The Transmission of Rayleigh Waves across an Ocean
Floor with Two Surface Layers, Part I1 : Numerical
Urs Hochstrasser and Robert Stoneley
“Under the water it rumbled on.”-S.
T . Culeridge, The Ancient Mariner
Summary
This investigation obtains values of the wave velocity and group
velocity of waves of Rayleigh type, following ocean paths corresponding to the structures indicated by experimental investigations using
explosions at sea. Eight different models have been considered, with
ocean depths 4 or 6 km, a sedimentary layer of depth I or 2 km, underlain by basic rocks of thickness 5 or 7 km; in all cases these structures
were supposed to rest on ultrabasic rocks of great thickness.
T h e wave velocity was found for a series of values of the wavenumber by solving an eleven-row determinantal equation ; the computations were performed by the use of the SEAC electronic computer of
the U.S. National Bureau of Standards, and from these values the corresponding values of the group velocity were obtained by numerical
differentiation. The general agreement of these dispersion curves with
observed values indicates that they should be useful in determining the
ray track and time of passage of a train of Rayleigh waves crossing an
ocean floor whose structure and depth are varying from place to place.
Introduction
T h e present paper constitutes Part I1 of an investigation of which Part I was
published three years ago (Stoneley 1957). Part I gave the derivation of the
wave velocity equation for periodic disturbances moving across an ocean of uniform depth H resting on a layer of sediments, of thickness TI,
beneath which is
a layer, in general basic, of rock of thickness T2, extending down to the MohoroviEiC
discontinuity, below which the material is denoted by the suffix 3. T h e wave
velocity equation is obtained as a determinantal equation of the eleventh order,
which, for an assigned structure, can be solved numerically to give values of the
wave velocity c corresponding to a series of values of the parameter K , where
2 7 ~ is
1 ~the wave length.
I.
Calculation of the wave velocity
T o agree with the layering of the ocean floor, so far as it is known, the values
H = 4, 6km; TI = I, 2km; Tz = 5, 7km have been chosen, corresponding to
eight different possibilities, so that in any actual basin of extent much greater
than the wave length of a surface wave linear interpolation (and probably extrapolation) would be permissible. T h e velocity of sound in sea-water is CO, and the
2.
I97
Urs Hochstrasser and Robert Stoneley
198
velocities of compressional and distortional waves in the elastic solid media are
written as a, P, with appropriate suffures. T h e density is denoted by p. T h e
constants used in the computations are
H = 4, 6km; co
Ocean:
Sediments Ti = I, 2 km; mi
TZ= 5, 7km; a2
Basic
c(3
Ultrabasic
= 1.53km/s,
PO
= I .8 km/s, /?I = I '0km/s, p i
= 6.5km/s, 8 2 = 3.7km/s, pz
= 8.1 km/s, 83 = 4.5 km/s, p3
= 1.03g/cm3
= I 23 g/cm3
= 2.9g/cm3
= 3qg/cm3
'The solution of the wave velocity equation was effected by use of the SEAC
electronic computer of the U.S. National Bureau of Standards, Washington, D.C.
A sequence of values of the parameter K Twas
~ assumed, and the equation solved
for c/,!?z. For values of KTZgreater than about 4 the accuracy of the solution
became doubtful. T h e solution is not complete, in the sense that only the fundamental type of oscillation was obtained; some values of higher modes were in
fact calculated, but a systematic evaluation of the many possible types of oscillation that may exist was not contemplated in the present research.
From the values of the wave velocity c the values of group velocity C were
calculated by numerical differentiation according to the formula C = c + K d c / d ~ .
T h i s process is a very unsatisfactory one; it involves considerable loss of accuracy,
and where both a forward difference formula and a central difference formula can
be used on the same set of values, spaced at equal values of the argument, the
discrepancy is notable. Thus the values of C quoted in the tables of this paper
are not trustworthy to the third decimal place, particularly for the larger values
of KTZ.
T h e period T can be found from the formula
2x
2rT2
I
KC
Pz
KTZ. clPz
T"-'-*
and is not affected by the inaccuracies of numerical differentiation. T h e values of
T and C calculated for the eight assumed structures are listed in the following
tables.
C//?Z,
H =6
Ti = I
Tz = 7
KTZ
c/Bz
0' 0
I 'I2347
0'1
1~11090
I so9886
1.08637
I '07 I 63
1'05075
1.01527
0.9565 I
0.88529
04
3 I 838
0.761 10
0.6732 I
0.61128
0.56641
0.53298
0.50744
0.43934
0.41 044
0' 2
0.3
0 '4
0.5
0.6
0 '7
0.8
0.9
I '0
1'2
I '4
1.6
1.8
2 '0
3 '0
4 '0
7
(s)
a,
76 '4
38.6
26 '0
19.8
16.2
13'9
12.7
12'0
H =6
Ti = 2
Tz
c( W S )
4.157
4.064
3'975
3'874
3'713
3'393
2.708
1 '707
I a206
11.5
1.017
11'2
0.834
0.874
0.910
0.959
1.014
I '045
I .087
?
10.5
9 '9
9 '4
8 '9
8 '4
6 '4
5 -2
KTZ
0 '0
0'1
0'2
0.3
0 '4
0.5
0.6
0 '7
0.8
0 '9
I '0
I '2
I
'4
I .6
I
.8
2 '0
3 '0
4 '0
c/Pa
1.12347
1.10906
I .09486
1.07959
I .06075
I.03253
0.98340
0.9I073
0.83460
0.76792
0.7 I232
0.62801
0.56875
0.52562
0.49320
0.468 10
0.39672
0.35900
~.~
=7
7
(s)
03
76.6
38 4
26.2
20'0
16.4
14.4
13.3
12.7
12.3
11.9
11.3
10.7
10'0
96
9'1
7'1
5.9
Transmission of Rayleigh waves across an ocean floor with two surface layers
H =6
Ti = 2
T2 = 5
H =6
Ti = I
Tz = 5
K
T2
0 '0
0'1
0 '2
0.3
0 '4
0.5
0.6
0 '7
0.8
0 '9
I '0
I '2
1'4
I
.6
I
4
2 '0
3 '0
4 '0
CIS2
7
1'12347
1.10827
1.09239
I '07219
I e03662
0.96 I44
0.86 I 83
0.77545
0.70778
0.65513
0.61368
0'55377
0.51351
0.485 18
0.46449
0.4887
0.40661
0.38512
(s)
00
76.6
38 '9
26.4
20'5
17'7
16.4
15.6
15.0
14'4
13.8
12.8
11.8
10.9
10'2
9'5
7 '0
5 '5
c(kmis)
KTZ
'CIS2
'12347
0 '0
1
0'1
I . I 1240
0'2
I '10227
0.3
0 '4
I .09265
0.5
0.6
0 '7
0 *8
0 '9
I '0
1'2
1'4
I .6
I .8
2 '0
3 '0
4 '0
.08303
.07267
I .06048
I '04472
I .02260
0.99067
0.94810
0.85128
0.76620
0.69890
0.64620
0.60452
0.48693
0'43499
I
I
7
1 '12347
0'1
1.10564
1.08630
I -06028
1.01098
'044
0.870
0.856
om0
0.903
0.977
1.038
I .087
1'115
1.148
?
?
0 *6
1
0 -2
0.3
0 '4
0.5
0 '7
0.8
0 '9
I '0
I '2
1 '4
I .6
I .8
2 '0
3 '0
4 '0
a,
4'157
4'077
4.006
3 '937
3.864
3'764
3.622
15.8
0.91529
0.81045
0'72589
0.66086
0.6 1050
0.57084
0/51319
0.47387
0.44556
0.424 I7
0.40729
0.35295
0.31570
(s)
co
76.8
39'1
26 '7
21'0
c (kmls)
4.157
4.022
3.865
3'578
2.682
18.6
1'255
I7 ' 5
16.7
16.1
0.854
0.7770)
0.780
0.738
0.802
0.864
0.902
0.910
15.2
14'9
13.8
12.8
I I '9
11'1
10.4
8 '0
6 '7
0.954
?
?
?
H =4
Ti = 2
Tz = 7
c (kmls)
13'3
11.6
10.4
9'5
9 '0
8 '3
7 '9
7 -6
7 '3
7.0
5 *8
4 '9
7
4'157
4.048
3 '903
3 '705
3 '027
I 461
(4
76 '3
38.5
25 '9
19.6
CIS2
0'0
H =4
Ti = I
Tz = 7
KTZ
199
KTZ
7
(4
0 '0
co
0'1
3'390
2'997
2'398
1 '723
I '042
0.864
0.826
0 '7
76 '4
38.7
26.1
19.8
16.0
13.6
I I '9
0 *8
10.8
0.9
10'1
0 44.4
I 43
0 '2
0.3
0 '4
0.5
0.6
I '0
I '2
1 '4
I *6
0.85I
2 '0
0.861
(0.877)
3 '0
9 '7
9'1
8 4%
8 '4
8.1
7 '9
6 *6
5 '7
c (kmls)
4'157
4.063
3 '975
3.885
3'775
3.629
3'399
3 '-9
2.382
I .682
I '26 I
0.773
0.68 I
0.671
0.649
0.723
?
?
Urs Hochstrasser and Robert Stoneley
200
rtTa
0'0
0'1
0'2
0.3
0'4
0'5
0.6
0.7
0.8
0.9
1'0
I '2
1'4
I .6
I
4
2 '0
3' 0
4'0
H =4
H =4
Ti = I
Tz = 5
7 (s)
Ti =
CIS2
1.12347
I '1 I 048
1.09801
I .08519
I .07046
I .05063
1.01888
0.96630
0.89826
0.83111
0.77230
0.68064
0-61527
0.56750
0.53160
0.50395
0*42821
0.39337
2
Tz = 5
c( W S )
co
KTZ
4'157
4.063
3 '974
3 -86I
3.649
3'434
2.840
I .966
I .264
1.104
04347
0.819
0.843
0.884
76-5
38.7
26.1
19.8
16.2
73'9
12'5
11.8
I 1 '4
I I '0
10.4
9 '9
9 '4
8 '9
8 '4
6.6
5 '4
CIS2
'12347
1.10787
I '09224
I -075I 8
1'05407
1.02316
0.97178
0.898 I 5
0.82131
0.75374
0.69713
0.61067
0.54925
0'50402
0'0
I
0'1
0'2
0-3
0 '4
0 '5
0.6
0 '7
0 43
0 '9
I '0
1'2
I
I
-4
.6
0.928
I
43
0.48402
0.972
?
?
2 '0
0 '44244
0 3 6 I 91
3 '0
4 '0
0.31754
7
(4
00
76.6
38.9
26.3
20'1
16.6
14.6
'3 '5
12.9
12.5
12'2
11.6
I 1'0
10.5
97
9.6
7.8
6 '7
c (km/s)
4'157
4 '04 1
3.915
3-756
3'535
3 Q44
2.175
I '307
0.873
0.799
0.694
0.658
0.65 I
0.710
?
?
?
?
3. Comparison with observations
A number of attempts have been made by various authors (e.g. Stoneley 1926;
Wilson & Baykal1948 ; Ewing & Press 1952) to compare the observed values of the
dispersion curves (i.e. Cvs T ) with those calculated according to an assumed constitution of the ocean and its underlying layers. Evidently the assumption of uniform
layers is an oversimplification, and will yield only an overall comparison. Berckhemer in 1956 considered the effect of variation in water depth on the time of
transmission; it was, in fact, along these lines that the present investigation was
I
I
20
I
I
40
Period
60
I
80
(5)
FIG. I
planned, so that not only might an overall comparison be made, but one in which
variation of oceanic structure could be allowed for (remembering, of course,
that the variation in surface wave velocity determines the departure of the path
from a great circle route).
Three of the dispersion curves are shown in Figure I. T h e seven black dots
on the diagram represent readings taken from a paper by Ewing & Press (1952)
Transmission of Rayleigh waves across an ocean floor with two surface layers 201
relating to the earthquake of 1950 July 29 in the Solomon Islands. While the
general fit is good, the readings do not fit closely any one of the three curves, and
the same is true of a comparison with the remaining calculated curves, to the
extent that these observations do not directly decide which is the better average
constitution of the ocean and floor. Some adjustments of the adopted values of
the density and elastic constants may be indicated by future researches. Figure 2
shows the observations of Wilson & Baykal and of Berckhemer compared with
four of the calculated dispersion curves. The two sets of observations, of course,
are not comparable, although both relate to the Atlantic Ocean, but the fit is sufficiently good to justify further attempts to reconcile the observed dispersion curves
of Rayleigh waves with the present knowledge of the ocean floor.
I
I
Ib
I
11
I
ia
I
ao
Period
I
I
24
Ib
(6)
FIG.2
In investigations, such as those of J. and M. Darbyshire (1956), on the transmission of the waves of microseisms from mid-ocean to a coastal station, the
tables given here will facilitate the computation of the ray path and the time of
transit.
This work, in part, was carried out at the U.S. National Bureau of Standards,
Washington, D.C., under a grant from the U.S. Office of Naval Research. We
wish to express our indebtedness to the Director of the National Bureau of
Standards for the facilities afforded to us, and for permission to publish this paper.
National Bureau of Standards,
Washington, D.C.,
16 Millington Road,
Cambridge.
1960 October.
U.S.A.
References
Darbyshire, J. & M., 1956. Mon. Not. R. Astr. SOC.Geophys. Suppl., 7 , 301.
Ewing, M. & Press, F., 1952. Bull. Seismol. SOC.A m . , p , 315.
Stoneley, R., 1926. Mon. Not. R. Astr. SOC.Geophys. Suppl., I, 349.
Stoneley, R., 1957. Bull. Seismol. SOC.
Amer., 47, 7.
Wilson, J. T. & Baykal, O., 1948. Bull. Seismol. SOC.Amer., 38, 46.