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SIMPLIFIED NUMERICAL MODELS FOR THEGENERATION OF SYNTHETIC REFLECTIONPROFILING SEISMOGRAMS AND SYNTHETICREFLECTION/REFRACTION SEISMOGRAMS
by
Charles David Lodge
United StatesNaval Postgraduate School
THESISSIMPLIFIED NUMERICAL MODELS FOR THE GENERATION OF
SYNTHETIC REFLECTION PROFILING SEISMOGRAMS AND
SYNTHETIC REFLECTION/REFRACTION SEISMOGRAMS
by
/ /3Z£4 3
Charles David Lodge
April 19 70
Tku> document hcu> been approved fan. public iz-lzcu>e. and 4>alt; lti> dUt/iibutlon JU> untmLttd.
Simplified Numerical Models for the Generation of
Synthetic Reflection Profiling Seismograms and
Synthetic Reflection/Refraction Seismograms
by
Charles David LodgeLieutenant Commander, United States Navy
B.A., Rice University, 1962
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN OCEANOGRAPHY
from theNAVAL POSTGRADUATE SCHOOL
April 1970
ABSTRACT
Two numerical models were developed which generate synthetic
seismograms in formats similar to those of records obtained in field
exploration. The models solve the hyperbolic wave equations in one
and two dimensions by finite difference approximations in initially
undeformed solution domains of transversely-isotropic layered media
subjected to a time varying, dilatational forcing stress applied at
the surface. A velocity attenuation term was included in the models
to inhibit strong boundary reflections.
The one-dimensional model produces synthetic reflection profiling
seismograms for arbitrary horizontal or dipping layers. The two-
dimensional model generates synthetic reflection/refraction seismograms
for horizontal layered media with arbitrary distribution of wave
velocities, Poisson's ratio and density. Several sample records were
produced for some representative velocity structures. The synthetic
seismograms were interpreted and gross correlation was carried out
as if they were actual field records.
OSTGKADUATE SCHOOLY, CALIF. 93940
TABLE OF CONTENTS
I. INTRODUCTION 11
II. THE SEISMIC REFLECTION PROFILING MODEL 14
A. THE MATHEMATICAL MODEL 14
B. THE SOLUTION DOMAIN 16
C. THE DIFFERENCE EQUATIONS - - 16
D. STABILITY AND CONVERGENCE 20
1. Stability 20
2. Convergence 21
E. DESCRIPTION OF THE PROGRAM 22
1. Distribution Parameters 22
2. Numerical Parameters 23
F. SCALING THE PROBLEM 27
III. THE REFLECTION/REFRACTION MODEL 30
A. THE MATHEMATICAL MODEL 30
B. THE SOLUTION DOMAIN 33
C. THE DIFFERENCE EQUATIONS 33
D. STABILITY AND CONVERGENCE 37
1. Stability -- 37
2. Convergence 38
E. DESCRIPTION OF THE PROGRAM 38
1. Distribution Parameters 39
2. Numerical Parameters 44
F. SCALING THE PROBLEM 45
IV. INTERPRETATION £7
A. THE REFLECTION PROFILING SEISMOGRAMS - 47
1. Variable Depth Records 51
2. The Seismograms 54
B. THE REFLECTION/REFRACTION SEISMOGRAMS - — 71
1. The Seismograms 74
2. The Contour Output --- 91
V. SUMMARY - - - 99
VI. FUTURE WORK - 101
COMPUTER PROGRAMS - 103
THE REFLECTION PROFILING MODEL 103
THE REFLECTION/REFRACTION MODEL - 109
SUBROUTINES - - 120
LIST OF REFERENCES 126
INITIAL DISTRIBUTION LIST - - 128
FORM DD 1473 - 129
LIST OF TABLES
TABLE PAGE
I. Sample Horizontal Station Distances 26
11. Sample Velocity Defining Array 26
III. Sample Attenuation Parameter Polygon
IV. Sample Forcing Function Polygon 27
V. Scales Available for C - 0.8 = 1.5 km/sec -- 29
VI. Sample Wave Velocity and Density Polygons 40
VII. Sample Forcing Function Polygon 40
VIII. Sample Attenuation Parameter Polygons 40
IX. Scales Available for C = 0.6 = 1.5 km/sec 46
X. Scale of Contoured Quantity 95
LIST OF FIGURES
FIGURE PAGE
1. The Solution Domain 17
2. Mesh Detail 18
3. Sample Velocity Structure for Seismic Profiling 24
4. Sample Distribution Parameters 25
5. The x-y Solution Space 34
6. Isometric Detail of Three-Dimensional Mesh 35
7. Sample Distribution of Wave Velocities and Density 41
8. Sample Distribution Parameters 42
9. Sample Distribution of the Attenuation Parameter 43
10. Velocity Structure C3H 48
11. Velocity Structure C4D 48
12. Velocity Structure C5B 49
13. Velocity Structure C6C 49
14. Velocity Structure C7A 50
15. Velocity Structure C8A 50
16. Forcing Function F7 52
17. Attenuation Parameter A14 52
18. Stress versus Time at Various Depths 53
19. Uncorrelated Reflection Profile C3H F7 A14 55
20. Reflection Profile C3H F7 A14 57
21. Reflection Profile C4D F7 A14 59
22. Reflection Profile C5B F7 A14 61
23. Reflection Profile C6C F7 A14 63
24. Reflection Profile C7A F7 A14 65
25. Reflection Profile C8A F7 A14 67
26. Precision Reflection Profile C4D F7 A14 69
27. Velocity Structure CA 72
28. Velocity Structure CB 72
29. Velocity Structure CC 73
30. Forcing Function F5A 75
31. Attenuation Parameter B3 75
32. Reflection Seismogram CA F5A B3 L2 76
33. Refraction Seismogram CA F5A B3 L2 81
34. Reflection Seismogram CB F5A B3 L2 83
35. Bottom Detector Seismogram CB F5A B3 L2 85
36. Reflection Seismogram CC F5A B3 L2 87
37. Theoretical Time-distance Curves for VelocityStructures CA and CB 89
38. Portion of the Printer Contour Plot Showing Generationof the Dilatational Wave 93
39. Portion of the Printer Contour Plot ShowingTransmission of the Dilatational Wave into theLower Media 94
40. Portion of the Printer Contour Plot Showing the
Development of a Dilatational Head Wave 96
41. Portion of the Printer Contour Plot ShowingGeneration of a Shear Wave at the Transition Zoneby the Dilatational Wave 97
42. Portion of the Printer Contour Plot ShowingDevelopment of an Interface Shear Wave 98
ACKNOWLEDGEMENTS
The author wishes to extend his sincere appreciation to Professor
Robert S. Andrews who provided invaluable assistance in delimiting the
scope of the problem and in interpretation of the results. Apprecia-
tion is also extended to the staff of the Naval Postgraduate School
computer Facility, especially Mr. David F. Norman, for assistance in
running and debugging the programs. Particular thanks are due to the
author's wife, Diana, whose moral support and secretarial skills aided
t lie project immeasurably.
I. INTRODUCTION
A major problem in exploration geology is the determination of
structural relationships for areas in which the rocks cannot be directly
examined. This is usually a subjective process, involving the extra-
polation of lithologic facies from a region where control is available,
such as surface outcrops, mines or wells, across an unknown or
inaccessible area to another point with known structure. Indications
of subsurface structure, including magnetic, seismic and gravity
measurements as well as interpretation of surface morphological
features, are frequently relied upon to aid this process.
The problem becomes particularly acute in oceanic regions where
the only available control may be continental outcrops several hundreds
of miles away, a few bottom grab samples and shallow sediment cores
and a rough bottom topography. Remote measurements may then become
the only method of initial structure determination.
Frequent use is made of the manner in which the ocean bottom and
sub-bottom rocks reflect and transmit acoustic or seismic energy, not
only for the correlation of near-shore with on shore structures, but
also for initial determination of sub-bottom structures. When con-
ducting seismic exploratory operations, the information obtained is
usually presented as a two-dimensional graph (seismogram) on which
three variables are shown. These variables are: horizontal distance,
time and intensity of energy received. Depth information may be
inferred from the time scale with a known or assumed vertical distri-
bution of sonic velocity. A high level of intensity is usually
assumed to represent a point or surface at which the physical
11
parameters of the media change more or less abruptly, reflecting energy
or providing a surface along which energy might be propagated.
Interpretation of the sei sinograms and the development of a sub-
surface structure can be aided by the use of a mathematical model of
the supposed structure which would produce a synthetic seismogram.
The synthetic seismogram could be compared with the field record for
correlation.
This paper presents the development of two numerical models which
produce synthetic seismograms for layered media subjected to an initial
time varying, compressional (dilatational) stress input applied at the
surface. The models were designed to have fairly wide variability
while remaining simple enough to preclude the use of inordinate
computer time and storage requirements. The one-dimensional model
produces a reflection profiling seismogram. It is applicable to the
case of vertically incident dilatational waves reflected and transmitted
by arbitrary bottom profile and sub-bottom layered structure of hori-
zontal or dipping interfaces. The two-dimensional model yields a set
of seismograms applicable to a reflection/refraction problem where the
receivers are distributed in a horizontal array. Both dilatational
and shear stresses are treated. The region modeled is one of horizontal
layered media with arbitrary distribution of shear and dilatational
wave velocities and of density.
No attempt was made here to include the wide variability of possible
media which is an integral part of the large SLAM (Stress waves in
LAyered Media) code model developed at the Illinois Institute of
Technology (IIT) [Constantino 1969] primarily to treat the neighbor-
hood of high energy (nuclear) explosively generated ground shocks.
12
The LIT model has the capability of simulating a wide variety of
conditions but requires a large amount of computing time and space in
order to produce a single trace. It is felt that the two models pre-
sented here have a greater applicability to marine and terrestrial
seismic exploration problems.
These models were developed on the basis of numerical approxima-
tions to the solutions of the wave equations in one and two dimensions
by finite difference methods. The computer programs were written in
the FORTRAN IV language and were run on a dual IBM 360/67 system. The
plotting subroutines used to drive a CALCOMP 765 plotter were prepared
by the staff of the United States Naval Postgraduate School Computer
Facility [l969j
13
II. THE SEISMIC REFLECTION PROFILING MODEL
The problem of synthetic seismic profile generation was approached
as one of repeated transmission of normally incident longitudinal
(dilatational) disturbances through transversely-isotropic media. The
seismogram was developed by changing the velocity structure in the
solution domain between pulse transmissions, corresponding to a change
in position of the shot point as the profiling vessel moves across an
area of changing water depth and sub-bottom structure. The solutions
are presented as successive traces as though from a pressure hydro-
phone located at any depth below the shot point.
A. THE MATHEMATICAL MODEL
It was assumed that the wave velocity (C) is continuous throughout
the domain and either varies linearly with depth (x) or is constant.
Additionally, it was assumed that in transition zones, where the wave
velocity does change, density remains constant. Under these assump-
tions, the equation of motion in one dimension for dilatational waves
can be written (modified from Wolf [1937J ) :
ht* by {?%)+ A fc . (i)
where u is longitudinal particle displacement and A is a velocity
attenuation parameter which is a function of x. Carrying out the
indicated differentiation of the right side of equation (1) yields:
E: *h> - cz s£u +. 1C 33l£ $£ 4- a ^V- . (2)H 1 c a** + 1C
^* ^y + A ^E U)
14
Four collateral conditions are required in order to complete a
well-posed problem. The following conditions were selected:
initial conditions:
II: u^,0) = 0'! (3)
12: ot (y ;o N
-= C , (4)
boundary conditions:
Bl: C(L,0 - O t (5)
B2 : UK (C,-t ) - fit-/}, t^ "ti •
(6a)
V (O, M =: ,t>ti , (6b)
where L is the distance to the lower boundary.
The initial conditions allow the solution domain to be initially
undeformed and at rest. The first boundary condition requires that
displacements are zero for all time at the lower boundary, which then
becomes perfectly reflective. The second boundary condition allows a
variable stress to be applied to the upper boundary for t( t,, For
t > t , normal stresses are not transmitted across this boundary. It
then becomes a free surface.
The second term on the right side of equation (1),A ^rr: , represe-
sents an attenuation factor, dependent on depth and particle velocity,
which was used to reduce the magnitude of the disturbance in the lower
regions of the domain so that reflections from the lower boundary would
not mask late arriving multiple reflections from points nearer to the
surface. The function A(x) is assumed to be zero in the region from
which reflections are to be studied.
15
B. THE SOLUTION DOMAIN
The solution domain (Fig. 1) is divided by N equally spaced mesh
points into N-l subintervals each of length £x. A sample wave
velocity distribution is shown in the figure. The upper boundary (x =
01 is shown on the left. In the region O^x^x., the wave velocity is
constant and equal to Cl. For x„4 x ^ L, the velocity is equal to C2.
In the transition zone, x. < x < x„ , the velocity varies linearly from
Cl to C2.
The time-depth mesh is an array of N by M mesh points. Each
rectangular mesh area is then of size S x by £t. This mesh is the
full solution domain and the objective of the model is to find a solu-
tion of equation (2) at each mesh point under the collateral conditions,
equations (3) through (6).
C. THE DIFFERENCE EQUATIONS
A central difference form for each of the partial derivatives in
equation (2) can be written in terms of the (i,j) mesh point [ Smith
1965] . Figure 2 shows the (i,j) and neighboring mesh points. The
required derivatives, in central difference form, become:
^S ± Cut- Ci-
16
12 3 4 5
t
VELOCITY = CI VELOCITY = C2
Xl
X2
Figure 1. The solution domain
N
t
u-l
17
\(j+1 A\ i.i+i
j
7T\-Pcnr—fti fen
j-1 c"n
Vij-i
i-1 L i+1
Characteristic curves
Figure 2. Mesh detail.
18
Substituting these into equation (2) gives the difference equation of
motion for the (i,j) mesh point:
"-)+ .-2cX,j-n> t|
j.i _
cf / ^i^-l^^u^jN
+ Lq (Cwi - O-A /O^J^U-y, A + fr.
(U^-O^-iN(7)
The collateral conditions become:
II: Util = o (8)
12:UL.a.- OC.o -
(9)
Bl: UMjj = O . (10)
B2: "\i "°°j = Fj ' (11)
where F. = F(t) , 0(ti t,
fj ° . •= > ti .
Solving equation (7) for u. .,-, gives:
E . o-tlj4i = [2UL,j (l- CiV2
-) Hr UL+x.j (C?fc* + ic^(c^,-c,.-i)c.a)
-cJ.r,(l+ i.A;St)]/[l-iALS^ , (i2)
where: £2 = ( ^-M *
Writing equation (12) for j=l and substituting u- n from equation (9)
yields the initial conditions:
II: L>-til = 6 (8)
12: 0„z - O . (13)
19
Writing equation (12) for i=l and substituting u . from equation (11)o
> J
yields the boundary conditions:
Bl: °^,j + i " ° » (10a)
B2: L,ri = [20 Lij(i-C 4V) + 2 a
,j(CfR 1
)
-:,r :(l^A i -t)^^(^)]/[l- i A t- (14)
The equations (12), (8), (13), (10a) and (14) then form a well-
posed problem for numerical solution. Initially, the j=l and j=2 rows
are set to zero following equations (8) and (13). The j+1 row can then
be solved utilizing equation (10a) for the lower boundary (i=N)
,
(14) for the upper boundary (i=l) and (12) for the interior points.
Note that the value of u anywhere along the interior of the j+1 row is
dependent only on the values of u at three points along the j row and
one on the j-1 row (shown as circles in Fig. 2).
D. STABILITY AND CONVERGENCE
1. Stability
Because of the large number of arithmetic operations involved
9(approximately 10 for the solution spaces used), it is necessary to
insure that there is a stable decay of errors introduced in the various
computations. Studies of the stability of the centered difference
formulation of the basic wave equation:
TZ- ' c -5** '(15)
have been carried out by several investigators [ Smith 1965j . Smith
investigated the stability by the Fourier series method and showed
20
that the centered difference formulation is stable for the mesh
ratio:
^ 4c
*< 16 >
The addition of the velocity gradient and velocity attenuation
terms in equation (2) alters the stability conditions somewhat. A
large velocity gradient, approaching discontinuous velocity change,
generates instability in the gradient region. Investigation of the
stability conditions showed that if the condition:
&C, - Ci+i - Ci-i 4 C'l , (17)
were imposed on the gradient regions, the requirement on the mesh ratio
[equation (16)1 becomes:
£t y0-89
.
bx * C (18)
The addition of a negative attenuation term tends to decrease
the right side of equation (18). This would lead to a more stringent
requirement on the mesh ratio. This effect was found to be quite small
for the values of A(x) necessary to damp lower boundary reflections.
Empirically, it was found that stability would be assured by requiring
that:
St y 0-S5- 4 -£- • (19)
2 . Convergence
A finite difference scheme is said to be convergent if the
exact solution of the difference equations approaches the exact
solution of the partial differential equations as Sx and St
21
approach zero together. Forsythe and Wasow [1967] have shown that
convergence of a finite difference scheme such as equation (12) with
boundary conditions such as equations (8), (13), (10a) and (14) is
convergent as long as the mesh ratio is such that the (i, j+1) mesh
point lies within the intersection of the characteristics through the
(i-l,j) and(i+l,j) mesh points (see Fig. 2). The equation defining
the characteristics for equation (2) is:
(£f ' <? (20)
This leads to:
6t
Convergence is then assured for:
= ±C •(21)
St > 1
4. -r * (16)
It is apparent then that as long as equation (19) holds, the finite
difference formulation will be both stable and convergent.
E. DESCRIPTION OF THE PROGRAM
The program requires 140K bytes of main core storage. An average
run with a 251x1800 mesh grid with 41 shot points requires approxi-
mately 13 min of main run time exclusive of output time required for
the plotting subroutines.
1 . Distribution parameters
The advantage of this model lies in the wide variability of the
conditions which may be easily simulated. The primary parameters used
to model various conditions are the distribution parameters. They are:
a. wave velocity distribution, C(x,t),
22
b. forcing function, F(t)
,
c. attenuation parameter, A(x).
The forcing function and the attenuation parameter arc put into
Liio model as single-valued, continuous, polygonal line functions of
the indicated dependent variable. The wave velocity is put into the
model as a two-dimensional array of points which describe the depths to
isovelocity layers at successive horizontal stations. Figure 3 shows a
sample distribution of isovelocity horizons with horizontal stations
and shot points indicated. Horizontal station distances and the
velocity defining array for this sample are shown in Tables I and II.
At each shot point, the wave velocity is a single-valued, continuous
polygonal line function of depth. An example of wave velocity distri-
bution at a single shot point is shown in Fig. 4 along with a sample
attenuation parameter and forcing function. The polygons for A(x) and
F(t) are defined in Tables III and IV.
2 . Numerical parameters
The numerical parameters, which remain constant throughout the
problem, are used to control the computation of the distribution para-
meters and to set the arrangement and size of the solution domain.
They are:
a. mesh size in the x direction, (DX)
,
b. mesh size in the t direction, (DT)
,
c. horizontal distance between successive solution stations
(shot point spacing), (DK)
,
d. number of total x mesh points, (N),
e. number of total t mesh points, (M)
,
f. number of solution stations, (KK),
23
o asPLH <in o2 2 COO M WM Z 2H i-i Oi—
i tn NCO U COO Q H S3P-c 53 O2& i—
i
Oi—
l
HO M P-, i—
i
M CJ> COH O H 2O
gCO > CO H
o—
A
HO<!HCO
<H53ONJI—
l
o
o— >
>
>
>
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t>
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t>
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t>
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t>
t>
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t>
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r-l IIICJ ///II //
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^ ///M / / /W // /o // /•J // /w // /> ///
/ / CN CO <tII CJ> o
II IIII
£ E>*HM (—
1
1—
1
C_> c_> OO o OhJ .J hJw w W> > >
I
H
60
4-1
O5-1
a
eCO
•HCU
w
J-i
Ou-i
aj
u
uo
•u
ex
ICO
CU
(-1
X
24
(a)
f
C3 —
CI
20.0
(b) 10.5 20.0
2.0
(O
-0.15 t-
t p-
Figure 4. Sample distribution parameters
(a) Wave velocity at one shot point,
(b) Attenuation parameter.(c) Forcing function.
25
g. number of points defining the distribution parameter
polygons
:
(1) wave velocity polygon, (NC)
,
(2) attenuation parameter polygon, (NA)
,
(3) forcing function polygon, (NF)
,
h. number of horizontal stations in the velocity defining
array, (NG)
,
i. mesh point at which the output stress is to be measured,
(IP),
j. horizontal distance to the first shot point, (EEE)
.
Additional numerical parameters are put into the program which
control the size, scale and arrangement of the output traces. For
details of their use see the explanatory notes in the program.
Table I.
Sample horizontal station distances
Station Distance
0.02.0
10.520.0
Table II.
Sample velocity defining array
Station Numbers12 3 4 Velocity
Depth 0.0 0.0 0.0 0.0 CI
to 2.0 2.0 4.5 9.8 CI
isovelocity 2.2 2.2 4.7 10.0 C2
horizons 4.7 4.7 4.7 10.0 C2
4.9 4.9 4.9 10.2 C3
20.0 20.0 20.0 20.0 C3
26
Table III.
Sample attenuation parameter polygon
Depth A(x)
0.0 0.010.5 0.020.0 -2.0
Table IV
Sample forcing function polygon
Time F(t)
0.0 0.00.10 0.00.40 -0.15
0.70 0.02.0 0.0
F. SCALING THE PROBLEM
All wave velocities and depths were put into the model in non-
dimensional form. The wave velocity in the upper layer and the depth
to the first transition zone were chosen as the definitive values for
non-dimensionalizing the problem. Non-dimensional time was then
determined by these values through the mesh ratio. Practically, the
scale of the problem is determined by the wave velocity values and one
of the mesh sizes. In general, the wave velocity is entered as C
distance units per time unit. For a mesh ratio of:
St _ 0-025 _ i- - -±- «t C??1
)
the stability condition [equation (19)1 allows the wave velocity to
vary from 0.0 to 3.4 distance units per time unit. If a value of C of
0.8 is to represent dilatational wave velocity in water (1.5 km/sec),
27
then the scale of the problem would depend on a choice of either time
or depth mesh size, chosen in view of the definition desired. Table V
shows some scales where the time mesh size was specified.
28
II
00«
o
oooCM
CM
cO
o t3
x: ou « /-v
a- 60) J-i
-^Q CD
o•J
eCO
o oo aj
B a to
tO CD w•H hJCU
WD
Oo00
<t-
oin
o om oo
00 <f
oCM
m
m•
m oo r^I-h v£> 0000 <J- i-l
oCM
o
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i—
i
X>co
H
CJ>
oM-l
cd
to
<D
4-)
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6CO
l-i
CO
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g -H CD
•H C WH 3 ^
CD
OCcO4-1
to p
<1-
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<l"O oo
mm r^
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m JO r-~ m 00
00in CO r^ i—*
oo
m
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oan
••^ mo m CM r-l
4J CU m o CM i—
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o oCD to CM i—
i
o o o oN ^-^ o o o o o o•HCO
x:CO ma> m r» mS *"> m r-~ m 00 r^
x e
oom oo
29
III. THE REFLECT ION/ REFRACTION MODEL
Synthetic reflection/refraction seismogram generation requires
solution of the wave equation in at least two dimensions. A rectangu-
lar solution space was chosen with three rigid, reflective boundaries
and one free boundary corresponding to the air-water or air-ground
interface. In order to produce a model with wide variability as well
as one that may be easily manipulated without inordinate computer time
and storage requirements, it was assumed that the media in the solution
space is horizontally isotropic. The y-derivatives of density and
Lame's constants (and hence of shear and dilatational wave velocities)
will then be zero.
A. THE MATHEMATICAL MODEL
It was assumed that density, o , and Lame's constants, A and ll ,
are continuous throughout the solution domain. Further, it was
assumed that density and the dilatational and shear wave velocities,
C and S, are either constant or vary linearly with depth. Under these
assumptions, the equations of motion in two dimensions for the propaga-
tion of shear and dilatational disturbances can be written (modified
from Officer [ 1958] and Takeuchi [l966J ):
6c
El
+ ¥"H^ + 4 5
E2:?
+ iY $15H
(23)
at(24)
30
where u and v are the particle displacements in the x and y directions
and B is a velocity attenuation factor. The stresses are:
Dilatational stress: VcT* ?>u" )
Shear stress: / \"d* bu~
These stresses are computed in the program and are used, in addition
to displacements, to form the seismograms.
Carrying out the indicated differentiation on the right side of
equations (23) and (24) and collecting terms yields:
El
i ^^ /SO . "&>A 2- ^A ^ .B^o
E2: Vv _ / V^A ^v
where it has been assumed that:
(25)
£rt (26)
31
The eight collateral conditions chosen to complete a well-posed
problem were:
initial conditions:
II: oCx^.o^) = O , (27)
12: ^0,^,0^ - O , (28)
13: ^^H,o) - O , (29)
14: \f^,L\,o) = O , (30)
boundary conditions:
Bl: L)L*,0,-\.) - O 9
«J (*,fA,0 = O ,
B2: v(K,0,-t) = O ,
V(L (H ,-t) = O ,(32)
B3: O x (0lH ,4r) + ^(O^.-fc) = F lH .t) * (33)
where: F(y,t) = for t > t
B4: °H (°iHi"t^ +vxl° 1 H.t) " O • (34)
The initial conditions allow the solution space to be initially unde-
formed and at rest. The first two boundary conditions (Bl and B2)
require that displacements are zero for all time at the subsurface
boundaries, which then become perfectly reflective. The other boundary
conditions (B3 and B4) allow the x=0 boundary to become a free surface.
Shear and dilatational stresses are not transmitted across the surface
for all time except for an externally applied, time-varying dilatational
stress, applied at a point or combination of points for t v. t, .
As in the one-dimensional model, an attenuation term was included
in the equations of motion (El and E2) in order to keep boundary
32
reflections from masking multiple reflections in the central regions.
The function B(x,y) is non-zero only in the vicinity of the three rigid
boundaries and is zero in areas where disturbances are to be studied.
B. THE SOLUTION DOMAIN
The solution domain is a three-dimensional rectangular array of L
by M by N mesh points dividing the domain into (L-l) by (M-l) by (N-l)
equal subdivisions, each of size ox by &y by £t. Figure 5 shows the
x-y solution space. The total solution domain is formed from N succes-
sive solution spaces or planes. A detail of the distance-depth-time
mesh in the vicinity of mesh point (i,j,k) is shown in Fig. 6.
C. THE DIFFERENCE EQUATIONS
As in the one-dimensional model, a centered difference form for
each partial derivative in equations (25") and (26) can be written.
Making use of the definitions for wave velocities:
c
ST
—^ /
P
(35)
(36)
finite difference forms of equations (25) and (26) may be written in
terms of wave velocities and their derivatives. Carrying out this
substitution and solving for the time leading (i,j,k+l) term for u and
v yields:
El: O-^.iw, =[iU-L.j,k(l-eRA-sfRc) + Ud+1i
jiL (ct+^ +^)KA
+ UKj ^(c':-^-^)KH^(;uLir ,,L-^U;|
- l|OS"Xc +VA(c\-s!)R£(37)
33-
—i cn ro <r 10
>"
I
I
o
1—
1
H ECu CO psi
II II II
H>-HH B
U U 1—
1
o o COhJ J 2fa w w> > QhJ Pd
1 1o x1—
1
COH<H
CNCM CMO CO §
II II ii
SK HM M My u COq o 55
g £ w> >
ga5 w
<iH
m
CN
0)
oCO
aen
Co
•r-l
4-)
i—l
oCO
>•.
I
X
u
(30
•i-l
fa
X
34
/
I
i,j,k+l
i-lf j»k
i» j-l»k
- +i
+ -
i.J+l.k
h J,k
* /-
i,j, k -l
t
i+l,j,k
+ -
Figure 6. Isometric detail of three-
dimensional mesh.
35
+ v/i-'.j.te. U*-^:)^ + W i
j+li k.-».M-li j-, 1 jc.)c"riCC + UA(c"-s")kS
v Co;lj+. 1
k-o:1
)-i,k)a&OM. ->/;,),k.^i (i + |r£%t)j / f i- §^&"
where: RA - (^ , *6 = ^fg_ , Re = /it
"
OA - Uf+i.JH,,!.. + pt ,,j. lik - O^.j-.k - U,, ifl| k
The collateral conditions become:
II: Ul(J1
- O ,
12: Vj,j (1 - ,
13: Ui/,,1 -
14:
Bl
V; P
B2:
Ufcj.k. -
^-W -
^uk -
^L,jk =
V
o
o
o
oo
o
o
o
B3: Od
B4: Vkj.t-M - i2N/i.j.k.(i-ci«C-SiRA) - v/,,j, k (* Si |2A)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
36
A well-posed problem for numerical solution is then formed by
equations (37) and (38) with collateral conditions (39) through (46).
Initially, the k=l and k=2 planes are set to zero following equations
(39) through (42). The k+1 plane can then be solved using equations
(43 ^ through (46) for the boundaries and equations (37) and (38) for
the interior mesh points.
D. STABILITY AND CONVERGENCE
1. Stability
Under the conditions that the attenuation and velocity gradient
terms are zero and that Sx = Sy, the difference equations (37) and
(38) reduce to those for which a stability analysis was carried out by
Alterman and Karal [ 1968 J , and Alterman and Rotenberg [1969
J
The former paper showed that the stability criterion is given by:
8* C(47)
In the latter paper, the authors indicated that stability was guaranteed
under the condition that:
g < °- 86' (48)
with the assumption that S/C = 0.55, corresponding to a Poisson's ratio
of 0.28. Poisson's ratio is defined in terms of wave velocities as:
<r - (49)
Empirically, it was found that the addition of attenuation and
velocity gradient terms reduced the right side of equation (47), while
37
an increase in Sy ( Sy > ox) reduced the left side. An approximate
stability criterion was established, given by:
b^b^-f< ^ wir ' (»)
under the condition that the restrictions on velocity gradients (for C
and S) given by equation (17) apply.
2. Convergence
Following the methods of Forsythe and Wasow [1967] and of
Fox et al. [ 1962 J , it was assumed that convergence would be assured as
long as the mesh ratios were sufficiently small to insure that the
(i,j,k+l) mesh point lies within the characteristic surfaces through
adjacent mesh points in the k plane. Fox equated the characteristic
surfaces with successive positions of a wave front. The most restrictive
condition on convergence is then given by the maximum dilatational
wave velocity in the solution space. This would give a less stringent
mesh ratio condition than the stability condition. Therefore, as
long as equation (49) holds, the method will be convergent as well as
stable.
E. DESCRIPTION OF THE PROGRAM
The program produces a set of four seismograms: shear and dilata-
tional stress and vertical and horizontal displacement. The data is
written on a temporary storage device (tape, disk or drum). A second
program then reads the data and plots the seismograms. An average run
with a 71x81 grid, 800 time mesh points and 25 detectors requires 278K
bytes of main core storage and a run time of 18 min. The output
requires 325K bytes of temporary storage space. The primary limiting
38
factor on a given problem is the time required for input/output to and
from the temporary storage device.
1. Distribution Parameters
As in the one-dimensional model, the distribution parameters
are used to model the physical conditions in the solution space. They
are:
a. dilatational wave velocity, C(x),
b. shear wave velocity, S(x),
c. density, RH(x),
d. forcing function, F(y,t),
e. attenuation parameter, B(x,y).
Wave velocities and density are put in the program as single-
valued, continuous, polygonal line functions of depth. The forcing
function is put in as was done in the one-dimensional model along with
horizontal stations at which F is to be applied. The attenuation
parameter is put into the program as two single-valued, continuous,
polygonal parameters. They are:
el. vertical attenuation parameter, BA(x)
,
el. horizontal attenuation parameter, BB(y).
The full attenuation parameter, B(x,y), is then computed as follows:
B(*.cO = MAX[BA(X), BB(H^] • (51)
Tables VI through VIII give sample numerical input used to
describe the various distribution parameters. Their distributions in
the solution domain are shown graphically in Fig. 7 through 9.
39
Table VI.
Sample wave velocity and density polygons
Depth C(x) S(x) RH(x)
0.0 0.6 0.0 1.03.0 0.6 0.0 1.0
3.3 1.2 0.82 2.17.0 1.2 0.82 2.1
Table VII.
Sample forcing function polygon
Time F(t)
0.0 0.00.1 0.00.7 -0.151.30 0.02.50 0.0
Table VIII.
Sample attenuation parameter polygons
lepth BA(x) Distance BB(y
0.0 0.0 0.0 -2.54.0 0.0 1.0 -1.0
7.0 -2.5 2.0 0.010.0 0.011.0 -1.0
12.0 -2.5
40
t
>^
o o o• • •
O o 1—1
II II II
u COPi
fa'
o
u
00
c•I-l
o(X
Co•HJ-l
CO
o *
•H >!i—l ua •Ha. CO
<o Ccu
c • X)o 0)
•H c XIu o SO N CO
ti
3 a CO
<-w o 0)
•r-l •H00 U 4-t
C •H •r-l
•r-l CO oo d oM m T-l
o H 0)
fa H >
fa H>CO
&<+-l
o
co•H4J
•HMUCO
exECO
CO
cu
u360
•r-l
fa
41
(a)
0.15
(b)
°l7.0
1
x —>
-2.5 BA
(c)
12.0
BB
•2.5 —
Figure 8. Sample distribution parameters.
(a) Forcing function(b) Vertical attenuation parameter(c) Horizontal attenuation parameter
42
43
2. Numerical Parameters
These parameters are used to control the computation of the
distribution parameters and to set the arrangement and size of the
solution domain. They are:
a. mesh size in the x direction, (DX)
,
b. mesh size in the y direction, (DY),
c. mesh size in the t direction, (DT)
,
d. number of total x mesh points, (L),
e. number of total y mesh points, (M)
,
f. number of total t mesh points, (N)
,
g. number of points defining the distribution parameter
polygons
:
(1) wave velocity and density polygons, (NC),
(2) forcing function polygon, (NF)
,
(3) attenuation parameter polygons:
(a) vertical, (NBA),
(b) horizontal, (NBB)
,
h. x mesh point at which the output is measured:
(1) stress, (IA),
(2) displacement, (ID),
i. number of horizontal output stations, (KK)
,
j. y index of the first output station, (JB)
,
k. mesh increment between output stations, (JJ),
1. shot point or area location:
(1) first shot mesh point, (MA),
(2) last shot mesh point, (MB),
m. last time mesh point at which F(y,t) is to be applied, (LL)
44
F. SCALING THE PROBLEM
The scale of the problem is determined by choices of wave velocity
values and either &t or ox and £y. For mesh ratios of:
St Q-Q£ _ i_
8* o.io 2. (52a)
fit - o^o| = i , (52b)
and under the condition on shear velocity that S = C/ V3" (0~ = 0.25),
the dilatational wave velocity may vary from 0.0 to 1.86 distance units
per time unit, and shear wave velocity may not exceed 1.07 distance
units per time unit. If a value of C = 0.60 is to represent dilata-
tional wave velocity in water of 1.5 km/sec, then the scale of the
problem would depend upon a choice of either time or distance mesh
size. Table IX shows scales available for various distance mesh sizes
specified.
45
o
o o oOJ ^~, o o CNCJ >-, £ o CN i—l
CTJ CM r-l
a .—1
oncu
ti No •r-l
•H CO4-1 o o O3 •-S o o r-
i—l X £ o r^o r~-
CO
Ln
oo00
CO
u jz
O MOg c <u
co cu co
•H J^CD
vOv£>
VD
Xo
r-( 5-i
,0 Ort m • •
H co
cu J-l
i—i CD
,a J-J
CO a)
.—l eH cO
cO J-i
> cO
cO CX
CO .—
(
a> CO
i—i oCO •Ho uCO cu
<U Og w a;
•h en
w•u•i-l
C1=>
^ e
a x e
o-
oo
ooo
<r oo o
o
oo
o•u a)
co CO
N
co
CO
CU
^ e
x e
CMCM o
CN O oO O o
mo
oo
LO
46
IV. INTERPRETATION
The synthetic seismic records produced by the two models presented
in this paper were interpreted first in the manner of actual field
records without direct reference to the structural profiles used.
Extensive use was made of gross seismic interpretation techniques pre-
sented by Grant and West [l965] , Dobrin [i960] , and Bullen [l959J ,
Interpretation of the reflection and transmission of incident pulses
was aided by the results of Alterman and Karal [1968j , Alterman and
Rotenberg [l969] , Gupta [ 1966a and 1966b] , Sengbush, Lawrence and
McDonal [ 1 9 6 1J and Tooley, Spencer and Sagoci [ 1965J .
A. THE REFLECTION PROFILING SEISMOGRAMS
Each reflection profile produced by the model is designated by an
alpha-numeric sequence. For example, C3H F7 A14 identifies the exact
structure (C3H), forcing function (F7) and attenuation parameter (A14)
used.
Six profiles were developed using three basic structures. The
structures, shown in Fig. 10 through 15, can be described in terms of
marine geology as follows:
(1) Shelf break with horizontal layers.
(a) Increasing velocity in each successive layer (C3H).
(b) With one low velocity layer (C4D).
(2) Flat bottom with double wedge.
(a) High velocity wedge (C5A)
.
(b) Low velocity wedge (C6C).
(3) Flat bottom, horizontal layering, penetrated unconformably by
a dipping bed.
47
distance > 22oo
5
10
o ^^C = 1.4
^. C - 0.8
° C = 2.2
=^\^o C = 2.8
ua-<u
•o 15
1 C = 3.420
25Figure 10. Velocity structure C3H
distance p-
10
a
15
20
25
C =
C =
C =
C = 3.0
Figure 11. Velocity structure C4D
22
48
distance *• 22
C = 0.8
C = 1.4
5
r - ? 2 ^====^10
x:ua.CD
1
15
C = 2.8-
1
C = 3.4
20
25
Figure 12. Velocity ^structure C5B
distance > 22
C = 0.8
C = 1.4
5
10
r - 1 ^"^=:::::::5=:
x:4-1
a0)
I15
C = 2.0
C = 3.0
20
25
Figure 13. Velocity structure C6C
49
15
J
20
25
distance
C = Q.8
?9
Figure 14. Velocity structure C7A
distance 22
20
C = 0.8
Figure 15. Velocity structure CSA
50
(a) High velocity bed (C7A)
.
(b) Low velocity bed (C8A)
.
All traces utilized forcing function F7 (Fig. 16) and attenua-
tion parameter A14 (Fig. 17).
1 . Variable Depth Records
As an aid to interpretation, several single traces were
developed with a single velocity structure, where the depth of the
detection point was varied between traces. The traces are shown in
Fig. 18 along with the mesh point at which each was computed. These
various detection points are shown as circles in Fig. 10. The traces
were generated using a source located at a horizontal distance of 0.5
units for the problem designated C3H F7 A14. The first disturbance in
each trace, designated d, is the direct wave. Pi and P_ designate the
reflected pulses from the first and second transition zones, while the
first primary multiple (P-, reflected from the free surface) is shown
by P,P . Other multiples are marked in a similar manner.
These traces show'clearly the development of a train of alter-
nating negative and positive pulses behind the main pulse. The change
in pulse shape is also quite evident. As the pulse progresses, the
amplitude decreases and the pulse width increases. Additionally, the
base and tip of the pulse form tends to become more rounded. These are
primarily mesh effects and are dependent on mesh size (DX) . When one
particle is unloaded as the pulse passes by, the adjacent particle is
still under stress, causing a reverse stress between the particles. In
gradient zones, where the derivative of C is non-zero, there is also an
elastic rebound effect which reinforces the pulse train.
51
0.1 0.4 0.7i •
i
0.03
Figure L6, Forcing function F7
52
p p P ? P <=+P
4P
<;P 1
\
*115
^Detection mesh point
time
6 10
Figure 18. Stress versus time at various depths
53
2 . The reflection profiling seismograms
Synthetic reflection profiling seismograms are shown in Fig. 19
through 26. Figure 19 is an uncorrelated version of Fig. 20, included
so that lineations may be observed without the influence of correlation
lines shown on the remaining figures. Solid lines on the seismograms
mark lineations which were correlated with first reflections from a
particular horizon. Dashed lines indicate multiple reflections which,
if used as primary reflections, might lead to erroneous interpretations
of structure. For each correlation, a particular phase was chosen and
followed laterally through successive traces. Figures 20 and 21 show
clearly the false dips introduced by the sloping bottom, increasing the
pulse travel time in the lower velocity upper layer. The bottom slope
causes a change in thickness of the top two layers (water and the first
subbottom layer). The false dip introduced is a function of the bottom
slope and the ratio of velocities involved. Note that there is a change
in dip at each point where a layer pinches out.
Figures 22 and 23 show similar false dips and horizons intro-
duced by variations in the thickness of a single layer. The high
velocity wedge causes a doming up to be introduced into a horizontal
transition zone below, while the low velocity wedge has the reverse
effect. The most obvious feature of the records for velocity structures
C7A and C8A (Fig. 24 and 25) is the relatively high amplitudes in the
records in the areas where the dipping beds pinch out against the flat
bottom. This is caused by a combination of factors. In velocity
structure C7A (Fig. 24), there is a sharp change in velocity contrast
across the bottom as the dipping bed outcrops. The steeper gradient
then reflects more energy, causing an increase in amplitude in both
54
Figure 19
Uncorrelated reflection profile C3H F7 A14
DX: 0.10 DT: 0025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
55
HU-A
Oucu
c
•H
o0)
56
Figure 20
Reflection profile C3H F7 A14
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
First reflections
Multiple reflections
57
A
O.' r A f
o
1
* fi
'*:
i : p
AftMWalii
'XT
fen.
srH<J
C_>
0)
iH•H>4H
OMaco•HJ-l
OQJ
QJ
oCM
OJ
M36C
58
Figure 21
Reflection profile C4D F7 A14
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
First reflections
Multiple reflections
• Location of precision trace (Fig. 26)
59
o1-1
ao•H4JOCD
<D
U3Ml•HP*
nWf\r\ptMAJ\
60
Figure 22
Reflection profile C5B F7 A1A
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
First reflections
Multiple reflections
61
H<!
u01
H•HMHO
CM
ao•H4-1
U
CMCM
d>
•H
62
Figure 23
Reflection profile C6C F7 A14
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
— First reflections
— Multiple reflections
63
64
Figure 24
Reflection profile C7A F7 A14
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
— First reflections
— Multiple reflections
65
-1-
<
Pn
<CJ
<U
H•HU-l
O
co
eg
0)
u3
•HEn
66
Figure 25
Reflection profile C8A F7 A14
DX: 0.10 DT: 0.025 DK: 0.50
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
— First reflections
— Multiple reflections
67
oMD
ao•H4-1
o0)
OJ
u
00
68
Figure 26
Precision reflection profile C4D F7 Al4
DX: 0.10 DT: 0.025 DK: 0.10 EEE: 4.0
N: 251 M: 1800 IP: 2
Horizontal distance (d)
versus
time (t)
— First reflections
— Multiple reflections
69
AT^AA*^
fa
Qu
oMO.
CO•HUOCD
CU
o•rl
CO
•HOCU
UP-
CN
CU
UP00•Hfa
70
first reflections and multiples. Additionally, there is a resonance
effect of the large number of multiples being generated. This resonance
effect is also present in velocity structure C8A (Fig. 25). There is
also a tuning effect due to the relationship between the thickness of
the transition zone and the primary period of the incident pulse
Sengbush, Lawrence and McDonal 1961 J which would lead to high ampli-
tude reflected pulses. The possibility that instabilities were intro-
duced in these high gradient areas was considered. All parameters,
however, are well within stability criteria established in Section II.
Figure 26 is an example of a "precision trace4 ' that may be
produced by the model. This seismogram presents a section of a record
obtained with the same velocity structure as that shown in Fig. 21
(C4D F7 A14) with an expanded horizontal scale. The horizontal
distance covered includes the region where the first sub-bottom layer
pinches out. This area (from 4.0 to 8.0) is marked on Fig. 21 with a
horizontal line in the record. The "precision trace" was obtained by
reducing the shot point spacing (DK)
.
B. THE REFLECTION/REFRACTION SEISMOGRAMS
Each set of synthetic seismograms developed by the model is
designated by a letter-number sequence similar to that used in the
reflection profiling model. CA F5A B3 L2 identifies the exact wave
velocity and density structure (CA) , forcing function (F5A) , attenua-
tion parameter (B3) and shot point location (L2) used to generate the
records. The seismograms presented in this section were developed
using the following velocity structures (Fig. 27 through 29):
(1) Two elastic layers, (CA) .
71
X
I
7
+—
F y —*-
JL 124. r> + o
C = 0.60 S = 0.35 RHO = 1.00
C = 1.40 S = 0.82 RHO = 2.10
-+ Reflection spread
o Refraction SDread
CAFigure 27. Velocity structure
o 4 +
C = 0.60
C = 1.40
S = 0.0 RHO - 1.00
S = 0.82 RHO =2.10
+ + Reflection spread• * Bottom spread
Figure 28. Velocity structure CB
12
72
1
C = 0.60 S = 0.35 RHO = 1.00
12
C = 1.10 S = 0.65 RHO = 2.10
C = 1.80 S = 1.05 RHO = 3.00
+ + Reflection spread
Figure 29. Velocity structure CC
73
(2) Water layer over elastic layer, (CB).
(3) Three elastic layers, (CC).
With the exception of the water layer in CB, all velocity structures
were designed with Poisson's ratio of 0.25.
All traces were produced utilizing forcing function F5A (Fig. 30),
attenuation parameter B3 (Fig. 31> and shot point location L2. At this
shot point location, the surface dilatational forcing stress is applied
at mesh points j=15 and j=16. This is marked by the letter "F" in
Fig. 27 through 29.
1. The Sei sinograms
Selected correlated synthetic seismograms are shown in Fig. 32
through 36. Figure 32 is a complete set of four synthetic seismic
records for a close (reflection) spread of 25 surface detectors for
velocity structure CA. The detector spread is shown in Fig. 27. Com-
parison of these records shows the interdependence of the displacements
and stresses. Actual land field records would include only vertical
and/or horizontal displacements, or particle velocity, while marine
records would only consist of the dilatational (pressure) record. The
additional synthetic seismograms enable the proper phase arrivals to be
quickly identified. A set of theoretical time-distance curves was
developed for velocity structures CA and CB (Fig. 37), where it was
assumed that the width of the transition zone was zero. There is good
agreement between these theoretical arrival times and correlated linea-
tions on the synthetic records.
The first trace on the dilatation record corresponds to a
normally incident wave. It bears a very close resemblance to traces
obtained with the one-dimensional model for similar velocity structures.
74
0.1 0.7 1.3
f
-0.15
Figure 30. Forcing function F5A
y ^
12
B =
v£>
XvO
CN
1
CNII
\II
pq
PQ
7
B = 2.6
Figure 3.L. Attenuation parameter B3
75
Figure 32
Reflection seismogram CA F5A B3 L2
Dilatational stress
Shear stress
Vertical displacement
Horizontal displacement
DX: 0.10 DY: 0.15 DT: 0.05
L: 71 M: 81 N: 800
IA: 2 ID: 1 JB: 16 JJ : 1
Horizontal distance (d)
versus
time (t)
Direct wave arrivals
First reflections
Multiple reflections
Shear-dilatation conversion wave
76
o o o
A*d_v . - Yui--\ ^ .ux/\^r^A.—^^a
J A!\iy wv/sj-^
—J-44- [A-p^ A_^p=r,._.A^
r
m
Ttm 1
J±Al 4- / -... v.-.^^ .^^-
-r ^.p-v^^^^---^^^-:,^-^ /-.. '-!_
—
k , vr_
t_ ^-^d^-'j
4=—/^Jv^/v
.'^v/v'-'^v p'^' '-
-••""H'v-"rr-"-^
t
^ V'/'-'f^^
-Mr-A
i^f \ /
X
J
M-V^f -X/-- -K- ^ ;/v-
r^^iV^N^t^t^
,^\/->
^v-^ivf-v r r -l J/- \ / X£ -r x - 'v
u I
vl f '
Af
A7+: - > ' W -V^- '-i-';i~t=p-"\-/ -
^ v -^vy-K
tl
-u—
=
u•uCO
oHCO
•uc8
CNI
hJ
mP3
<CLO
<u6cd
u00oeCO
•H0)
CO
co•H•uo0)
0)
<!CMn0)
u360•H
77
CO
<mPm
<Ju
u
iCO
•H0)
CO
CO•HUod)
rHM-l
a)
paog
•H
78
<l \ I
Kt'-,
A 'A'
LI TVY
i /i \i^\
tf . '
; A &
I "7 T ''-- \<
/ U"1, /"X^
I 1.4/ • / -. ' I•
i /
J iv /
J
Si A-
hZ y£
\\\\ I J-S
-->
k:. r ;... ^.cd£
^// : k—T^
• ly"
4/
V
4-
/
1' " - Xr v. • ^1-' .'"v.V^ . A
I - 1
Kk-fi.
1
c0)
a0)
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Figure 33
Refraction sei sinogram CA F5A B3 L2
Shear stress
DX: 0.10 DY: 0.15 DT: 0.05
L: 71 M: 81 N: 800
IA: 2 ID: 1 JB: 22 JJ: 2
Horizontal distance (d)
versus
time (t)
Direct wave arrivals
First reflections
Refracted arrivals
81
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Figure 34
Reflection seismogram CB F5A B3 L2
Dilatational stress
DX: 0.10 DY: 0.15 DT: 0.05
L: 71 M: 81 N: 800
IA: 2 ID: 1 JB : 16 JJ : 1
Horizontal distance (d)
versus
time (t)
o Direct wave arrivals
First reflections
- — — Multiple reflections
83
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Figure 35
Bottom detector seismogram CB F5A B3 L2
Dilatational stress
DX: 0.10 DY: 0.15 DT: 0.05
L: 71 M: 81 N: 800
IA: 20 ID: 20 JB: 16 JJ: 2
Horizontal distance (d)
versus
time (t)
Direct wave arrivals
Surface reflected multiples
Interface wave conversion
85
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Figure 36
Reflection seismogram CC F5A B3 L2
Dilatational stress
DX: 0.10 DY: 0.15 DT: 0.05
L: 71 M: 81 N: 800
IA: 2 ID: 1 JB: 16 JJ: 1
Horizontal distance (d)
versus
time (t)
o Direct wave arrivals
First reflections
Multiple reflections
Shear wave conversion
87
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° Dilatational arrivals (CA and CB)
^ Shear arrivals (CA only)
Subscripts:direct wavefirst reflected waverefracted wave
Refraction spread
Reflection spread
25
20
t15
10
-35
Figure 37. Theoretical time- distance curves for
velocity structures CA and CB.
89
Elastic rebound is present, causing the development of a cyclic wave
train. Phase reversal is observed for events reflected from the free
surface on all records. These features are particularly apparent in
the contour plots discussed in the next section.
Although no refracted arrivals are shown on any of the records
of Fig. 32, a close examination of the traces revealed their presence
after a refraction record was obtained. Figure 33 is the shear stress
refraction record for velocity structure CA. The surface detector
spread (shown in Fig. 27) extended across most of the solution space
into the attenuation region on the right.
In addition to the arrival of the refracted shear wave, Fig. 33
shows the development of a small shear stress in advance of the first
shear arrival. This is generated by the passage of the dilatational
wave across the mesh, a result of the coarseness of the mesh relative
to the wave velocity. As the dilatational wave passes a mesh point at
an oblique angle relative to the x or y axis, shear strains are set up
between mesh points. In fact, it is just this effect which causes the
generation of the initial shear wave at the forcing function applica-
tion point, or, more precisely, between one application point (where
the applied dilatational stress causes some displacements and the
neighboring mesh point where displacements are different. The shear
wave is then generated from two centers. For shot point location L2,
these centers are (fictitious) mesh points (%, 14%) and(%, 16%).
Figure 34 is the dilatational stress reflection record for
velocity structure CB. This structure (shown in Fig. 28) is identical
to CA except that, in the upper layer, shear wave velocity (and hence,
Lame's constant/^) was set to zero. The record is, of course, very
90
similar to that obtained from structure CA except that the amplitudes
of the first arrivals are somewhat greater. This is caused by the lack
of rigidity in the upper layer leading to a loss of the attenuation
effect of shear strains on dilatational strains.
An example of a seismogram obtained with a spread of detectors
on the bottom of a water layer is shown in Fig. 35. Because of the
relationship between shot point location and detector location, the
only lineations available for correlation, except for the direct wave,
are multiple reflections between the bottom and the surface, and re-
fracted waves. An additional effect observed on this record is the
generation of a head wave in advance of the direct wave. This wave is
generated at the interface between the two layers by the dilatational
wave travelling through the lower medium at a high velocity.
A three-layer reflection seismogram is shown in Fig. 36. The
main feature of this record, not seen in the two-layer seismogram, is
the mutual interference between the first multiple from the first
transition zone and the reflected wave from the second transition zone.
As these two waves are out of phase with one another, the initial
effect is one of destructive interference. As distance from the shot
point is increased, the waves arrive from somewhat different directions,
altering the phase relationships, leading eventually to constructive
interference.
2. The Contour Output
An on-line printer contour plotting subroutine was developed so
that the reflection and transmission of elastic waves might be observed
in a qualitative manner. At chosen intervals, the values of horizontal
or vertical displacement, dilatational or shear stress or vorticity are
91
computed for each mesh point, converted to an alphabetic representation
and printed as a rectangular contour plot. Figures 38 and 39 show
portions of such plots, dilatational stress plotted for the problem
CB F5A B4 L2 at time mesh levels K=16 and 91 respectively. Figure 38
shows the generation of the initial dilatational wave by the forcing
function. Figure 39 shows the progress of the dilatational wave as it
is reflected and transmitted at the transition zone. Some minor modifi-
cation of the main two-dimensional program was necessary to produce this
form of output. Examination of the subroutine shows what additional
parameters were necessary. Table X shows the scaled values represented
by the plotted characters.
An animated motion picture was made of dilatational and shear
stress for this problem. This presentation enables the progress of
wave fronts throughout the solution space to be readily observed,
including phase changes where they occur. This output allowed the
efficacy of the attenuation parameter to be determined quite easily
during the development phase of the model. A copy of this motion
picture on Super 8mm film is on file at the Naval Postgraduate School,
Department of Oceanography (Code 58Ad), and is available to interested
investigators on a loan basis.
Some effects were noted on the contour outputs before they were
identified on the seismograms. The dilatational plots first showed the
development of a head wave in the upper layer (Fig. 4CH . Other pre-
sentations clearly show the development of a shear wave in the lower
medium due to the incidence of a dilatational wave against an elastic
transition zone (Fig. 41) and the generation of an interface wave
travelling along the fluid-elastic boundary in advance of the shear wave
(Fig. 42).
92
DILATATION: CB F5A B4 L2 K=16
i
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2 GGGi OCCL'lCC'KN ZZ* CCOCCGGGGGGCGGGGUGJGGX CLCCCCCCCCG // *v CGCCGCCCCCCCCCCCGGCGu lCUGCClOGL 3 JZZn CGOCGGOGCGGCCGCGGGGCG-> LCl-CCCCCCCCC t- KGCCOCCCGGGGGCCGGOGO'JuC6 CiuQGuUGGGGGGCCK.KCCGGGGGGGCCCGCCCCGGCGQGG7 GGGuOGGGGuGCGLDGCGGGGJuCGOOGaGCCGCGGGGGG5 gggccccccccgcccgOcucccgccccccgccgcgoogcci GCQGCGCCGCCCGtGGfcGOCCOGCOCCGCOCCCCCGGGGC
: j GCGGOGGCOCGGGuGGGGGGGOGCQCGGOGOCCOOGOOCGLI CGCCCCCCCCDCCrCOCCJt GCCCCGCCOCeCCCCCCCGCG12 GGuGuGGGGCGCOUGGGUGGOGGGQCOGOOGCCGGCCQCGi 3 CGCClClGGCGCGCGCCCLGGCGCGCCGGGCCCCOGGGGGI -t GGGGGtCCCC GCGCGCCGCCCCCCCCCCGCCCCCGCCCGG
1 15 GGGCGGGGGCCGGCGGCGGGGGGCOCGGOOCCCGOQGGGG1
i Q GG'uCGi I CGCCGGCGGt CGCCGCCOCCOCCOCCQSQOUGG17 GGGGCGCGCCGGGCGl GUOGGGGCOCGGOOCCCCGGGGGC
1 ] ^ GGOGGICGGCCCCCCGOCOGCOGCQOGOOOOGOGOGGGOG j
* 19 CCCCCCCCCCCCGCCi*CCGCCCCCGCCGCCCCCCCCCGGO21 GGUGGGUGGGGGuLG{3GGUdGGGGGOQGOGGCOGG-GGGGO\21 CCGGGCGCCGCCnCCCCCCCGGCCGCCOGOCCGCGCCGOG \ '
22 GGGGGCGGnGGCGCGdGGGt CCCCLCCrCGCGCCCCCuCG \;2 GGOCGGGGGaCCGC GCCdOGC.GCCGGCGOGGCGCUGGOQG^^JX2h GuJL CGGGLCGCCGGCCOGCGGCCOCGOOOGCGGGGGGOG T? 5 LGuGGGCGuGGCGCCGCGGGOOCCGQGOGQQCCCCGCGGC2o CCGGGGGC.GLCCCCGQCCGGOOCCOOOQCaOGQGOQG'JGa2 7 GGGGCC.CCCGGGCGCGCGOCCCCCCCGCCGCCCCCGCGGG
•< GGGGGGGQGGGGGCGC CGC GC GGCGCGQGGCCCCCCGGGOZ 9 CCGGGGCGCCGCGCCCCCGCCOCCGCCCGGCCCGGGCGGC_ - GGGGGGGGCCOGGGGGGCCCCOCCGOGGGGCCCCOGCOCCH lull CI GGGG'CCGCGGGGl.'CCGCCOGGGGGGCC0GGGGGC •
- 2 GGGCCGGCCCCCCCCl CGGCl GCCGCCGGGGGCOOGGGGG_ - GCCcCLLLL"CCCLLGGCL. r CGGCCCCCGCCCCCCCCCGGC3<* GGGCGGCCCCGCGCCC CCLGGGCGOGCOOOOGCQQQGGGO33- LGCCCCCCGCGGGCG.C CCl llCC CCCCGCCCCCCCGbGGL36 GGGCGGCGGCCCGCGnCOGGGOGCGGOOGCGCCGGGGG'CC37 GGGCGCCCCCCCGXC LCCGCCGCCCCCGOCCCCGGGGOQGit1 GUuGGCGCGGCCCf CCCGGCCGCCCCCCCQCCCCCGCGCC3 T CCQGGCGuGGCCGGGi GGCCGCCCGOCGOGCGCOGGOGGC4c GCGCGGGCC CC CCGGC CGGCCGC C GGGGGGGCCOOGGaGG4 1 OCUGCCCCCGGCGGGOCGGQC GCCGGCGGGCCGCOG'CGCCh2 GLGCCGCGGGCCGGGOCGGCl GCCGGCCGGCCGGGGGGOO-» 3 CEOCCCCCGCCCOGCGCCQCC CCCCGCCGCCCCCGGCGCC*4 UGJCGGGuGCGGGCGGCOGGGUCCCOCOGGCGCCGCCGGC+ b GGGCCCCCCCGCCCGCCGGCCCCCOCGCCGCCGGGGGGGG4 c GGGGGGGGGCaCGCGCCCGCCGCCCCCCCCCCCGGOGOCC47 CCGCCCCUOGCCGCCGCGGCCGCGGCGOG.CCCCCGCCGGC4 8 GGOGGCGOGCGCGCGGCGGGCCCCfJCGCGGGCGCQOGGO.G4 ; c.rGCCLGi:;'GGG ! JLJGGC n GCCGLGCGCCOCCCCCCLQGGo G L GGGCCCGGCGCGCGC CGGCCOGCOOGGGOGOGOGGOOOG12 3 4
: c c c c
i —**
T Transition zoneF Forcing function application point
Figure 38. Portion of the printer contour plot showinggeneration of the dilatational wave. .
... ...
(see Table X for scaled values)
- - . .
93
DILATATION: CB F5A B4 L2 K=91FF
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T Transition zoneF Forcing function application point
Figure 39. Portion of the printer contour plotshowing transmission of the dilatationalwave into the lower media
(see Table X for scaled values)
8^
94
Table X.
SCALE DF CCNTCURED QUANTITY
I SYM(I) VALUE
1 A -0.00854172 -0.00812503 B -0.00770834 -0.00729175 C -0.00687506 -0.00 645837 -0.00604178 -0,0056250Q E -0,0052083
10 -C, 004791711 F -0.004375012 -0.003958313 G -0.003541714 -0.003125015 H -0.002708316 -0,002291717 J -0,001875018 -0,001458319 K -0.001041720 -0.00 0625021 -0.000208322 0.000208323 M 0.000625024 0,001041725 N 0.001458326 0,001875027 P 0,002291728 0.002708329 Q 0,003125030 0.003541731 R 0.003958332 0.004375033 S 0.00479 1734 0,005208335 T 0,005625036 0.006041737 U 0,006458338 0,006875039 V 0.007291740 0.007708?41 w OVEPSCALE42 z UNDERSCALE
95
DILATATION: CB F5A B4 L2 K=200
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if pqrg e 9mh gh k oggoonopogoooogoooogooppp11 pppj p g ggh k gcggogogngonogogogpqgqogg12 mkj qh gh pogggoooogggoooooogoooopooH M R JH p K G CGGGPPGGGGGGGOPOPOQOGOOOCO14 M K P GG K OGGGGPGGnOOOGGOGGGOGOGGGG1 ^ QM G N G K GG nPGnGGnnGPnGOGPGGOQHC©«-16 N M M M 0J HJ K GOGGOGGGOGGPrGGJ^efn^riQPGn17 K P NNNM HG JKK 00^O0G0O<mp^lTO0CPnpG0dP01q (< p p n G GGGC^fTDPPGOGOOOGPGGOPP
]q p rphhFFG 000 OPPOOPOOOOOOOCCnOPOOP?0 G S n FF OPnpp 0GO000OO0O00OPPnOC0d?A K K N mm QO 00 GOOOOGOOPOPGG2? JM W PH H C MM GO QfOOCOOOPGn-5 QnctM J J n 000 000 OPOPOOOGGOOO-74 nnriOO KK K OPG Ppp GOP OOOOGOGGOOGG? c GGOGO KKKK 0000 GGO GPP OOOOCOCGOOOP~>h PGOPP KK OOO n 0P PGP POO OOOPOPOOnPOO2 7 POOOO KK OPOOGG GOO GOG GOPOCOGOOOGO?3 00000 nnngO0PPO0O npp OOPPGPGOOOPn?o nnnOG GnnccCGOOOG OHO ObOGOOOGOCOn3,: QPGOGO OGGGGCGGPnP POO OGGGGOGOOOGnn GGGGPP POOGPOnOGOGG GOG GGOGOOGOGPnp2? OGOOOO CPGOCCGGOHGr1 PGG OOGGGOOnGGGn*"* GOOOOO GGGGOGOnGGGG riGO OOnoCOGOOGGG^4 nOOPOP OGPGGppPOO^G n GPG GGGOOOOGOGOGG3^ OOHGOG OOGOOCCGGPnGG POG OOOCOCOOOOOGG16 CnoOOO GGnOGGPOGOGOGGG GGOO POOOOOGGGGPGP^7 POOOOOOOOOOOOCOOOOOOOO 0000 GGOGOPOGGGGCG38 n-<nGGGOGGPGPG nGGnGG rinG 0000 GGOnOGOGGOOGn'2 000000000000000000000 GOGP OOOPOCOGGOGGG40 GGO GOOGG OOCCOGGGOG 000 GGGOOCOOOGPGG41 GG GOPG OOOOOGOOOn GIG GOOOnCOOOOOOn42 GO OPG 0000000000 POO OOOOnnoOOOOOOG4* P P nGOOnOHGHG GPG GGGOOGGOGOOGGG44 000000000 OOnp fiodGOOGOOGOnGG45 000000000 GGO PGOOQCOGOOPC^n46 OOOOOQOPO GOP noOGOOOOGnGOnn47 g PGGGdnnG PGG QOOOdOOnoOOOCO40 on nnnnppnn f]DQ 00O0O nP00nG0O0040 000 00000000 OPO OdnooddcOOOOOOnSQ nno npnooGG O^H OGPOOOOOOPGGGOn
u 5 6 7 800000
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T Transition zoneH Head wave
Figure 40. Portion of the printer contour plotshowing the development of a dilatationalhead wave
.
(see Table X for scaled values)
96
SHEAR: CB F5A B4 L2 K=84FF
1 OOOCOGOCCCaCOCOOOOOCCOCCOCOOOOOCOOOOOOOO2 00000000000000000000000000000000000000003 OOOCOCOCCGOCOOOOCOOOOOOOOOGOOOOOOOOOOOOO4 OOOOOOOOOQOCOOOOCOOOOOOCOOOOOOOOOOOOOOOO5 ooooooooooo noooooonncooGoooooooooooooQOo6 noocooooococoooooocoooocoooooooooooooooo7 0OOOO00O0C00OGOOCOOC000GO00O0000GOG00OO08 OCOOOOOOOOOOOOOOGOOOCOOCOOOOOOOOOOOOOOOO9 OOOGOOOOOOOCOOOOGCGOOOOCOOOOOOOOOOOOOOOO
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T Transition zoneF Forcing function application point
Figure 41. Portion of the printer contour plotshowing generation of a shear wave atthe transition zone by the dilatationalwave.
(see Table X for scaled values)
97
SHEAR: CB F5A B4 L2 K=1781?
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T Transition zone
I Interface wave
Figure 42. Portion of the printer contour plot
showing development of an interface
shear wave.(see Table X for scaled values)
98
V . SUMMARY
The two seismic models presented here were developed primarily as
tools to assist in marine geophysical field exploration. The models
produce synthetic seismograms similar in format to that of actual field
records in order to facilitate direct comparison. Although both models
present two-dimensional records (horizontal distance versus time) , the
reflection profiling model is made up of a series of one-dimensional
model solutions. The complete solution simulates a moving source of
vertically incident pulses where the velocity structure is slightly
altered between transmissions. In contrast, the reflection/refraction
model treats the propagation of energy in two dimensions. Each complete
solution simulates a single pulse travelling through the solution space.
The synthetic seismograms are produced as records from a series of
detectors in a horizontal array.
The models are based on numerical solutions to the wave equations
in one and two dimensions in bounded domains. The equations are
expressed in unsimplified forms which are not amenable to direct
analytical solution. The stability of these numerical solutions and
their convergence to the analytical solutions was investigated. Sta-
bility and convergence criteria were established in terms of mesh
ratios and wave velocities.
In order to provide a wide variability of physical simulation, the
solution domains are modeled by a number of easily varied non-dimen-
sional parameters. These parameters are used to set the size and sub-
division of the solution space as well as the distribution of elastic
parameters and the shape and extent of the forcing function.
99
The structures modeled in this paper were, in general, based upon
realistic distributions of the non-dimensional elastic parameters in
marine environments. Some of these models have inherent resonance with
the type and width of the forcing function used and the mesh sizes.
Records are produced with some features which reflect this resonance.
In particular, there is some tuning of the pulses in the narrow transi-
tion zone in some of the models. Care should be taken when trying to
extrapolate features found in these records to cover a more general
case.
100
VI. FUTURE WORK
The applicability of these models to a wide variety of studies was
the primary consideration in their design. The interpretation of various
velocity structures in this study was limited to a gross overview of
simpler problems that might be encountered in field exploration. Al-
though a full presentation of the use of all of the variable parameters
was not attempted here, there are areas in both models where improvements
could be made which would increase their variability.
A first step in improvement of the one-dimensional model might be
the inclusion of a variable depth source. It was felt that this was not
warranted in the model due to the complexity of a forcing function at
depth that could not be included as part of one boundary condition.
The same improvement might be applied to the two-dimensional model.
However, this would involve an additional order of magnitude of com-
plexity and the model could no longer be called "simplified." A more
reasonable improvement could be made which would remove the requirement
that all transition zones be horizontal. The wave velocities and
density could be put into the model as two-dimensional arrays instead
of vectors. Terms involving the y-derivative of Lame's constants and
of density would then be retained. This improvement would approximately
double the main core storage required (to 550K bytes) and increase the
main program run time by about one-third (to 24 min)
.
Although the attenuation parameters used in both models are con-
sidered adequate for the purpose for which they were included (elimi-
nation of boundary reflections), they are by no means definitive.
101
These models could be used to study the effects of various velocity
attenuation functions by entering A(x) or B(x,y) as functions of other
physical parameters (e.g. overburden pressure, Lame's constants or
particle displacement).
Only a single forcing function is presented for each model. Several
functions were examined in the development of the models including a
flat-topped pulse, spike pulse and "sine" function. The models could
be used to study the reflection and/or refraction of a wide variety of
forcing functions, limited only by the size of the time mesh increment
(DT).
In Section III, it was noted that an interface wave was observed in
shear stress on the contour plot output. This model could certainly be
used to study the various classes of such waves under a variety of
conditions. Some preliminary work was done in generating particle
motion plots from the vertical and horizontal displacement outputs
available. Although elliptical particle motions were observed as pre-
dicted I Alterman and Rotenberg 1969 J , limitations of distance from
the generating point did not permit identification of the interface
phases.
Neither model is by any means limited to the homogeneous layer-
transition zone structure presented in this paper. The velocity
structure could consist entirely of transition regions of various
gradients. This could be applicable to investigation of the effects of
the gradients on reverberation.
A collection of records is on file at the Naval Postgraduate School,
Department of Oceanography (Code 58Ad)
.
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LIST OF REFERENCES
Alterman, Z.S. and Karal, F.C., 1968, Propagation of Elastic Waves in
Layered Media by Finite Difference Methods, Bulletin of the Seismo-logical Society of America, v. 58, n. 1, pp. 367-398.
Alterman, Z.S. and Rotenberg, A., 1969, Seismic Waves in a QuarterPlane, Bulletin of the Seismological Society of America, v. 59, n. 1,
pp. 347-368.
Bullen, K.E., 1959, An Introduction to the Theory of Seismology,Cambridge University Press.
Constantino, C.J., 1969, Two Dimensional Wave Propagation through Non-linear Media, Journal of Computational Physics, v. 4, n. 3, pp. 147-
170.
Dobrin, M.B., 1960, Introduction to Geophysical Prospecting, McGraw-Hill,
Forsythe, G.E. and Wasow, W.R., 1967, Finite Difference Methods forPartial Differential Equations, John Wiley.
Fox, L., and others, 1962, Numerical Solution of Ordinary and PartialDifferential Equations, Pergamon Press.
Grant, F.S. and West, G.F. , 1965, Interpretation Theory in AppliedGeophysics, McGraw-Hill.
Gupta, R.N., 1966a, Reflection of Elastic Waves from a Linear Transi-tion Layer, Bulletion of the Seismological Society of America, v. 56,n. 2, pp. 511-526.
, 1966b, Reflection of Plane Elastic Waves from TransitionLayers with Arbitrary Variation of Velocity and Density, Bulletin ofthe Seismological Society of America, v. 56, n. 3, pp. 633-642.
Officer, C.B., 1958, Introduction to the Theory of Sound Transmission,McGraw-Hill.
Sengbush, R.L., Lawrence, P.L. and McDonal, F.J., 1961, Interpretationof Synthetic Seismograms, Geophysics, v. 26, n. 2, pp. 138-157.
Smith, G.D., 1965, Numerical Solution of Partial Differential Equations,Oxford University Press.
Takeuchi, H. , 1966, Theory of the Earth's Interior, Blaisdell.
Tooley, R.D., Spencer, T.W. and Sagoci, H.F., 1965, Reflection andTransmission of Plane Compressional Waves, Geophysics, v. 30, n. 4,
pp. 552-570.
126
United States Naval Postgraduate School Computer Facility, 1969, Techni-
cal Note 0211-03, Plotting Package for NPGS IBM 360/67, by P. C.
Johnson.
Wolf, A., 1937, The Reflection of Elastic Waves from Transition Layersof Variable Velocity, Geophysics, v. 2, n. 4, pp. 357-363.
127
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 20
Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 93940
3. Professor R. S. Andrews 5
Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940
4. Department of Oceanography 3
Naval Postgraduate SchoolMonterey, California 93940
5. Oceanographer of the Navy 1
The Madison Building732 N. Washington StreetAlexandria, Virginia 22314
6. LCDR C. D. Lodge 2
P.O. Box 274Pebble Beach, California 93953
7. Professor J. J. von Schwind 1
Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940
128
Security Classification
DOCUMENT CONTROL DATA -R&DSecurity « lassification of title, body of abstract and indexing annotation must be entered when the overall report is classified)
on oin A ^ ing AC Ti vi ty (Corporate author)
Naval Postgraduate SchoolMonterey, California 93940
it. REPORT SECURITY CLASSIFICATION
Unclassified2b. GROUP
REPORT TITLE
Simplified Numerical Models for the Generation of Synthetic Reflection ProfilingSeismograms and Synthetic Reflection/Refraction Seismograms
DESCRIPTIVE NOTES (Type ol report and, inctusi ve dates)
tester's Thesis; (April 1970)*u tmoRiSi (First name, middle initial, last name)
Charles David Lodge
REPOR T D A TE
Ipril 1970
7a. TOTAL NO. OF PAGES
128
7b. NO. OF REFS
17• . CONTRACT OR GRANT NO
h. PROJEC T NO
9a. ORIGINATOR'S REPORT NUMBER(S)
9b. OTHER REPORT NOISI (Any other numbers that may be assignedthis report)
3 DISTRIBUTION STATEMENT
:his document has been approved for public release and sale; its distribution.s unlimited.
I SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITY
Naval Postgraduate SchoolMonterey, California 93940
ABSTRACT
Two numerical models were developed which generate synthetic seismograms in:ormats similar to those of records obtained in field exploration. The modelsiolve the hyperbolic wave equations in one and two dimensions by finite differenceipproximations in initially undeformed solution domains of transversely-isotropic.ayered media subjected to a time varying, dilatational forcing stress applied athe surface. A velocity attenuation term was included in the models to inhibittrong boundary reflections.
The one-dimensional model produces synthetic reflection profiling seismogramsor arbitrary horizontal or dipping layers. The two-dimensional model generatesynthetic reflection/refraction seismograms for horizontal layered media withrbitrary distribution of wave velocities, Poisson's ratio and density. Severalample records were produced for some representative velocity structures. Theynthetic seismograms were interpreted and gross correlation was carried out asf they were actual field records.
D ,r»"..1473N 0101 -807-681
1
(PAGE 1)
Security Classification
129A-31408
Security Classification
KEY WORDS
Numerical
Seismic
Model
DD .'£" .1473 < BA«>S/N 0101-807-6821
ROLE W T
130Security Classification A- 31 409
Thes i s
L7912
I
Thesis
L7912c.l
117688
^"Simplified numerical
Jels for the 9«ne«• „ rtf synthetic « c
f,ection P"* 1 ' 1 "*
seismograms and syn
thetic reflection/
^fraction seismograms
117688
L°d9S^plHied numerical
models for the genera-
tion of synthetic re
flection P"-°f,V
n?n-
seismograms and syn-
thetic reflect. on/
refractionseismograms
,hesL7912
S^num^msaSJJiS
3 2768 002 12610 4
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