Enhancing tidal prediction accuracy in a deterministic model using chaos theory S.A. Sannasiraj a, * , Hong Zhang b , Vladan Babovic c , Eng Soon Chan d a Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600036, India b School of Engineering, Griffith University, Gold Coast Campus, QLD 4215, Australia c Tectrasys AG, Sihleggstrasse 23, Wollerau 8832, Switzerland d Tropical Marine Science Institute, National University of Singapore, Singapore 119223, Singapore Received 22 January 2003; received in revised form 18 March 2004; accepted 25 March 2004 Available online 9 June 2004 Abstract The classical deterministic approach to tidal prediction is based on barotropic or baroclinic models with prescribed boundary conditions from a global model or measurements. The prediction by the deterministic model is limited by the precision of the prescribed initial and boundary conditions. Improvement to the knowledge of model formulation would only marginally increase the prediction accuracy without the correct driving forces. This study describes an improvement in the forecasting capability of the tidal model by combining the best of a deterministic model and a stochastic model. The latter is overlaid on the numerical model predictions to improve the forecast accuracy. The tidal prediction is carried out using a three-dimensional baroclinic model and, error correction is instigated using a stochastic model based on a local linear approximation. Embedding theorem based on the time lagged embedded vectors is the basis for the stochastic model. The combined model could achieve an efficiency of 80% for 1 day tidal forecast and 73% for a 7 day tidal forecast as compared to the deterministic model estimation. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Embedding theorem; Genetic algorithm; Tidal forecasting; Local model; Time delay 1. Introduction In ship navigation and harbour operations, tidal information is extremely important. In many cases, tidal prediction of a forecast horizon from 6 h to a few days would be exceptionally helpful in the finalization of work schedules inside harbours or, indeed, for many other coastal activities. The current practice of tidal prediction is undertaken by either using a tidal predic- tive deterministic model or by a time series forecasting model. Each of the above two approaches has its own capabilities of prediction. Numerical models can predict the physics of the tidal movement. However, even if the tidal flow governing system of equations can model the prediction framework with good aptness, there are many factors that diminish the prediction capability of the model. These delimiting factors are the initial conditions and external forcing, such as boundary fluxes and sur- face driving wind forces. If the initial state of the system and boundary conditions were not predicted with a good accuracy, the prediction at any later time would become questionable irrespective of the higher prog- nostic capability of the model. These classes of problems which are sensitive to initial and boundary conditions, thus, show chaotic behaviour in its predictive state variables. Alternatively, statistical forecasting models can be either linear such as auto-regressive integrated moving average (ARIMA) models or nonlinear models based on chaos. However, the linear models would fail to retain higher accuracy for more than a few hours in the fore- cast horizon in the class of nonlinear dynamical prob- lems and chaotic systems. Different classes of methods are called upon to bring out the chaotic behaviours. The nonlinear models based on chaos theory, such as neural networks and embedding theorem, are based on the correlation between different state variables of the * Corresponding author. Fax: +91-44-22578625. E-mail addresses: sasraj@iitm.ac.in (S.A. Sannasiraj), hong.zhang@griffith.edu.au (H. Zhang), vb62@bluewin.ch (V. Babo- vic), tmsdir@nus.edu.sg (E.S. Chan). 0309-1708/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2004.03.006 Advances in Water Resources 27 (2004) 761–772 www.elsevier.com/locate/advwatres