A model-based approach to the computation of area probabilities for precipitation exceeding a certain threshold

In order to provide consistent weather warnings it is essential to estimate the probabilities for certain weather events occurring. There are a number of operational numerical and statistical methods for estimating the probability that precipitation occurs at a fixed location (a point probability). However, there is a growing interest in the computation of probabilities for precipitation occurring somewhere in a geographical region (an area probability). An example would be the area of responsibility of a fire department, where an emergency occurs when there is intense rain anywhere within that area. The derivation of such area probabilities is difficult and no applicable method for their computation is known. In literature, some theoretical relationships between point and area probabilities are discussed under very simplifying assumptions. Furthermore, several stochastic models for precipitation cells have been proposed with none of them being suitable for our purposes due to restrictions such as e.g. spatial and temporal stationarity or inappropriate (or missing) model fitting. Recently, we proposed a (less restrictive and more robust) model-based approach to the computation of area probabilities for the occurrence of precipitation. This method is based on a non-stationary stochastic model for precipitation cells (basically a Cox germ-grain model with circular grains), which is characterized by a sequence of local intensities for the formation of precipitation cells and a cell radius. Furthermore, we proposed an approach for the algorithmic computation of all model characteristics from a sequence of point probabilities and described how area probabilities can be computed by repeated simulation of that model. In the present paper, we discuss an extension of this model by additionally including precipitation amounts. The available data contains sequences of point probabilities for precipitation exceeding various thresholds. At first, a gamma distribution is fitted to obtain moments of point precipitation amounts. Then, we briefly recall our model for the representation of precipitation cells. The most important and most recent results address the spatial stochastic modeling of precipitation amounts, which is done by assigning a response function to each precipitation cell. The response functions are then multiplied by random scaling factors and summed up to obtain precipitation amounts. We derive formulas for the expectation and variance of random point precipitation amounts in our model. Comparing the computed moments with those obtained from the data allows us to fit the distributions of the random scaling variables. Finally, area probabilities for the occurrence of precipitation exceeding a given threshold are estimated by repeated simulation of the precipitation model. We evaluate our results by analyzing how the choice of the response function and the distribution of scaling variables influence estimated point and area probabilities. The novelty of the presented approach is that, for the first time, a widely applicable estimation of area probabilities is possible, which is based solely on predicted point probabilities (i.e., no precipitation observations are needed). Furthermore, the method works in a quite reasonable computation time, which makes it suitable for applications in modern weather prediction.