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OCR for page 114

ON THEORIES OF THE ORIGIN OF MICROSEISMS
by J. G. Scholte
In the last decennium the concept that
many microseisms are generated by a storm at
sea has been more and more generally accepted;
detailed investigations as for instance by Ber-
nard (1941) as well as the successful detection
of hurricanes by means of tripartite stations
prove the validity of this view beyond any doubt
(Gutenberg 1952).
It is however still uncertain by which proc-
ess these seismic movements come into exist-
ence; the observations often point in different
directions and it is therefore not possible to
formulate a theory covering all observed data.
Perhaps the most useful way to treat this
matter theoretically is to ignore various mete-
orologic and oceanographic circumstances and
to start from the undisputed fact that a dis-
turbance at the surface of the ocean causes at a
distance of the order of 108cm. microseisms
with an amplitude of say 5 n and a mean period
of about 6 seconds.
The movement of the ocean in the vicinity
of the storm area has an amplitude which is of
course several times greater than 5 (i and as
the vertical motion at the bottom has to be con-
tinuous the same is true for the movement of
the water. In view of the well known fact that
the amplitude of gravity waves decreases ex-
ponentially with the depth, it is evident that
the motion of the water which generates at the
bottom the seismic waves is not a gravitational
but a compressional wave and that we may
neglect the effect of gravity on this process.
Consequently we have to consider waves of
compression in a purely elastic system consist-
ing of a fluid layer of finite depth h covering a
solid body. Consider a cartesian coordinate sys-
tem with the x axis in the free surface paral-
lel to the direction of propagation and the z
axis vertically downward. In order to avoid
complications which are irrelevant to this prob-
lem we suppose this body to be semi-infinite.
Denoting the horizontal component of the move-
ment by u and the vertical one by w,
»• x
Figure 1
114

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ON THEORIES OF THE ORIGIN OF MICROSEISMS 115
a compressional wave travelling in the liquid in
the direction y is described by
o o ,
u = -—, w = -—, (j>0 = A exp
ox oz
I /x sin y + Z cos y \ v
|ivl r -*)/
where v = the frequency, c = the velocity of At the bottom z = h three other waves are
sound in water, and y is an angle of incidence excited:
measured from the vertical.
(i v(X Sin Y - '
\ \ c
the reflected wave *i = RA exp
and two refracted waves:
the longitudinal one: u = —, w = ——, with the potential
dx dy
/x sin a + 2 cos a h cos y h cos a
and the transverse wave: u = ——, w = ——7
oz ox
with the vector-potential ^ in the y-direction:
'x sin p + Z cos p h cos y h cos
r />
f = DtAexp / i vl -
a and b are the velocities of longitudinal and tion (R and D) are determined by the condition
transverse waves in the solid. The quantities that the tensions TZ7, Tzx and the vertical
(i and ft are angles of refraction for compres- motion w have to be continuous across the
sional and sheer waves in the solid respectively. plane z = h;
The coefficients of reflection and refrac-
- R —T^ = Dl

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116 SYMPOSIUM ON MICROSEISMS
p + Rp = Di Pi cos 2( 3 + Dt pi sin
sin 2a cos 2(3
0 = D -- - Dt --
where p and p^re the densities of the liquid and solid respectively.
The solution is:
(b2 pc cos a \ 1
cos2 2p + — sin 2a sin 20 - - - I -
a2 pj^cosy/A
2p cos 2(3 2pb2 sin 2p
D = - , Dt = -
pi A pi Aa2
b2 pc cos a
with: A = cos2 2(3 + — sin 2a sin2p +
a2 pia cos y
Arriving at the surface z = O the reflected wave , gives rise to a new wave fa in the + y
direction; as the normal pressure caused by #, and fa has to disappear at z = O we have
lx sin Y + z cos y + 2 h cos y ^
exp •< i v I t
Thus the twice reflected wave fa is equal to i£„ multiplied by —R exp (2 iq), with q = (vh
cos Y)/C; it follows that the wave *i travelling in the Y direction is given by
oo
«—r I I
Hence
n—o
/x sin Y + Z cos
exp*'
0 =A * : -' (i)
+ " 1 + R exp (2iq)

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ON THEORIES OF THE ORIGIN OF MICROSEISMS 117
and in the same way the reflected wave system 0. is
x sin Y - Z cos Y + 2h cos Y
. . RA "P
1 + R exp (2iq)
similar expressions for the refracted waves are easily obtained.
From these potential-functions we derive the movement of the ocean's bottom:
u = w tan a (cos 2(3 - 2 -cos a cos p) and
p cos a exp
w = 2iv A 1 1C It
Pi a
A jexp (-iq) + R exp (iq)|
which after some reduction can be written as
u = w sinS 4 cos 2&\— - sin28) - 2 cos(3 V and
1 \«« /
where
N =1 cos22p(— - sin2p) 2+ 4 sin^e cosS Lcos q - 2i —f sinss)
|^ \a2 / J pi\ C2 7
(2)
sin q
vh\/b2 \i/
and q = "J\C^" " sin2(3) (3)
The primary wave qp() which excites this sure TM of the secondary waves \ cancel out at
whole wave system, causes a pressure Ta at z = o. Supposing 4>,, to be generated by a
the free surface z = o equal to — pv2i£,,; the pres- pressure
P = P expJ iv(— si

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118
SYMPOSIUM ON MICROSEISMS
uniformly applied on the plane z = o, the ampli-
tude A of <#'„ is equal to
In actual circumstances this periodic pres-
sure — which is in any case necessary to obtain
waves of compression — is confined to a finite
area; in order to obtain a function which de-
scribes the actual conditions better than the
function
P exP
( x ^
/ iv(— siny - t)>
I C J
we change this into
pJ0(v - sinyj exp (-ivt)
where J,, is a Bessel function and
The motion of the bottom is then given by
the same expressions (2), if we change the fac-
tor exp (ivx/c sin y) into iJ^vr/c sin y) for
the horizontal (radially directed) component
and into J0(vr/c sin y) for the vertical one.
Remembering the discontinuous factor of
Weber
r
'0 ifr>rc
It will be seen that the pressure function p J,,
(vr/c sin y) exp (-ivt) changes into a func-
tion which is equal to p exp (-ivt) for r < r0
and vanishes for r > r0 if we apply the oper-
ator
remains constant (= Q). In the limiting case
r,, = 0 the normal force Q is evidently concen-
trated in the point 0 and is expressed by
f°°T ( r° \ , / r°
I Jt Iv — sin yl d Iv — sin y
271
•e
J J
da.
The parameter r,, is arbitrary; following
Lamb's procedure (1904) we diminish r,,, at the
same time increasing p in such a way that the
total forcenpr£ exerted on the plane z = 0
Consequently by applying the operator
•o
0 f v sin y /v sin y\
^J —r- dl-r-J

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ON THEORIES OF THE ORIGIN OF MICROSEISMS
119
to the expressions for u and w we obtain the
motion at the bottom of the ocean, generated
by a force Q concentrated in one point of the
surface. We readily find
u = -
vO
W —
•ivt f sinap f /b2 „ \
—M—^cos 2P I — - sinzpl
41 i!i' '
J i I v - sin p] d sin p
ivt [ sinp / r \
I -TT- Jo (v ~ sinpl d sin p
-2 cos
It can be shown that the main value of
these types of integrals is for large values of r
contributed by the residues of the integrand at
the zeroes of N. The microseismic movements
at large distance from the generating force
Qe"lVt are therefore:
Wm *
- 2
(sinpm)'^ iv I 5 sinp.-tj + | Tti
(9N/3 sinp). *
(4)
From (3) we see that the equation N = 0
determines for each value of vh/c one or more
values f^,; writing this equation in the form of
the denominator in (1) :
motion is caused by constructive interference
of the plane waves in which the spherical wave
originating at the origin can be decomposed
(Press and Ewing 1948).
—R exp ( 2 i q) = 1
(5)
it is obvious that the values YI» corresponding
to the roots ^ determine the directions Y for
which the reflected elementary wave 4'\ is iden-
tical to ^i-2 (the phase shift caused by the
two reflections at the boundaries cancels the
difference in ohase due to twice transversing
the layer). Therefore the main part of the
It follows from (5) that, as | R I = 1, y,,, has
to be greater than the angle of total reflection
for the tranvserse wave; hence b > c and sin
Again if sin 0 < b/c the quantity q is real;
for each value of sin between 1 and b/c we
obtain therefore an infinite series of values
vh/b satisfying N = 0. In the diagram
(fig. 2) several of these modes are shown (we

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120
SYMPOSIUM ON MICROSEISMS
have used the numerical values a/b
b/c = 2 andp/p!= 0.4).
- VS.
Figure 2
With the exception of the first mode each
curve starts at sin 0=1 and approaches sin
fj = b/c asymptotically. The curve of the first
mode shows two peculiarities:
if h = <{, the equation N = 0 changes into the
simple Rayleigh equation for the suboceanic
medium
cos2 2p + 4 sinap cos p (
sn
p)^ = 0
with the root sin 3 = b/S,, where Si — the
velocity of Rayleigh waves. At smaller values
of sin |3 (in fig. 2 for sin B <-1.0788) the equa-
tion N = 0 yields a negative value of vh/b.
In the second place: if h = oo the equa-
tion changes into the equation for Stoneley
waves in the two semi-infinite media (water
and suboceanic rock) :
/b2 r% p /b2 \v /b* \ v
cos2 2p + 4 sin2 p cos p I — - sin2 p I = 2i — ( sin2 p I/2 ( — - sin* p }' -
\a2 / p1\a2 /\c2 /

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ON THEORIES OF THE ORIGIN OF MICROSEISMS
121
with the root sin (3 = b/S-j, where S-j = the
velocity of these waves. Hence the first curve
starts at sin jS = b/S, > 1 and tends asymp-
totically to sin /3 = b/Sj > b/c.
0-75-
0-50-
0-25-
60
Figure 3
The values of Cm =
./-*_)
\dsinp/m
have been calculated by Longuet-Higgins (1950) and are plotted in fig. 3 against h v b. In fig.
4 the ratio
/
— = sinp < cos
I
- 2 cosp
>
is shown as a function of sin 3.

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122
SYMPOSIUM ON MICROSEISMS
U/
/W
1.4
0.2 -
1.0
1.2
14
1.6
1.8
2.0
sin
Figure 4
The maximum of the first mode appears
atvh/b« 0.85; supposing b = 2.8 km/sec and
h = 3 km we have v — 0.79 (period of about
8 sec.). At a distance of 3000 km the ampli-
tude of the vertical component = Q X 1.5 X
10"' cm. In order to obtain a microseismic
amplitude of 5 n the total force Q has to be
about 3.10 '"' dynes; assuming the radius of the
storm area ss> 18 km it appears that a mean
pressure variation of 1/3 mb is necessary to
produce the observed microseisms.
In this calculation it has been assumed that
the pressure variations at widely separated
points are correlated; as this will not be the case
in actual circumstances, the obtained value of
1/3 mb has to be interpreted as the effective
pressure variation.
Supposing the phases of these pressure os-
cillations at points separated by a distance
greater than a to be incorrelated the effective
pressure of the storm area b- is about a/b
times the mean pressure.
An explanation of such a pressure varia-
tion has to be found either in the atmospheric
or in the hydrodynamical circumstances during
a storm. With regard to the first and most
obvious explanation we refer to the observa-
tions at Wei-ka-wei by Gherzi (1921) ; if the
atmospheric pressure in the "eye" of the ty-
phoon changes periodically with an amplitude
of 0.5 mb this would be sufficient to generate
microseisms at large distances of the track of
the storm (Scholte 1943). It is perhaps pos-
sible to obtain more data about this phenome-
non by placing a network of microbarographs
in the regions where typhoons often occur.
Recent observations by Donn (1951) may
also elucidate the connection between atmos-
pheric disturbances and microseisms in the
western hemisphere.
REFERENCES
, J., 1904, The cyelones of the Far East, Manila.
i

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ON THEORIES OF THE ORIGIN OF MICROSEISMS
123
BERNARD, P., 1941, Etude sur I'Agitation Microseismique
(These).
DONN, W., 1951, Joitrn. Meteor. 8, 406.
GHERZI, E., 1921, Etudes sur les microseismes, Note de
Sism. Obs. Wei-ka-wei.
GUTENBERG, B., 1952, Observations and theory of micro-
seisms, Cal. Inst. Techn.
LAMB, H., 1904, Phil. Trans. London A, SOS, 1.
LoNGUET-HicciNS, M. S. & URSELL, F. 1948. Nature,
162, 700.
LoNGUET-HiGGiNS, M. S., 1950, Phil. Trans. London A,
243, 1.
MlCHE, M., 1944, Ann. Ponts et Chanss, 2, 42.
PRESS, F. and EWING, M., 1948, Trans. Am. Geoph. Un.
29, 163.
SCHOLTE, J. C., 1943, Versl. Akad. Amsterdam 52, 669.
STONELEY, R., 1926, M.N.R.A.S. Geoph. Suppl. 1, 349.
WHIPPLE, F. J. W. and LEE, A. W., 1935, M.N.R.A.S.
Geoph. Suppl. 3, 287.
(c) The Mode of Transmission Over Conti-
nental Paths—It has been observed that short
period surface waves (with periods in the mi-
croseism range) propagate with surprisingly
little attenuation over large continental paths.
(Press and Ewing 1952). Although no
theory has been presented to account for the
details, it is apparent that the continental crust
behaves as an homogeneous sialic plate for
these waves. It seems possible that micro-
seisms, once past the barrier at the continental
margin, may well be transmitted in a manner
similar to these earthquake phases.
REFERENCES
Discussion
FRANK PRESS
Columbia University
Dr. Scholte is to be complimented for his
concise discussion of Theories of the Origin of
Microseisms. We can only add a few remarks
on certain aspects of the problem. It is con-
venient to discuss separately (a) the nature
of the source (b) the mode of transmission
over oceanic paths (c) the mode of transmis-
sion over continental paths.
(a) The Nature of the Source—The wave in-
terference theory of Longuet-Higgins (see his
paper in this volume) is the only published
treatment which can quantitatively account
for energy transfer from the atmosphere to
the ocean. Little work has been done on the
possibility that pressure fluctuation and gusti-
ness present in turbulent air masses can trans-
fer energy directly to compressional waves in
the ocean. In this connection the impulsive
modification of sea waves by wind gusts
(WHIPPLE AND LEE, 1935) may be impor-
tant.
(b) The Mode of Transmission Over Oceanic
Paths—There seems to be general agreement
among investigators on the manner of trans-
mission of elastic energy across the oceans
(Stoneley 1926, Press and Ewing 1948,
1950a, b, Scholte 1943, Longuet-Higgins
1950). That transmission peaks oc-
cur for certain periods has been pointed out on
several occasions. In order for these trans-
mission peaks to develop fully, propagation
paths of the order of 100 wavelengths or more
are needed, a condition not fulfilled by many
microseismic situations. It seems therefore
that these transmission peaks can play only a
secondary role in any theory on the origin if
microseisms. It is particularly disturbing that
surface waves from oceanic earthquakes con-
tain comparatively little energy in the micro-
seism period range.
LoNGUET-HiGGiNS, M. S., A Theory of the Origintof
Microseisms, Phil. Trans. Roy. Soc. Ser. A., 243,
pp. 1-35, 1950.
PRESS, F., and EWING, M., A Theory of Microseisms,
with Geologic Applications, Trans. Amer. Geoph.
Union, 29, pp. 163-174, 1948.
PRESS, F., EWING, M., and TOLSTOY, I., The Airy Phase
of Shallow-Focus Submarine Earthquakes, Bull.
Geol. Soc. Amer., 40, pp. 111-148, 1950.
PRESS, F., and EWING, M., Propagation of Explosive
Sound in a Liquid Layer Overlying a Semi-Infinite
Elastic Solid, Geophysics, 15, pp. 426-446, 1950.
EWING, M., Two Slow Surface Waves Across North
America, Bull. Seism, Soc. Amer., 43, pp. 219-228,
1952.
SCHOLTE, J. G., Over Het Verband Tussen Zeegolven
en Microseismen, I and II, Verslag Ned. Akad.
Wet., 52, pp. 669-683, 1943.
STONELY, R., The Effect of the Ocean on Rayleigh,
Waves, Mon. Not. Roy. Astron. Soc., Geophys.
Suppl., 1, pp. 349-356 1926.
WHIPPLE, F. J. W., and LEE, A. W., Note on the Theory
of Microseisms, Mon. Not. Roy. Astron. Soc.,
Geophs. Suppl., 3, pp. 287-297,1935.
Discussion from the Floor
(Dinger asked if the absence of periods
of one to seven seconds across the western At-
lantic was also true of the Pacific, and Press,
replied yes, on the Aleutians to Hawaiian path.)
Bath. Dr. Scholte mentioned that the ratios
of horizontal and vertical amplitude of micro-
seisms at De Bilt were larger than could be
explained by his theory. The reason is obvi-
ously the very loose ground at De Bilt with
around 9 km. of sediments (in accord with the
theory of A. W. Lee).
(Haskell pointed put that where there is a
big contrast in velocity between the surface
layer and the underlying medium, there may

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124
SYMPOSIUM ON MICROSEISMS
be large ratios of horizontal to vertical am-
plitudes. Romney pointed out that there are
some microseisms with periods as great as 20
seconds, but with much lower amplitudes than
those of shorter period.)
Longuet-Higgins. The occurrence of micro-
seisms in "groups" appears to be an example
of a very general phenomenon which is ob-
served whenever a disturbance can be con-
sidered as the sum of a large number of dis-
turbances of about the same frequency. Such
a sum was first considered by Rayleigh in 1880
in connection with sound from many different
sources—this is sometimes called "the bee-hive
problem." He showed that the probability
P(a)da of the sound wave amplitude being be-
tween a and a-da was given by
P(a) =
±2.
a 2
where a was the r.m.s. amplitude. The height
of sea waves (defined as the difference in ele-
vation between a crest and the preceding
trough) has been shown to obey the same sta-
tistical law. In this case the generating area
of the swell can be considered as divided up
into regions, each large compared with the
length of a sea wave, and each giving a sinus-
oidal contribution of independent phase. A
similar concept probably applies to micro-
seisms. An analysis of the statistical distribu-
tion of microseismic amplitude over a fairly
short interval of time would be of interest.
(Dorm commented that frontal micro-
seisms are weaker in the summer than in the
winter. Carder replied that recently there
have been some intense summer cold front mi-
croseisms in Washington.)
Bath. Cold fronts cannot be located sufficient-
ly accurately by interpolation from weather
maps, especially not when they pass over
oceans. Hydrographic records of pressure and
temperature have to be used.
There is no microseismic effect observed
when cold fronts pass the limit of the conti-
nental shelf outside the Norwegian coast,
whereas there is generally rapid increase of
the microseisms when the cold fronts pass the
coast itself.
Jardetzky. It is yet difficult to understand in
all details the mechanism of transmission of a
disturbance in the air to be a recorder of mi-
cioseisms. There are three media involved.
The wave propagation in the ocean bottom (or
coast) does not present any difficulty, but there
is no agreement about the kind of disturbance
at the sea surface and the behaviour of the sea.
The observations are interpreted in different
ways and the surf, the strong wind in the cold
front of a cyclone, the atmospheric pulsations,
the ocean swell or the interference of gravity
waves are made responsible for microseisms in
different theories. Neither of them seems to
be convincing when all observations are taken
into account, but each one might be true for a
corresponding group of microseisms. It is dif-
ficult to see whether a sufficient amount of ener-
gy is communicated to the sea by the air masses
directly in the form of compressional waves or
it is transmitted or increased by the action of
gravitational waves. There is no doubt more
that the movement of air masses in a cyclone
has to be taken as a primary disturbance. The
atmospheric pulsations can be one of charac-
teristics of this movement. On assuming the
existence of these pulsations (or some other
cause producing compressional waves) at the
interface air-water, one can determine the part
played by the latter. The signed has com-
puted from the theory of propagation of a
plane wave in the vertical direction a curve
mentioned by Dr. Press. This curve repre-
sents the amplitude of the vertical displace-
ment at the sea bottom in terms of § H , where
H is the depth, ot the velocity of sound in
water, u the circular frequency). The shape
of this curve suggests that the ocean acts as a
filter. For example, if a = 1.430 m/sec., H
= 1000 or 3000 m. periods of waves, which will
reach the bottom with amplitudes not chang-
ing essentially, will vary from 1.5 to 4 sec.
Making clear such an interpretation of the
behaviour of the water layer, this result does
not explain the conditions at the sea surface.
It seems that far more systematized data should
be correlated with each of the factors involved
in order to clarify those conditions.
(Longuet-Higgins brought up the subject
of gusts again. He made a plea for measure-
ment of their intensity at sea. Van Straten
pointed out that there is a strong land-sea
breeze on the East coast, and suggested it
might make a difference between the day and
night frontal microseisms.) Deacon agreed
with the remarks about the possibility of mi-
croseisms of all periods up to a certain maxi-
mum being caused by one source, although they
were ground oscillations of short and long pe-
riods produced by other causes such as traffic,
wind on mountains, buildings, etc. Sixteen
second ocean waves produced 8 second micro-
seisms at Kew, and it seemed very likely that
the 2 second microseisms observed by Father
Lynch might be caused by 4 second waves on
the Great Lakes. Dr. Longuet-Higgins had
used 1/2 second waves to reproduce 14 second
microseisms in the bottom of his tank.
With regard to the common explanation
of microseisms associated with weather dis-
turbances it was likely that when the explana-
tion was found it would be universally satis-
factory. The position at present was very
difficult to understand. On the eastern side of
the ocean the microseism records looked like
L
i

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ON THEORIES OP THE ORIGIN OF MICROSEISMS
125
the frequency spectrum of the waves if the
scale was divided by two. There was also a
theory which appeared to be very satisfactory.
On the western side of the ocean great empha-
sis was placed on the possible effect of micro-
barometric oscillation which looked nothing
like the microseism records; if they had any
period it was the wrong one, and the theory
used to explain the energy transfer required
confirmation at many points. The theory used
on the eastern side of the ocean was held to be
unsatisfactory.
It might be useful to concentrate more ef-
fort on the study of the phenomenon when it
appeared to be simplest. It would be very use-
ful if a seismologist from the United States of
America could come to England to work on
wave and microseism recordings made in the
United Kingdom; he would be sure of a warm
welcome and plenty of material to work on.

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