUT
2539
-
Pie
Print errors and sums of squares
2539
-
Pl
and scope display sums of squares
Print sums of squares and scope
DEIATED
PRINTOUT
display sums of squares
(MAGNETIC TAPE)
2539
-
P12
Print errors and sums of squares
2539
-
P13
Print sums of squares
If one of the operators on X were badly prepared,
it might happen that machine overflow would occur causing
the machine to stop while computing for that operator.
If
this does happen, starting the machine over at 166 will have
the effect of ignoring the bad operator and proceeding to
the remaining ones.
Preparation of Data Parameter
is prepared as
Each set of data x~t y , zi, and u
a separate parameter and then the four parameters are combined into one long one.
Octal
Address
1054
1055
x
X
Octal
Address
1054
5 Y 1055
The form of each is identical.
Octal
Address
1054
y
1056
1056
2037
X499
Start at 1033
2037
Y499
Start at 1033
2
10 6
0
z
z
z
2037
Z499
Start at 1033
Octal
Address
1054
1054
10 6
u
u1
2037
u4 99
Start at 1033
Notes:
It is not necessary that the series contain 499
members.
However, there must be four traces.
If less than
four are to be used, short dummy traces must be inserted.
For consistency with the operator tape, the order of
combination of the separate parameters must be x , yi,
Zi,
u .
The data must appear as integers in the range
-99 through +99.
Pkrenration of 0perator Parameters
Up to 8 operators may be prepared on a single tape
in the following fashion.
Octal
Address
+N
+.XXXX
1055
1056
1057
1060
1061
1062
Explanation
Contents
-0,-1,-2 or -3
+k
&no. on tape
of oerators
N
Ident. no. for first operator
-0 if xi, -1 if YA
-2 if zi, -3 if U1
+M
+.XXXX
t.XXXX
+_.x~XX
a7a-1
x1.l
b x 10
Constants
for yi
0 f
0
1100
1101
00 x 10 '-
0
P
E
A
T
Constants
7
for z.
10.
a
0 x
1110
1111
.
d0 x 10
Constants
1
for u.
d x 10
I~ent. no. for second operator
etc.
1120
1121
1165
registers.
Constants
for x.
R
1070
1071
Notes:
1 rst
a x 103
2*
0
P,
etc.
It is not necessary to put anything into irrelevant
For example, if
the first operator had an M of
3 registers 1061-1064, 1071-1074, 1101-1104,
would be considered irrelevant.
did not use the u
Again, if
and 1121-1124
this operator
series in its prediction mechanism,
registers 1111-1120 would be irrelevant.
w
(L)
I
LC)
2
&
daN
el Is
+
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If888 25 =",2
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8
o c. ..
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t:
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-
8
8
-
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-TV
.
84,
6.
1
N
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~
t
a
8r 11
OH I
Nt
ttt11t
-
IoIAH
,
1
0
I
E
t
0
280
00
0 9L
-o
3*-3
MO-1
400
ol
4-
02
A
9 03
104
2',
-
06
/07
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-
J
222
523
23
I24 J"y
4
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DIGITAL COMPUTER
OCTAL PROGRAM
PR~.LCIONhLWINDEX
AUTOR
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TAPE1m4,
NUMBER
LABORATORY
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TITLE
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M
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AUTHOR SIMPSICIN
TAPE NUMBER
2539
LABORATORY
FORM
INDEX
DATE
!1
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2
APPENDIX C
APPENIX C
ITEBATION I (2615 m 2) -Decriotion and Use
Of
This program was written with the purpose
of obtaining least squares fits for linear operators
as described in Part IV.
It
computes essentially as
described, but has provisions for changing its inorement after cycling for a certain prescribed number
of times.
The program was designed to be run in conjunction with the Prediction XV program described in
Appendix B, and to illustrate the conveniences which
The data to which the linear
programs can include.
operator is to be fitted is prepared in the same fashion
as in Prediction XV.
The information about the operators
to be found (Iteration I solves up to eight operators
one after the other) is prepared as a single tape.
The
operator coefficients once fQrmed are printed out, and also
are left in the machine in a form to be used directly with
the prediction program.
The output of Iteration I was designed to
eliminate identification problems.
In addition to
printing out the coefficients identified, it prints out
the operator number, the operator parameters including
which set of data is predicted, and the variance and
minimum sums of square errors.
These last two numbers
allow a rapid computation of the percent reduction
R = 1 -
min va,
which is
the least squares fit.
a measure of the goodness of
A sample of the output appears below.
Variance sum = 004953
Minimum pum = 001840
Operator No. -1010
N 066
n = 050
k = 002
M = 003
T4 predicted
a4000 = +0292
a340
= -0000
+0565
a2 40A
a140 = -0000
aO 40 = +0005
d3/0
d2 40
dl40
= -0341
= -0117
= -0410
dO 40
= +0708
The Program may be used to print out all the
values of I as they are computed.
A plot of these values
for a particular operator appears in Part IV.
One other feature in this program is a *roll
back" procedure.
This permits us to avoid having to
start from scratch if the machine fails in the middle of
the long computations.
Every fifteen seconds during the
computation, all of electrostatic storage is transferred
to the magnetic drums which are very reliable.
Then if
electrostatic storage is destroyed, we can call back the
program from the magnetic drums and start over where we
left off not more than fifteen soonds ago.
Use of Iteration I
If the operators are prepared as described on
the next page with a tape number X then the instructions
for the operation of this program would be:
Erase storage, Sil switch down
Read in 2615 m 2
Read in 2615 P
Read in
(control tape)
(data tape)
Y
Place X in photoelectric reader
Start over at 145
The control tapes control the output and serve
the following functions:
2615 P 0
Print out operator and identification (direct printer)
2615 P 1
Print out operator, identi-
fication, and all values of I
(delayed printer).
Prenaration of Onerator Parameters
The information for each operator is prepared
as a short separate tape and the tapes are then combined
in any order.
Octal
Address
1001
1002
1003
100
1005
1006
1007
1010
12
1112+X3
1113
The form of each operator is identical.
Contents
+N
Explanation
First member op. interval
+n
Length of op. interval
-1 if X not used
+0, or -1
+0, or -1
-1if y
+0, or -1
+0, or -1
+.XXX
-1 if uz
-1 if
-~0,-1,-2,or -3
+k
+.XXXX
Start at 147
a
1A
First o4erator no.
-0 if xj, -1 if yi
-2 if zi, -3 if ui
mean of pred. series x 10-
The first of the separate tapes must have one
additional register, register 1000, which contains + no.
of operators on the combined tape.
Register 2123 contains +.0010 which is the
first increment to be used for the a term.
Register 2223
contains +.0100 which is the first increment to be used with
the remaining constants.
Register 3421 is the counter for
the cycles at these increments.
The second set of increments
is 1/10 the first set, and appears in registers 2124 and 2224.
The counter for this set is register 3433.
These registers
may be changed to adapt to the particular problem.
The "roll back" proced.ure in case of electrostatic
storage failure is
Erase storage
Read in 2615 P 13
Start over at 145
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TITLE
APPENDIX D
APPENDIX P
AUTO CROSS-CORRELATION I 2559 mo, ml) - DescritijnQand UseOf
This program was written for the WWI Digital
Computer to compute the unnormalized sample correlations
N+n-1
:
i+N
3x 0, 1, 2 ....m
the conventions for preparation of the data
x
and Y are identical with those desribed for Prediction
XV in Appendix B, with the exception that the data tapes
need not be combined after preparation. A short tape is
prepared containing the information N, n, and m as follows
Register
1047
+ N
(First data point in blook)
(Octal)
1050
+ n
(No. data points in block)
1051
+ m
(No. lags)
2559 mo handles individual data tapes and is used
as follows
Erase storage, put Sil down
Read in 2559 mo
Read in Z
(tape for N, n, m)
Read in X
(X data tape)
Read in Y
(y
data tape)
Start at 770
If X and Y are not identical we get half of the
cross-correlation curve (for j
0).
To get the other half,
we repeat the instructions interchanging the order of read-in
for X and Y. If X and Y are the same tape, we get the
entire auto-correlation curve, since since auto-correlations
are symmetric about the zeroth lag.
The correlations are printed out by the direct
printer as seven-place numbers, ten per line, the 09 lag
being the first no. on the first line, the 1
leg being
the second no. on the first line, etc.
2559 ml performs the same functions as 2559 mo,
but is adapted for handling the combind tapes used with
Prediction XV. It assumes there are 3 real data sets plus
a dummy set and forms the nine correlations representing the
permutations of the 3 real sets.
The correlatios are over
380 values of the data, and are taken to 100 lags.
The
output is the delayed printer, and requires one minute for
each 100 lag: block.
At this rate the program can perform
8 or 10 million multiplications in 4 hours of machine time.
A sample of the output is shown on the next page.
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The instructions for use are
Erase storage, Sil switch down
Read in 2559 ml
Read in W
(combined data)
Start at 677
If the combined tape has 4 real data sets, and we
want the 16 permutations of correlation, then an additional
tape is used and the instructions are
Erase storage, Sil switch down
Read in 2559 ml
Read in 2559 p4
Read in
W
Start at 677
2559 ml is equipped with the same *roll back"
procedure that Iteration I is (Appendix C).
In case of
machine failure
Erase storage
Read in 2559 p13
Start at 677
Traveinpg Correlations
With the aid of tape 2559 p1O we can use 2559 mo
to obtain correlations from highly overlapping blocks of the
data.
The correlations are over blocks 50 in length and the
number of lags is taken to be 20.
The first reading in each
block has an index (N) equal to k x 10 where k = 3, 4, .
44.
The procedure for using 2559 mo in this way is
Erase storage, put Sil down
Read in 2559 mo
Read in X
Read in Y
Read in 2559 plO
Start at 770
leave in P.E.T.r.
(21 lags are printed for N = 30)
Read in
Start at 770
(21 lags are printed for N = 40)
Read in
Start at 770
(21 lags are printed for N =50)
etc.
Start at 770
(21 lags are printed for N = 440)
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OCTAL PROGRAM
TITLF~~la
AbTHOR
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LABORATORY
FORM
DEX
SIMPSONl
NUMBER
DATE
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A2PEI~DIX E
APPENDIX E
BIOGRAPHICAL NOTE
Stephen Milton Simpson, Jr.
Attended Yale University September 1946 - June 1950,
receiving B0S. in Physics.
Entered the Massachusetts
Institute of Technology in the Department of Geology in
September 1950.
Member of Phi Beta Kappa and 6igma Xi.
Presently under appointment as Instructor in the
Department of Geology and Geophysics at the Massachusetts
Institute of Technology.