Loading effect of a self-consistent equilibrium ocean pole tide on the gravimetric parameters of the gravity pole tides at superconducting gravimeter stations

The gravimetric parameters of the gravity pole tide are the amplitude factor δ, which is the ratio of gravity variations induced by polar motion for a real Earth to variations computed for a rigid one, and the phase difference κ between the observed and the rigid gravity pole tide. They can be estimated from the records of superconducting gravimeters (SGs). However, they are affected by the loading effect of the ocean pole tide. Recent results from TOPEX/Poseidon (TP) altimeter confirm that the ocean pole tide has a self-consistent equilibrium response. Accordingly, we calculate the gravity loading effects as well as their influence on the gravimetric parameters of gravity pole tide at all the 26 SG stations in the world on the assumption of a self-consistent equilibrium ocean pole tide model. The gravity loading effect is evaluated between 1 January 1997 and 31 December 2006. Numerical results show that the amplitude of the gravity loading effect reaches 10⁻⁹ m s⁻², which is larger than the accuracy (10⁻¹⁰ m s⁻²) of a SG. The gravimetric factor δ is 1% larger at all SG stations. Then, the contribution of a self-consistent ocean pole tide to the pole tide gravimetric parameters cannot be ignored as it exceeds the current accuracy of the estimation of the pole tide gravity factors. For the nine stations studied in Ducarme et al. [Ducarme, B., Venedikov, A.P., Arnoso, J., et al., 2006. Global analysis of the GGP superconducting gravimeters network for the estimation of the pole tide gravimetric amplitude factor. J. Geodyn. 41, 334–344.], the mean of the modeled tidal factors δ_m = 1.1813 agrees very well with the result of a global analysis δ_CH = 1.1816 ± 0.0047 in that paper. On the other hand, the modeled phase difference κ_m varies from −0.273° to 0.351°. Comparing to the two main periods of the gravity pole tide, annual period and Chandler period, κ_m is too small to be considered. Therefore, The computed time difference κ_L induced by a self-consistent ocean pole tide produces a negligible effect on κ_m. It confirms the results of Ducarme et al., 2006, where no convincing time difference was found in the SG records.     

Scale factor for lengths of the geopotential model

Summary Using the data in[1], the scale factor for lengths is derived of the geopotential model R ₀ =GM/W ₀ (W ₀ is the potential on a generalized geoid). The resulting value, R ₀ ==6 363 672.9 m, which is2 m less than the original value[5], is practically the same as that in[6].     

The diurnalK 1 tide in the world ocean — A numerical investigation

Summary Based on a special numerical method (HN method) hydrodynamical processes in the sea were studied with an aim at their prediction. Within this framework, constantly checking the reliability of the numerical models, tides and wind driven motions have been investigated in estuaries, in marginal and adjacent seas, and in the oceans (seeH. Friedrich [1], W. Hansen [2–4], H. P. Schmitz [5]). Encouraged by the promising results, these numerical investigations have been extended by the author to the world ocean, initially to the semi-diurnalM ₂ tide ([8]) and now to theK ₁ tide. Results of computations carried out for theK ₁ tide are given in this paper.     

Determination of vertical displacements over the coastal area of Korea due to the ocean tide loading using GPS observations

This paper describes the GPS applicability for detecting the vertical displacements of ground stations caused by ocean tide loading effects. An experiment was carried out using 12 permanent GPS stations located in the coastal area of Korea using data in the period 1 July until 26 August 2003. The relative height differences were calculated from hourly DGPS data processing based on the carrier-phase observation. The power spectra of the M₂ and N₂ constituents of ocean tide loading were derived using the CLEAN algorithm. The differential vertical displacements generated by the ocean tide loading effect are typically 3–25 mm in coastal area of the Korea. We compared the results from GPS with those of the ocean tide models, NAO.99Jb regional model and GOT00.2, FES99 global models. The M₂ (N₂) amplitude differences of vertical displacements between GPS and GOT00.2 is 1.22 ± 3.61 mm (1.01 ± 1.48 mm), and that of the M₂ (N₂) amplitude difference between GPS and FES99 is 0.04 ± 4.64 mm (0.64 ± 1.75 mm), whereas the M₂ (N₂) amplitude difference between GPS and NAO.99Jb are 0.05 ± 1.03 mm (0.86 ± 1.18 mm). The highest vertical displacements at the PALM station are found for 24.5 ± 0.7 mm from GPS observation, and 22.9 mm from the regional model NAO.99Jb and 13.17 and 10.00 mm from the global models GOT00.2 and FES99, respectively. These values show that the vertical displacements derived from GPS are in good agreement with those of the regional model NAO.99Jb around Korea, more than with the global models. This result indicated that GPS is an effective tool to measure the vertical displacement caused by the ocean tide loading effect in the coastal area, and we need to use the NAO.99Jb ocean tide model rather than the global ocean tide models in and around the Korean peninsula for position determination with permanent GPS installations. This work demonstrates that vertical displacement caused by the M₂ and N₂ constituents of ocean tide loading can be measured by carrier-phase DGPS.     

On the problem of determining the vertical gradients of gravity anomalies

Summary Green's theorem on harmonic functions makes it possible to determine the integral relationship between the harmonic function and its derivative with respect to the normal on a closed Lyapunov surface. The conditions of solvability are given by Fredholm's theory of integral equations. The solution for a sphere was presented by Molodenskii[3] and the general solution with the help of Molodenskii's parameter k by Ostach[4]. The present paper indicates a possibility of solving this problem with the help of a system of linear algebraic equations, a simplified modification of the Ostach-Molodenskii solution and, finally, a method, based on Eremeev's solution of the fundamental integral equation[5].     

Parameters of the fluid core resonance inferred from superconducting gravimeter data

We have estimated the parameters of fluid core resonance (FCR) due to the nearly diurnal free wobble of the Earth's core based on the superconducting gravimeter (SG) data obtained at the following four observation sites; Esashi and Matsushiro in Japan, Canberra in Australia and Membach in Belgium. By fitting the tidal admittances normalized with the O1 wave at each site to a model of the damped harmonic oscillator, we obtained values of 429.66 ± 1.43 sidereal days, 9350–10,835, −4.828E−4 ± 3.4E−6, −3.0E−5 ± 4.5E−6 for the eigenperiod, the Q-value and the real and imaginary parts of the resonance strength, respectively. Our values obtained from only using the gravity data are very consistent with those inferred from the VLBI nutation data. Our study strongly indicates that the systematic difference between two estimations from the gravity and the nutation in particular for the Q-value, which has been shown in previous works, is mainly caused by the inaccurate correction for the ocean tide effects. The error in the ocean tide correction is discussed based on the comparison among four global ocean tide models; Schwiderski model (1980), NAO.99b (Matsumoto et al., 2000), CSR4.0 (Eanes and Bettadpur, 1994) and GOT99.2b (Ray, 1999).     

A note on the effect of the ground on the convective temperature field in the atmospheric surface layer

Summary The paper presents the solution of the equation of heat conduction with density of heat sources given generally. For two special cases the computed central temperature excesses of model[5] are compared with the results of some authors[1, 6] who deal with convection in the surface layer.     

On the triaxiality of the earth on the basis of satellite data

Summary The evolution of the opinions as to the problem of the triaxiality of the Earth in the period prior to satellite geodesy can be seen, e.g., in[1–18]. Recently the opinion has been voiced that triaxiality is a result of the mathematical treatment of data rather than reality[19–21], especially since this is a comparatively small parameter. This opinion is not in contradiction with the results of satellite observations[22–28], but the non-zero values of the harmonic coefficients of the second degree and second order are a reality, they yield a value of the equatorial flattening of about1/90 000, and the representation of the equatorial section by an ellipse is justified even if the harmonics n=3, k=1 and n=3, k=3 have amplitudes only about half as small, and some other parameters might occur with just as much justification besides triaxiality.     

Comparison of satellite and terrestrial gravity data

Summary The integral mean values of gravity on the surface W=W ₀ , obtained from satellite observations with the use of harmonic coefficients[3, 7] and from terrestrial gravity measurements[12], are compared. The squares and products of the harmonic coefficients were neglected, with the exception of [J ₂ ⁽⁰⁾ ] ² , which was taken into account. The Potsdam correction and the geocentric constant are being discussed. The paper ties up with[13–15] and the symbols used are the same. The given problem was treated, e.g., in[2, 4, 6, 8–10]; in the present paper the values of gravity are compared directly.     

Parameters of the normal gravity field deduced from satellite observations

Summary The parameters of the normal gravity field were deduced from the harmonic coefficients[3, 4] upto n=6 and compared with the parameters used hitherto. The symbols used are the same as in papers[5, 6, 8] with which this paper connects up.     

Seasonal variation of the <Emphasis Type="Italic">M</Emphasis><Subscript>2</Subscript> tide

The seasonal cycle of the main lunar tidal constituent M ₂ is studied globally by an analysis of a high-resolution ocean circulation and tide model (STORMTIDE) simulation, of 19 years of satellite altimeter data, and of multiyear tide-gauge records. The barotropic seasonal tidal variability is dominant in coastal and polar regions with relative changes of the tidal amplitude of 5–10 %. A comparison with the observations shows that the ocean circulation and tide model captures the seasonal pattern of the M ₂ tide reasonably well. There are two main processes leading to the seasonal variability in the barotropic tide: First, seasonal changes in stratification on the continental shelf affect the vertical profile of eddy viscosity and, in turn, the vertical current profile. Second, the frictional effect between sea-ice and the surface ocean layer leads to seasonally varying tidal transport. We estimate from the model simulation that the M ₂ tidal energy dissipation at the sea surface varies seasonally in the Arctic (ocean regions north of 60°N) between 2 and 34 GW, whereas in the Southern Ocean, it varies between 0.5 and 2 GW. The M ₂ internal tide is mainly affected by stratification, and the induced modified phase speed of the internal waves leads to amplitude differences in the surface tide signal of 0.005–0.0150 m. The seasonal signals of the M ₂ surface tide are large compared to the accuracy demands of satellite altimetry and gravity observations and emphasize the importance to consider seasonal tidal variability in the correction processes of satellite data.     

Verifying the body tide at the Canary Islands using tidal gravimetry observations

Gravity tide records from El Hierro, Tenerife and Lanzarote Islands (Canarian Archipelago) have been analyzed and compared to the theoretical body tide model (DDW) of Dehant el al. (1999). The use of more stringent criterion of tidal analysis using VAV program allowed us to reduce the error bars by a factor of two of the gravimetric factors at Tenerife and Lanzarote compared with previous published values. Also, the calibration values have been revisited at those sites. Precise ocean tide loading (OTL) corrections based on up-to-date global ocean models and improved regional ocean model have been obtained for the main tidal harmonics O₁, K₁, M₂, S₂. We also point out the importance of using the most accurate coastline definition for OTL calculations in the Canaries. The remaining observational errors depend on the accuracy of the calibration of the gravimeters and/or on the length of the observed data series. Finally, the comparison of the tidal observations with the theoretical body tide models has been done with an accuracy level of 0.1% at El Hierro, 0.4% at Tenerife and 0.5% at Lanzarote.     

The series computation of the gravitational perturbation due to an ocean tide

The series expansion of the gravity perturbations due to an ocean tide is severely hampered by the computationally necessary truncation of the series. The source of this difficulty is investigated and found to be easily remedied by modifying the gravimetric factor slightly so that the series converges much faster. The improved series is tested with a spherical cap load and a published M₂ ocean tide model, and is found to give more reliable results.     

Berechnung der Funktionen einfacher Streuung auf Trübungsteilchen in der Atmosphäre

Summary In 1967, a series of observations were carried out at Lomnický tít of the intensity of light of the clear sky. Using the de Bary method[1], the observations were used to determine the function of simple dispersion on turbind particles in the atmosphere and compared with theoretical functions, which hold for Jung's power distribution of the particles according to size[7] and for the logarithmic Gauss distribution of the particles[8].     

LAGEOS Sensitivity to Ocean Tides

Satellite Laser Ranging (SLR) to LAGEOS has a remarkable contribution to high-precise geodesy and geodynamics through deriving and validating various global geophysical models. This paper validates ocean tide models based on the analysis of satellite altimetry data, coastal tide gauges, and hydrodynamic data, i.e., CSR3.0, TOPEX4.0, CSR4.0A, FES2004, GOT00.2, and the CSRC Schwiderski model. LAGEOS orbits and SLR observation residuals from solutions based on different ocean tide models are compared and examined. It is found that LAGEOS orbits are sensitive to tidal waves larger than 5 mm. The analysis of the aliasing periods of LAGEOS orbits and tidal waves reveals that, in particular, the tidal constituent S₂ is not well established in the recent ocean tide models. Some of the models introduce spurious peaks to empirical orbit parameters, which can be associated with S₂, S_a, and K₂ tidal constituents, and, as a consequence, can be propagated to fundamental parameters derived from LAGEOS observations.     

Geoidal curvature radii from satellite data for different degrees of smoothing

Summary Radii of curvature and their anomalies of a smoothed geoidal surface are computed using Stokes's constants J _n ^(k) , S _n ^(k) of the Earth's body, obtained from satellite orbit dynamics[2]. Different degrees n of smoothing are used (n = 8, 12, 21). The notations are the same as in[4, 5].     

Earth tides observed by gravity and GPS in southeastern Alaska

We analyzed gravity data obtained in Juneau and global positioning system (GPS) data obtained from three PBO sites in southeastern Alaska (SE-AK), which are part of a US research facility called ‘EarthScope’, and we compared the obtained tidal amplitudes and phases with those estimated from the predicted tides including both effects of the body tide and ocean tide. Global tide models predict the ocean tides in this region of complex coastline and bathymetry. To improve the accuracy of prediction, we developed a regional ocean tide model in SE-AK.Our comparison results suggest: (1) by taking into account the ocean tide effect, the amplitude differences between the observation and the predicted body tide is remarkably reduced for both the gravity and displacement (e.g. for the M₂ constituent, 8.5–0.3 μGal, and 2.4–0.1 cm at the AB50 GPS site in Juneau in terms of the vector sum of three components of the north–south, east–west and up–down), even though the ocean tide loading is large in SE-AK. (2) We have confirmed the precise point positioning (PPP) method, which was used to extract the tidal signals from the original GPS time series, works well to recover the tidal signals. Although the GPS analysis results still contain noise due to the atmosphere and multipath, we may conclude that the GPS observation surely detects the tidal signals with the sub-centimeter accuracy or better for some of the tidal constituents. (3) In order to increase the accuracy of the tidal prediction in SE-AK, it is indispensable to improve the regional ocean tide model developed in this study, especially for the phase.     

Determination of the parameters of a selenocentric reference system and the deflections of the vertical at the lunar surface

Summary Stokes' constants and, the selenocentric constant, and the angular velocity of the rotation of the Moon define the shape of the external equiselenopotential surfaces, generalized in dependence on the degree N of the harmonics preserved. The scale factor for lengths was computed on the basis of absolute gravity measurement made by the first lunarlanding mission Apollo11 at the landing site[1] under the assumption of a sufficient accuracy of the Stokes' constants used[2, 15]. Anyway, the numerical solution here is only to be considered as an example of the application of the outlined theoretical method, inclusive of the parameters of the lunar reference system, which will be made considerably more accurate when gravity measurements at more points of the lunar surface are available.Presented at the XVth IUGG General Assembly, Moscow, July 30 – August 14, 1971.     

On one generalization of trochoidal waves

Summary The concept of the generalized trochoidal waves discussed in[1] is revised and modified. A new formula defining the auxiliary function (b, c) was found with the aid of the results derived in[3] and some physical considerations.     

An observational study of energy balance in the atmospheric lunar tide

Analyses of satellite orbit-perturbation and altimeter data have been used in the past few years to evaluate sea-tide dissipation. A value of about 2.5 TW for the M₂ tide is emerging from this work, which for the first time has placed our knowledge of sea-tide energy balance on a firm observational basis. A comparable improvement for the air tide is not yet possible, but an energy-balance estimate of M₂ air-tide dissipation is made here from the best available spherical-harmonic analysis of the lunar barometric tide, namely that of 1969 byHaurwitz andCowley. Full account is taken of the flux of tide energy from the ocean, by means of sea-tide elevation derived from satellite data, and effects of sea-tide attraction and load are included. The result of this observational assessment is an M₂ air-tide dissipation of about 10 GW maintained almost entirely, on the average, by the sea tide.