The Effect of Ocean Tide Loading on Tides of the Solid Earth Observed with the Superconducting Gravimeter

Transcription

1 Geophys. J. R. astr. (1975) 43, The Effect of Ocean Tide Loading on Tides of the Solid Earth Observed with the Superconducting Gravimeter Richard J. Warburton, Christopher Beaumont* and John M. Goodkind (Received 1975 May 30)t Summary The superconducting gravimeter has been calibrated to an accuracy of 0-2 per celit and was used to measure the influence of the ocean tides on the tides of the solid earth at two locations in southern California. These measurements, which show a signal to noise ratio of 70db in the 1 and 2 cycle per day band, and the accurate calibration of the gravimeter make possible a quantitative test of ocean load calculations for the 0, and M, constituents and the ocean models upon which they are based. We have computed the ocean load effect by using Farrell's Green's functions and modifications of previously published ocean co-tidal charts. The agreement between observed and computed earth tides is within 0.2 per cent and corroborates the placement of an M, amphidrome at 1500 km southwest of San Diego. If the amplitudes of the computed tides are correct it also sets an upper limit of 0.1" on the phase shifts of the solid earth tides. We have also observed the loading effect on the coast due to the nonlinear MI ocean tide. 1. Introduction Measurement of the tides of the solid earth with gravimeters are contaminated by the effects of ocean tides. Thus, the extraction of the geophysical information about the structure of the Earth contained in records of Earth tides, if it is not limited by instrument noise, will be limited by the accuracy to which ocean effects can be removed (Melchior 1966; Kuo el al. 1970; Jackson & Slichter 1974; Robinson 1974). In order to improve this accuracy over what has been available previously, three requirements must be met. First, an accurate theoretical model must be available for computing world-wide tides. Second, sufficiently low noise measurements must be made. Third, an accurate, absolute calibration of the instrument must be made since the ocean loading is at most 10 per cent of the full tidal signal. In this work we have calibrated an improved version of the superconducting gravimeter to & 0.2 per cent and have measured the tidal amplitudes at two locations in southern California, La Jolla and Pifion Flat. We have also computed the effect of ocean loadiug for the 0, and M, tidal constituents at these locations using Farrell's (1972a) Green's functions and global models for the ocean tides (Hendershott & * Present address: Gravity Division, Earth Physics Branch, Department of Energy, Mines and Resources, Ottawa, Canada K1A OE4. t Received in original form 1975 April

2 708 R. J. Warburton, C. Beaumont and J. M. Goodkind Munk 1970; Hendershott 1973) modified to conform with the results of Munk, Snodgrass & Wimbush (1970). Agreement between theoretical tides, using the computed ocean loads, and the observed tides is better than 0.2 per cent in all cases. Furthermore, we have compared the observed and theoretical differential 0, and M, load tides, which are determined only by the local tidal distribution. 2. Instrument The basic features of the superconducting gravimeter have been reported earlier (Prothero & Goodkind 1968, 1972). The instrument noise reported in that work resulted from operating the temperature and pressure sensitive magnetic field coils, which are used to levitate the superconducting ball, in direct contact with the liquid helium bath. Although the effect of variations in the bath temperature and pressure partially cancelled each other, the temperature dependence of the magnetic field was still AH/E-IAT z 10-6/K. The helium bath at best could be controlled to a few mk so that an instrument noise of Ag/g z was present. Also, transfers of liquid helium produced much larger variations of temperature and pressure which in turn caused large disturbances, and usually resulted in offsets in the magnetic field and gravimeter signal. In the present instrument the entire system, including the superconducting ball, capacitor plates, superconducting magnet coils, and superconducting shield are suspended in a vacuum and its temperature is regulated to a few pk. In this case the temperature coefficient is AHIHAT E lo-'/k so that the rms instrument noise is reduced to Ag/g lo-". The instrument is now insensitive to bath temperature and pressure and no offsets occur during transfers. Fig. 1 shows the power spectrum of the gravimeter signal for 100 days (1972 October 3 to 1973 January 12) at La Jolla. The spectrum has a signal to noise ratio of over 70 db at the largest tidal constituents, a ratio which is almost 20 db larger than that reported for the earlier instrument (Prothero & Goodkind 1972). In fact, in the present instrument, signal to noise is limited by variations in gravity resulting from fluctuation in atmospheric pressure and in the nearby ocean level. Strong correlations between the gravity signal and the barometric pressure exist between 0.1 and 0.7 cycles/day and at 4, 5,6 and 7 cycles/day. These effects will be analysed in a later paper. A strong correlation with the local ocean is present at 3.86 cpd (M4) where the gravimeter signal is about 10db above the predicted tide of the solid earth. This large M, signal is known to be a consequence of the non-linear response of the nearby shallow coastal waters and will be discussed in more detail below. Instrumental noise due to temperature fluctuations is db below barometric and local ocean signals and can be further reduced by direct measurement and subtraction from the observed signal. 3. Calibration The gravitational force from a 14-in diameter, mercury filled, hollow steel sphere weighing 714k 0.1 Ib was used to calibrate the instrument. The mercury sphere was periodically (every 15 min) rolled under the gravimeter into t in. dimples drilled at accurately determined locations in an aluminium base plate located under the instrument. The dimples were arranged in the pattern of a cross with 11 holes in two perpendicular directions with 2.0 k in. spacing between their centres. The calibration superimposed a square wave with a maximum amplitude of lopgal on the tide signal. In order to measure the square pulses accurately it was necessary to remove the tides. Ocean loading, geophysical noise, and manmade noise

3 Effect of ocean tide loading on tides of the solid earth 709 d G E

4 710 R. J. Warburton, C. Beaumont and J. M. Goodkind Table 1 Values for the parameters of equation (1) used for the calibration Required Fitting R, Parameters accuracy xo, Yo, c M 0.2 per cent *1 lb 0.15 per cent XO cm cm yo * 0.6 cm cm R cm A0.15 cm C -t0*2 per cent Fitting xo, Yo, c *1 lb 0.15 per cent cm cm cm O.OOO1 set a limit on the precision of this subtraction and, thus, on the calibration. A substantial improvement could be obtained by using a second instrument to measure the actual tide signal which would then be subtracted from the signal of the instrument being calibrated. The parameters determining the calibration constant C are the mass M of the mercury sphere, the position (xo, yo) directly beneath the gravimeter, and the vertical distance R between the centres of mass of the mercury sphere and the levitated gravimeter ball. These parameters were estimated from a least squares fit of 300 measurements to the equation: where V is the height of the voltage step obtained from moving mass M to position (x, y) and G is the gravitational constant. The least-squares estimates are shown in Table 1, column 2. Column 1 shows the accuracy required for a 0.2 per cent calibration. The errors represent a 95 per cent confidence level. It was found that the confidence of the calibration could be improved by measuring R directly and using the measured value in the above equation. The measured value of R is well within the uncertainty of the least squares fit value. Column 3 of Table 1 shows the least squares fit value of C using the measured value of R. It is accurate to 0-2 per cent. We also measured the response of the instrument to tilting. For small angular changes 0 the voltage change AV is given by where go is local gravity and CT is the ' tilt ' calibration coefficient. CT volts/ pgal, a value approximately 8 per cent larger than the calibration coefficient found using the direct gravitational attraction. Thus, tilt provides an inaccurate method of calibration for this instrument. A tiltmeter mounted on the gravimeter pier was used to measure tilting caused by rolling the mercury sphere into position. The maximum tilt signal produced by the mercury sphere resting on the edge of the concrete pier was 2 x radians. This corresponds to a change in gravity of only 2 x pgal, a change that can be neglected when compared to the 10 pgal signal from the mercury sphere itself. The instrument was calibrated at La Jolla. Since we could not repeat the calibration at Piiion Flat, we have attempted to use a consistency check to test its stability. We measured the capacitive feedback force necessary to recentre the superconducting ball when a magnetic force of about tidal amplitude is applied to it. The magnetic force is generated by changing the current in the lower magnet coil by a known amount. The ratio of these two forces will remain constant if the instrument

5 Effect of oceao tide loading on tides of the solid earth 71 1 calibration, the capacitor plate and ball geometry, and the magnetic flux configuration remain constant when the instrument is moved. After moving the gravimeter to Piiion Flat this ratio increased by 0.46 per cent. It is unlikely that the calibration constant has in fact changed since the instrument was kept at 4 K while being moved. However, in moving the instrument it was necessary to reduce the magnet current to zero and then to relevitate the ball. We assume that there has been a small change in the magnetic flux configuration between Piiion Flat and La Jolla and that the magnetic force applied by changing the current in the lower magnet coil has increased correspondingly. If this assumption is correct, the calibration constant would be unchanged. Therefore, although we can only check the stability of the calibration to 0.46 per cent, we assume that it is significantly better than this. 4. Theoretical ocean load tide We have calculated the theoretical gravitational load tides of two major tidal constituents, M, (principal lunar semi-diurnal) and 0, (principal lunar diurnal) for comparison with the observed tidal gravity signal at La Jolla and Piiion Flat. These constituents were chosen because they have the best known global distributions, they are well removed in frequency from the effects of radiational tides, and there are no predicted anomalies in the Earth tide admittance due to core resonance at these frequencies. The load tides were calculated by convolving Farrell's Gutenburg-Bullen A gravity load Green's functions (Farrell 1972a) with theoretical models of the M2 and 0, global ocean tide distribution taken from Hendershott & Munk (1970) and Tiron, Sergeev & Mirchurin (1967), respectively. These global tide models are numerical solutions of the Laplace Tidal Equations (LTE) using coastal observations as boundary conditions. The models are in general agreement with the few existing deep ocean pressure observations (see, for example, Munk et al and Zetler et al. 1975) and island observations. However, the predicted tides in some areas are at variance with the known ocean tide distributions, mainly because significant features in the ocean bottom topography cannot be described in sufficient detail on the coarse finite difference grids (4" to 6") used in the calculations. Smaller errors also arise because the theoretical results represent incomplete solutions of the LTE, and because the gravitational self-attraction of the tidal water mass has been ignored. More recent attempts (Farrell 1973; Hendershott 1972) to include these factors in the calculations are inconclusive due to a numerical error. Fortunately, the ocean load tide in southern California is dominated by the tide in the north-east Pacific Ocean. In this area regional models for the M, and 0, tides based on coastal and ocean bottom pressure gauge observations have been developed by Munk et al. (1970). Their results were used to modify the M, and 0, global tide models in the north Pacific Ocean. The modifications to the 0, model are minor because the Tiron er al. chart is in good (perhaps fortuitous) agreement with the observations, Fig. 2(a). The modification of Hendershott's M, co-tidal chart was, however, more drastic and involved moving his amphidrome, located north-west of Hawaii, to the location proposed by Munk et al km south-west of San Diego, Fig. 2@). The convolution integrals were evaluated by dividing the co-tidal charts into approximately equi-phase, equi-amplitude triangles, convolving the Green's functions with the tide in each triangle and summing the results. This method has been described in more detail by Bower (1971), and Beaumont & Lambert (1972). The resulting load tide vectors shown in Table 2 are divided into two parts, the north-east Pacific, for which the Munk et al. models are used, and the ' Rest of the World's Oceans' based on the global tidal models. The latter vector is the sum of

6 712 R. J. Warburton, C. Beaumont and J. M. Goodkind a FIG. 2. (a) Modified co-tidal chart for O1. (b) Modified co-tidal chart for Mt. approximately equi-amplitude irregularly oriented vectors from each of the oceans, no one ocean dominating. Its accuracy is dependent on the accuracy of the low order coefficients in a spherical harmonic expansion of the global tide models. Calculation of the global load vector is further complicated because the Tiron et al. and Hendershott models do not conserve tidal mass. Conservation of mass is unimportant in these solutions of the LTE where a semi-permeable coastline has been used to simulate frictional energy dissipation on the continental shelves. However, for the gravity convolution, tidal mass must be conserved (Farrell1972b). Because the distribution of non-conserved tidal mass is not known, an artifice such as distributing this mass equally over all the oceans must be used. This mass is small (7.01 g cm-' " for Mz and 0.47 g ~m-~-28.30" for 0,) but not negligible. In order to estimate errors incurred by this enforced mass conservation, results are presented for three different methods. In the first method the non-conserved mass is distributed uniformly over all oceans. In the second method each of the major oceans is forced to conserve its own tidal mass. The two methods subtract significantly different tidal masses from some oceans, in particular the Indian Ocean, but the total load vectors are similar. These vectors should be compared with the results of the third method where no attempt is made to conserve mass. We have no way of determining an accurate method of mass conservation but we assume that the errors incurred by this lack of knowledge are no larger than the difference in the vectors computed by methods 1 and Results The observational results presented here were calculated from three sets of gravimeter data. Runs 1 and 2 were taken at La Jolla from 1972 July 6 to August 29, and from 1972 October 3 to 1973 January 12, respectively. Run 3 was taken at Piiion Flat for 365 days starting on 1973 September 8. The gravimeter signal was filtered using a low-pass Butterworth filter, and then recorded at l-min intervals on magnetic tape. A measured phase shift of 0~464"~0~0l"/cpd and an amplitude attenuation of less than 0.02 per cent were induced by the Butterworth filter. Disturbances due to earthquakes, power failures and helium transfers were interpolated and

7 a f P (D 0 p P a 9 U p Table 2 Computed ocean load vectors. I, gives results of conserving total mass of all oceans; 2, conserving mass of oceam individually; and 3, without conserving mass N.E. Pacific 1 3 Other oceans Effect of height above ocean Total load vectors Total tide includ- 1 ingoceanload 2 Total gravimetric 1 factor including 2 ocean La Jolla Piiion Flat OX M2 01 M2 Amplitude Local epoch Amplitude Local epoch Amplitude Local epoch Amplitude Local epoch (Pl?al) (degrees) f * i la I- 1 *35

8 714 R. J. Warburton, C. Beaumont and J. M. Goodkind Table 3 Gravimetric factors and local epoch for O1 and M, measured at La Jolla, California (32.87" N, " W) Gravimetric factor 6 Measured load vector Local epoch Amplitude Local epoch Run Tide 181 (degrees) I AS1 ball* (degrees) (days) Run Run Run Run2 54 M Run 1 M Run Run Run2 54 *The amplitudes of the local theoretical rigid-earth tide used to scale 1AS1 are sin (2 X 32.87") = pgal for O1 and cosz (32.87") == pgal for M2 at La Jolla. Table 4 Gravimetric factors and local epocli for O1 and M, measured at Pifion Flat, California (33.59" N, " W) Gravimetric factor 6 Measured load vector Local epoch Amplitude Local epoch Run 3 Tide 181 (degrees) I A61 (psal)* (degrees) (days) : : : M : : : * The amplitude of the local theoretical rigid-earth tide used to scale 1AS1 are sin (2 X 33.59") = pgal for O1 and cos' (33-59") = pgal for M2 at Piiion Flat.

9 Effect of ocean tide loading on tides of the solid earth 71 5 the one minute data digitally filtered and decimated to hourly samples. The interpolated intervals contribute less than 3 per cent of the total data record. The complete data records were cosine tapered and Fourier analysed to determine the observed amplitudes and phases of the 0, and M, tidal constituents. An identical analysis was performed on the theoretical rigid-earth tide which was generated for the same time period as the observed data. The results of the Fourier analysis are shown in Tables 3 and 4 and in Fig. 3. The amplitude of gravimetric factor 161 is the ratio of the observed tidal amplitude to the theoretical amplitude and the local epoch is the phase difference between the observed and theoretical tides. The normalized amplitude \AS\ and the local epoch of the ' measured load vector ' were calculated as the vector difference between the observed gravimetric factor 6 and an assumed theoretical gravimetric factor of with zero phase lag (at the local equilibrium potential). The amplitude of the ' measured load vector ' is also given in pgal and is calculated by scaling ]AS1 using the local amplitude of the theoretical gravity tide of the rigid-earth. The stability of the ' measured load vectors ' was estimated by dividing the Piiion Flat record into six sections each having a duration of 54 days 15 hr, and also into three sections of 109 days, 7 hr duration. Run 2 at La Jolla was similarly divided into two 54 day, 15 hr sections. These sections were cosine tapered and analysed in the same way as the complete data sets. The maximum deviation of the gravimetric factor in the 54 day segments at Piiion Flat is 0.65 per cent for 0, and 0.27 per cent for M,. This is a larger deviation than would be expected from the background noise level between tidal peaks. The largest peaks on the background noise spectrum have an amplitude of 0.1 per cent of 0,. Interference from neighbouring tidal peaks cannot account for the variation since to a first order approximation this contamination is eliminated by normalizing the tidal estimates by the theoretical tide estimates for an identical time interval. Slow variations in the calibration constant of the instrument also fail to explain the phenomenon since the variations in 6 for O1 and M, (and other constituents) are only partly correlated. We therefore conclude that there are real variations of the tidal amplitudes at this level which require further study. The observational results can be compared with the theoretical load vectors and gravimeter factors computed in Section 4 in a number of ways. The sum of the ocean load vector and the computed earth tide, described above, should equal the measured Earth tide if the ocean load model is correct, and no other effects contribute to the loading. Using the results of method 1 (conserving total mass of all oceans) the observed 0, tide is larger than this theoretical tide by 0.15 per cent at Piiion Flat and smaller by 0.10 per cent at La Jolla. Using method 2 (mass of oceans conserved individually) for M, the observed tide is smaller by 0.20 per cent at Piiion Flat and by 0.05 per cent at La Jolla. These differences are within our estimated calibration error. The differences between 0, and M,, however, cannot be a consequence of calibration error. The calculation by method 2 appears better than by method 1 since the latter produces discrepancies of 0.49 per cent at Piiion Flat and 0.33 per cent at La Jolla for M,. 0, is little affected by oceans other than the north Pacific and has therefore only been calculated using methods 1 and 3. See Table 2. A more revealing comparison of the small differences between observation and theory is made by comparing only the ocean load part of the tide, Fig. 3. In this figure only the small load tide part of the total vectors (the boxed region in Fig. 3(a)) is illustrated in greater detail in Fig. 3(b) and (c). The origin in Fig. 3(b) and (c) is therefore the end point of a vector representing the theoretical tide of the solid earth with gravimeter factor and zero phase lag. The arrows represent the ocean load vectors calculated by method 1 for 0, and method 2 for M,. In general the agreement between observation and theory is excellent. The calculated phases of the 0, load vectors are within the scatter of the data with a maximum deviation of 5" for the 54 day sections and 1" for the complete runs 1,2 and 3.

10 716 R. J. Warburton, C. Beaumont and J. M. Goodkind HC wo NEP NEP k b LA JOLLA M2 HC 1 pgal wo I 3c PlfiON FLAT FIG. 3. Polar plots of ocean loading part of measured and theoretical O1 and M2 earth tide at La Jolla and Piiion Flat. The origin (0) is the end point of a vector representing the local theoretical tide of the solid earth with gravimetric factor and zero phase lag. The load tide amplitude is represented by the distance from the origin to the observations (0, A, +, 0 ) or to the tips of the theoretical load vectors. The phase lag is measured clockwise. At La Jolla 0 is run 1, A is run 2, and The calculated amplitudes of the 0, load vectors fall outside the scatter of the data points and appear to be approximately 5 per cent too small. For the M, load vectors at both locations the maximum deviation between the calculated and observed phase is 8" for the 54 day sections and 3" for the full length runs. At Piiion Flat both the M, calculated phase and amplitude are within the data scatter. However, at La Jolla the phase is outside the scatter (but only by 3") and the amplitude is 5-10 per cent too small. This result suggests that the ocean tide model may be inadequate for M,, at least in part, because of some variability in its distribution in the neighbourhood of La Jolla. It should be noted that an error in the calibration constant would displace data points in a nearly horizontal direction, while an error in the phase correction, or a NEP

11 Effect of wean tide loading on tides of the solid earth 717 phase shift in the theoretical solid earth tide would move the points vertically. It is, therefore, clear that neither a calibration error nor a phase correction will significantly improve the agreement between both the 0, and M, observations and calculations at both locations. We can therefore conclude that the remaining small differences between observation and theory can be attributed to the inexact description of the ocean provided by the ocean model used in the load calculation. If the model as presented is correct then the phase shift remaining on the tides after correction for ocean loading is at most 0.1". Since the model is certainly not perfectly accurate, this can be interpreted as the limit to which the phase of the tides of the solid earth can be determined by this work. In principle this sets a lower limit on the Q of the Earth. In order to estimate that limit we assume that the semi-diurnal tide (P,' spherical harmonic) is driving a damped simple harmonic oscillator at the frequency of the 54-min period normal mode. In this case the lower limit is 45, substantially lower than typical Q's measured directly by the decay of normal modes. - I Phase Lag FIG. 4. Polar plot illustrating the differential gravity vectors between Pifion Flat and La Jolla. The amplitude difference is represented by the distance from the origin (0) to the observations (0, x), or the theoretical points (@, m). The differential phase lag is measured clockwise. For the theoretical represents the differential load when the ocean tide models do not conserve mass, and represents the equivalent result when the mass of each Ocean is conserved separately. The x 's and 's represent the observational results involving either run 1, or run 2 from La Jolla. X and T are the results from the complete ' runs ' and x and, shown in the enlarged insets, are the individual ' 54 day ' differences illustrating the scatter in the observational results.

12 718 R. J. Warburton, C. Beaumont and J. M. Goodkind Therefore, it is unlikely that the phase shift is actually a measure of dissipation in the solid earth. A further discriminant is afforded by a comparison of the observed and theoretical differential gravity tides between La Jolla and Piiion Flat. The differential tide is simply the vector difference between either the theoretical, or observed, tide from the two locations. It has the principal advantage of being sensitive only to tidal loading from nearby oceans, provided that the station separation is not large. The differential tide has this property because the gravitational load Green's function is a slowly varying function of distance at large distances (Farrell 1972a). Accurate observations are required for the calculation to be meaningful because the observational error is now equal to the sum of the individual observational errors. Fig. 4 is a polar plot illustrating the differential tide between Piiion Flat and La Jolla. The results for the O1 constituent are in total agreement if it is assumed that the La Jolla run 2 results are correct. The M, theoretical results are not in total agreement with the observational results even if run 1 at La Jolla is assumed to be the correct result. It is more appropriate perhaps to use run 2 from La Jolla in this comparison because during this interval the gravimeter calibration was performed. Using this assumption the differential load tides for the 0, and M, constituents are respectively 0 and 28 per cent too small. It is unlikely that the M, tidal model in the vicinity of La Jolla is sufficiently in error to account totally for the discrepancy. A full explanation must await further observations from La Jolla. The differential tide calculation also reinforces evidence that the Earth tide per se is totally insensitive to crustal structure (Beaumont & Berger 1974). Seismological results (Thatcher 1972) suggest that the Mohorovicic discontinuity beneath Piiion Flat is at least 10 km deeper than it is near La Jolla. If the Earth tide were sensitive to this difference, the observed vector would be rotated with respect to the theoretical vector. The vectors in Fig. 4 demonstrate that differences greater than 0.003, or 0-26 per cent, in the relative gravimetric factors are inadmissible. 6. M4 tide In addition to the large scale loading effects described above we have observed loading due to the non-linear local M, tide. In the spectrum of the gravimeter signal (Fig. l), taken at La Jolla, there is a peak at 3.68 cpd which is db above the background noise and is 10db larger than the expected M, earth tide. A similar anomalously large peak appears in the spectrum of the ocean level measured at Scripps Pier in La Jolla cove (1.6 km away) and is presumably a consequence of the non-linear response of the shallow coastal waters to the M, tide. The measured ocean level and gravimeter signal are in phase, the correct phase relationship if the signal results either from direct gravitational attraction or from a loading effect. The absence of the M, peak in the Piiion Flat data is consistent with the non-linear tide being confined to a relatively narrow coastal strip. Since these measurements set some constraints on models of the non-linear coastal tides, we have attempted to determine the admittance of the gravimeter to the ocean level. Three different methods have been used. (1) The ratios of the peaks on the two spectra yield 3-58 and 3-22 x lo-, pgal cm-' for runs 1 and 2 at La Jolla. (2) The admittance as a function of frequency was computed from cross spectra averaged over 14 blocks of data each of 7.5 days length. There is a peak in the admittance at 3-86 cpd with a value of 2-85 x pgal cm-'. The coherence at the peak is 0.9 and is approximately 0.3 in adjacent bands. (3) The records were band passed between 3 and 5cpd and then least squares fit to each other yielding admittances of 1.1 and 2.0~ lo-, pgalcm-' for the runs 1 and 2 at La Jolla. The difference in the method 2 and 3 results may be a consequence of the noise on the ocean level at frequencies adjacent to the M, frequency. Jf this noise is

13 Effect of ocean tide loading on tides of the solid earth 719 associated with variations which are more localized than the enhanced M, tide phenomenon it would have a correspondingly smaller effect on gravity. Since the least squares fit yields an average admittance for the entire pass band it would then be smaller than that appropriate to the M, peak. For this reason we assume that method 2 yields the best estimate of the admittance. The gravimeter at La Jolla was located 123m above the mean water level and 1.6 km inland. A uniform rectangular strip of water along a linear shore line at these distances would produce a change in gravity of 1.0 x lom2 pgal cm-' of water. Thus about 1-9 pgal cm-i must be accounted for by the loading effect. This is a reasonable value for the loading as can be demonstrated by a simple model. Using the Boussinesq solution (Farrell 1972a), a strip of water lo3 km long by 50 km wide would result in a gravity change of 1-3 x lo-' pgal cm-'. 7. Conclusions By calibrating the superconducting gravimeter to an accuracy of 0-2 per cent we have been able to provide a quantitative test of computed ocean loading effects on the earth tides in southern California. The consistency of the predictions and observations is evidence for the overall validity of the ocean model, the loading Green's function and the assumed gravimetric factor of for the solid earth. On this basis the results show that present ocean models can predict load vectors for 0, and M, to within 5 and 10 per cent, respectively, for southern California. There does, however, remain an apparent signal of up to 0.6 pgal in amplitude in the M, differential load tide that is not accounted for by the present model. Further observations from La Jolla are required to confirm its existence. Extension of this work to other tidal constituents will determine the frequency dependence of the gravimetric factor of the solid earth, with ocean effects removed. The one year record from Piiion Flat is being studied in this manner for evidence of a resonance in the core and other causes of frequency and time dependent gravimetric factors. Acknowledgments This work was supported by the National Science Foundation, and by the National Aeronautics and Space Administration. One author, Christopher Beaumont, was supported by a Green Fellowship from the La Jolla Foundation, during the course of this work. The authors wish to thank Dr J. Berger for support in operating the superconducting gravimeter at Piiion Flat, Richard Reineman for technical and operational assistance, and Dr W. E. Farrell for critically reading this manuscript. Richard J. Warburton and John M. Goodkind: Department of Physics, University of California, Sun Diego, La Jolla, California Christopher Beaumont: IGPP, University of California, Sun Diego, La Jolla, California

14 720 R. J. Warburton, C. Beaumont and J. M. Goodkind References Beaumont, C. & Lambert, A., Crustal structure from surface load tilts, using a finite element model, Geophys. J. R. astr. SOC., 24, Beaumont, C. & Berger, J., Earthquake prediction: modification of the Earth tide tilt and strains by dilatancy, Geophys. J. R. astr. SOC., 39, Bower, D. R., Measurement of the Earth tide and regional heterogeneity due to the ocean tide, PhD Thesis, University of Durham, England. Farrell, W. E., 1972a. Deformation of the Earth by surface loads, Rev. Geophys. Space Physics, 10, Farrell, W. E., 1972b. Global calculations of tidal loading, Nature, 43, 238. Farrell, W. E., Earth tides, ocean tides, and tidal loading, Phil. Trans. R. SOC. Lond. A., 274, Hendershott, M. C., The effect of solid earth deformation on global ocean tides, Geophys. J. R. astr. SOC., 24, Hendershott, M. C., Ocean tides, EOS Trans. AGU, 54(2), 76. Hendershott, M. C. & Munk, W., Tides. Ann. Rev. Fluid Mech., 2, 205. Jackson, B. V. & Slichter, L. B., The residual daily Earth tides at the South Pole, J. geophys. Res., 79, Kuo, J. T., Jachens, R. C., Ewing, M. 8z White, G., Transcontinental tidal gravity profile across the United States, Science, 168, 968. Melchior, P., ze Earth tides, Pergamon Press Ltd., Oxford. Munk, W. H., Snodgrass, F. & Wimbush, M., Tides off-short: transition from California coastal to deep-sea waters, Geophys. Fluid Dyiz., 1, Prothero, W. A. & Goodkind, J. M., A superconducting gravimeter, Rev. Sci. Instrum., 39, Prothero, W. A. & Goodkind, J. M., Earth tide measurements with the superconducting gravimeter, J. geophys. Res., 77,926. Robinson, E. S., A reconnaissance of tidal gravity in southeastern United State<, J. geophys. Res., 79, Thatcher, Wayne R., Surface wave propagation and source studies on the Gulf of California region, PhD Thesis, California Institute of Technology. Tiron, K. D., Sergeev, Y. & Michurin, A., Tidal charts for the Pacific, Atlantic, and Indian Oceans (trans.), Vestn. Leningrad Univ., 24, Zetler, B., Munk, W., Mofield, H., Brown, W. & Dormer, F., M.O.D.E. tides, J. Phys. Oceanography, in press.