Phase delay of the solid earth tide

1970, Phys. Earth Planet. Interiors 2, 233—238. North-Holland Publishing Company, Amsterdam

PHASE DELAY OF THE SOLID EARTH TIDE*

STEWART W. SMITH and PIERRE JUNGELS Seismological Laboratory, California Institute of Technology, Pasadena, California 91109, U.S.A.

Received 10 November 1969

Tidal strain data from Isabella, California and ~afia, Peru have been analyzed to determine the phase shift of the M 2 solid tide and the Love number combination (h—31). The phase advance of the tidal bulge is 3.0 ± 10, and (h—31) = 0.475. Tidal strain meters appear to be more suitable for such measurements than do gravirneters. Indirect tidal effects due to ocean and atmosphere loading of the crust are minimized by performingthe tidal analysis

on the areal dilatation rather than on individual strain compo-

1. Introduction

produces 3.2 ±5.8°.Most of this uncertainty is attributed to indirect tidal effects. In the present paper, strain seismographs rather than gravimeters are used to determine the tidal phase shift. Their use appears preferable because they do not respond directly to the attraction of the moon as does a gravimeter, but only to the elastic distortion produced on the Earth. For this reason the expected phase shift is almost seven times larger on a tidal strainmeter and is thus easier to measure. A second reason is that the sum of two orthogonal strainmeter outputs is proportional to areal dilatation which is to first order unaffected by ocean and atmosphere loading of the crust.

The yielding of the Earth to tidal forces is not instantaneous, but lags slightly due to energy dissipation within the solid Earth, in the oceans and in the atmosphere. The early work in recognizing and interpreting this effect has been reviewed by MACDONALD (1964). The current view based on ocean tide dissipation estimates of MILLER (1964) and solid Earth dissipation estimates from free oscillations of SMITH and ANDERSON (1969) clearly calls for oceanic dissipation to be the controlling factor in determining the solid tidal phase shift. LAGUS and ANDERSON (1968) reached this conclusion after showing that one can, in fact, compare the dissipation function for tides and free oscillations when elastic and anelastic properties are functions of depth. Great difficulty has been encountered in experimentally measuring the phase delay of the solid Earth tide, and the wide variations in results have been discussed by MACDONALD (1964). For example, HA.uRISON et al. (1963) in a worldwide study of Earth tides using gravimeters show a mean value of the gravity tide phase shift of 0.8°,but the standard deviation of all their observations is 2.8°. Converting this to the actual phase of the tidal bulge as is discussed later, Contribution 1692, Division of Geological Sciences, California Institute of Technology, Pasadena, Calif. *

nents. The lunar retarding torque calculated from this result is (5.7 ± 1.7) x 10-23 dyne cm which is in agreement with the value obtained from discrepancies in the lunar orbit. The tidal effective Q determined here is 10, which is entirely controlled by losses in the ocean and atmosphere and has no bearing on the anelasticity of the solid Earth.

2. Instrumentation As is well known, tidal signals are more easily observed on gravimeters and tiltmeters than they are on strain seismographs. The basic difference being that all classes of pendulums, including both vertically sensing gravimeters and horizontally sensing tiltmeters respond directly to the gravitational attraction of the Moon as well as to the gravitational effects of the tidal distortion of the Earth, whereas strain measuring devices are sensitive only to the Earth’s tidal distortion. The relative size of these effects can easily be calculated from the consideration of Love’s numbers. Considering only the semi-diurnal tidal potential due to the Moon, and

233

234

STEWART W. SMITH AND PIERRE JUNGELS

using the results of MELCHIOR (1966) and (1964)wefind: Ag~= (1 + Ii

4k) 29X cos 2t, 1 cos 2t, Ag~ = (1 + k—I) 2g A = (h—31) 2X cos 2t, —

MAJOR

et al.

~mr

—

=

—~

2

.

gravirneter, tiltmeter, areal strain,

2

cos 2 Sill 0,

with =

in R 2

o

= = =

=

gravitational constant, mass of Moon, radius of Earth, declination, colatitude.

These expressions are derived on the assumption that the tidal bulge of the Earth is in phase with the gravitational attraction of the Moon. If we permit a small elastic shift e between the two, as would be expected if there is energy dissipation during a cycle of tidal yielding, we see that only that part of the observed signal which is due to the tidal yielding is shifted in phase. For example: Ag~= 2gX cos 2t + (h 4k) cos (2t 2e), A = 21(h —31) cos (2t 28). —

—

—

When the phase of such signals is calculated by means of Fourier analysis we find

Fig. 1.

=

arctan

Z

where h, k, / are Love numbers, t is local hour angle, g is the surface value of the Earth’s gravity field and X

arg (Ag)

(It 4k) sin 2e 1+(h—4k)cos26 — —_____________

—1 A ~ 2e 2 1 + (h 1 4k~ —

—

—

arg ( ) 8. For representative values of h = 0.6 and k = 0.3, we see, as was also demonstrated by SLICHTER (1960), that the experimentally determined phase shift from gravimeter data is reduced by a factor of 0.13. The expected phase shift for strain is thus approximately seven times larger than for a gravimeter, and five times larger than for a tiltmeter. Since e is already a small number and a difficult one to extract from tidal data, use of strain sensing equipment appears to offer the best hope for its measurement. Unfortunately, good sites for tidal strain measurements are difficult to find. Principal difficulties are due to secondary tidal effects such as crustal loading produced by atmospheric and oceanic tides, anisotropic tidal yielding due to inhomogeneities in the crust near the site, and long period noise or drift due to circulation of ground water and the thermoelastic strains that accompany it. There is occasionally a direct thermoelastic effect where strains produced by near surface temperature changes are transmitted elastically to great depth. For most sites, conduction of heat from the surface and the resulting thermal effects on the instruments present no problem. Although all of the factors pertinent to evaluating a strain site are not well understood, water stands out as a key problem. The success of a strain installation for both tidal arid longperiod work seems to be inversely related to the amount of water present in the surrounding rock. Of the strain

Tidal strain records from ~

—

l’eru.

235

PHASE DELAY OF THE SOLID EARTH TIDE ~ANA TIDAL STRAINS

20

Fig. 2.

H.C

= -

~

Fig. 3.

Aug.

25

I

Sep.

5

1963

10

15

Typical monthly sample of tidal strains and resulting areal strain for ~afia, Peru.

STRAIN ACCUMULATION 1ANA. p(~ II

AUG.

II~lII

JAN. 962

111111

JULY

liii

Long term variation of secular strain at ~aña,

JAN. 963

II

JULY

liii

II JAN 1964

I

I•I•l JULY

liii

Peru. Note that areal strain does not show large annual variation.

sites we currently operate in California, Peru and Hawaii, the location at ~afia, Peru is the driest. It also turns out to be the most stable for secular strain and the quietest for tidal and long-period noise. Examples of tidal strains observed at ~lafia are shown in fig. 1, where actual recordings are reproduced to illustrate the signal to noise ratio. In fig. 2 a replotted section of one month’s recording is displayed to illustrate the long term stability at this site. Also illustrated in fig. 2 is the function areal dilatation used in the analysis described below. Going a step further in considering the long term stability, secular strains observed over an interval of two years are shown in fig. 3. A strong almost seasonal effect is seen here, which is substantially reduced by forming the areal dilatation. The inference here is that much of the seasonal effect can be represented as either a surface load or a surface temperature perturbation as will be discussed in the next section.

3. Analysis The experimental problem is to isolate a single tidal constituent of the theoretically determined equilibrium tide and compare its phase with actual tidal deformation of the solid Earth. The tidal constituent most appropriate is the M 2 lunar semi-diurnal tide, because it is the largest and because its period of 12.42 hours is enough different from the harmonics of daily 24 hour effects to be easily separable with a month’s recording. The phase of the Earth’s tidal response can be determined directly by Fourier analysis of the data. The phase of the equilibrium tide can be determined by evaluating the analytic expression for the tidal potential, which will have the same phase as the gravity variation, the vertical deflection or the areal dilatation. If one needs to calculate the equilibrium tidal strain in a particular direction it is necessary to assume values for the Love numbers h and 1. An alternative procedure

236

STEWART W. SMITH AND PIERRE JUNGELS

is to generate theoretically the entire tidal function for the time interval of interest, and then subject it to the same numerical Fourier analysis that is used for the experimental data. One advantage of this process is that errors in the phase shift for a particular tidal line that are caused by the use of a finite length of data will be the same for both equilibrium and actual tides and thus tend to cancel. Another way of stating this is to note in the experimental data there is contamination of the M2 line due to some ofthe other tidal harmonics. The amount of the contamination, for a fixed record length, depends on the starting time, that is, where in the tidal cycle the sample begins. By generating the complete tidal function and then numerically analyzing it we produce a similar effect for the equilibrium tide, thus when the phase of the observed tidal strain and the equilibrium tide are subtracted we hope to minimize this effect. Another advantage in having the tidal function available as a time series, is that it can be compared directly with the experimental data in order to estimate the extent of the various indirect tidal effects such as daily temperature and pressure fluctuations and ocean loading effects. In a recent analysis of tidal strains, Kuo (1969) chose to work with the areal dilatation as calculated from a three-axis strain gage. For similar reasons we have also chosen the areal strain as the most appropriate for analysis. It is not strongly dependent on the Love number 1, the value of which is not well known, and more importantly, as shown by OZAWA (1957), it is unaffected to first order by ocean or atmospheric loading. This is a result of the fact that when an arbitrary vertical load is applied to the surface of a half space, surface strains measured at right angles are always equal and opposite. It can be easily shown that a surface temperature distribution produces thermoelastic strains which have the same property as those produced by a normal load. As a result, close-in loading effects due to the nearby ocean should be cancelled when one forms the areal dilatation. For loads applied at such great distances that the sphericity of the Earth cannot be neglected, the resulting strains are quite different than those for a halfspace. Using the results of LONGMAN (1963) we can see that significant areal dilatation can be produced by normal loads on a sphere at distances beyond about 5°.The mass of water displaced in the ocean produces normal loading, and in addition,

shear stresses are developed as a result of the flow of water at the ocean bottom. As a result, Earth strain gages respond to both direct and indirect tidal effects, and our technique of using areal dilatation can only eliminate the effects of close-in loading. One difficulty encountered in using the areal dilatation is stability of calibration of the strain gages. The phase of the areal dilatation is affected directly by errors in the relative calibration of the two orthogonal strain gages. In the case of vertical motion or gravity variation, although the observed amplitude of a tidal constituent is directly affected by the instrument calibration, errors in calibration will not affect the phase, at least to first order. Operating sensitive strain extensometers in remote locations makes for some difficulty in maintaining accurate calibration. As a check, for each month subj6cted to analysis we assumed the Love numbers h and I, and used the amplitude of the tidal spectrum to calculate the strain sensitivity of each component. By doing this we could eliminate those time intervals when the instrumental calibration appeared to have changed from its nominal value. In the course of doing this calculation we discovered a consistent discrepancy in the ratio of sensitivities of the two strain gages at the Rafla station. The only explanation for this appears to be that inhomogeneous structure at the site is causing anisotropic tidal response in the two tunnels. The simplest kind of such an inhomogeneity would be the existence ofjoints or cracks with a preferred orientation in the immediate vicinity of the tunnel. We cannot comment further on this problem. However, we have corrected the Safla data to make the relative sensitivities consistent with the expected Love numbers. It is interesting to note that careful measurement of the amplitude of long-period G waves at Nafla did not reveal this same anisotropic response, so it appears that the effect is frequency dependent, only becoming important for periods longer than several hundreds of seconds. As a result of the uncertainties present in this kind of data, the best way of estimating the expected errors in the phase shift is by internal consistence of the data for a number of intervals at different times and for several stations in widely differing localities. In our case we have nine monthly analyses from Safla, Peru and two from Isabella, California. The results are displayed in table 1 and in fig. 4. The mean value for

237

PHASE DELAY OF THE SOLID EARTH TIDE

independently from individual strain components. Be-

TABLE 1

Date

Station

Sept. 1963 Oct. 1963 Nov. 1963 May 1964 June 1964 July 1964 Aug. 1964 Sept. 1964 Oct. 1964 Oct. 27— Nov. 26 1966 June 14— July 14 1968

~aAa

Phase lag (2e)

cause of the anisotropic regime at Safla, values of

—

5.22 6.48

—

5.58

(h —31) are considered to be unreliable at this site. Values for (Ii 3/) from the Isabella data are listed in table 2, and their average value is 0.475. Using the

(h—31)

— — — — — —

—

4.07 4.40 1.40 3.96 4.32 4.68

accepted value of h 1 is 0.04.

TABLE 2

Isabella

Station 0.563

8.48

0.388

7.56

0.6, we can thus esttmate that

=

Isabella ~1aña

Location

Elevation (m)

35°39.8’NI l8°28.4’W li°59.8’S 76°51.OW

800 580

Direction N38°W, N52°E N30°W, N60°E

4. Interpretation

_________________________________________ I I I I I I .1 I PHASE DELAY OF AREAL STRAIN (2 el

4

-

—

3

-

—

The results of the previous section are compared with other experimentally determined tidal phase shifts in table 3. In this table, the observed phase delay of the

0

2

-

—

gravitational force is converted to the spatial phase ad-

Z

I

-

c

0

:~

NA

TABLE 3

—

•I

•2

I

I

4

I

I

6

‘~-lSA8ELL4”. a

Locality Worldwide

Degrees

Fig. 4.

Distribution of phase delay (2e) from areal strain at ~aña, Peru and Isabella, California.

Stations Reference 12 HARRISON eta!. (1963)

Worldwide

would be data from other strain seismographs located in regions where the indirect tidal effects are less important, such as the interiors of large continents, The spectral analysis of areal strain reported here yields an estimate of the quantity (h 3/). Since indirect tidal effects are clearly important at both the Isabella and Safia localities, we did not attempt to determine 1 —

MELCHIOR

—1.3 ± 1.3 4.3 ± 4.3

(1964) 4

PARu5KII

—0.7 ±0.3 2.2 ± 1.0

7

(1963, 1967) PERTSEV (1966)

—0.1 + 0.7 0.3 ±2.2

United States

I

Kuo (1969)

—

1.2*

North and South America

2

this paper

—

3.0 ± 1.0

Asia

Saña is 4.5°and that for Isabella is 8.0°.The standard deviation for the nine measurements at Safia is 1.3°, which is a measure of the experimental error at a single site. The difference between Isabella and Safla mean values is a measure of the error introduced by indirect tidal effects not completely eliminated in the analysis. We thus estimate the true phase shift 2e to be the average of the Isabella and Rafla values, that is to one significant figure 6 ±2°.The internal consistency of the data indicate that further analysis at these two stations would improve the accuracy ofthe estimate of the true phase shift. What would be needed to do this

27

Aço, e (bulge) —1.0 ± 1.8 3.2 ± 5.8

Asia

Northeastern

__________________________________________________ *

Single measurement only

vance of the tidal bulge using the factor 0.52ö/(o I) = 3.2 (SLICHTER, 1960), where ö, the gravitational factor, is assumed to be 1.19. In the case of the tidal strains presented here, the factor is just 0.52, which is the ratio of the period of the M2 tide to the period of rotation —

of the Earth. Uncertainties listed are the standard deviation times 0.67. The table is not intented to be an exhaustive compilation, but only to represent some typical values of tidal phase shift and illustrate the uncertainties that exist. A good deal of the phase data published in Earth tide studies is of little use since it is not clear if instrumental phase characteristics have

238

STEWART W. SMITH AND PIERRE JUNGELS

been included. With the exception of some of the values in MELCHIOR (1964), all of the data listed in this table have been corrected for instrument phase shift. Only in the results of PERTSEV (1966) arid the present paper was made an attempt to correct for indirect tidal effects. An average of these results weighted with the probable error listed for each determination is 2.4°.This illustrates that to one significant figure, the spatial phase advance of the tidal bulge for the M2 component is 3° with an uncertainty of less than 1°.Following MACDONALD (1964) we find that the tidal phase shift of 3° determined here implies a lunar retarding torque of 5.7 x 1023 dyne cm. This is to be compared with the result of MuNK and MACDONALD. (1960) of 3.9 < 1023 dyne cm derived from discrepancies in the lunar orbit over the last 270 years. In deriving their result, Munk and MacDonald assumed a value for the torque exerted by the solar atmospheric tides. Since tidal friction losses may be non-linear, it is not possible to separate lunar and solar effects. They note that different models for frictional losses in the oceans and atmosphere can give rise to a 20 % difference in the computed lunar discrepancy thus it is reasonable to assume that their resulting tidal torque is uncertain by at least this amount. The previous discussion of the uncertainty in the phase shift of tidal strains indicates that this value may be uncertain by as much as 30%. Thus the values from astronomical data and from tidal strain data appear to be in agreement. As a result, no significant changes are suggested for the lunar orbit calculations such as are reviewed by RUSKOL (1966). The phase shift reported here can be converted into a tidal effective Q for the Earth-ocean-atmosphere systern according to the relation Q~’= tan 2e (MACDONALD, 1964). The result is Q = 10. LAGUS and ANDERSON (1968) reported a Q = 70 by correcting the results of PERTSEV (1966) for instrument phase shift; however, they misinterpreted his phase shift as 2s, the actual phase advance of the tidal bulge, wher~the value he reported was actually A p5 the phase lag of the force of gravity (PARIISKII, 1963). Their Q should thus be divided by 3.2 as described earlier. LAGUS and ANDERSON (1968) showed that strain energy in the Earth for a semi-diurnal tide and for the gravest mode of free oscillation 0S2 is distributed in a similar manner. Thus

the reported

Q

fo 380 for 0S2 should be appropriate for solid Earth tides, unless Q becomes very frequency dependent between 1 cycle per hour and 2 cycles per day. The values reported for tidal effective Q, including the one in the present paper, are so low, however, that they most certainly are controlled by ocean and atmophere losses. Acknowledgments Both of the tidal strain observatories used in this study were established under the direction of the late Professor Hugo Benioff. The operation of the station at Rafla has been in cooperation with the Instituto Geofisico del Peru, and it is a pleasure to acknowledge the continuing support of its director Dr. Giesecke in this program. Support for the analyses described in this paper was provided by NASA Grant NGL 05-002-069 and AFOSR Grant AF62-42l. References GOLDREICH, andN.S.NESS, SomRI. (1966), Icarus 375. HARRISON, J.P.C., LONGMAN, R.5,FORBES, E. KRAUT and L. SLICHTER (1963), J. Geophys. Res. 68, 1497.

Kuo, J. T. (1969), J. Geophys. Res. 74, 1635—1644. LAGUS, P. L. and D. L. ANDERSON (1968), Phys. Earth Planet.

Interiors 1, 505.

LONGMAN, I. (1963), 3. Geophys. Res. 68, 485—496. MACDONALD, 0. J. F. (1964), Rev. Geophys. 2, 467. MAJOR, M. W., G. H. SUTTON, J. OLIVER and R. METZGER (1964),

Bull. Seismol. Soc. Am. 54, 295.

MELCHIOR, P. (1966), The Earth tides (Pergamon Press). MELCHIOR. P. (1964), Res. Geophys. 2, 163. MELCHIOR, P. (1964), Contribution apportée par les marbes

terrestres dans l’étude de larotation de la terre. In: Markowitz and Gumod, eds., Continental drift, p. 71.

MILLER, G. R. (1966), J. Geophys. Res. 71, 2485. MUNK, H. (Cambridge and 0. J. F. MACDONALD (1960), The rotation of the W. Earth University Press). OZAWA, I. (1957), Study on elastic strain of the ground in

Earth tides, Disaster Prevent. Res. Inst. Kyoto Univ. Bull. 15, 36.

PAIUISKII, N. N. (1963), Izv. Akad. Nauk SSSR Ser. Geofiz. 193. PARIISKu, N. N., S. BARSENKOV, V. VOLKOV, D. GRIDNEV and

M.

KRAMER (1967), Izv.

Akad. Nauk SSSR Ser. Fiz. Zemli 2,60. Geofiz. 10, 25.

PERTSEV, B. P. (1966), Izv. Akad. Nauk SSSR Ser. RUSKOL, E. L. (1966), Icarus 5, 221.

SLICHTER, L. B. (1960), Mar6es terrestres, Bull. Inform. Sci.

Tech. Paris, 21. S. W. and D. L. ANDERSON (1967), Attenuation of the Earth’s free oscillations (abstract), Intern. Union Geol.

SMITH,

Geophys.