Download Reflected-diffracted waves in fracture zone models

Geophys. J. R. astr. Soc. (1984) 19, 353-361
Reflected-diffracted waves in fracture zone models
Marek Grad
Institute of Geophysics, liniversity of Warsaw, 02--OY3 Warsaw,
Pasteura 7, Poland
Accepted, in revised form, 1983 December 23
Summary. The paper presents the results of modelling of diffracted and
reflected-diffracted waves in fracture zones. The Berryhill method was used
and the calculations were made for a profile perpendicular t o the diffracting
edge. Several homogeneous models of the Earth’s crust, characterized by
different values of crustal thickness, velocity and horizontal distance between
shot point and diffracting edge were considered. A dependence of the relative
amplitude of diffracted waves on the location o f the diffracting edge is given.
The pattern of the seismic wavefield depends upon the dimensions of the
fracture zone. Amplitude curves of reflected-diffracted waves are presented
for a series o f models o f fracture zones. The possibility of applying the amplitudes of reflected-diffracted wave trains to the interpretation of the structure
of fracture zones in the Earth’s crust is andysed for different types of
fracture zones.
1 Introduction
The wavefield observed during studies of the Earth’s crust using the deep seismic sounding
method is of a very complex nature. This indicates the complex structure of the Earth’s
crust, where there are not only horizontal discontinuities, but also lateral inhomogeneities
related t o the boundaries o f crustal blocks, as well as inhomogeneities within the blocks
themselves. In addition t o regular refracted and reflected waves, several waves with anomalous apparent velocities are observed, e.g. diffracted waves. They originate in the Earth’s
crust at inhomogeneities of different types: deep fractures forming block boundaries, intrusions o f upper mantle material, areas o f contact and overlap o f layers etc. The kinematic
properties and methods of interpretation of travel times of diffracted waves have been
described in several papers (Oblogina, Piip & Kochai 1962; Berzon 1977; Grad & Perchud
1978). The travel times of other seismic waves, however, could also have similar properties,
e.g. waves reflected from steep or curvilinear discontinuities. The kinematic criterion alone,
therefore, is often insufficient for defining the nature of a seismic wave. Thus the dynamic
properties of diffracted waves become of great importance. In earlier studies of the deep
structure of the Earth’s crust diffracted waves were treated as disturbances, or at best they
were interpreted qualitatively. It appears, however, that a fu!l quantitative interpretation
12
354
M . Grad
o f the kinematic and dynamic properties of diffracted and reflected-diffracted waves
provides information on the structure o f fracture zones. The results of theoretical modelling
o f diffracted and reflected-diffracted waves are presented in this paper. The calculations
were carried out using the Berryhill (1977) method for the case when the profile line is
perpendicular to the diffracting edge.
2 Modelling of diffracted waves
-
Berryhill method
The dynamic properties of diffracted waves have been considered in a number of studies,
both theoretical (Krey 1951: Oblogina 1952, 1964: Trorey 1970: Hilterman 1975; KlernMusatov. Kovalevsky & Tokmulina 1972; Klem-Musatov. Kovalevsky & Cherniakov 1975;
Berryhill 1977; Fertig & Muller 1979) and experimental (Ten Chi-Wen 1963; Hilterman
1970: Kovalevsky, L,oktsyk & Averko 197 1 ). The application o f the Kirchhoff retarded
potential niethod t o diffracted waves for non-zero separation of the source and receiver
has been described by Berryhill (1977). The basic equation in this method is the Kirchhoff
formula (Officer 1958: Bith 1'169). The acoustic pressure at the point g can be described
b y the formula
where P(t--a/u)/a is the equation o f the spherical wave. F ( t ) the pulse at the source s, r, is
t h e distance between the source s and the element d A , r p is the distance between the receivelg and CIA,and 0, is the angle beLween the normal to the surface A and the direction determined by the radius rg (Fig. I ) . Equation ( I ) can be transformed into the following form:
p,(t)
=F(t)
* D(t)
(2)
where t h e function D ( r ) depends on the medium geometry. An expression describing this
function has been given by Berryhill ( 1977). Calculations o f theoretical relative amplitudes
o f diffracted waves have been carried out for various models o f the Earth's crust using this
method (Grad 1979). The calculations were made for the case of a profile line perpendicular
t o the diffracting edge, for a constant velocity and for a constant reflection coefficient.
Figure 1. Locati(in of t h e w ~ i r c c5 a n d rcccivcr p in relation to tlic surf'acc A . 'The \pinhols arc the u t i i c as
in e q u a t i o n ( 1 ) .
355
Rejlected-diffracted waves
-1.5
-1.0
03
-0.5
0
0.5
'ii
i
-
Figure 2. Theoretical relative amplitude curve%o f diffracted waves as a function of normalized distance.
The point of osculation of reflected and diffracted travel-time curves delimits the
boundary o f the shadow zone for the reflected wave, and is equal x,, = 2xd, where xd is a
horizontal distance between the shot point and the diffracting edge. At the osculation point
the amplitude of the diffracted wave is equal to half the amplitude o f the reflected wave
( A D = %A,O). With increasing distance the amplitude rapidly decreases and near the minim u m point of the travel-time curve of the diffraction it is only about 0.1 of the amplitude
o f the reflected wave. The amplitude decrease is approximately of exponential character. A
phase reversal of the diffracted wave occurs at the osculation point x o s . In general. the
relative amplitude of the diffracted wave depends on the location of the diffracting edge in
relation to the source, i.e. upon the coordinates xd and h . The results are presented in Fig. 2 ,
where the amplitudes are shown as a function o f a normalized distance X = (x-x,,)/h to
make the comparison possible for models with considerably different coordinates xd and h
of the diffracting edges. For a constant depth h the relative amplitude increases with increasing xd. For a constant xd, however, with increasing depth o f the diffracting edge the relative
amplitude o f the diffracted wave decreases. Although the velocity appears in a direct form in
equation ( I ) , its influence can be neglected for the velocity range from 5.0 to 6.4 km s-',
corresponding to the average velocities in the Earth's crust (Grad 1979). The shape of the
diffracted wave pulse differs from the shape of the incident wave pulse. The difference
increases with increasing distance from the xos point. A spectrum shift towards lower
frequencies is then observed, as well as a decrease of the dominant frequency. These main
results of theoretical modelling are in good agreement with the results of laboratory investi-
356
M. Grad
gations o f t h e amplitude and spectrum o f diffracted waves (Kovalevsky & Averko 1967;
Ten Chi-Wen 1963; Hilterman 1970; Kovalevsky P t al. 1971).
3 Wave pattern in fracture zone models
The kinematic properties of travel times of waves reflected and diffracted in a fracture zone
depend o n the size o f the fault throw A h , the fault width A x and the position of the wave
source in relation t o the whole fracture zone, The minimum points and the osculation points
to the reflected wave travel times (Fig. 3) are characteristic points o f the diffracted wave
travel times. The minimum points of the diffracted wave travel times are located vertically
above the edges o f the reflecting boundareis, while the osculation points determine the
boundaries of geometrical shadow zones for the reflected waves. The shadow zone width
depends on the fracture zone width. In this zone only diffracted waves occur. In the viciqity
o f the osculation points interference of reflected and diffracted waves takes place. The size
o f the interference zone is a function of the parameters xd, h and u, the fracture zone dimensions Ax, A h and t h e duration f of the reflected wave pulse. A theoretical seismogram for a
fracture zone model (xd = 20 km, h = 30 km, u = 5.8 km s-', A x = 4 k m , A h = 1 k m ) is
shown in Fig. 4(a). The wavefield has a complex structure. In the distance interval of
x < 30 k m and x > 60 km the wavefield is of a regular character: apart from reflected waves.
diffracted waves with small ampfitudes also occur. In the interval x = 30-60 kni slightly
increased and strongly decreased amplitudes are observed (Fig. 4b). In t h e vicinity of the
points x = 35 and 55 km the reflected-diffracted wave amplitude reaches its maximum value,
~-
I
1
I
I
1
T, s
17
Xd+lX
x,j
model
x d 2 ~0 k m h ~ 3 k m
0
a x . 5 k m b h - 2 km
v 1 5 8 kmls
16
h
:-
h+bh
15
14
\
\
13
12
11
10
I
0
I
I
I
20
I
I
I
I
I
I
x.km
I
10
30
40
50
60
Figure 3. Theoretical travel times of reflected and diffracted waves for a fracture zone model.
70
Reflected-diffracted waves
357
C
1
00
.
'
30
'
'
'
*
' . ' . .
40
'
50
'
'
'
'
'
60
'
.
b)
3
'
'
'
x km
Figure 4. Theoretical wavefield of a fracture zone (a) and the maximum amplitude as a function of
distance (b). (1) Reflected wave amplitude, (2) reflected-diffracted wave amplitude, (3) diffracted wave
amplitude, T, = reduced time [ T , = t - ( l / u ) d = ] .
exceeding by about 20 per cent the 'pure' reflected wave amplitude. At the osculation points
(x = 40 and 48 km) the reflected-diffracted wave amplitude is half of that o f the 'pure'
reflected wave. The smallest amplitudes are observed in the geometrical shadow zone
(x = 40-48 km), in which the amplitude drops t o about 0.2. The wavefield pattern as well
as the shape of the curve A ( x ) strongly depend on the values of A x and Ah. For a fracture
zone with large dimensions compared with the wavelength, a distinct displacement of the
reflected-diffracted wave travel times is observed, as well as a marked decrease in amplitude.
These effects decrease with decreasing values of A x and Ah. For small dimensions of the
fracture zone the time shirt of signals is insignificant, and in practice the only indication that
the fracture is present is a local decrease in the amplitude of the reflected-diffracted wave
train.
4 Amplitudes of reflected-diffracted waves
The calculations of amplitude curves of diffracted waves using the Berryhill method were
obtained for several models (e.g. results presented in Fig. 2 ) . From these results an approximate formula for the relative amplitude of the diffracted wave was found:
where
358
M. Grad
05
0 4
03
0 2
01
0
0.5
1 0
Figure 5 . Comparison of relative amplitudes of diffracted waves as a function of 2. (1) amplitudes determined by t h e Herryhill method. ( 2 ) from equation (3).
A comparison o f t h e amplitudes obtained for the exact solution and those obtained with the
use o f equation ( 3 ) is shown in Fig. 5 . At the relative distances shown and for the ratio
x d / h = 0.6--2.0, t h e differences d o not exceed a l'ew per cent. Thus, equation (3) is
sufficiently accurate to be used for modelling the diffracted waves in the fracture zones of
t h e Earth's crust. The rriodel types are shown in Fig. 6. and the parameters of the fracture
zones are listed in Table I . For all the models xd = 20 kni and h = 30 km. Calculations were
made for the incident wave pulse F ( t ) = sin 27rft, of frequency f = 10 Hz and duration 0.2 s.
The results. that is the maximum amplitudes of the reflected-diffracted wave train, are
shown in Fig. 7. Characteristic dependences of amplitude o n distance were obtained for the
individual types of fracture zone models, as discussed in the following.
Model of a reflecting half-space limited by an edge ( A ) . This corresponds t o a fracture
zone of large throw, in which a reflected-diffracted wave from the upthrow does not affect
h
;-"
X
@
x
x
x
x
x -x
X
I
h
X
+?-t
h
X
X
X
X
x-
h
Figure 6. Types of fracture zone models.
Reflected-diffructed waves
359
Table 1. Values of paratneters of the fracture zone
mvdels
_-
-
__
iodel
h X.1._
-
Lhi
__
~
A1
_-
El
82
83 __
c1
c2
c3
-
-
~
10
10
1__
0
00
20
40
~
40
05
0 25
40
0
40
75
~
20
C'2
I0
C'3 ._30
C I
10
C"2
20
10
c"3
c"4
~
~
01
02
03
04
05
06
10
40
20
10
0 58
0 29
015
~
~
a 33
C'l
~
~
05
0 75
0 25
0 33
03
03
03
- -
03
-_
10
10
20
10
0 25
0 25
0 25
0 25
0 25
0 25
10
._
~
-0~.
5
__
the wavefield of the downthrow. In this case we observe an increase of the reflecteddiffracted wave amplitude b y about 20 per cent in the region o f interference of the reflected
and diffracted waves. at a distance at which the difference in the arrival times o f these waves
is half the period. As the distance increases, the amplitude of the 'pure' diffracted wave
decreases exponentially.
Fault model (B). Two reflecting half-spaces are bounded by fractures between which no
reflecting elements are present. Two maxima o f the reflected-diffracted wave train are
observed, associated with the edges of the two half-spaces. The distance between the maxima
depends on the fault zone dimension. The greater the width A x , the greater the distance
between the maxima and t h e smaller the amplitude of the reflected-diffracted wave train in
the geometrical shadow zone for the reflected waves.
Fault zone model (C). Between the edges of the reflecting half-spaces there occurs a
reflecting element placed at different depths within the range h, h t A h . An additional
maximum o f amplitude, related to a wave reflected from the element inside the fault, is
observed. The position of this maximum is practically independent of the depth of the
reflecting elements in relation to the adjacent reflecting half-spaces.
Fructure zone nzodel (C'). between the diffraction edges of the half-spaces limiting the
fracture zone there occur t w o reflecting elements. Fig. 7 shows t h e amplitudes of t h e
reflected-diffracted waves for different lengths and depths of the reflecting elements. Two
marked maxima connected with the edges of the half-spaces are observed. The amplitudes
between the maxima are different for different models, and their level is higher than in a
similar model o f type (B).
Fracture zone model [C"). Between the edges of the half-spaces there are three reflecting
elements of different lengths placed at different depths. The amplitude changes are less
diversified than those of model C'. The minimum amplitude in the geometrical shadow zone
is of the order of0.4-0.5, being slightly smaller than in the C ' type models.
D$yraaction element model (0).The reflecting elements are at the same depth but have
different lengths. A distinct decrease of the amplitude with decreasing length of the reflect-
360
M.Grad
20
30
40
50
60 x . k m
20
30
50
40
6 0 x.km
.. . .. . .
20
30
40
50
60 x . k m
20
30
50
40
1
I
20
60 x.km
.. .. . . . .
C"1
C
C'4
---
'.
I
0
C'3
... ....
"
30
40
50
60 x.km
20
30
40
50
60 x . k m
Figure 7. Relative amplitudes o f the reflected-diffracted wave train for models of a fracture zone.
ing element is observed. For small A x , which corresponds to the transition to the 'diffraction point', the amplitude is close t o zero.
5 Conclusions
The properties of diffracted and reflected-diffracted waves have been discussed. In deep
seismic sounding investigations the interpretation of these waves is o f vital importance for
o u r knowledge of the deep structure of the Earth's crust. An analysis o f t h e kinematic
properties, as well as the amplitudes and spectra of reflected-diffracted waves makes it
possible t o determine the structure of a fracture zone and tectonic disturbances (Grad
1984).
In the case of a fault of large throw (type A) it is possible to determine unequivocally
the position of the edge. If no reflecting element is present in the fracture zone (type B),
it is possible to determine the width of the zone, as long as the distance between the maxima
of the amplitudes o f the reflected-diffracted wave train is known (Grad 1981). The complex
structure o f the fracture zones manifests itself in the presence of additional maxima (types
C, C', C"). Short correlation sequences of relatively large amplitude decrease with distance
can be interpreted as reflecting elements, e.g. of the nature of intrusions (type D). Their
dimensions, however, must be many times greater than the wavelength.
Rejlected-d@,~racted
waves
361
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