Vertical Ocean-loading Deformations Derived from a Global. Mark S. Schenewerk, J. Marshall and William Dillinger

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1 Journal of the Geodetic Society of Japan Vol. 47, No. 1, (2001), pp Vertical Ocean-loading Deformations Derived from a Global GPS Network National Geodetic Survey Mark S. Schenewerk, J. Marshall and William Dillinger (Received August 30, 2000; Revised October 30, 2000; Accepted October 30, 2000) Abstract The proliferation of permanent Global Positioning System (GPS) tracking sites and improvements in estimating the vertical component of a site's coordinates from GPS measurements present an opportunity to directly observe the crustal deformation caused by loading from ocean tides. The long, continuous record from these sites enables the accumulation of observations coherently with respect to the driving tidal forces, and estimates of the ocean-loading signal can be made from these accumulated observations. Techniques like this defeat atmospheric, geometrical, and multipath effects which can have a comparable magnitude, but do not have the same period as ocean-loading. A project to estimate ocean-loading effects at sites globally was undertaken at the Geodetic Research Division of the National Geodetic Survey. The technique used in and results of that project are presented here. Emphasis will be placed on the diurnal and semi-diurnal tidal signals. Although excellent agreement with existing models is found for most sites, a large fraction, generally those at higher latitudes where the complex coastlines and poorer ocean tidal data begin to dominate, show significant differences from standard ocean-loading models. 1. Introduction The ebb and flow of ocean tides have a geodetic/geophysical effect on sites located near the coast: changes in weight caused by the changing quantity of water deform the supporting crust of the Earth. The magnitude of this deformation and its phase relative to the driving potential, i.e. relative to the apparent position of the Moon and Sun, depend upon the shape of the coast, sea-floor topography, latitude and other factors. Models of the tides and deformation caused by ocean-loading from tides (OLT) are available but can suffer from significant errors in some locations, particularly at high latitudes because of lack of detailed data on ocean currents and sea-floor topography, and in regions with complex coastlines because of limitations in the grid size used to approximate the coastline. In these more troublesome locations, direct measurement may be the most feasible means of gen erating OLT amplitude and phase parameters. Given the "explosion" of GPS sites, GPS would seem to be the most practical tool, but this begs the question of how best to use GPS data to measure OLT with the highest accuracy in shortest time. The technique of coherently averaging data to reinforce a desired signal by including unknowns with periods

2 238 Mark S. Schenewerk, J. Marshalland and William Dillinger corresponding to common ocean tides in GPS processing software seems to offer the oppor tunity to directly detect subdaily variations in site positions caused by OLT. A project to estimate OLT vertical displacements at permanent, GPS tracking sites using this technique was undertaken at the National Geodetic Survey (NGS). Brief descriptions of the project and results are given here. A more complete discussion can be found at the NGS web page ( listed as a Project under Geosciences Research. 2. Data and Processing The fundamental processing tool was the PAGES software developed at the NGS (Sche newerk et al., 2000) and used for GPS satellite orbit and earth orientation parameter (EOP) estimation as well as the more mundane site coordinate, velocity and neutral atmospheric (tropo) delay estimation. In this project, double-differenced, ionosphere-free phase data were used as observables. The Saastamoinen (Saastamoinen,1972) zenith delay models and NMF mapping functions (Niell, 1996) with a seasonal model for surface temperature, pressure and relative humidity were used to correct for neutral atmosphere path delays. Solid earth tidal signals were modeled (Cartwright and Tayler,1971; Cartwright and Edden,1973) and removed. The Schwiderski tidal model (Schwiderski,1983) was used to correct for OLT effects. At this time, no atmospheric loading correction is available in this program. PAGES was modified to optionally also estimate vertical amplitude and phase param eters for most commonly significant OLT signals with periods corresponding to the tidal terms: Darwinian designations K2, M2, N2, 52, K1, O1, P1, Q1, M f, Mm and S3 a. When "turned on", these OLT parameters would be part of the normal equation matrix created by PAGES during data processing. A partially reduced form of this matrix is stored to a file before any constraints are applied. At some later date, this file can be read, the contents possibly combined with similar matrices stored in other files, and the resulting matrix in verted and solved. GPSCOM, also developed at the NGS, was used for this type of matrix manipulation. Creation of matrix pieces, or Helmert blocks, in this manner enables the pro cessing of a large data set in logical or convenient subsets without sacrificing the complete, self consistent solution. For this project, the complete data set was broken into daily subsets which were, in turn, broken into regional subsets. The only drawback to breaking up the data processing in this manner is the assumption that each block of the matrix, that is to say each piece of the data processed separately, is uncorrelated from the others. Certainly within each day, this assumption is false; weakly false, but false nonetheless. However, em pirical evaluation indicates that this is a small effect which is further reduced when multiple days are combined. Monte Carlo simulations indicated that only a few weeks of data would be needed to separate and estimate the eight daily and subdaily OLT components. Pragmatically however, concerns about multipath, diurnal and seasonal environmental variations local to each site, and other unmodelled effects demanded that longer time spans be used. These con cerns were validated by two trial projects (Schenewerk et al., 1995, Schenewerk et al. 1999) indicating that several months of data would be needed if OLT parameters for all sites were

3 Vertical Ocean-loading Deformations Derived from a Global GPS Network 239 Fig. 1. The meaning of arrow symbols. to be estimated. Although explicitly unnecessary, data from every third day from , were included thereby averaging over three entire seasonal cycles and sidereal years. This extended time span also permitted the self-consistent estimation of a reference frame, expressed as station coordinates and velocities, satellite orbits, and EOPs, as well as OLT parameters. Data from 353 sites were included, more than half of which are located in North America providing a relatively dense network on that continent suitable for more detailed evaluation of this technique and results. Phase ambiguities were not fixed to their integer values, and the phase ambiguity estimates were discarded as nuisance parameters. Satellite orbits, EOPs and two hour, piece-wise, linear zenith wet delay corrections were estimated, but also discarded as nuisance parameters once all data from a day were combined. All station coordinates, velocities, and OLT parameters were estimated and the portion of the matrix containing these parameters was saved for further processing. These Helmert blocks, derived from three years of GPS data, ultimately were combined into this single, self consistent solution. 3. Results These OLT results, when displayed graphically, uniquely use a single arrow symbol to show both amplitude and phase relative to the driving potential (refer to Figure 1). The length of the arrow corresponds to the amplitude of the signal, i.e. the longer the arrow, the larger the amplitude. The orientation of the arrow indicates the phase. Arrows pointing straight up, the "12 o'clock" position, imply 0 phase lag. Arrows at 90 clockwise from straight up, the "3 o'clock" position, imply a +90 phase lag, and so forth. Similarly, when differences between these estimates and a model are shown, the length of the arrow represents the absolute value of the difference in magnitude, the orientation of the arrow represents the difference in phase. Remember that these arrows do not show vector displacements on the surface of the Earth, but rather the amplitude and phase of a periodic vertical signal. Figure 2 shows OLT estimates corresponding to the M2 or principle lunar semi-diurnal tide, typically the largest signal at a site. More useful is Figure 3 showing the differences

4 240 Mark S. Schenewerk, Fig. 2. Fig. 3. GPS derived GPS J. Marshalland derived and William Dillinger M2 OLT estimates. M2 OLT estimates minus Schwiderski model values. between these estimates and OLT values derived from the Schwiderskitide model which will be used as a standard for comparison. While most sites show a good match between the estimate and model, 90% agree with the Schwiderski model amplitudes by 5 mm or less, a few regions do not: sites around the Gulf of Alaska and Hudson Bay in North America, near the Drake Passage between South America and Antarctica, and, to a lesser extent, sites in or near the Malaysian Archipelago and the Strait of Gibraltar. Similar comparisons to OLT values from newer tide models, which incorporate more complete data, give better results, and so the poor matches cited here represent extreme cases. Table 1 summarizes the comparison of the GPS derived estimates to five common OLT models (Francis, 1999) for 112 sites along the coasts of North America. Sites in the interior of North America have virtually no tidal signal and are excluded from these comparisons. The overall mean and standard deviation of the differences are listed by model and tidal

5 Vertical Ocean-loading Deformations Derived from a Global GPS Network 241 Table 1. Comparisons of common OLT models to GPS derived values. component. Comparing the amplitude difference standard deviations for the M2 component, for example, shows that the newer models do indeed provide improved OLT values when compared to those derived from the Schwiderski model. Table 1 also reveals two limitations of the GPS derived OLT parameters. Note that the K2 and K1 parameters compare poorly against all models. These tides have periods very close to a GPS satillite's orbital period and twice that period respectively. It is believed that strong aliasing occurs in the processing between the K2 and K1 parameters and the satellite orbits or, perhaps, other signals such as site multipath which will repeat day to day with this same characteristic period. This aliasing effectively makes it impossible to estimate these OLT signals from GPS observations. Fortunately, the covariance matrix generated from this processing shows little correlation between these and the other tidal parameters implying no deleterious effects from retaining the K2 and K1 terms in the solution. Also note that the Q1 phase difference standard deviations are significantly larger than other tidal terms although the amplitudes seem to compare well. The Q1 and K2 signals are the smallest of the OLT signals considered here, typically 1 mm or less in amplitude. The GPS derived estimates correctly give near zero amplitudes for the Qi signal, but contain no phase information. This implies a sensitivity limit from this technique for tidal signals of approximately 1 mm. 4. Summary Ocean-loading vertical deformation parameters corresponding to eight semi-diurnal and diurnal tides were computed for 353 globally distributed, permanent, GPS tracking sites using the GPS data themselves. The overall comparison of these results to the Schwiderski and other, newer models is good for the M2, N2, S2, 01, and P1 signals with amplitude dif ferences 5 mm for 90% of the locations. Estimates for the K2 and K1 tidal signals compare

6 242 Mark S. Schenewerk, J. Marshalland and William Dillinger poorly because of aliasing between these terms and GPS satellite orbits and related signals. The Q1 estimates are appropriately small but contain no phase information implying a sen sitivity limit of approximately 1 mm for this technique. Universally small signal amplitudes and negligible correlations between the K2 and K1, Q1, and other tidal terms indicate that retaining the model values, or, in the future, constraining these three parameters to their a priori values or omitting them from estimation altogether should not incur unacceptable errors. Estimates for sites at high latitudes or in regions with complex coasts compare more poorly to model values, probably because of limitations in the models. For example, sites located on the Gulf of Alaska coast have 3 cm to 5 cm tidal signals with 1 cm to 2.4 cm residual signal remaining after correction using the Schwiderski model. This should be considered an extreme example, however. Recent, more sophisticated ocean-loading models based upon newer, more complete data compare better. Acknowledgment This project would have been impossible without the Crustal Dynamics Data Information System at NASA's Goddard Space Flight Center, the Scripps Orbit and Permanent Array Center, and the National Geodetic Survey's CORS data center. All figures were created using the Generic Mapping Tools (Wessel and Smith, 1991). References Cartwright, D. E. and R. J. Tayler (1971): New Computations in the Tide-generating Potential, Geophys. J. R. astr. Soc., 23, Cartwright, D. E. and A. C. Edden (1973): Corrected Tables of Tidal Harmonics, Geophys. J. R. astr. Soc., 33, Francis, 0. (1999): personal communication. Neill, A. E. (1996): Global Mapping Functions for the Atmosphere Delay at Radio Wavelengths, J. Geophys. Res., 101, Saastamoinen, J. (1972): Atmospheric Correction for the Troposphere and Stratosphere in Radio Ranging of Satellites, in The Use of Artificial Satellites for Geodesy, Geophys. Monogr., 15, Schenewerk, M., W. Dillinger and S. Hula (2000): On-line Documentation for the PAGES Suite of Processing Software, Schenewerk, M. S., G. L. Mader and T. M. vandam (1995): An Estimate of Ocean-Loading Effects on Baseline Repeatability from GPS Observations, IGS Workshop Proceedings, "Special Topics and New Directions", May 15-18, 1995, Potsdam, Germany, GeoForschungsZentrum, Schenewerk, M., J. Marshall, W. Dillinger and N. Weston (1999): Vertical Ocean Loading Deformations Derived from a Global GPS Network, EOS, Trans., Amer. Geophys. U., 80, supp., 262. Schwiderski, E. (1983): Atlas of Ocean Tidal Charts and Maps, Part I: The Semidiurnal Principal Lunar Tide M2, Marine Geod., 6, Wessel, P. and W. H. F. Smith (1991): Free Software Helps Map and Display Data, EOS, Trans., Amer. Geophys. U., 72, 441,