The M 2 ocean tide loading wave in Alaska: vertical and horizontal displacements, modelled and observed

Crustal deformations caused by surface load due to ocean tides are strongly dependent on the surface load closest to the observation site. In order to correctly model this ocean loading effect near irregular coastal areas, a high-resolution coastline is required. A test is carried out using two GPS sites located in Alaska, where the ocean tide loading effect is large and consequently observed easily by relative positioning with GPS. The selected sites are Fair (Fairbanks) and Chi3 (located on an island that separates Prince William Sound from the Gulf of Alaska). Processing of hourly baseline solutions between Fair and Chi3 over a period of 49 days yields a significant ocean tide loading effect. The data are processed using different strategies for the tropospheric delay correction. However, the best results are obtained when 1-h ZTD (Zenith Tropospheric Delay) parameters for hourly solutions are used. In this case ocean tide loading is not absorbed into the ZTD parameters. Hence, ocean tide loading can be well resolved in the GPS data analysis. In addition, the M ₂ ocean tide wave in the Gulf of Alaska has a very large amplitude. Although the horizontal M ₂ ocean tide loading amplitude in general is only about 1/4 of the vertical M ₂ ocean tide loading amplitude, the differential horizontal M ₂ ocean tide loading displacements are nevertheless measurable using differential GPS (DGPS). When using the GOT99.2 ocean tide model and taking the coastal structure into account, the predicted differential vertical M ₂ amplitude and Greenwich phase lag due to ocean tide loading are 19.3 mm and 110.2 degrees respectively, while GPS measurements yield 21.3 ± 1.0 mm and 99.7±2.8 degrees. Similarly, the predicted differential horizontal M ₂ amplitude and Greenwich phase lag (in the north–south direction) are 4.5 mm and –77.0 degrees, while GPS yields 5.4 ± 0.3 mm and –106.3±3.3 degrees. Only the north-south component of the differential horizontal M ₂ ocean tide loading wave is considered, because the east–west component is too small for the processed baseline and not detectable using DGPS.     

Application of a spherical FFT approach in airborne gravimetry

The basic idea of this paper is to modifyPoisson's integral for harmonic downward continuation into a convolution formula in the space domain. In this manner, the Fast Fourier Transform can be applied. The method is applied to airborne gravimetry, motivated especially by the Greenland survey. The accuracy of data continuation from the flight-level to the ground is analysed. In particular, the influence of latitudinal extension is investigated, since the introduced convolution formula is exact only for the mid-parallel of the analysed area. The results obtained justify the conclusion that the introduced method is applicable to processing of real data. Extended quadratic areas (up to 2500km × 2500km) in equatorial areas and up to 500km × 500km in regions with latitudes about 75°) can effectively be processed in one single procedure.     

Photogrammetric tracking of tracer particles in modelled ocean flows

Three-dimensional particle tracking velocimetry (3D-PTV) has been used to quantify simulated ocean circulation and internal waves, in a 5 m diameter rotating tank. According to hydrostatic theory the water surface in the rotating tank will attain the surface of an elliptic paraboloid of revolution. Three “off the shelf” video cameras mounted above the water surface, rotating with the tank, monitored the 3D displacements of sparsely distributed surface and neutrally buoyant submerged particles (diameter of 0.01 m) inside a water volume of ca. 0.5 × 0.7 × 0.5 m³. A numerical procedure for the calibrations of the photogrammetric system is described. The particles were automatically tracked through a time series of consistent image triplets using an epipolar approach. The experiments gave a standard error σ_o = 0.33 pixels and a repetition accuracy, RMSE = 0.15 mm in X, Y and 0.5 mm in Z (the height).     

Excitation of non-atmospheric polar motion by the migration of the Pacific Warm Pool

Changes in the annual variation of the Earths polar motion are found to be largely caused by the variation of the atmospheric angular momentum (AAM). Recent simulation results of oceanic general circulation models further suggest global oceanic effects on the annual polar motion in addition to the atmosphere. In comparison with previous model studies of global oceanic effects, this research particularly singles out a large-scale ocean anomaly and investigates its effect on the annual polar motion, determined from satellite observations of the movement of the Western Pacific Warm Pool (WPWP). Although the scale of the warm pool is much smaller than that of the solid Earth, analysis of the non-atmospheric polar motion excitation has shown that the WPWP contributes non-negligibly to the annual polar motion. The analysis consists of over 30 years of WPWP data (1970–2000) and shows values of polar motion excitation for the x-component of (2.5 mas, –79°) and for the y-component of (0.6 mas, 173°). Comparison of this result with the total geodetic non-atmospheric polar motion excitation of (10.3 mas, 59°) for the x-component and (10.6 mas, 62°) for the y-component shows the significance of the WPWP. Changes in the Earths polar motion have attracted significant attention, not only because it is an important geodetic issue, but also because it has significant value as a global measure of variations within the hydrosphere, atmosphere, cryosphere, and solid Earth, and hence global changes.Tel: 86–21–64386191 Fax: 86–21–64384618Acknowledgments. The authors are grateful to Dr. R. Gross (JPL) and two anonymous reviewers for providing invaluable comments. They also thank Dr. J.L. Chen (CSR) for helpful discussions. Y. Zhou, D. Zheng and X. Liao were supported by the National Natural Science Foundation of China (10273018, 10133010) and Key Project of Chinese Academy of Sciences (KJCX2-SW-T1). X-H. Yan was supported by the National Aeronautics and Space Administration (NASA) through Grant NAG5–12745, and by the National Science Foundation (NSF) through the Presidential Faculty Fellow award to X-H. Yan (OCE-9453499). W.T. Liu was supported by the NASA Physical Oceanography Program.     

Harmonic maps

Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L₀ and relative isometric latitude q=Q–Q₀ gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B[–40°,+40°], L[–31°,+49°], B₀= ±30°, L₀=9° and (2) B[46°,56°], L{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°,13.5°]; [13.5°,16.5°]}, B₀= 51°, L₀ {6°,9°,12°,15°}.     

Precise alpine geoid determination without digital terrain models

The short wavelength geoid undulations, caused by topography, amount to several decimeters in mountainous areas. Up to now these effects are computed by means of digital terrain models in a grid of 100–500m. However, for many countries these data are not yet available or their collection is too expensive. This problem can be overcome by considering the special behaviour of the gravity potential along mountain slopes. It is shown that 90 per cent of the topographic effects are represented by a simple summation formula, based on the average height differences and distances between valleys and ridges along the geoid profiles, δN=[30.H.D.+16.(H−H′).D] in mm/km, (error<10%), whereH, H′, D are estimated in a map to the nearest 0.2km. The formula is valid for asymmetric sides of valleys (H, H′) and can easily be corrected for special shapes. It can be used for topographic refinement of low resolution geoids and for astrogeodetic projects. The “slope method” was tested in two alpine areas (heights up to 3800m, astrogeodetic deflection points every 170km ²) and resulted in a geoid accuracy of ±3cm. In first order triangulation networks (astro points every 1000km ²) or for gravimetric deflections the accuracy is about 10cm per 30km. Since a map scale of 1∶500.000 is sufficient, the method is suitable for developing countries, too.     

World-wide synthetic tide parameters for gravity and vertical and horizontal displacements

The response of the Earth’s crust to the direct effect of lunisolar gravitational forcing is known as the body tide. The body tide is superimposed by surface-loading forces due to the pressure of the periodically varying ocean tide acting on the Earth, called ocean tide loading (OTL). Both body tide and OTL can be decomposed into components of the same frequency known as tidal parameters. However, OTL is more complicated than body tides because of the dynamic effects of the ocean. Estimating OTL requires a model of the ocean tides and knowledge of the elastic properties of the solid Earth. Thus, synthetic tide parameters (amplitude factors and phase leads) have been developed here on a world-wide grid for gravity and positional displacements. The body tide contributions were added to the oceanic contribution to provide the Earth tide response. The accuracy and reliability of the synthetic tidal parameters have been estimated by comparing observed gravity and vertical-displacement tide parameters with those interpolated from our synthetic model, which shows good agreement. Tests also indicate that the synthetic tide parameters provide realistic gravimetric and displacements for practical use in tidal prediction.     

Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree n _max = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), Δr _CF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of Δr _CF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.     

Changes in the position of the geocentre due to variations in sea-level

The hydrostatic equation is used to show that load departures on the ocean floor due solely to the oceans may be determined from the excess of recorded over steric sea-level and a representative density. The effect of these departures on the position of the geocentre is estimated using formulae derived by Stolz (1976) and data compiled from Pattullo et al. (1955) and Pattullo (1960; 1963). The resulting maximum seasonal position change turns out to be about 2mm. Addition of the results calculated here and those published earlier by Stolz (1976) indicates that the maximum geocentre motion due to seasonal variations in air mass, ground water and sea-level is about 5mm, which is about an order smaller than the point-coordination accuracy modern high-precision geodesy aims to achieve.     

Separation of residual ocean tide signals in a collinear analysis of Geosat altimetry

The effects of residual ocean tide signals on altimetric mesoscale variability estimates are analyzed in the north east Atlantic region using one year of Geosat Exact Repeat Mission data. The variability is evaluated along collinear tracks and covariance functions are determined. Initial results show that the average variability is about 10 cm. Variability values of 15–20 cm are found between Iceland and the Faeroe Islands and values up to 1/2 m are found near the coast of the UK. Then a procedure for estimation of ocean tide signals combined with the collinear analyses was tested. A separation of the estimated tide residuals associated with M₂, S₂, and N₂ resulted in a significant reduction of the variability. The average variability decreased to 5 cm and the temporal correlations vanished. A reduction of the large variability values from 15–50 cm to 5–10 cm demonstrates that significant residual ocean tide signals can be estimated and removed efficiently in a collinear analysis of the mesoscale ocean variability.     

Assessing the accuracy of predicted ocean tide loading displacement values

The accuracy of ocean tide loading (OTL) displacement values has long been assumed to be dominated by errors in the ocean tide models used, with errors due to the convolution scheme used considered very small (2–5%). However, this paper shows that much larger convolution errors can arise at sites within approximately 150 km of the coastline, depending on the method used to refine the discrete regularly spaced grid cells of the ocean tide model to better fit the coastline closest to the site of interest. If the local water mass redistribution approach is implemented, as used in the OLFG/OLMPP software recommended in the IERS 2003 conventions, OTL height displacement errors of up to around 20% can arise, depending on the ocean tide model used. Bilinear interpolation only, as used in the SPOTL and CARGA softwares for example, is shown from extensive global and regional comparisons of OTL displacement values derived from the different methods and softwares to be more appropriate. This is verified using GPS observations. The coastal refinement approach used in the OLFG/OLMPP software was therefore changed in August 2007 to use bilinear interpolation only. It is shown that with this change, OTL displacement values computed using OLFG/OLMPP, SPOTL and CARGA invariably agree to the millimetre level for coastal sites, and better than 0.2 mm for sites more than about 150 km inland.     

FFT-Evaluation and applications of gravity-field convolution integrals with mean and point data

Due to the fact that the spectrum of a convolution is the product of the spectra of the two convolved functions, the convolution integrals of physical geodesy can be evaluated very efficiently by the use of the fast Fourier transform (FFT) provided that gravity and/or terrain data are available on a regular grid. All Fourier transform-based methods usually treat the gridded data as point values despite the fact that these discrete values may have been obtained by averaging and they represent mean values over the whole area of a grid element. In the frequency domain, this fact can be taken into account very easily, because the spectra of mean and point data are related via a two-dimensional (2D) sinc function. The paper shows explicitly this relationship using the convolution integrals for the evaluation of geoid undulations, deflections of the vertical, and gravity and gradiometry terrain effects. Numerical tests are presented, indicating that the differences in the two approaches may become significant when highly accurate results are wanted. The application of the2D sinc function in the evaluation, update, and inversion of other convolution integrals is briefly discussed as well.     

Bringing GPS into harsh environments for fully automated deformation monitoring

Engineering projects that require deformation monitoring frequently utilize geodetic sensors to measure displacements of target points located in the deformation zone. In situations where control stations and targets are separated by a kilometer or more, GPS can offer higher precision position updates at more frequent intervals than can normally be achieved using total station technology. For large-scale deformation projects requiring the highest precision, it is therefore advisable to use a combination of the two sensors. In response to the need for high precision, continuous GPS position updates in harsh deformation monitoring environments, a software has been developed that employs triple-differenced carrier-phase measurements in a delayed-state Kalman filter. Two data sets were analyzed to test the capabilities of the software. In the first test, a GPS antenna was displaced using a translation stage to mimic slow deformation. In the second test, data collected at a large open pit mine were processed. It was shown that the delayed-state Kalman filter developed could detect millimeter-level displacements of a GPS antenna. The actual precision attained depends upon the amount of process noise infused at each epoch to accommodate the antenna displacements. Higher process noise values result in quicker detection times, but at the same time increase the noise in the solutions. A slow, 25 mm displacement was detected within 30 min of the full displacement with sigma values in E, N and U of ±10 mm or better. The same displacement could also be detected in less than 5 h with sigma values in E, N and U of ±5 mm or better. The software works best for detecting long period deformations (e.g., 20 mm per day or less) for which sigma values of 1–2 mm are attained in all three solution components. It was also shown that the triple-differenced carrier-phase observation can be used to significantly reduce the effects of residual tropospheric delay that would normally plague double-differenced observations in harsh GPS environments.

Gravimetric examination of Hagia Sophia's subsurface structure

The subsurface structure of Hagia Sophia, one of the oldest sacred monuments in the world built between 532–537 under the reign of Justinian in today's Istanbul, has been investigated by using two relative LaCoste-Romberg gravimeters in order to detect hidden cavities which have also served as earthquake dampers in similar constructions. On the building's ground floor a grid of 100 points with a grid size of about 4.m was measured. The mean gravimetric point error was ± 3.10^–8 ms^–2. The result of the examination is that cavities were not detected in the inner central part of Hagia Sophia with a larger diameter than 8.m down to a depth of about 20.m, and Hagia Sophia's foundation was found to be a slope of natural rock with a downward inclination to the East that has a small crest symmetrical to the building's East-West axis.     

Estimates of ocean tide loading displacements and its impact on position time series in Hong Kong using a dense continuous GPS network

Three-dimensional ocean tide loading (OTL) displacements of eight diurnal and semidiurnal constituents at 12 sites in Hong Kong were estimated using 3–7 years of continuous global positioning system (GPS) observations. OTL displacements were estimated using the precise point positioning (PPP) technique on a daily basis and then combined. The OTL displacements obtained by GPS were compared with predictions using seven recent global ocean tide models. The effect of OTL displacements on GPS position time series was also investigated. The study shows that the GPS-derived OTL displacements (excluding K1 and K2 constituents) agree best with those predicted by the GOT4.7 and NAO99b models. The GPS/model agreement is generally at the sub-millimeter level, except for S2, K1, and K2 constituents with relatively large errors. After systematic biases between the GPS and model values are removed, the misfits of all sites for M2, S2, N2, O1, P1, and Q1 are less than 0.5 and 1.0 mm in the horizontal and vertical components, respectively, while larger misfits (within 2.5 mm) are observed for K1 and K2. Integer ambiguity fixing slightly improves the east component of OTL displacement estimates. The study also finds that GPS-derived OTL corrections, instead of model predicts, can be used in daily data processing with the exception of K1 and K2. Including K2 corrections, a secular vertical rate of up to 1 mm/year in position time series can be induced, which needs to be confirmed by further studies.     

Ocean loading effects on the prediction of Antarctic glacial isostatic uplift and gravity rates

The effect of regional ocean loading on predicted rates of crustal uplift and gravitational change due to glacial isostatic adjustment (GIA) is determined for Antarctica. The effect is found to be significant for the ICE-3G and ICE-5G loading histories (up to ?8 mm/year and ?3 mm/year change in uplift rate and ?3 cm/year and ?1 cm/year equivalent water height change (EWHC) of surface mass, respectively). The effect is smaller (+1 mm/year; +0.25 cm/year) for the IJ05 loading history. The impact of ocean loading on the rate of change of the long-wavelength zonal harmonics of the Earth’s gravitational field is also significantly smaller for IJ05 than ICE-3G. A simple analytical formula is derived that is accurate to about 3% in a root-mean-square sense that relates predicted or observed gravitational change at the surface of the Earth (r = a) to the EWHC. A fundamental difference in the definition of the load histories accounts for the differing sensitivities to ocean loading. IJ05 defines its surface load history relative to the present-day surface load, rather than specifying an absolute loading history, and thus implicitly approximates the temporal and spatial mass exchange between grounded ice and open ocean. In contrast, ICE-3G and ICE-5G specify an absolute load history and explicit regional ocean loading substantially perturbs predicted GIA rates. Conclusions of previous studies that used IJ05 predictions without ocean loading are relatively robust.     

Surface gravitational field and topography changes induced by the Earth’s fluid core motions

Using a Love number formalism, the elastic deformations of the mantle and the mass redistribution gravitational potential within the Earth induced by the fluid pressure acting at the core–mantle boundary (CMB) are computed. This pressure field changes at a decadal time scale and may be estimated from observations of the surface magnetic field and its secular variation. First, using a spherical harmonic expansion, the poloidal and toroidal part of the fluid velocity field at the CMB for the last 40 years is computed, under the hypothesis of tangential geostrophy. Then the associated geostrophic pressure, whose order of magnitude is about 1000 Pa, is computed. The surface topography induced by this pressure field is computed using Love numbers, and is a few millimetres. The mass redistribution gravitational potential induced by these deformations and, in particular, the zonal components of the related surface gravitational potential perturbation (J₂, J₃ and J₄ coefficients), are calculated. Overall perturbations for the J₂ coefficient of about 10^–10, for J₃ of about 10^–11 and for J₄ are found of about 0.3×10^–11. Finally, these theoretical results are compared with recent observations of the decadal variation of J₂ from satellite laser ranging. Results concerning J₂ can be described as follows: first, they are one order of magnitude too small to explain the observed decadal variation of J₂ and, second, they show a significant linear trend over the last 40 years, whose rate of decrease amounts to 7% of the observed value.     

Estimating atmospheric pressure loading regression coefficients from GPS observations

The loading exerted by atmospheric pressure on the surface of the Earth causes deformations, mainly in vertical direction. Consequently, these deformations are also subject to pressure variations. At present this effect is only modeled by a few research groups in the post-processing of very long baseline interferometry (VLBI) and global positioning system (GPS) observations. As the displacements may clearly exceed the accuracy goals, we implement vertical pressure loading regression coefficients as a new estimable parameter type in the Bernese GPS software. This development is applied to a network of 60 European permanent GPS stations extending from 35 to 79° northern latitude. The analysis comprises 1,055 days of observations between January 2001 and February 2004. During that period pressure variations as large as 80 hPa occurred at high latitude sites. A least squares solution including all observations and all relevant parameters yields significant regression coefficients for all stations but reveals also some critical issues with regard to the capability of this geodetic approach to verify results based on the geophysical convolution method.An erratum to this article can be found at     

High accuracy in short ISS missions

Traditionally Inertial Surveying Systems (ISS) are used for missions of30 km to 100 km length. Today, a new type ofISS application is emanating from an increased need for survey control densification in urban areas often in connection with land information systems or cadastral surveys. The accuracy requirements of urban surveys are usually high. The loss in accuracy caused by the coordinate transfer betweenIMU and ground marks is investigated and an offsetting system based on electronic tacheometers is proposed. An offsetting system based on a Hewlett-PackardHP 3820A electronic tacheometer has been tested in Sydney (Australia) in connection with a vehicle mountedLITTON Auto-Surveyor SystemII. On missions over750 m (8 stations,25 minutes duration,3.5 minuteZUPT intervals, mean offset distances 9 metres) accuracies of37 mm (one sigma) in position and8 mm in elevation were achieved. Some improvements to theLITTON Auto-Surveyor SystemII are suggested which would improve the accuracies even further.     

Geodetic surveys in Messina straits area

By using the fit of theoretical displacements on a fault according toMindlin andCheng (1950) dislocation theory, the vertical deformation field (maximum subsidence of 70 cm) associated to the Dec. 28 1908 MessinaM=7.0 earthquake is compared to the results of a spirit levelling survey that we performed in the Messina Straits area in 1981–1982. The differences in level from the last levelling campaign in the area, completed in 1970, show that the coastlines have undergone a moderate differential subsidence of about1 mm/year in the last decade. This is in agreement with the trend observed around 1908 and with the data of the Messina tide-gauge relative to the period 1897–1908 and 1910–1918. The lack of data in the period 1918–1970 does not allow definite conclusions about the vertical tectonic deformations in the area. Recent data on planimetric deformations have shown a South-North relative motion of Sicily. A comparison with the theoretical displacements indicates that a pure dip-slip of4 cm on the 1908 fault system does explain the observed vertical deformation, but not the horizontal.