Some statistical considerations associated with the data assimilation of precipitation observations

Bayes's theorem is applied to the problem of analysing temperature and mois ture in a volume of air given a single observation of precipitation amount, utilizing a model of non-convective precipitation and prior estimates of t he fields. Results using different statistics and shapes of probability dis tributions are examined. These include normal, truncated normal, and log no rmal distributions with special treatment of the value zero. The uncertaint y of the model's formulation is considered in addition to uncertainty of ob servations. The posterior distribution is multi-modal due to the model's fo rmulation using a conditional expression. The dominant mode may be predicte d as a non-precipitating state by the model, although the observation indic ates precipitation is present. Means and modes of posterior distributions d epend sensitively both on the assumed statistics and the shapes of the unde rlying distributions. The results suggest that the usual minimization of a cost-function should not be used cavalierly to assimilate precipitation obs ervations.