Test of theoretical solid earth and ocean gravity tides

Geophys. J. Int. (1996) 125, 106-1 14 Test of theoretical solid earth and ocean gravity tides John M. Goodkind University ofcalifornia, Sun Diego, 9500 Gilrnan Drive, La Jolla, CA 92093-0319, USA Accepted 1995 October 14. Received 1995 October 14; in original form 1995 January 30 SUMMARY Data from three superconducting gravimeters at four different locations are analysed for agreement with theoretical computations of gravity tides and of the influence of ocean tides on gravity. The absolute value of the measured tide is determined by an independent absolute calibration of the gravimeters. The method differs from previous tests of tide theories in that it compares the entire predicted time series rather than specific harmonic constituents. In this way it is possible to test the solid earth and ocean tide effects independently. We find that the ratios of observed to predicted solid earth tides at different locations are the same within an uncertainty of about ko.2 per cent. However, the magnitude of the tides deduced by the absolute calibration of the gravimeters is 0.6 per cent larger than the theoretical solid earth tide. We point out that an accurate determination of the solid earth tide in this way provides a direct comparison between the gravitational constant, G, at laboratory scales and at the distance to the moon. Key words: gravity, tides. INTRODUCTION The properties of the interior of the earth are deduced primarily from seismic data and from low-frequency normal modes excited by deep earthquakes. Therefore these properties are determined from the response of the solid earth at frequencies higher than roughly Hz. Very low-frequency rheological properties are determined from glacial rebound. The tides of the solid earth present the only means for measuring the response of the earth between these two frequency ranges. Measurement of the solid earth tides has been limited by the superposition of the influence of the ocean tides. The ocean effect amounts to a few per cent of gravity tides at most locations, and the elastic response of the solid earth amounts to 16 per cent. From this one can deduce the accuracy with which the ocean effect must be known in order to achieve a desired accuracy of measurement of the elastic response of the earth. During the past decade, improvements have been made in the theory of the solid earth response at tidal frequencies and in the computation of the ocean effect based on global maps of the ocean tides. At the same time, the precision of tidal measurements has been substantially improved through the use of superconducting gravimeters. The present work examines the accuracy to which the solid earth tide response can be determined with these improved tools. We utilize records from three of our superconducting gravimeters which have been moved among locations and have been calibrated by a direct measurement of the signal from a sphere of mass 323.87 f 0.10 kg moved under the gravimeter. Simultaneous measurement of tides by two or more instruments at La Jolla for periods up to three months, and in Miami for 18 months, have determined that the relative calibration factors of the instruments are constant to within the random noise limit of fo.o1 per cent. The calibration factor depends only on the geometry of the levitated sphere, the levitating coils and the feedback coil, so we expect that it should remain constant through warm ups to room temperature or transport to new locations. The data presented here are consistent with that expectation and indicate that the solid earth gravimetric factor is the same at the locations measured to within about f0.2 per cent. The computed solid earth gravity tide has been compared to measured values in several previous publications (see the complete list of references in Scherneck 1991, especially the work of E. W. Schwiderski, Global ocean tides, parts I to VII, on which this work and the work of Agnew are based; Baker, Edge & Jeffries 1991; and Melchior 1994). The first comparison to use the theory of Wahr (1981), which includes ellipticity and rotation of the earth, found that the tides computed for the 1066A earth model were smaller than measured values by 0.6 per cent (Dehant & Ducarme 1987). In that work, the comparison between theory and observations was made, as for all previous tidal studies, in the frequency domain. The gravimetric factor, 6(w), was defined as the frequencydependent transfer function between the computed tidal forcing function and the measured variation in gravity, exclusive of the influence of the ocean tides. We shall call this the solid 106 0 1996 RAS

Solid earth tides 107 earth tide. ~(cu) was measured for the four largest tidal constituents, 01, P,, K1 and M2. The influence of the ocean tides on gravity tides was obtained by convolving the Farrell (1972) Green s function with the Schwiderski cotidal maps (see Scherneck 1991) for the oceans. This has become routine for the analysis of gravity time series. In this way, a similar transfer function, which we will denote ~(w), can be found which is the ratio of the ocean load portion of the gravity tide to the tidal forcing function. We shall call this the ocean load tide. However, ~(w) must be complex since the ocean response is not in phase with the tidal driving forces. Dehant & Ducarme (1987) then subtract the computed values for ~(w) from the measured spectral amplitudes, and the remainder is assumed to yield 6(w). Independent determination of 6(w) and ~ (w) from the data is not possible by this method. A more recent test (Baker et al. 1991) found that the gravimeter calibrations for the data used by Dehant & Ducarme (1987) were inaccurate. This test was limited to the 0, and M2 terms, and measurements were made only in Europe. The computed ocean effect on O1 is only 0.4 per cent in Europe, so this constituent was used to test the theoretical solid earth tide. It was found to agree with the Dehant & Ducarme (1987) theoretical 6(w) to within 0.2 per cent. The M, term was then used to demonstrate that the Schwiderski map needed correction by 7 per cent around the Iberian peninsula. More recent re-examination of the calibration factors (Melchior 1994, 1995) has shown that disagreements between the various calibrations of the data bank at Brussels ranged between 0.3 and 1 per cent, and that the Schwiderski maps are accurate around the Iberian peninsula. Thus the accuracy of calibration of gravimeters has been re-examined a number of times, since it is critical to the evaluation of theoretical models of the earth. However, testing for a possible latitude dependence requires only precise relative calibration of gravimeters, which is much easier to achieve. The relatively narrow range of latitudes in the European study of Baker et al. did not allow a significant test for the possible latitude dependence of 6(w). Successive corrections to the Wahr computer codes have yielded successively smaller latitude-dependent terms, which are now thought to be significantly below the limits of measurability. Further corrections to the computer codes are still in progress and could lead to additional small changes of 6(w) (Dehant 1995, private communication). Global tests of the theoretical tides will not benefit from the small influence of the oceans on 0, that exists in Europe. Therefore, when testing the theories to an accuracy of 0.5 per cent or better one cannot assume that the ocean effect is correct as computed. In the work presented here we use records from three of the superconducting gravimeters constructed at the University of California, San Diego, at four widely separated locations to test both the solid earth and ocean load tides simultaneously by working in the time domain. We construct the full time series for the solid earth tide from 6(w) computed by Ducarme & Dehant (1987), and the harmonic development of the tidal forcing function derived by Tamura (1987). For the fits described below we use the full latitude dependence included in the 1987 computation, but we also compare the results with the most recent corrections to the Wahr computer codes. These are independent of latitude within the precision of the data and use the PREM earth model (Dehant 1995). Similarly for the ocean effect we construct a full time series. We use computations provided by D. C. Agnew (1994, private communication) which determine ~(w) from the Schwiderski maps convolved with the Farrel Green s function. Our procedure is to fit these two theoretical tide time series plus the local barometric pressure time series to the gravimeter signal. Thus the fit determines only three scalar parameters: (1) the ratio of gravimeter output voltage per pgal of computed solid earth tide, which we label E; (2) the ratio of gravimeter output voltage per pgal of predicted ocean tide effect, 4; and (3) the output voltage per mbar of atmospheric pressure change, y. (Note that E and 4 for a given instrument are inversely proportional to the amplitudes of the computed tides.) If both the solid earth and ocean effect computations were consistent with the data, we would find that E = 4. The potential shortcoming of this method is that all of the same frequencies are present in both theoretical time series, so the covariance of the solid earth and ocean effects might be large. At the locations reported here, the ocean effect is between about 2 and 5 per cent of the total tidal signal. 4 appears to differ from E by as much as 20 per cent and is time-dependent. However, due to the phase differences between ocean and solid earth tides, at all locations other than Miami, the off-diagonal terms of the covariance matrix are equal to (within about 50 per cent) or less than the diagonal term for the solid earth tide. At Miami, the square root of the covariance of E and 4, CT+ is five times greater than the standard deviation of E, CT~, and there is a clear correlation between E and 4 as a function of time. In Hawaii, the correlation function between the computed solid earth and ocean effect time series passes through zero for a lead of 14 min and is a maximum for a lead of 20 hr 40 min. We find, therefore, that ae+ 5 6, for most of the data. A further test of the influence of possible errors in the computation of ocean effects on E was made by (1) leaving the ocean effect out of the fitting procedure and (2) forcing the value of 4 to equal E. In the former case, at La Jolla, E was increased by 0.6 per cent and in the latter case by 0.1 per cent. The variance of the residual was increased by a factor of 2 in the former case and by 1.5 in the latter. In the work that follows, we show that the values of E determined for a given instrument are consistent at the various locations but that the values of 4 and oe+ differ substantially. On this basis we conclude that the method yields correct values for E, even though the computed ocean load appears to be inaccurate. GRAVIMETER CALIBRATION AND COMPARISON TO E In order to compare the absolute value of the theoretical tides with the measured tides, absolute calibration of the gravimeters is needed. For the present work, this was obtained as a by-product of a prior experiment to test the gravitational inverse-square law (Goodkind et al. 1993). In that experiment, a mercury-filled spherical steel shell weighing 323 865 f 100 g and a solid steel sphere weighing 184 441 f 1.0 g were placed, at different times, on a moveable platform under the gravimeter. The spheres were moved every 10 mins between pairs of vertical positions so that a square wave was superimposed on the tidal signal of the gravimeter. Data were obtained for 1 yr with each sphere. About 1000 cycles of these square waves were then used to determine the signal voltage corresponding to the difference in gravitational force from the sphere between each pair of positions. The calibration constant was determined by fitting the data to the computed gravitational force differ-

108 J. M. Goodkind ence between all pairs of locations. The force was computed assuming the inverse-square law to be correct with the gravitational constant measured by Luther & Towler (1982). The platform was moved horizontally on an x-y motion table to determine the position at which the centre of mass of the spheres was along the vertical passing through centre of mass of the gravimeter. It was moved vertically by a 3.81 cm diameter precision screw of pitch 1.5748 turns cm-' between pairs of positions separated by 31 cm over a total range of 1 m. The relative vertical position was monitored independently by an optical linear encoder and by the angular position of the precision screw to yield an estimated precision of f0.006 cm. The absolute distance between the centre of mass of the spheres and the centre of mass of the gravimeter test mass was determined to an accuracy of ko.020 cm. The largest potential for systematic error in the experiment was expected to be in this distance, but it would need to be in error by 0.1 cm to produce a 0.5 per cent error in the calibration constant. We can also attempt to determine both the calibration constant and the absolute value of this distance by adjusting an offset - % 0.01 296 0.01294 0.01 292 (II 0.0129 3 0.01288 0.01 286 0.01 284 0.01282 4190 53190 73090 92890 112790 12691 Figure 1. SGA at La Jolla. The heavy line is the value of the absolute calibration factor, and its standard deviation is marked by the light lines. The values of E shown in Table 1, obtained by fitting theoretical tides to the data, are represented by open circles with 1u error bars. The solid squares are the values obtained by fitting to the latest computations of theoretical tide amplitudes (Dehant 1995, private communication). to the absolute distance so as to minimize the departure from the inverse-square law at all distances. The precision of both are, of course, reduced but this yields a calibration constant 0.5 per cent smaller than that determined from the directly measured distance. That means an even greater disagreement with the theoretical tide. The calibration factor for the instrument used in this experiment, SGA, and using the measured absolute distance, was found to be 0.012849 1.3 x Volt pga1-i. The data series was frequently interrupted for the purposes of the experiment, so records that were useful for fitting the tides were relatively short and of varying lengths. The values of E determined from the tide fits are shown along with the absolute calibration factor in Fig. 1. The average of the fits to the tides yields E = 0.012908 8.0 x Volts pgal-'. The values E and 4 as functions of time, along with the calibration determined by the laboratory experiment, are also shown in Table 1. The calibration factor differs by 4.5~7 from E. Using the most recent correction to the Wahr-Dehant tides (Dehant 1995, private communication) yields the points shown as solid squares in Fig. 1. They correspond to tides that are 0.6 per cent smaller than measured according to our calibration. In the final section of the paper we discuss possible explanations, but first we present time and location dependences that identify some of the factors that limit the accuracy of the determination of E. TIME DEPENDENCE IN ALASKA The record from Fairbanks, Alaska, using instrument SGB, is longer and more complete than that at La Jolla, so a better test of the stability of values of E and (I can be made. Fit parameters for mostly 1-month long records are shown in Fig. 2 and Table 2 for a period of two years. We find larger apparent variations than for La Jolla, even though the records are longer. In particular, there is a relatively large swing of E during the winter of '93-'94. In order to try to determine the ultimate precision that could be obtained for E, we examine possible explanations for these time dependences. A simple explanation could be that coastal loading effects of ocean tides vary with sea-level changes resulting from weather patterns. The effects would vary in both phase and amplitude Table 1. Solid earth, E, and ocean load, 4, fit parameters for gravimeter SGA at La Jolla, California. The standard deviations given for individual records are determined from the least-squares fits. Those for the averages are for deviations of the individual records from the average. dates E 9 @ 0, E (Dehant 95) 4-12 t016, 1990 0.012921 5.94E-06 0.016565 9.27E-05 0.012943 4-28 to 5-2,1990 0.012908 4.03E-06 0.015771 4.99E-05 0.012939 5-18 to 5-25, 1990 0.012916 2.95E-06 0.015380 5.21E-05 0.012933 6-2 to 6-8, 1990 0.012898 3.70E-06 0.015807 6.16E-05 0.012918 6-15 to 6-22, 1990 0.012896 4.48E-06 0.015846 7.58E-05 0.012917 7-21 to 8-4, 1990 0.012908 3.39E-06 0.016080 5.42E-05 0.012926 8-22 to 8-28, 1990 0.012919 5.1 1E-06 0.015929 1.17E-05 0.012934 9-14 to 9-24, 1990 0.012903 4.45E-06 0.014971 8.3%-05 0.012907 10-29 to 11-16, 1990 0.012909 3.87E-06 0.015058 7.02E-05 0.012925 1-5 to 1-14, 1991 0.012909 2.66E-06 0.015451 3.94E-05 0.012932 2-3 to 2-9, 1991 0.012904 1.87E-05 0.015749 2.74E-04 0.012927 average 0.012908 8.02E-06 0.015662 0.000457 0.012927 absolute calibration 0.012849 1.30E-05

Solid earth tides 109 0.01 381 T 0.01 7 0 4- \ =L 0.01379 0 0.01377 > W w 0.01 375 0.016 0.015 s. u) 4-0 5 0.01373 0.014 1011192 411193 1011 IS3 411194 1011194 Figure2. Solid earth, E, (0) and ocean, 4, (x) parameters computed for 1-month periods in Fairbanks, Alaska. Data are plotted at times corresponding to the start of the period computed. Table 2. Fit parameters for data from gravimeter SGB at Fairbanks, Alaska. Standard deviations are as described for Table 1. is the square root of the covariance of E and 4. u, is the standard deviation of the residual signal from its mean after fitting and subtracting the three time series. y is in units of gravimeter signal voltsbarometer signal volts. The barometer yields 7.6 V atmosphere-' so y must be multiplied by 7.5 x to obtain the pressure admittance in pgal mbar-'. date E 0, 4 Y (JV Gv (J5t NOV-92 0.01376242 2.98E-06 0.015481 5.29E-05 0.593101 1.32E-03 8.64E-03 5.27E-06 Dec-92 0.01377616 3.1 OE-06 0.015311 5.81 E-05 0.627852 1.50E-03 6.77E-03 6.54E-06 Jan-93 0.01376691 1.33E-05 0.016684 2.33E-04 0.303187 5.73E-03 2.17E-02 2.04E-05 Mar-93 0.01377511 3.08E-06 0.015391 5.14E-05 0.535561 1.16E-03 5.42E-03 8.24E-07 Apr-93 0.01378294 3.08E-06 0.015239 4.63E-05 0.514735 2.72E-03 4.35E-03 3.22E-06 May-93 0.01376632 4.56E-06 0.015008 7.95E-05 0.477019 2.81 E-03 8.02E-03 6.60E-06 Jun-93 0.01377491 3.02E-06 0.015457 5.60E-05 0.410479 2.94E-03 8.19E-03 5.93E-06 Jut-93 0.01378087 2.59E-06 0.015377 4.69E-05 0.524762 1.72E-03 6.92E-03 4.42E-06 AUQ-93 0.01377783 2.60E-06 0.015437 4.11 E-05 0.530559 1.47E-03 5.77E-03 9.92E-07 Sep-93 0.01375045 3.55E-06 0.015110 5.21 E-05 0.582102 1.83E-03 6.65E-03 4.63E-06 OCt-93 0.01374243 2.99E-06 0.014940 4.65E-05 0.590186 1.64E-03 6.53E-03 1.21E-06 NOv-93 0.01377393 2.39E-06 0.015146 4.25E-05 0.614569 1.51 E-03 6.19E-03 4.13E-06 Dec-93 0.01375668 3.44E-06 0.015377 6.28E-05 0.527868 1.96E-03 9.55E-03 6.71 E-06 Jan-94 0.01378703 3.24E-06 0.016005 5.69E-05 0.498321 2.67E-03 4.40E-03 5.91 E-06 Feb-94 0.01381003 3.18E-06 0.015634 5.00E-05 0.538458 1.40E-03 6.65E-03 2.12E-06 Apr-94 0.01377765 2.82E-06 0.015012 4.50E-05 0.548332 1.62E-03 6.28E-03 1.98E-06 May-94 0.01377586 2.78E-06 0.015593 4.80E-05 0.543832 3.1 OE-03 6.86E-03 4.36E-06 Jun-94 0.01378814 3.20E-06 0.015269 5.68E-05 0.519572 4.1 2E-03 8.06E-03 5.50E-06 OCt-94 0.01376379 2.74E-06 0.014847 4.06E-05 0.606966 1.02E-03 3.75E-03 1.87E-06 averages 0.013774 0.015297 0.539517 0 0.000017 0.00031 0.05141 relative to the computed solid earth tide and ocean load effect, and would lead to variations in both E and 4. However, we do not have the ocean tide and other data required to test this hypothesis. Another possible explanation is found in the gravity signal that remains after fitting and subtracting the three time series. We call this the residual. In Table 2, the standard deviation, u,, of the residual from its mean is a measure of periodic signals that are not fitted by the theories, as well as of non-tidal gravity variations. There is no apparent correlation between ov and the deviations of E from its mean value, so the determination of E does not appear to be affected directly by these other signals. None the less, periodic terms that appear in the residual do so because they are imperfectly correlated with the computed solid earth tides and ocean effects computed from the Schwiderski maps. However, some variable portion of these terms will be correlated with the tides and therefore will affect E and 4. We argue that the dominant causes of the fluctuating periodic terms of the residual also cause fluctuations in E. The periodic signals remaining in the residual are examined in Fig. 3. This shows the amplitudes of sinusoids, fitted to the residual and to the theoretical solid earth tide, at the frequencies of the indicated tidal constituents. The ratio of residual amplitudes to tide amplitudes is largest at frequencies close to S1 and S,, and smallest at 0, and M,. This implies that solar radiation is responsible for the largest periodic terms remaining

110 J. M. Goodkind 0.9 I V 911 192 3120193 10693 42494 1111 0194 Figure 3. Data from Fairbanks, Alaska. W are the average of amplitudes of sinusoids fitted to the residual signal of 1-month records after subtraction of the theoretical tides and the pressure. 0 are the same average of fits of sinusoids to the full theoretical solid earth tide. The ratio of the amplitudes of the residual to those of the theoretical tide are largest at frequencies close to S, and S,. in the residual. However, the amplitudes of the residuals at these frequencies are approximately 0.3 pgal for S2 and 0.8 pgal for Pi and K,. These values are too large to be explained by the direct influence of variable global atmospheric tides (Haurwitz & Cowley 1973; Warburton & Goodkind 1977; Crossley & Hinderer 1994). Local variations in atmospheric tides or atmospheric admittance can also be ruled out since there is no apparent correlation between the fit parameters, y, and E. Indeed, if the pressure is left out of the fit entirely, the values of c are changed by at most about 0.5 per cent. Thus a very substantial misfit of the pressure would be required to account for the variations in E. The values of y in Table 2 do not indicate such large departures from the mean, and the largest observed departures are not at the same time as those in I:. Other causes for the periodic terms in the residual might be radiation-induced ocean tides, local distortion of the ground due to temperature or pressure changes, and temperature changes of the support structure of the gravimeter. The latter two alternatives are unlikely since the periodic variation of indoor and outdoor temperature is a minimum during the winter months and maximum in the summer, whereas the largest deviation of c from its mean occurred during the winter of 93-94. Local barometric pressure exhibits very small diurnal and semidiurnal variations at Fairbanks relative to those at lower latitudes, and these also reach a maximum during the summer and a minimum during the winter. Fig.4 shows that in February 1994 there was an unusually sharp increase in the diurnal component followed by a sharp decrease before reaching the summer-time maximum. This is a departure from an otherwise smooth seasonal variation. There is no significant seasonal variation of E or y, so regular seasonal variations of radiation tides appear to be accounted for, using our three parameter fits. This suggests that their was an anomalous response to radiation tides in February 1994. If this appeared in the ocean as well as in the atmosphere, it could be responsible for the apparent change in F. Regardless of the physical mechanism for these temporal variations of E and 4, they do set the limit of accuracy to Figure 4. W are amplitudes of I and 0 are 2 cycle per day sinusoids, fitted to month-long barometric pressure records from Fairbanks. The points are plotted at the start times of the months that are fitted. which E can be determined. If the ocean is responsible for the problem, as suggested in the arguments above, then further improvement of the measurement of c will be possible when adequate data concerning ocean tides and sea-level becomes available. If the variations contain some regular seasonal periodicities, then improvements in the determination of c will be possible simply by obtaining longer gravity records. TEST FOR LOCATION DEPENDENCE OF E The gravimeter labelled SGB has been operated at three locations. Their coordinates are: longitude latitude elevation (m) La Jolla - 117.27 32.87 50 Miami -80.385 25.61333 2 Fairbanks - 147.4986 64.97806 308 The gravimeter labelled SGC has been operated at La Jolla and at the Hawaiian Volcano Observatory, the coordinates of which are: longitude latitude elevation (m) Hawaiian Volcano Observatory - 155.29166 19.42433 1239.3 We next examine the data for differences in e between these widely separated locations and then for limits on any possible latitude dependence of E. The latitude separation between Fairbanks and the other locations is sufficient to provide nearly the maximum difference predicted by the earlier versions of the theory. SGC We first examine the data from SGC at La Jolla and Hawaii. No observable latitude dependence is expected between these sites, but the differences between mid-ocean and continental crust could, in principle, yield differing tidal responses. SGC was operated at the Hawaiian Volcano Observatory from December 1989 to April 1992. It was stored at room temperature from April 1992 until January 1994, when it was set into operation at La Jolla.

Solid eurth tides 11 1 Most of the data taken in Hawaii and in Miami were slightly degraded by a software error in the real-time digital filter. It resulted in clock offsets of a few tens of seconds when data acquisition was interrupted. For most of the data this occurred once per day at 00 : 00 hr. In order to determine the influence of these offsets we fitted the tides and atmospheric pressure to the data in one-day segments for several months at each of the two locations and for SGC at La Jolla, where the time has been maintained via GPS satellite. The tides were generated for each day with a sequence of starting times at 10-s intervals. The start time that yielded the smallest variance of the residual was then assumed to yield the correct tidal amplitudes. These were then compared to the values of E and 4 obtained by fitting to month-long segments. The results at La Jolla, where the time was accurate and the gravity signal relatively quiet, demonstrated that the time can be determined in this way to better than $_20 s. The differences between ES at the best-fitting times and those 20 s on either side differ by at most 0.02 per cent, and when averaged over one month by less than 0.01 per cent. For June 1994, the average of E for the 1-day records with start times determined in this manner was 0.013878 4.8 x V pga1-l. For a fit of the entire month with the known start time it was 0.013857 1.8 x V pga1-i. Thus the uncertainty in E is increased by a factor of 30 by fitting to 1-day segments but no bias is introduced. Parts of the record from Hawaii contained sufficiently few and small time errors that there was no need to analyse single-day segments. In other segments, E and 4 were computed by both methods and found to agree to well within the error limits of either method. The data for Hawaii in Fig. 5 show much larger variation than the data from Fairbanks. They also show much larger variation than the data from the same instrument at La Jolla in Fig. 6. The variation appears to have a seasonal periodicity. This means that more precise measurement of the solid earth tide in Hawaii will require several years of data to make certain that seasonal variations are properly averaged. The overall averages for SGC operated at the two sites are summarized in Table 3. The two locations yield the same E to within close to the uncertainty of 0.25 per cent imposed by the noisier 0.01390 1 I 0.01 6 0.01 5 0.01 4 0.01 3., 11291 4122191 731191 11891 211692 Figure 5. Values of the tidal fit parameters for SGC at the Hawaiin Volcano Observatory. 0, E; 4. 0.013872 0.01 2 0.01 72 - h 0 %. 5 W 0.013868 0.013864 0.013860 \ \ ' d \ o 0.017-8 0.0168 2 > 0.0166 Y 8., 0.013856 0.0164 4130194 5130194 6129194 72994 8128194 9127194 Figure6. Data from SGC at La Jolla after its return from Hawaii and after storage at room temperature for 2 years. E; 0, 4.

112 J. M. Goodkind Table 3. Tidal parameters for La Jolla and Hawaii. E 0 8 4 04 Y by 0" SGC Hawaii 291 to 492 0.013819 3.49 x lo-' 0.01464 5.19 x -0.3310 0.58 0.0206 SGC La Jolla 594 to 1094 0.013863 4.99 x 0.01668 2.39 x 0.7081 0.16 0.0093 Hawaii data. However, 4 differs by almost 4a. In all locations but Miami, 4 > E and the actual ocean load departs from the theory by as much as 20 per cent. SGB The La Jolla data for SGB were obtained from three short time segments obtained just prior to shipment of the instrument to Miami. However, the stability of the tidal amplitudes at La Jolla determined by the records from SGA and SGC indicate that these records should yield results accurate to within about ko.2 per cent. The parameters for these three segments of data are shown in Table4. The data from all locations are summarized in Table 5. In Miami, the correlation between solid earth and ocean tides is strong. Consequently, ae is large. Data for the month of May 1991 are shown in Fig. 7. The figure shows the typical strong anticorrelation between 4 and E and the relatively large variations in E that result from fitting 1-day records. Although the long-term average of these records shown in Table 5 has a greater uncertainty than at the other sites, it is consistent with them. TEST FOR LATITUDE DEPENDENCE The data shown in Table 5 were analysed using the latitude dependence of the Dehant-Ducarme (1987) computation. We have also computed a solid earth tide by setting all of the latitude-dependent terms for 6(w) equal to zero. By fitting the two time series computed for Fairbanks to each other for the months December 1992 to July 1993, we find that the tides without latitude dependence are larger than those that include it by the factor 1.002332 & 9.0 x Similar computations at La Jolla and Hawaii yield differences of less than which are too small to be measured with the results presented here. The ratio of E measured at Fairbanks to that measured at La Jolla, using the theory including latitude dependence, is 1.0004 f 2.1 x If there were no real latitude dependence this procedure would have yielded a ratio of 1.0023. The indicated standard deviation is somewhat misleading since it is not associated with normally distributed random noise. It results from the time dependence of the tidal residual which has periodic components. None the less, the present data cannot distinguish between the two cases. If the measured fluctuations in E are partly seasonal, then results for longer records will improve more rapidly than for random fluctuations, and tidal records of several years will be important. Tidal records of this length are available from superconducting gravimeters in Europe. However, a comparison to records at high latitudes can only be accomplished to the required precision if relative calibration factors of the various instruments are measured by simultaneous recording of tides at the same location. Relative calibration of superconducting gravimeters can be determined in this manner to fl x with records of only a few day's length. Table 4. Fit parameters for SGB operated at La Jolla, California. 1989 e 0, 4 Y =, Ov G+ 4f27-430 0.013790 8.75~10-0.015684 1.1 lxlous.594 2.06~10'" 2.28~10" Y.6Gx10 7 62-67 0.013760 Fx10 6 0.016511 4.81~10"~.I99 1.31~10~~ 1.86~10"~ 3.66~10" 616-6E5 0.013760 2.01~10" 0.016807 3.03~10"~,370 4.25~10~ 1.?4x10 2 2.55~10" averages 0.013770 1.75~10~' 0.016334 5.82~10~.388 0.019791 Table 5. Summary of data from all locations. E 0- $IE 4) SGC VoltspGal VoWpGal La Jolla May-Oct, 1994 0.013863 5.0~10-~ 1.20 0.0167 Hawaii Feb 91 -Apr 92 0.013819 3.5~10" 1.06 0.0146 - SGB Miami Dec 90 - Aug 91 0.013713 3.3~10.' 0.99 0.0136 La Jolla apr-june, 1989 0.013774 1.7~10' 1.19 0.0163 Fairbanks Nov '92-Nov '94 0.013773 1.5x10-' 1.11 0.0153 - SGA La Jolla Mar'9Oto Mar'91 0.012908 8.0~10.~ 1.178 0.0152 absolute calib. 0.012849 1.3x10-' 61 Y 0 w pgmb 2.4~10~.383.087 5.2~ lo4.18.32 1.6~10.~ 247.308 5.8~ 1 O4 2.5~10~,294.028 5.8~ 1 O4

Solid earth tides 11 3 0.014-0.025 0.0139-0.0136-0.0135 - - 0.005 0.01 34 ' 0 51 1 5'6 5111 516 521 5126 5131 Figure7. Tidal fit parameters for 1-day records from Miami during the month of May 1991. W, E; 0, 4. ABSOLUTE AMPLITUDE OF THE TIDES At locations where the ocean loading effect is not strongly correlated with the solid earth tide, the solid earth tide amplitude is determined to within 0.1 per cent. However, the amplitude of the tides determined from our absolute calibration of a gravimeter in the laboratory are larger than the theoretical tides by 0.6 per cent. We have been able to construct only three possible explanations for this disagreement. (1) There is an undetected systematic error in the absolute calibration. (2) The solid earth tide theory is in error. (3) G is 0.6 per cent smaller at the distance to the moon and sun than it is at 1 m. The first possibility can be tested only by additional laboratory experimentation, by comparison to an absolute meter as described below, or by the use of another independent means of calibration such as applying a known acceleration to the instrument (Richter et al. 1995; Van Ruymbeke 1989). The second possibility may be resolved by computations now in progress at several laboratories. The 6(w) computed from the PREM model is only 0.2 per cent smaller than that for the 1066A model, so changes in the earth model are unlikely to account for 0.6 per cent. However, the inelastic response of the mantle has been variously estimated to increase 6(w) by 0.1 to 0.7 per cent (Lambeck 1988), 0.35 per cent (Wahr & Bergen 1986) and 0.1 to 0.2 per cent (Dehant 1987), so it might account for at least part of the discrepancy. The third possibility can be ruled out by other experiments only if a specific functional dependence of G on distance is assumed. The absolute value of g at the surface of the earth yields the product G(RE)ME, where G(RE) is the value of G at a distance equal to the radius of the Earth. The orbit of the moon yields G(R,)ME. These are the same to within about one part in lo6 (Stacey et al. 1987). Thus G(R,) = G(RM). A measurement of G at a length scale of order 5 km was made by measuring g as a function of depth in the ocean (Zumberge et al. 1991). They found agreement with the laboratory measurements of G (Luther & Towler 1982) within about 0.2 per cent, so G( 10 cm) = G(5 km) to this accuracy. There are no direct comparisons to the laboratory value of G at ranges between 5 km and RE, and thus none between the laboratory scale and RM. However, experiments to test for departures from the inverse-square law of gravity have all been interpreted in terms of a Yukawa-type potential added to the Newtonian inversesquare law (Stacey et ul. 1987; Adelberger et ul. 1991) so that the gravitational potential is assumed to be Assuming that the scale length, 1, is between 5 km and RE, then at 1 m the potential is very accurately given by (1 + tl)gmrn %II,ter = - r Our results would require tl = 0.005. Experiments interpreted in terms of this potential have set upper limits on tl at all distances greater than a few cm to tl - 0.001. Therefore, if our results truly indicated a difference between G( 1 m) and G(RE) the difference could not result from a potential of the Yukawa type. Thus there is motivation for using a variety of methods for absolute calibration of superconducting gravimeters. A different method for doing so has been attempted by recording tides simultaneously at the same location with a superconducting gravimeter and an absolute gravimeter (Goodkind et al. 1991; Richter 1995). However, the signal-to-noise ratio for the tides measured by the absolute meter was such that the record lengths were not long enough to yield a calibration as accurate as the one used here. A repeat of this experiment by one or more of the groups owning both absolute and relative gravimeters should be able to equal or improve upon the accuracy of the calibration by Goodkind et al. (1993). Using an applied acceleration, Richter et al. (1995) report an accuracy of 0.02 per cent. CONCLUSIONS We have demonstrated that the solid earth tide amplitude determined by fitting theoretical computed tides to observed

114 J. M. Goodkind gravity variations is consistent to within 0.2 per cent at four different locations. We have argued that the uncertainty is a consequence of a real-time dependence of the tide signals that probably originates in the oceans. These may consist, in part, of a seasonal periodicity, so the accuracy of determination of the tidal amplitudes could be improved with records of several year s length. An absolute calibration of the gravimeters indicates that the solid earth tide amplitudes differ from the theoretical prediction by 0.6 per cent. The cause of this discrepancy is one of three alternatives which cannot be distinguished with presently available data. ACKNOWLEDGMENTS The author wishes to thank Conrad Young for his work in installing the instruments at Miami, Hawaii and Fairbanks, Gary Puniwai for maintaining the instrument at Hawaii, and Knute Berstis for providing data from an alternate acquisition system at Fairbanks. We thank Mark Zumberge and Robert Parker for comments on the manuscript. Special thanks are due to Veronique Dehant for extensive, helpful comments on the manuscript and for providing current values of the theoretical tide. This work is supported in part by NOAA under contract no. 50-DGNC-2-00065. REFERENCES Adelberger, E.G., Heckel, B.R., Stubbs, C.W. & Rogers, W.F., 1991. Searches for new macroscopic forces, Ann. Reo. Nucl. Part. Sci., 41, 269-320. Baker, T.F., Edge, R.J. & Jeffries, G., 1991. Tidal gravity and ocean tide loading in Europe, Geophys. J. Int., 107, 1-11. Crossley, D. & Hinderer, J., 1995. Effective barometric admittance and gravity residuals, Phys. Earth plunet. Inter., 90, 221-241. Dehant, V., 1987. Integration of the gravitational motion equations for an elliptical uniformly rotating earth with an inelastic mantle, Phys. Earth planet. Inter., 49, 242-258. Dehant, V. & Ducarme, B., 1987. Comparison between the theoretical and observed tidal gravimetric factors, Phys. Earth planet. Inter., 49, 192-212. Farrell, W.E., 1972. Deformation of the Earth by surface loads, Rev. Geophys. Space Phys., 10, 761-797. Goodkind, J.M., Young, C., Richter, B., Peter, G. & Klopping, F., 1991. Comparison of two superconducting gravimeters and an absolute meter at Richmond Florida, Cahiers du Centre Europien de Giodynamique et de Siismologie, 3, 91-98. Goodkind, J.M., Czipott, P.V., Mills, A.P., Jr., Murakami, M., Platzman, P.M., Young, C.W. & Zuckerman, D.M., 1993. Test of the gravitational inverse-square law at 0.4 to 1.4 m mass separation, Phys. Rev. D, 47, 1290-1297. Haurwitz, B. & Cowley, A.D., 1973. The diurnal and semidiurnal barometric oscillations, global distribution and annual variation, Pure & appl. Geophys., 102, 193-222. Lambeck, K., 1988. Geophysical Geodesy. The slow d&rmutions ofthe earth, Clarendon Press, Oxford. Luther, G.G. & Towler, W.R., 1982. Redetermination of the Newtonian gravitational constant, G, Phys. Rev. Lett., 48, 121-123. Melchior, P., 1994. A new data bank for tidal gravity measurements, Phys. Earth planet. Inter., 82, 125-155. Melchior, P., 1995. A continuing discussion about the correlation of tidal gravity anomalies and heat flow densities, Phys. Earth planet. Inter., 88, 223-256. Richter, Bernd, Wilmes, H., Nowak, I. & Wolf, P., 1995. Calibration of a cryogenic gravimeter by artificial accelerations and comparisons with absolute measurements, IUGG meeting, Boulder, Colorado. Scherneck, H.G., 1991. A parametrized solid earth tide model and ocean tide loading effects for global geodetic baseline measurements, Geophys. J. Int.. 106, 677-694. Stacey, F.D., Tuck, G.J., Moore, G.I., Holding, S.C., Goodwin, B.D. & Zhou, R., 1987. Geophysics and the law of gravity, Rev. Mod. Phys., 59, 157-174. Tamura, Y., 1987. A harmonic development of the tide generating potential, Bull. d inform. Maries Terr., 99, 6813-6855. Van Ruymbeke, M., 1989. A calibration system for gravimeters using a sinusoidal acceleration resulting from a vertical periodic movement. Bull. Gtodtsique, 63, 223-235. Wahr, J.M., 1981. Body tides on an elliptical rotating elastic and oceanless earth, Geophys. J. R. astr. Soc., 64, 677-703. Wahr, J.M. & Bergen, Z., 1986. The effects of mantle anelasticity on nutations, earth tides, and tidal variations in rotation rate, Geophys. J. R. astr. Soc.. 87, 633-668. Warburton, R.J. & Goodkind, J.M., 1977. The influence of barometric pressure variations on gravity, Geophys. J. R. astr. Soc., 48,281-293. Zumberge, MA., Hildebrand, J.A., Stevenson, J.M., Parker, R.L., Chave. A.D., Ander, M.E. & Spiess, F.N., 1991. Submarine Measurement of the Newtonian Gravitational Constant, Phys. Reo. Lett., 67, 3051-3054.