測地 学会 誌,第47巻,第1号 Journal of the Geodetic Society of Japan (2001),243-248頁 Vol. 47, No. 1, (2001), pp. 243-248 GOTIC2: A Program for Computation of Oceanic Tidal Loading Effect Koji Matsumoto , Tadahiro Sato, Takashi Takanezawa and Masatsugu Ooe National Astronomical Observatory (Received October 16, 2000; Revised January 27, 2001; Accepted January 27, 2001) Abstract Oceanictidal loading signals are in general smallerthan solid Earth tide signals, but they can be measuredby geodetictechnique and need to be accuratelycorrected for because they are in general a sourceof noise. In order to meet this requirementwe have developeda program to compute the loading tide most accurately in Japanese regionby using a combinationof global and regionalocean tide modelsand fine-scale land-sea grids. The program computes six kinds of loading (radial and horizontal displacements,gravity, tilt, strain, and deflectionof the vertical) for 21 constituents includinglong-period tide. 1. Introduction It has been well known that the loading of the Earth by ocean tides disturbs the mea surements of the solid Earth tide. Thus accurate loading determination is important for studies of the Earth tides. Not only for the Earth tide study, the loading tides may not be negligible in modern geodetic measurements with increasing precision such as absolute gravimetry, super-conducting gravimetry, satellite altimetry, GPS, VLBI and so on. Accu rate loading tidal correction is one of the important factors for deep interpretation of such geodetic measurements. There has been a FORTRAN program for computing such loads named GOTIC (program for Global Oceanic Tldal Correction) which was developed by Sato and Hanada (1984). GOTIC computes loads based on Farrell's (1972) convolution integral in which needed are an ocean tide model, a land-sea data base, and a mass loading Green's function. Schwiderski (1980) model, covering 11 constituents, is used as the input tide model. The loading tide outputs from GOTIC can be improved by using state-of-the-art ocean tide model and fine- scale land-sea data base. Because the local tides generally contribute much of the load for sites near coast, it is preferable to combine global and regional ocean tide models. We redesigned GOTIC to be most accurate in Japanese region by incorporating both global ocean tide model and regional one around Japan and by using fine-scale computational meshes generated from high-resolution digital elevation map. The new program is more straightforward to use and can predict tides in time domain. In this paper we describe the outline of GOTIC2, a revised version of GOTIC, and input ocean tide models. 244 Koji Matsumoto, Tadahiro Sato, Takashi Takanezawa and Masatsugu Ooe 2. Methodology Ocean load tide L is given by a global integral where p is the ocean water density, H is the ocean tidal height , GL is the mass-loading Green's function of interest (displacements, gravity, tilt, strain, and deflection of the verti cal), and T is the combination of trigonometric functions of azimuth (a) which is necessary to compute a vector or tensor load. H and L are complex-valued. There are two choices of the Green's function for GOTIC2, one is given by Farrell (1972) based on Gutenberg-Bullen A model and the other is given by Endo and Okubo (personal communication, 1983) based on 1066A Earth model. GL is function of the angular distance (ƒÓ) between the estimation point with coordinates (colatitude, longitude) = (9', A') and the loading point (9, A). For example, the Green's function for radial displacement is expressed as where Me and R are mass and radius of the Earth (assumed spherical), h'n is loading Love number, and P, is the Legendre function of degree n. Fig. 1. Green's function for radial displacementgiven by Farrell (1972). Note that the Green's function value is multipliedby Rq51012where R is radius of the Earth and q is angular distance. Shown in Figure 1 is CD given by Farrell (1972), which indicates that the Green's function generally has larger absolute value with smaller angular distance. It means that loading contribution from adjacent sea is of importance. Then there may be three points for accurate loading estimation, in particular when the estimation point is located at near coast; (1) accurate ocean tide model in the coastal region (see section 3), (2) high-resolution land-sea data base (see section 4), and (3) integration of Green's function in equation (1). We integrate Green's function over the finite area dS in (1) when q5is smaller than 30 degrees. If dS is small enough we may approximate the Green's function as F(ƒÓ) CL(ƒÓ) =a+bƒÓ+cƒÓ2 (3) •¬ G0TIC2: A Program for Computation of Oceanic Tidal Loading Effect 245 where the normalizing factor F(ƒÓ) = RƒÓfor displacements and gravity, (RƒÓ)2 for tilt and deflection of the vertical (strain is computed from Green's function of displacements; see Sato and Hanada (1984)). Then the loading contribution from a ocean grid centered at (9s, A~) with size of DB by Da is expressed as By approximating the integration area in equation (4) as a rectangle area of size ƒ¢x by Dy, we can project the spherical coordinates on a plane which contacts with the Earth at the center of the rectangle area (xe, yc). For example, equation (4) for radial displacement (F(q) = RƒÓ, T(a) = 1) is transformed into•¬ The explicit expression of the integral (5) can be found in Sato and Hanada (1984) as well as those for other load components. Total loading effect is finally obtained by summing up ƒ¢ Lt over the whole grids. 3. Ocean Tide Models For the loading computation Schwiderski (1980) model had been used as a standard ocean tide model until mid 1990's, but this model was not sufficient for accurate load calculation due to its coarse resolution (1•‹) and error in open ocean. Since the launch of satellite altimeter TOPEX/POSEIDON (T/P) in 1992, many global ocean tide models are developed and the accuracy of ocean tide model has significantly improved in the open ocean, but the models are problematic in shallow water and coastal region (e.g., Shum et al., 1997). This is partly due to the fact that T/P ground track spacing is too wide (2.83•‹ at equator) to resolve short-wavelength feature of tides in shallow water. In order to improve tides in shallow seas Matsumoto et al. (2000) applied fine-scale along- track tidal analysis to 5 years of T/P data to conserve the high-wavenumber component of ocean tide. The T/P tidal solution are then assimilated into barotropic hydrodynamical model. As for ocean self-attraction/loading effect, the hydrodynamical model does not use classical linear approximation which does not hold in the coastal region, but uses one rigorously estimated from spherical harmonic expansion of global ocean tidal field. They developed a global ocean tide model (NAO.99b model) and a regional model around Japan (NAO.99Jb model) for major 16 constituents in diurnal and semi-diurnal bands, i.e., M2, S2, N2, K2, 2N2, u2, v2, L2, T2, K1, 01, P1, Q1, M1, 001, and J1. The regional model has higher resolution (5 arc minutes) than the global model (0.5•‹) and assimilates 219 Japanese coastal tide gauge data in addition to T/P data. The accuracy assessment using independent 80 coastal tide gauge data ensures better accuracy of NA0.99Jb model than •¬ 246 Ko ji Matsumoto, Tadahiro Sato, Takashi Takanezawa and Masatsugu Ooe other global models, for example, Matsumoto et al. (2000) reports that RMS misfits of MZ constituent are 1.648 cm for NAO.99Jb, 3.968 cm for NAO.99b, 4.911 cm for CSR4.0 (improved version of Eanes and Bettadpur,1994), and 4.173 cm for G0T99.2b (Ray, 1999). GOTIC2 is designed to combine the global and the regional model so that loading estimates around Japan is more accurate than any other region on the globe. GOTIC2 can handle with long-period ocean tide model of Takanezawaet al. (2001). This purely hydrodynamical (no T/P data are assimilated) global model is named as NAO.99L model. The loading tides for 5 constituents (Mtm, Mf, Mm, Ssa, Sa) can be estimated by GOTIC2 based on this model. GOTIC2 covers therefore 21 constituents in total. 4. Land-sea Data Base As was mentioned in section 2, high-resolution land-sea data base is one of the impor tant factor for the accurate loading estimation. This is particularly of importance when estimation point is near coast or in a small island. GOTIC2 uses land-sea data base which is generated from 50m-resolution digital elevation map, covering all over Japan, compiled by Geographical Survey Institute (GSI) Japan. GOTIC2 reads this data base when the est- Table 1. Size of computational meshes used in GOTIC2. Fig. 2. Example of land grids describing the Miyake-jima. These grids are generated from 50m resolution digital elevation map compiled by Geographical Survey Institute Japan. The map shown here consists of 3rd- and 4th-order meshes. GOTIC2: A Program for Computation of Oceanic Tidal Loading Effect 247 imation point is located inside Japan, otherwise reads land-sea grid derived from the coast- line data base of Wessel and Smith (1996). We use four different-sizedgrids in the loading computation, the smaller grid is used for the smaller angular distance. Table 1 summarize the size of the grids and Figure 2 shows the example of the grids depicting the Miyake-jima, a small island of about 9 km diameter. 5. Importance of Regional Ocean Tide Model Hatanaka et al. (2001) analyzed GPS data of Japanese dense network and showed that the correction by GOTIC2 effectivelyeliminates loading tide signals in displacements.