where,? where Fi is a prediction by model i and the weights wi are normalized so that their sum is unity. For equally reliable models, wi = 1/N.

Table 1: A contingency table of forecast performance. Letters H, M, F, R represent the total numbers of occurrences of each contingency in the verification sample.

A way of defining the weights associated with the different models making up the multi-model ensemble is to relate the weights to any kinds of skill scores (ai) of interest at each grid point.

As an example, ai can be defined as the subtraction of the false alarm rate FARi from the hit rate HRi for the ith model over the training period.

where the HR and the FAR are defined from a 2 X 2 contingency table shown in Table 1 for forecasts of a binary event as follows:

The contingency table is a useful concept for verifying the occurrence of rainfall. In the Table 1, H (hits) denotes the frequency of correctly predicted occurrences, F (false alarms) the frequency of forecasts where no rain occurred, M (misses) the frequency of rain occurrences that were incorrectly predicted, and R (non-events) the frequency of non-rain occurrences correctly predicted. When analyzing forecasts for the presence of rain in the correct location, it is desirable to obtain a high HR in conjunction with a low FAR. Empirically, the best choice for is usually 0.5. An alternative way of defining the weights is that ai in the equation (3.3) is defined as the Brier skill score at each point during the training period. This approach gives a different (better) weight to each model in terms of its corresponding skill score. The probabilistic multi-model ensemble forecast, therefore, depends chiefly on how to define . It must be noted that the weights wi are varying in space, that is, the different models have varying relative contributions to the total probability depending on the spatial location of the point in question. Another possible way to construct probabilistic multi-model ensemble forecasts can be achieved as follows; first, develop a set of deterministic multi-model ensemble forecasts from each model separately, which will give member multi-model ensemble forecasts, Si. We use these Si in place of Fi in the equation (3.2) in order to make a probability of precipitation (POP).