The most accurate determinations of the global ocean tides are currently based on altimeter measurements made by the Topex/Poseidon satellite. The error spectrum corresponding to the M₂ tidal solution is here estimated, primarily by inverse methods and secondarily by simple differencing in flatter than tidal signal spectrum, and it exceeds 10% of the signal at spherical harmonic degree 15 and above. The tide error also exceed the anticipated sensitivity of the upcoming GRACE gravity mission for all degrees below 40, and possibly below 50.Copyrighted by American Geophysical Union

Eanes, R. J.

Egbert, Gary D.

Pavlis, N. K.

Ray, Richard D.

Undetermined

ScholarsArchive@OSU

GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 1, PAGES 21-24, JANUARY 1,2001 Error spectrum for the global M2 ocean tide R. D. Ray, • R. J. Eanes, 2 G. D. Egbert, 3 and N. K. Pavlis 4 Abstract. The most accurate determinations of the global The tidal error spectrum is estimated here by two com- ocean tides are currently based on altimeter measurements plementary methods: by computing differences in existing made by the Topex/Poseidon satellite. The error spectrum models and by estimating errors through global inverse cal- corresponding to the M2 tidal solution is here estimated, pri- culations where reasonably realistic a priori error covari- marily by inverse methods and secondarily by simple differ- ances are employed. Because we are interested in the effect encing of several of the best tidal models. The tidal error of tide model errors on the gravity field as determined by spectrum is flatter than the tidal signal spectrum, and it ex- ceeds 10% of the signal at spherical harmonic degree 15 and above. The tide errors also exceed the anticipated sensitiv- ity of the upcoming GRACE satellite gravity mission for all degrees below 40, and possibly below 50. 1. Introduction satellite geodesy, the error is expressed in terms of equiva- lent geoid height error, as was recently done by Wahr et al. [1998]. Two possible systematic errors in the conversion to tidal geoid variations are also examined: the common as- sumptions of a constant seawater density and of a spheri- cal earth. In this note we confine ourselves to the dominant semidiurnal lunar tide M2. Since the launch of the Topex/poseidon satellite in 1992 considerable progress has been made in our knowledge of the global ocean tides [e.g., Le Provost et al., 1995]. This progress has been quantified in several ways: by numer- ous comparisons with in situ tide determinations, primarily with coastal tide gauges at small islands and bottom pres- sure recorders on the sea floor; by variance reduction tests with independent satellite altimeter data; and by compar- isons with tidal estimates from satellite tracking [e.g., An- dersen et al. , 1995; Shum et al., 1997; Desai et al. , 1997]. (Tests of tidal current velocities from some of the new tide models have also been made by Dushaw et al. [ 1997] and by others, although tidal currents are not the topic of the present paper.) Another approach to understanding accuracies of ocean tide models is to determine their error spectra--that is, their error amplitudes as a function of the degree n of a spherical harmonic expansion. In addition to characterizing model ac- curacies, reliable error spectra have many applications [e.g., Tsaoussi and Koblinsky, 1994]. One application (prompting the present study) involves estimating the effect of aliased tide model errors on monthly determinations of the Earth's gravity field as anticipated to occur during the upcoming GRACE satellite mission. The first step of this work is to determine the error spectrum of the best available models. 1 Laboratory for Terrestrial Physics, NASA Goddard Space Flight Center, Greenbelt, Maryland. 2Center for Space Research, University of Texas, Austin, Texas. 3College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon. 4Raytheon ITSS, Greenbelt, Maryland. Copyright 2001 by the American Geophysical Union. Paper number 2000GL011674. 0094-8276/01/2000GL011674505.00 2. Formalism We employ the spherical harmonic normalization com- monly used in physical geodesy [Heiskanen and Mortiz, 1967], where the Legendre functions Pnm(cos O) satisfy .;_• [Pnm(•t)] 2 dtt - 2(2 - Sm,0). 1 The contribution of the ocean tides to the geoid undulation N is expressed as •SN(O, qo, t)- a • • Pnm(cos0)[•SCnmcOsmqo+•SSnmsinmqo] n= 1 m=0 (1) where a is the earth's mean radius and (O, 9) are spherical polar coordinates. The time-varying Stokes coefficients are computed from the tidal elevation •(O, q0, t) by [cf Wahr et al., 1998] { •Cnm(t) } a2pw(l +k•n) f/ { cosmq0 } Snm(t) -- Me(2n+ 1) •Pnm sinmq0 dg• (2) where Pw is the mean density of the ocean, Me is the mass of the earth, and k•n is a loading Love number of degree nwhich accounts for the geoid perturbation from ocean tidal loading of the solid earth. Define the Stokes coefficients •Cnm and •Snm as the root mean squares over a complete tidal cycle T (12.42 hours) of the time-varying coefficients. Then the tidal contribution to the variance of the geoid, taken over the entire globe, is 1 //f•N2dtd•--a2E(•2nmq-•nnm). 4r•T We take the amplitude spectrum to be the quantity •Nnm-a(•2nmq-•2nm) 1/2 (3) 21 22 RAY ET AL.' ERROR SPECTRUM FOR THE M2 TIDE The degree amplitude spectrum is [Wahr et al., 1998] (Z aXn n --2 •Nn -- a •Cnm q- rn \m-O (4) 3. Tidal error spectra Figure 1 shows the spectrum of the ocean tide computed from three recent global models of the M2 ocean tide. (The figure strictly shows the tide elevation spectrum, not the geoid spectrum; the two are related by the factor preceding the integral in (2).) The three models are all deduced from Topex/Poseidon altimetry using different methods of analy- sis, including (for TPXO.4) formal data assimilation into a hydrodynamic model. All are improvements to earlier tide models that Shum et al. [1997] found were among the most accurate of those available for the entire global ocean. Solu- tion GOT99.2 is described by Ray (1999); CSR4.o is similar in design but different in some details, including the method used for tidal analysis. Assimilation solution TPXO. 4 is an update to that described by Egbert et al. [1994] and Egbert [ 1997]. Both CSR4.O and GOT99.2 default to a purely hydro- dynamic solution [Le Provost et al., 1994] in polar regions above the Topex inclination of 66 ø, while TPXO.4 extends polewards via its own linearized hydrodynamic model. The three spectra are clearly in close agreement, although careful examination reveals minor differences. The largest variation in the spectra is a function of degree n, not order m. The cor- responding degree amplitude spectrum is shown as the top curve in Figure 2 (based on TPXO.4). The degree amplitude spectra of the tide model errors, based on different approaches to estimating the errors in the coefficients •SC, m and •SS, m, are shown as the colored curves labeled 1-4 in Figure 2. ']['hey stem from two complementary methods. 3.1 Error estimates from inverse methods The assimilation solution TPXO. 4 can be viewed as an op- timal interpolation or smoothing of the T/P altimeter data, using a covariance determined by the Laplace tidal equa- tions (LTE) and prior statistical assumptions about the er- rors in these equations [Egbert et al., 1994]. Posterior errors can thus be calculated to characterize uncertainty in the esti- mated fields arising from incomplete altimeter coverage and errors in the data. We use a Monte Carlo approach [Dushaw et al., 1997], which we summarize here. A series of ran- dom realizations of the LTE forcing error (which accounts for approximations in the equations, bathymetry errors, and numerical grid truncation) are generated using the prior co- variance derived in Egbert et al. [ 1994]. For each realization i the perturbed LTE are solved numerically to yield synthetic tidal elevations •i, i = 1,..., N. (For the results reported here we use N -- 20.) These elevation fields are each sampled with the spatial and temporal pattern of the altimeter, random data errors (representative of noise and non-tidal oceanogra- phy in the T/P data) are added, and the synthetic data are inverted (using the same procedure applied to real T/P data to generate TPXO.4) for estimates •i. The differences •i- •i are representative of errors in the assimilation estimates of tidal elevations. The error fields are found to be largest in the polar seas (beyond the satellite data coverage) and in the shallow seas surrounding Indonesia. The root mean square amplitudes of spherical harmonic coefficients of the N error fields are used to estimate the tidal error spectrum. Parameters of the prior covariance can be adjusted so that the amplitude and spatial structure of the random realiza- tions •i are grossly consistent (in amplitude and spatial struc- ture) with the T/P data, but it must be admitted that signifi- cant uncertainties remain about the "true" prior covariance. We have thus performed calculations for several different (consistent) prior covariances to test the sensitivity of our conclusions to poorly constrained parameters. Curves 1 and 2 in Figure 2 give tidal error amplitude spectra calculated using decorrelation length scales of 300 and 600 km, re- spectively, for the LTE forcing errors. The two curves are in good agreement for degrees less than 10. For higher de- grees the shorter dynamical error decorrelation length scale results (quite sensibly) in somewhat larger errors. But dif- ferences between the two cases are not so great; both cross the GRACE sensitivity curve near degree 50. GOT99.2 • O -," TPXO.4a • Figure 1. Spherical harmonic spectra of three recent global ocean tide models as a function of degree n and order m. Vertical axis represents pectral amplitude in sea surface height. RAY ET AL.: ERROR SPECTRUM FOR THE M2 TIDE 23 10 ø - E ß •. 10 '•- . E o 10 =, . . 10 • - ., 0 10 20 30 40 50 60 degree Figure 2. Degree amplitude geoid spectra of the global M2 tide, various estimates of the present-day M2 errors, and the anticipated sensitivity from 3 months of data from the GRACE satellite gravity mission. Error curves are shown in color and are numbered as follows: (1) inverse estimate corresponding to TPXO. 4 assuming a 300-km decorrelation length for the dynamical error covariance; (2) same except with 600-km decorrelation scale; (3) estimate from the dif- ference in models TPXO. 4 and CSR4.o; (4) estimate from the difference in models CSR4.o and CSR3.0; (5) error caused by the neglect of earth flattening; (6) error caused by use of a constant density for seawater, with the larger error employ- ing density 1025 kg m -3 and the smaller the more realistic 1035 kg m -3. derestimate the real errors since identical methods of anal- ysis were used, and indeed curve 4 falls below curves 1-3 everywhere except for the very lowest degrees where it rises above curves 1 and 2. Except for these very low degrees, curves 3 and 4 are consistently below curves 1 and 2. 3.3 Other systematic errors Curves 5 and 6 in Figure 2 explore the effect of two possi- ble systematic errors in the estimates of the tidal geoid vari- ations. For this purpose, both assume that there is no error at all in the tidal elevations • (which are taken from model TPXO.4). Curve 5 shows the error caused by the neglect in Eq. (2) of the earth's flattening. Specifically, it is the spec- tral difference between a harmonic expansion based upon a spherical earth surface (radius 6371 km) and one based on a "serrated" ellipsoidal surface [Jekeli, 1981 ]. The error is seen to peak around degrees 5 to 7. Curve 6 (actually comprising two curves) shows the er- ror caused by the common assumption of constant seawater density, which allowed Pw in Eq. (2) to be outside the double integral. The error in such assumption has been computed by including Pw within the spatial integration where it then rep- resents the average density within the water column at any location. This density has been computed from the Levims et al. [ 1994] climatological densities. The two error curves correspond to assuming Pw is either a constant 1025 kg m -3 or 1035 kg m -3. The latter is more representative of the true global mean seawater density. The former is more repre- sentative of surface waters, and it induces the larger errors, actually crossing several of the other curves below degree 6. 3.2 Error estimates from model differences A more simplistic approach to estimating the error in the tidal coefficients is by computing the differences among our three adopted models. Since all are based on the same al- timeter data, this approach likely underestimates the errors somewhat, but it is still a useful check on the inverse results. Figure 3 shows the maximum elevation difference be- tween TPXO.4 and CSR4.o, computed at every point of the global ocean. Throughout most of the open ocean where Topex/Poseidon data are of greatest benefit, the differences between these two models are small, generally less than 2 cm. In coastal regions where the tidal wavelengths are shorter and the tide is less easily mapped by relatively widely spaced satellite observations, model differences are signifi- cantly larger. Some of these differences are also in regions of extremely large tidal amplitudes where even a small per- centage error can produce large discrepancies. As expected, model differences are also large in polar latitudes outside the satellite limits; results in these regions rely completely on hydrodynamic modeling and (possibly) a sparse set of tide gauge measurements. Figure 3 certainly suggests locations where future modeling efforts should be directed. Curve 3 in Figure'2 shows the degree amplitude spectrum computed from Figure 3. A similar result is obtained from differences involving the GOT99.2 model. Curve 4 is a spec- trum computed from the difference between CSR4.o and its earlier version CSR3.0. This curve is especially prone to un- 4. Discussion Curves 1-3 in Figure 2 represent our best estimates of the error spectrum for present models of the global M2 ocean 30'$ 80'S 60'E 120'E 180' 120'W $O'W o' 0 i 2 4 6 8 10 20 oll Figure 3. Maximum sea surface height difference between the CSR4.o and TPXO. 4 models of the M2 global ocean tide, in cm. Note the color scale is nonlinear. The maximum difference (in Hudson Strait) exceeds 4 meters. Some basin boundary errors (not in TPXO.4) are discussed by Smith and Andersen [ 1997]. 24 RAY ET AL.: ERROR SPECTRUM FOR THE M2 TIDE tide. Curve 4 is, as expected, an underestimate. The er- ror spectrum is considerably flatter than the signal spectrum. Above degree 15 or so, the error exceeds 10% of the signal. Because curve 5 everywhere falls below curves 1-4 we conclude that errors caused by neglecting the earth's flat- tening are smaller than the errors induced by incomplete knowledge of the tidal elevations •. Until further signifi- cant improvements in ( are forthcoming, the spherical earth approximation appears justified (although the "serrated" el- lipsoid approximation is nearly as simple to implement as the spherical approximation). Similarly, errors induced by using a constant seawater density in (2) can be ignored if one adopts a sufficiently representative density near 1035 kgm-3; a density of 1025 kg m -3 induces errors exceeding curves 1 and 2 when n < 5. In the context of the upcoming GRACE gravity field mis- sion, we observe that the expected GRACE sensitivity curve in Figure 2 crosses above the tidal signal curve only at de- grees above 56. This implies that ocean-tide force modeling on the satellite must employ much higher degree spherical harmonic expansions than is traditionally done in satellite geodesy. More importantly, we observe that present-day tide model errors exceed the expected GRACE sensitivity atall spheri- cal harmonic degrees below about 40 or 50. It is important to acknowledge, however, the temporal mismatch in these curves. The GRACE sensitivity corresponds to a long-term (3-month) estimate of the gravity field; the tidal error curves correspond to mean errors over 12.4 hours. To some extent the effect of the tide errors will be mitigated by temporal averaging. To what extent is still unclear. The tide errors will certainly alias into longer, non-tidal periods (depending on the satellite sampling characteristics). Whether the errors could seriously hinder the estimation of the gravity field at the level of the instrument precision or hinder the geophys- ical interpretation f the gravity field temporal variations ian object of ongoing study. References Andersen, O.B., P. L. Woodworth, and R. A. Flather, Intercompar- ison of recent ocean tide models, J. Geophys. Res., 100, 25261- 25282, 1995. Desai, S. D., J. M. Wahr, and Y. Chao, Error analysis of empirical ocean tide models estimated from Topex/Poseidon altimetry, J. Geophys. Res., 102, 25157-25172, 1997. Dushaw, B. D., G. D. Egbert, P. F. Worcester, B. D. Cornuelle, B. M. Howe, and K. Metzger, A Topex/Poseidon global tidal model (TPXO.2) and barotropic tidal currents determined from long-range acoustic transmissions, Prog. Oceanogr., 40, 337- 367, 1997. Egbert, G. D., Tidal data inversion: interpolation and inference, Prog. Oceanogr., 40, 53-80, 1997. Egbert, G. D., A. F. Bennett, and M. G. G. Foreman, Topex/Posei- don tides estimated using a global inverse method, J. Geophys. Res., 99, 24821-24852, 1994. Heiskanen, W. A. and H. Moritz, Physical Geodesy, Freeman and Co., San Francisco, 1967. Jekeli, C., The downward continuation to the Earth's surface of truncated spherical nd ellipsoidal harmonic series of the gravity and height anomalies, Rep. 323, Dept. Geodetic Science, Ohio State Univ., 1981. Le Provost, C., M. L. Genco, F. Lyard, P. Vincent, and P. Canceil, Spectroscopy f the world ocean tides from a finite element hy- drodynamic model, J. Geophys. Res., 99, 24777-24797, 1994. Le Provost, C., A. F. Bennett, and D. E. Cartwright, Ocean tides for and from Topex/Poseidon, Science, 267, 639-642, 1995. Levitus, S., R. Burgett, and T. Boyer, Worm Ocean Atlas 1994, U.S. Dept. Commerce, Washington, 1994. Ray, R. D., A global ocean tide model from Topex/Poseidon al- timetry: GOT99.2, NASA Tech. Memo. 209478, 58 pp., 1999. Shum, C. K., et al., Accuracy assessments of recent global ocean tide models, J. Geophys. Res., 102, 25173-25194, 1997. Smith, A. J. E. and O. B. Andersen, Errors in recent ocean tide models: possible origin and cause, Prog. Oceanogr., 40, 325- 336, 1997. Tsaoussi, L. S. and C. J. Koblinsky, An error covariance model for sea surface topography and velocity derived from Topex/Posei- don altimetry, J. Geophys. Res., 99, 24669-24683, 1994. Wahr, J., M. Molenaar, and F. Bryan, Time variability of the Earth's gravity field: hydrological nd oceanic effects and their possible detection using GRACE, J. Geophys. Res., 103, 30205-30229, 1998. R. J. Eanes, CSR, Univ. Texas, 3925 W. Braker Lane, Austin, TX 78759-5321 G. D. Egbert, COAS, Oregon State University, Corvallis, OR 97331-5503 N. K. Pavlis, Raytheon ITSS, 7701 Greenbelt Rd., Greenbelt, MD 20770 R. D. Ray, NASA/GSFC, Code 926, Greenbelt, MD 20771; richard.ray @ gsfc.nasa.gov (Received April 9, 2000; accepted October 4, 2000.) 

Error spectrum for the global M₂ ocean tide

https://ir.library.oregonstate.edu/downloads/xk81jq72d

Error spectrum for the global M₂ ocean tide

Abstract

Similar works

Full text

Available Versions

ScholarsArchive@OSU