Abstract
The problem of deriving tidal fields from observations by reason of incompleteness and imperfectness of every data set practically available has an infinitely large number of allowable solutions fitting the data within measurement errors and hence can be treated as ill-posed. Therefore, interpolating the data always relies on some a priori assumptions concerning the tides, which provide a rule of sampling or, in other words, a regularization of the ill-posed problem. Data assimilation procedures used in large scale tide modeling are viewed in a common mathematical framework as such regularizations. It is shown that they all (basis functions expansion, parameter estimation, nudging, objective analysis, general inversion, and extended general inversion), including those (objective analysis and general inversion) originally formulated in stochastic terms, may be considered as utilizations of one of the three general methods suggested by the theory of ill-posed problems. The problem of grid refinement critical for inverse methods and nudging is discussed.
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References
Andersen, O. B., Woodworth, P. L. and Flather, R. A.: 1995, 'Intercomparison of recent tide models', J.Geophys.Res. 100, 25, 261–25,282.
Bennett, A. F. and McIntosh, P. C.: 1982, 'Open ocean modelling as an inverse problem: Tidal theory', J.Phys.Oceanogr. 12, 1004–1018.
Bennett, A. F.: 1985, 'Array design by inverse methods', Progr.in Oceanogr. 15, 129–156.
Bennett, A. F.: 1990, 'Inverse methods for assessing ship-of-opportunity networks and estimating circulation and winds from tropical expendable bathythermograph data', J.Geophys.Res. 95, 16, 111–16, 148.
Bennett, A. F.: 1992, Inverse Methods in Physical Oceanography, Cambridge University Press, Cam-bridge.
Cartwright, D. E., Edden, A. C., Spencer, R. and Vassie, J. M.: 1980, 'The tides of the northeast atlantic ocean', Phil.Trans.Roy.Soc.London 298, 87–139.
Cartwright, D. E. and Ray, R. D.: 1990, 'Ocean tides from geosat altimetry', J.Geophys.Res. 95, 3069–3090.
Cartwright, D. E. and Ray, R. D.: 1991, 'Energetics of global ocean tides from geosat altimetry', J.Geophys.Res. 96, 16, 897–16, 912.
Dushaw, B. D. Cornuelle, B. D. Worcester, P. F. Howe, B. M. and Luther, D. S.: 1995, 'Barotropic and baroclinic tides in the central North Pacific Ocean determined from long-range reciprocal acoustic transmissions', J.Phys.Oceanogr. 25, 631–647.
Egbert, G. D., Bennett, A. F. and Foreman, M. G. G.: 1994, 'TOPEX/POSEIDONtides estimated using a global inverse model', J.Geophys.Res. 99, 24, 821–24, 852.
Egbert, G. D. and Bennett, A. F.: 1996, 'Data assimilation methods for ocean tides', in P. Malanotte-Rizzoli (ed. ), Modern Approaches to Data Assimilation in Ocean Modeling, Elsevier Press, Amsterdam, pp. 147–179.
Eknes, M. and Evensen, G.: 1997, 'Parameter estimation solving a weak constraint variational formulation for an Ekman model', J.Geophys.Res. (in press).
Gjevik, B., N¨ost, E., and Straume, T.: 1994, 'Model simulation of the tides in the Barents sea', J.Geophys.Res. 99, 3337–3350.
Gjevik, B. and Straume, T.: 1989, 'Model simulation of the M2 and the K1 tide in the Nordic seas and the arctic ocean', Tellus 41A, 73–96.
Gotlib, V. Yu. and Kagan, B. A.: 1980, 'Spectrum of free oscillations in the World Ocean,' (in Russian), Dokl.AN SSSR 262, 974–977.
Gotlib, V. Yu. and Kagan, B. A.: 1981, 'Numerical simulation of tides in the World Ocean. 1. Parame-terization of the shelf effects', Dt.Hydrogr.Z. 34, 273–283.
Gotlib, V. Yu. and Kagan, B. A.: 1983, 'A reconstruction of the spatial structure of diurnal tides in the World Ocean with the use of the eigen functions of the Laplace tidal operator', (in Russian), Okeanologiya 23, 538–542.
Filloux, J. H., Luther, D. S., and Chave, A. D.: 1991, 'Update on seafloor pressure and electric field observations from the North-Central and Northeastern Pacific: tides, infratidal fluctuation, and barotropic flow', in B. Parker (ed. ), Advances in Tidal Hydrodynamics, John Willey, New York, pp. 617–640.
Foreman, M. G. G., Delves, L. M., Barrodale, I., and Henry, R. F.: 1980, 'On the use of the Proudman-Heaps tidal theorem', Geophys.J.Roy.Astron.Soc. 63, 467–478.
Hadamard, J.: 1932, Le probl` eme de Cauchy et les ´ equations aux d´ eriv´ ees partielles hyperboliques. Paris, Hermann.
Hansen, P. C.: 1992, 'Analysis of discrete ill-posed problems by means of the L-curve', SIAM Rev. 34, 561–580.
Hendershott, M. C.: 1973, 'Ocean tides', Trans.Amer.Geophys.Union 54, 76–86.
Jourdin, F., Francis, O., Vincent, P. and Mazzega, P.: 1991, 'Some results of heterogeneous data inversion for oceanic tides', J.Geophys.Res. 96, 20,267–20,288.
Kantha, L. K.: 1995, 'Barotropic tides in the global oceans from a nonlinear tidal model assimilating altimetric tides, I. Model description and results', J.Geophys.Res. 100, 25, 283–25,309.
Kagan, B. A., Kivman, G. A. and Voronkov, K. L.: 1995, 'Modeling global tides with data assimilation by means of a combined adjoint-inverse method', Ann.Geophys. 13, Part II, Suppl.II, 228.
Kagan, B. A. and Kivman, G. A.: 1995, 'A necessary condition for representativeness of island tidal measurements', J.Geophys.Res. 100, 11,047–11,050.
Kivman, G. A.: 1996, 'Assimilation of sea-level data into a global tidal model by means of the generalized method of residuals', (in Russian), Okeanologiya 36, 835–841.
Kivman, G. A.: 1997, 'Weak constraint data assimilation for tides in the the arctic ocean', submmitted to Prog.Oceanogr.
Knudsen, P.: 1994, 'Global harmonic degree models of the seasonal variability and residual ocean tides from TOPEX/POSEIDON altimeter data', J.Geophys.Res. 99, 24, 643–24,656.
Koblinsky, C. J., Gaspar, P., and Lagerloef, G.: 1992, The Future of Space-born Altimetry: Oceans and Climate Change, Joint Oceanographic Institution Inc., Washington, D. C.
Kowalik, Z. and Proshutinsky, A. Yu.: 1995, 'Topographic enhancement of tidal motion in the Western Barents sea', J.Geophys.Res. 100, 2613–2637.
Lattes, R. and Lions, J.-L.: 1967, M´ ethode de Quasi-R´ eversibilit´ e et Applications, Dunod, Paris.
Le Provost, C., Genco, M. L., Lyard, F., Vincent, P. and Canceil, P.: 1994, 'Spectroscopy of the World Ocean tides from a finite-element hydrodynamic model', J.Geophys.Res. 99, 24, 777–24,797.
Le Provost, C., Bennett, A. F., and Cartwright, D. E.: 1995, 'Ocean tides for and from TOPEX/POSEIDON', Science 267, 639–642.
Luyten, J. R. and Stommel, H. M.: 1991, 'Comparison of M2 tidal currents observed by some deep moored current meters with those of Schwiderski and Laplace models', Deep Sea Res. 38, Suppl. 1, pp. S573–S589.
Lyard, F. and Genco, M. L.: 1994, 'Optimization methods for Bathymetry and open boundary conditios in a finite element model of ocean tides', J.Comput.Phys. 114, 234–256.
Mazzega P.: 1985, 'M2 model of the global ocean tide derived from SEASAT altimetry', Marine Geodesy 9, 335–363.
Mazzega, P. and Jourdin, F.: 1991, 'Inverting SEASAT altimetry for tides in the Northeast Atlantic: Preliminary results', in B. Parker (ed. ), Advanances in Tidal Hydrodynamics, John Wiley, New York, pp. 569–592.
McIntosh, P. C., and Bennett, A. F.: 1984, 'Open ocean modelling as an inverse problem: M2 tides in Bass strait', J.Phys.Oceanogr. 14, 601–614.
Munk, W. H. and Zetler, B. D.: 1967, 'Deep sea tides: A program', Science 158, 884–886.
Platzman, G. W. Curtis, G. A., Hansen, K. S., and Slater, R. D.: 1981, 'Normal modes of the World Ocean: Part II. Description of the modes in the period range 8-80 hours', J.Phys.Oceanogr. 11, 579–603.
Proudman, J.: 1918, 'On the dynamical equations of the tides', Proc.London Math.Soc., Ser. 2, 18, 1–68.
Proudman, J.: 1925, 'A theorem in tidal hydrodynamics', Phil.Mag. 49, 570–579.
Sanchez, B. V. and Cartwright, D. E.: 1988, 'Tidal estimation in the pacific with application to SEASAT altimetry', Marine Geodesy 12, 81–115.
Sanchez, B. V.: 1991, 'Proudman functions and their application to tidal estimation in the World Ocean', in B. Parker (ed. ), Advanances in Tidal Hydrodynamics, John Wiley, New York, pp. 27–39.
Sanchez, B. V., R. D. Ray and D. E. Cartwright: 1992, 'A Proudman-function expansion of the M2 tide in the mediterranean sea from satellite altimetry and coastal gauges', Oceanology Acta 15, 325–337.
Schwidersky, E. W.: 1980, 'Ocean tides, II, A hydrodynamic interpolation model', Marine Geodesy 3, 219–255.
Tarantola, A.: 1987, Inverse Problem Theory.Methods for Data Fitting and Model Parameter Esti-mation, Elsever, Amsterdam.
Ten Brummelhuis, P. G. J., Heemink, A. W., and van den Boogaard, H. F. P.: 1993, 'Identification of shallow sea models', Int.J.Numer.Methods Fluids 17, 637–665.
Tikhonov, A. N. and Arsenin, V. Ya.: 1977, Solution of Ill-Posed Problems. W. H. Winston and Sons, Washington, D. C.
Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., and Yagola, A. G.: 1990, Numerical Methods for the Solution of Ill-Posed Problems (in Russian), Nauka, Moskow.
Voronkov, K. L. Kagan, B. A., and Kivman, G. A.: 1996, 'A global tidal model with assimilation of coastal sea-level data', (in Russian), Izv.Ross.Akad.Nauk, Fizika Atmosph.& Okeana 32, 630–634.
Wahba, G.,: 1990, Spline Methods for Observational Data, Society for Industrial and Applied Math-ematics, Philadelphia.
Woodworth, P. L. and Cartwright, D. E.: 1986, 'Extraction of the M2 ocean tide fromSEASAT altimeter data', Geophys.J.Astron.Soc. 84, 227–255.
Zahel, W.: 1991, 'Modeling ocean tides with and without assimilating data', J.Geophys.Res. 96, 20,79–20,391.
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Kivman, G.A. Assimilating data into open ocean tidal models. Surveys in Geophysics 18, 621–643 (1997). https://doi.org/10.1023/A:1006535821489
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DOI: https://doi.org/10.1023/A:1006535821489