M. S. Schenewerk, J. Marshall, and W. Dillinger
National Geodetic Survey
1315 East-West HWY
Silver Spring, MD, 20910, USA
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.
This information can also be found at the NGS web page, http://www.ngs.noaa.gov listed as a Project under Geosciences Research.
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. For example, sites along the east coast of North America rise and fall 2 centimeters twice per day driven by the principal, semidiurnal lunar tide.
Models of the deformation caused by ocean-loading from tides (OLT) can suffer from significant errors in some locations because of:
In these more troublesome locations, direct measurement may be the most feasible means of generating OLT parameters.
- the lack of detail on the shape of the ocean floor and currents,
- limitations in the grid size used to approximate the coast.
Given the "explosion" of GPS sites, GPS would seem to be the most practical tool. NGS' fundamental GPS processing tool, the PAGES software (Schenewerk et al., 2000), was modified to optionally also estimate vertical amplitude and phase parameters for the most commonly significant OLT signals (refer to Table 1). When data from many days are processed and combined, these subtle, periodic signals are reinforced while other "noise" averages to zero.
Here, only the eight diurnal and semidiurnal tidal terms will be discussed.
Table 1: Tidal Harmonics Included in this Project
|M2||Principal lunar||12.42 h|
|S2||Principal solar||12.00 h|
|N2||Major lunar elliptical||12.66 h|
|K2||Luni-solar declinational||11.97 h|
|O1||Principal lunar||25.82 h|
|P1||Principal solar||24.07 h|
|Q1||Major lunar elliptical||26.87 h|
|K1||Luni-solar declinational||23.93 h|
|Mf||Lunar fortnightly||13.66 d|
|Mm||Lunar monthly||27.55 d|
|Ssa||Solar semiannual||182.62 d|
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.
Data from every third day from 1997 - 1999, were included thereby averaging over several 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.
HELMERT BLOCKING STRATEGY
The large amount of data to be used in this project and the large number of parameters needed for a strict least squares adjustment required that the problem be divided into manageable sized pieces using the Helmert blocking technique. The data were broken into:
Creation of matrix pieces, or Helmert blocks, in this manner enables the processing of a large data set in logical or convenient subsets without sacrificing the complete, self-consistent solution.
Separately, each subnetwork for each day was processed with PAGES using the following models and options:
The following were estimated:
The matrix pieces from each day were combined using GPSCOM, also developed at the NGS, creating a daily solution. At this point the phase ambiguities, neutral atmosphere, GPS satellite orbits and EOPs were discarded as "nuisance" parameters by Gaussian elimination.
The portion of the matrix containing all station positions, velocities and OLT parameters was saved for further processing.
These daily pieces were then combined into yearly and multiyearly solutions like the one presented here.
The 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 degrees phase lag. Arrows at 90 degrees clockwise from straight up, the "3 o'clock" position, imply a +90 degree 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 between these estimates and OLT values derived from the Schwiderski tide 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 Straight 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.
Summary figures showing the OLT estimates and differences to the Schwiderski derived values for all sites.
Table 2: GPS - Model Statistics for 112 North American Sites
The poorer performance (presumably by the models) is accentuated in Table 3, which shows similar statics for the M2 term only with the sites grouped into latitude bands.
Table 3: GPS - Model Statistics for 112 North American Sites Grouped into Latitude Bands
|20 - 30 deg||0.14+/-0.13||0.15+/-0.11||0.12+/-0.12||0.13+/-0.12||0.22+/-0.22|
|30 - 40 deg||0.31+/-0.14||0.26+/-0.20||0.32+/-0.22||0.32+/-0.25||0.26+/-0.15|
|40 - 50 deg||0.36+/-0.23||0.44+/-0.20||0.45+/-0.20||0.43+/-0.16||0.40+/-0.15|
|50 - 60 deg||0.99+/-0.41||1.25+/-0.52||1.18+/-0.44||0.96+/-0.41||1.48+/-0.65|
|60 - 70 deg||0.64+/-0.33||0.81+/-0.42||0.68+/-0.32||0.70+/-0.40||1.00+/-0.53|
|20 - 30 deg||12.1+/-46.3||2.6+/-44.5||10.9+/-44.9||9.7+/-44.3||5.5+/-45.0|
|30 - 40 deg||18.0+/-28.4||16.4+/-24.6||17.8+/-26.5||17.5+/-26.0||11.6+/-27.7|
|40 - 50 deg||23.2+/-12.8||25.1+/-14.3||16.7+/-13.8||14.3+/-16.0||12.2+/-12.0|
|50 - 60 deg||-1.7+/-27.8||4.2+/-23.9||10.4+/-36.9||0.0+/-25.2||4.5+/-31.4|
|60 - 70 deg||1.0+/-17.8||5.0+/-16.5||4.7+/-16.7||4.6+/-15.7||4.5+/-14.7|
Table 2 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 satellite orbits and the K2 and K1 parameters effectively making 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 Q1 signal, but contain no phase information. This implies a sensitivity limit from this technique for tidal signals of approximately 1 mm.
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, O1, and P1 signals with amplitude differences 5 mm for 90% of the locations. 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.
Estimates for the K2 and K1 tidal signals compare poorly because of aliasing to the GPS satellite orbits.
The Q1 estimates are appropriately small but contain no phase information implying a sensitivity limit of approximately 1 mm for this technique.
Negligible correlations between the K2 and K1, Q1, and other tidal terms indicate that retaining these parameters did not result in unacceptible errors for the other estimated parameters. However, when applying an OLT correction, it is recommended that the K2, K1, and Q1 model values be used rather than these GPS estimates.
GPS derived OLT values from a preliminary 1994 - 1999 solution.
Cartwright, D. E. and R. J. Taylor (1971): New computations in the tide-generating potential, Geophys. J. R. Astron. Soc., 23, 45-74.
Cartwright, D. E. and A. C. Edden (1973): Corrected tables of tidal harmonics, Geophys. J. R. Astron. Soc., 33, 253-264.
Francis, O. (1999): personal communication.
Neill, A. E. (1996): Global mapping functions for the atmosphere delay at radio wavelengths, J. Geophys. Res., 101, 3227-3246.
Ray, R. D. and E. J. O. Schrama (1997): personal communication.
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. Ser. 15 edited by S. W. Henriksen et al., AGU, Washington, D.C., 247-251.
Schenewerk, M., W. Dillinger, and S. Hilla (2000): On-line documentation for the PAGES suite of processing software, http://www.ngs.noaa.gov/GRD/GPS/DOC/toc.hmtl.