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Application of Evidence Theory to Quantify Uncertainty in Forecast of Hurricane Path

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Uncertainties and errors in computational results on hurricane forecasts originate from various sources -- failure of a climate model to describe correctly the atmospheric physics and the interaction between the atmosphere and the ocean; stochastic nature of model parameters; errors associated with the discretization and algorithmic approximations, to mention just a few. Depending on the origin, the uncertainties can be categorized as aleatory (random, stochastic) and epistemic (due to lack of knowledge) uncertainties. In reality, the interaction among the uncertainty sources and the lack of knowledge make it generally impossible to identify and separate the sources to quantify their individual contribution to the total forecast uncertainty. In the present study, we define an appropriate measure to quantify the total uncertainty in forecasts of the given climate model, computer code and grid. Such a measure would allow one not only to compare quantitatively the performance of different climate models, but also evaluate the effectiveness of modifications introduced in a single model.

As both aleatory and epistemic uncertainties are intricately interwoven in a hurricane forecast, one needs a statistical theory that could handle them together to quantify their impact on forecasts. Whereas the probability theory could deal with the aleatory uncertainty and the possibility theory with epistemic uncertainty, the evidence theory [1] provides a systematic framework for such a study, and it is relatively well developed among other related theories. Unfortunately, there are very few practical applications [2] of evidence theory and they differ considerably from the one addressed here. Previously, we developed [3] an approach based on evidence theory to quantify uncertainty in turbulence computations. Results of testing the approach in a turbulent flow encourage us to apply similar approach to hurricane path forecasts with appropriate extension and modification.

Evidence theory provides the necessary tools not only to quantify the forecast uncertainty, but also to fuse the results of different forecasts. The idea of improving the overall credibility of hurricane path predictions by combining results of several forecasts is not new. Multimodel superensemble technique [4] is an example of the successful implementation of the idea. However, multimodel forecasts like the single model forecasts do not provide information on the forecast accuracy. The present approach provides the quantitative assessment of the forecast accuracy and differs completely from other multimodel techniques in its mathematical foundation.

Briefly stated, the procedure involves first quantifying the uncertainty in results using each of several climate models under controlled conditions, where observation data for hurricane paths are available. Uncertainties in the spatial coordinates describing the hurricane position are quantified separately and assumed to be time dependent. This information is then combined with the results of simulations using each model to forecast a hurricane path for which observational data are not available. For each model at each instant of the forecast, we construct a grid centered on the model prediction of the hurricane position. Each grid interval in latitude and longitude directions is characterized by the degree of support (or belief) that the real hurricane position falls within the interval. The degree of support is the measure of model uncertainty. Its interval values are found during the previous step of the procedure. Then, the results from all models are fused using Dempster's rule of evidence theory to create a new prediction. The final prediction is also characterized by degrees of support. The details of the procedure will be discussed in the final paper.

The database for hurricanes of 1998-2001 in the Pacific Ocean is used to quantify uncertainty in forecasts by global models from two operational centers -- the U.S. Navy Operational Global Atmospheric Prediction System (NOGAPS) and European Centre for Medium-Range Weather Forecasts (ECMRWF). The performance of these two models is compared for each year. We also track the effectiveness of annual modifications in each model. The data for hurricanes of year 2000 are used to evaluate the approach we developed to fuse different forecasts. Observational data for three hurricanes from the South Pacific region, three hurricanes from the East Pacific region, and six hurricanes from the West Pacific region are used solely for the evaluation of the quality of predictions obtained with the new technique. The data for other hurricanes of year 2000 are used to quantify the model uncertainty during the step preceding forecasts fusing.

Authors express their gratitude to Professor T. N. Krishnamurti and Dr. V. Kumar (Department of Meteorology, Florida State University) for providing observational and model data necessary for this study.

References 1. Shafer, G., “A Mathematical Theory of Evidence,” Princeton, NJ: Princeton University Press, 1976. 2. Oberkampf, W. L., Helton, J. C., and Sentz, K., “Mathematical Representation of Uncertainty,” AIAA 2001-1645, 2001. 3. Poroseva, S. V., Hussaini, M. Y. & Woodruff, S. L., “On Improving the Predictive Capability of Turbulence Models Using Evidence Theory”, AIAA-2005-1096, 2005. 4. Williford, C. E., Krishnamurti, T. N., Torres, R. C. et al., “Real-Time Multimodel Superensemble Forecasts of Atlantic Tropical Systems of 1999,” Monthly Wetaher Rev., v. 131, 2003, pp. 1878-1894.