All numerical models are subjected to errors and results are hence incomplete. Specifically with respect to tide models there may be many causes for concern, including unknown dissipation parameters [Taguchi et al., 2010] inexact bathymetry, uncertainty in initial and boundary conditions, missing physics, especially in shallow seas [Taguchi, 2004; Setiawan, 2007]. Constraining tide models through geophysical data reduce this deficiency and can help to determine missing physics or uncertain parameters. However conventional data assimilation methods are afflicted with difficulty in handling the rapid growing number of altimeter data and above all in reproducing the non-linear tides in shallow water regions. The tide model HAMTIDE (Hamburg direct data Assimilation Methods for TIDEs) of the Institut für Meereskunde of the University Hamburg was developed to overcome those issues.
It is based on the generalized inverse methods for tides developed at the University of Hamburg [Zahel, 1995]. The principle of the HAMTIDE is the direct minimization of the model deficiency and the inaccuracy of the recordings in a least square sense, resulting to solve the over determined algebraic equations and so called normal equation respectively. The equations are then solved by a memory saving iterative method for the given sparse matrix and the model is corrected simultaneously by inferring the physics from data. The dynamic residuals are then used for the detection of possible model errors such as bathymetry, parameterization of dissipation, loading and self-attraction (LSA) and so on [Taguchi et al., 2010].
The direct method differs from usual initialization problems like quasi-geostrophic ocean model, where the initial conditions are varied as free parameters (controls) while the model is utilized as a strong constraint, and the variational technique is usually used for the solution approach (Fig. I). On the other hand the direct method puts the truth of both dynamics (weak constraint) and data in to question (Fig. I). Hence all variables x (for the whole tidal period) of the model are taken into account and adjusted till the distances of the measurements and the model-counterparts reach the minimum and the exremum of the weighted sum of dynamic and data residuals is achieved.
HAMTIDE is run with spatial resolution 7.5’. Solutions were constrained by data from the past on ongoing satellite altimeter missions as compiled by the DGFI altimeter data bank obtained from 15 years time series observed by TOPEX and Jason-1 [Bosch et al., 2009]. Altimeter data were provided in form of an empirical analysis of cross-calibrated multi-mission satellite altimeter data providing constituents such as M2, S2, N2, K2, K1, O1, Q1, P1 und 2N2.
Amplitude (AMPL) and phase (PHAS) of water elevation ζ(x,y) are denoted by a complex number like: ζ(x,y) = RE + i * IM with RE = AMPL * cos(PHAS) and IM = AMPL * sin(PHAS) and they are given in the following order:
LON LAT RE IM AMPL PHAS
324.000 83.625 0.4718 3.1347 3.17 81.44