Geophys. J. R. asfr. SOC.(1967) 13, 187-196.
The Amplitude-Distance Curves for Waves Reflected at a
Plane Interface for Different Frequency Ranges
VIastisIav Cervenf
Summary
The amplitude-distance curves for compressional waves reflected a t a
plane interface between two elastic half-spaces differ quite appreciably
for different frequency ranges (in the case of a symmetrical point-source).
At some distances from the source, the amplitude-distance curves deviate
considerably from those computed by the methods of the geometric ray
theory even at very high frequency. At lower frequencies the characteristic
shape of the amplitude-distance curves implied by the geometric ray theory
may be completely changed.
A whole series of models of the interface was examined numerically.
The results presented were obtained by exact numerical integrations
along suitably chosen contours of integration in the complex plane,
which suppress the oscillatory character of the integrand. The method
can be generalized virtually without difficulties to cover also the more
general types of seismic waves propagating in a layered medium.
1. Theoretical background
The amplitude-distance curves of compressional waves reflected at a plane interface between two elastic half-spaces (for a harmonic point-source) are investigated.
The exact numerical integrations were made by using a suitable transformation of
the contours of integration in the complex plane to suppress the oscillatory character
of the integrand. The case which we have studied can be generalized without difficulties to cover also other types of waves propagating in multi-layered media.
We use the cylindrical system of co-ordinates (r, 4, z). Consider two perfectly
elastic, homogeneous and isotropic half-spaces which are in contact at a plane interface
z =O.
We put the symmetrical harmonic point-source of compressional waves at
the point r = 0, z = H and the receiver at the point r, z =H. The velocities of compressional and shear waves in the medium with the source are denoted by a , and b,, and
in the second medium by a2 and b2; similarly, the densities are denoted p1 and p 2 .
We then obtain for the horizontal and the vertical components of displacement of
the compressional reflected waves, u and w, the formulae (Brekhovskikh 1960,
Cerveng 1959):
.i
A(q)H,'(krq)exp [2ikHJ(1 -q2)]q2(1-q2)-3dq,
u= -+kexp(-iwt)
-03
m
s
w=%k exp (- iot) A(q)H,'(krq) exp [2ikHJ(1 -q2)]qdq,
1
I
I
-00
187
13
188
V. Cervenp
where k is the wave number in the medium containing the source (k = 2 n / l = 2nf/a,,
A is wavelength, f is frequency) and A(q)-the reflection coefficient of plane compressional waves-is
given by the formula
A(q)=A2(q)lA,(q),
,,
+
A 2(q)= -+q2[n12(p- I ) - 2q2(m- 1)12 k J(n2 - q 2 )J(n2n22- q 2 ) [nI2 2q2(rn - I)]'
+4J(1 --q2)J(n2 - q 2 ) J ( n 1 2 -q2)J(n2n22 -q2)(m- 1)'q2
kJ(n2 -q2)J(n, - q 2 ) n14p+ J (1-q2> J(n2n2* -q 2 )n , "P
+J(1-q2>
n=a,la,,
J h 2- q 2 ) [ m 2 - 2 q 2 h - Ill2,
n,=a,lb,,
n2=a2/b2,
m=P,b2'1Pl b12, P=P2IP1*
In A l ( q ) the positive, in A2(q) the negative option is taken throughout. HO1(krq)
and H,'(krq) are Hankel functions. The Riemann sheet of integration is given by
relations:
for q> 1,
argJ(1 - q 2 ) = + n
argJ(nZ-q2)=Jn
for q>n,
argJ(n12-q2)=+n
for q>n,,
argJ(n2nZ2-q2) =+n for q>nn2.
We have been mainly interested in the properties of reflected waves in the neighbourhood of some singular points where the results of the geometric ray theory are
burdened with a large error, such as the neighbourhood of the critical points. As
these singular points do not lie in the immediate proximity of the source, it was possible
to use for the Hankel function several terms of its asymptotic expansion and still
ensure the required accuracy. Accordingly we could rewrite the formulas for u
and w in the following form:
u=(k/2nr)* exp(-iot+in/4) \A(q)(l +Z(krq))exp (ikB(q))q'(l -q2)-*dq,
C
(
S (
w = (k/2nr)* exp (- iot+ in/4) A(q) 1 +Z(krq)) exp ikB(q))qfdq,
C
\
FIG. 1. Contour of integration of saddle point method.
The amplitude-distancecurves for reflected waves
189
where B(q)=rq+2HJ(1 - q 2 ) , q=ql+iq, and Z(krq) gives several terms of the
asymptotic expansion of the Hankel function. C is the new contour of integration
which will be discussed later.
The problem is to find the exact values of u and w from (2). The above stated
integrals for u and w and integrals of a similar nature have already been discussed
many times with the help of approximate methods. An approximate evaluation of
those integrals rests in that the contour of integration is replaced by another one in
the complex plane, e.g. by the contour of steepest descent which passes through the
saddle point q =qo given by the formula qo = r / J ( r 2+ 4H2)= sin i , where i is the angle
of incidence. This contour of integration forms an angle of - 71/4 with the real axis
at the saddle point (Fig. 1). Along this contour the modulus of the integrand decreases
rapidly with increasing distance from the saddle point, and its oscillations are suppressed. It is then possible to limit the integration to the effective part of the contour
in the vicinity in the saddle point. There the function A(q) can often be replaced by
a constant or a suitable superposition of simpler functions (Brekhovskikh 1960,
Cerveng 1959, Honda & Nakamura 1953, Smirnova 1964). However, in many
instances, particularly at low frequencies, the approximate methods lead to results
burdened by a considerable error.
FIG. 2. Contours of integration of alternatives A and B (see text). Above:
alternative A, contour of integration D. Beneath: alternative B, contours of
integration D1 and D2.
190
V. Cenenf
2. Distortion of contours to facilitate computatjon
The integrals for u and w can be evaluated exactly, however. To achieve this we
should choose a contour of integration coinciding with the effective part of the contour
of steepest descent in the vicinity of the saddle point. We can then be sure that the
integrand will decrease rapidly with increasing distance from the saddle point and
will not oscillate to any appreciable extent. The numerical integration can be effected
b y standard quadrature methods.
We have chosen several alternative contours of integration suitable for different
cases. We shall present two of them, designated A and B respectively (see Fig. 2).
In alternative A the contour of integration is given by parametric equation
J(l-42)=J(1-qo2)+Y(l-i),
with real parameter y, - co < y < co. At the saddle point y =0. The evaluation of
the integrals along this contour was highly convenient for qo approaching unity
(1 >qo>0.7); otherwise, on the left-hand branch of the contour of integration D the
integrand decreased but slowly with increasing y (and even became divergent at
qo<O-7).
In alternative B the substitute contour of integration consisted of two parts, D ,
and D 2 . Part D2 corresponds to alternative A, part D 1 is given by parametric equation
with real parameter y. O<y<co.
q,,<0.7.
E
4%
'
- Y (1 -0,
The contour can be used for any qo, even for
1.0
0.4
0.2
00
'
100
u
1'
30'
40'
!
0'
0.4
A.
0.3
-
~
____
_--_
0.2
0.1
FIG.3. Large velocity contrast (ul/az=0.4). Above: modulus of reflection coefficient of plane waves 1 A ! as a function of angle of incidence i. Arrows indicate
positions of critical angles. Beneath: Asymptotic amplitude-distance curve of
vertical component of displacement of reflected wave A, (geometric ray theory).
Arrows indicate critical distances.
The amplitude-distance curves for reflected waves
FIG.4. Large velocity contrast (Ul/a2=0*4). Exact amplitudedistance curves of
vertical component of displacement of reflected wave A, for different H/h: HJh=3;
10; 30; 100 (full lines). Asymptotic amplituds-distance curves (geometric ray
theory) are given for comparison (dashed curves).
191
192
V. Clervenjl
The singular points of the function A ( q ) will lie on the real axis. In some cases it
is necessary to bypass some of the branch points of the Riemann surface. The configuration of the branch points will differ one from another for different ratios of
velocities a,, b,, u2, bz. It is again possible to choose several alternative contours
which will bypass (if necessary) the branch points. Along all the integration paths
we can carry out without difficulties numerical integration.
The computations were made on computers ELLIOTT 503 and SIRIUS. The
values of the integrals were always ascertained at the same time for the whole spectrum
of frequencies (or for the spectrum of values Ifin). On the ELLIOTT 503 computer
evaluation of one integral for one frequency took about 0.3 s of machine time. Computations concerning all types of the resulting head waves were made at the same
time. The paper will not consider the head waves unless they interfere with the
reflected wave. The regions of the interference were calculated for the sinusoidal
pulse of the length of one period. Whenever the head wave interferes with the reflected
wave, the resultant reflected wave is naturally considered as an interference reflectedhead wave. Similarly, we shall not discuss here residue contributions (Stoneley
waves).
The finding that the amplitude-distance curves of reflected waves depend very
strongly on frequency, can be taken to represent the main result of those computations.
As is well known, the amplitude-distance curves of reflected waves do not depend
on frequency in the zero order of approximation of the geometric ray theory. In the
case of finite frequencies, the deviations from the geometric ray theory are, however,
quite considerable for many intervals of epicentral distances.
0.3
0.4
0.5
0.6
0.7
r/2H
FIG. 5. Large velocity contrast (al/az=@4). Amplitude-distance curves of
verticaI component of displacement of reflected wave A , in neighbourhood of
first critical point. Numbers near curves denote H/X. Also the asymptotic
amplitude-distance curve (geometric ray theory) is given (dashed line).
193
The amplitude-distance curves for reflected waves
80"
r/2H
FIG. 6. Small velocity contrast (al/az=O.8). Above: modulus of reflection
coefficient of plane waves / A / in dependence on angle of incidence i. Arrow
indicates position of critical angle. Beneath: asymptotic amplitude-distance
curve of vertical component of displacement of reflected wave A, (geometric
ray theory). Arrow indicates critical distance.
3. Results of computations
The amplitude-distance curves of reflected compressional waves change their
shape very appreciably owing to the effect of frequency. Let us note the results of
computations made for two characteristic models of the interface: interface with a
large velocity contrast and interface with a small velocity contrast.
Large velocity contrast. We shall assume that the refractive index n takes the
value of 0.4 (n=a,/a2) and that a,/b, =a2/bz=J3, p= 1. For this model of the
interface, the modulus of the reflection coefficient ofpZane waves IAl has a characteristic
shape shown in Fig. 3. The arrows point to the position of two critical angles:
il*=sin-'ul~az and i,*=sin-'a,/b,.
At the first critical angle, / A ]has a sharp
peak while the other (very broad) maximum is attained beyond the second critical
angle. Between the two critical angles, I A I displays a deep minimum. An analogous
situation is obtained with the amplitude-distance curve of the vertical displacement
component of the reflected wave A , calculated on the basis of geometric ray theory: a
sharp maximum at the first critical point, a broad maximum beyond the second
critical point and a deep minimum in between (see Fig. 3).
The situation will change substantially if the results of exact numerical integrations
for finite frequencies are considered. Fig. 4 shows the exact amplitude curves of the
vertical component of reflected waves for four values of H i 1 : H/A = 3 ; 10; 30; 100.
The asymptotic amplitude curves (of the geometric ray theory) are shown in thin
dashed lines in all figures.
194
V. Cerrenf
r/2H
FIG. 7. Small velocity contrast (ul/uz=O.8). Exact amplitudedistance curves
of vertical component of displacement of reflected wave A , for different H / k
H/A=l ; 3; 10; 30 (full lines). Asymptotic amplitude-distancecurves (geometric
ray theory) are given for comparison (dashed curves).
The amplitude-distancecurves for reflected waves
195
For very high H / I the amplitude-distance curve preserves in general the shape
resulting from the geometric ray theory. Only the position of the maxima in the
vicinity of critical points changes somewhat. At lower H/A, the whole character of
the amplitude-distance curves changes. The maxima are shifted to considerable
distances beyond the critical points, the sharp peaks are smoothed. In some cases,
the amplitude-distance curve has a maximum just where the geometric ray theory
curve has a deep minimum, and vice versa. In the region where head waves
separate from reflected waves the amplitude curve is dashed. The amplitude curve
may oscillate there.
In Fig. 5 the region of the first critical point is given in detail. The numbers near
the curves give the values of H / 1 .
Small velocity contrast. We shall assume that the refractive index n=0.8 (and
that a l / b i= a,/b2= 43, p = 1). Interfaces with small velocity contrast are of great
importance in seismic practice, e.g. in deep seismic soundings of the Earth's crust.
The properties of waves reflected on similar interfacesJave already been studied by
approximate methods for high H / A (H/A= 15-50) (Cerven); 1961, Cervenjl 1966,
Smirnova 1964). The method of exact numerical integrations lends itself without
difficulties to studies of the reflected wave even in the range of seismological frequencies (H/A- 10).
For this model of the interface the modulus of the reflection coefficient of plane
waves I A I has a characteristic shape shown in Fig. 6. The arrow points to the position
of the critical angle i,*=sin-'a,/a,.
The modulus of the reflection coefficient has
again a sharp peak at the critical angle where it reaches its local maximum, whereas
beyond it, it changes very slowly. The amplitude-distance curve of the vertical
displacement component of reflected wave A , computed on the basis of geometric
ray theory, is shown in Fig. 6. The amplitude-distance curve displays a prominent
maximum at the critical point (critical reflections).
The amplitude-distance curves of reflected waves determined on the basis of
exact numerical integrations are given in Fig. 7 (for H/A= 1; 3; 10; 30). For comparison the asymptotic amplitude-distance curves (of the geometric ray theory) are
given there in thin dashed lines.
As Fig. 7 indicates, a shift of the maximum of the amplitude-distance curve from
the critical point to greater distances from the source is typical of all H/A. For seismological frequencies this shift can amount to considerably high values.
The knowledge of the frequency is thus an indispensable condition for the computation of the amplitude-distance curve of reflected waves. Equally indispensable
is the knowledge of the frequency whenever we wish to arrive at some conclusions
about the parameters of the interface from the amplitude-distance curves of reflected
waves.
Geophysical Institute,
Charles University,
Prague 2, K e Karlovu 3,
Czechoslovakia.
References
Brekhovskikh, L. M., 1960. Waves in Layered Media. Academic Press.
Cervenjl, V., 1959. On the reflection of spherical waves at a plane interface with
refractive index near to one, Studia geophys. geod., 3, 116.
196
V. Cervenf
Cervenl, V., 1961. The amplitude curves of reflected harmonic waves around the
critical point. Stitdia geophys. geod., 5, 319.
cervenl, V., 1966. On dynamic properties of reflected and head waves in the
n-layered Earth’s crust, Geophys. J. R. ustr. SOC.,11, 139.
Honda, H. & Nakamura, K., 1953. On the reflection and refraction of the explosive
sounds at the ocean bottom, Scient. Rep. Tohoku Univ., Ser. 5, 4, 125.
Smirnova, N. S., 1964. Calculation of wave fields in the neighbourhood of singular
points, Report at 1st Symposium on Geophysical Theory and Computers,
Moscow-Leningrad, 1964.