Symposium on Microseisms: Held at Arden House, Harriman, N.Y. 4-6 September 1952, Sponsored by the Office of Naval Research, and the Geophysical Research Directorate of the U.S. Air Force. (1953)

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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

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

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 <t>, 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)

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 <iv(— sin Y-t]> 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 <t>\ 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

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

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

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 /

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.

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

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

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

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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