Rainfall estimation using raingages and radar - A Bayesian approach: 1. Derivation of estimators

Procedures for estimating rainfall from radar and raingage observations are constructed in a Bayesian framework. Given that the number of raingage measurements is typically very small, mean and variance of gage rainfall are treated as uncertain parameters. Under the assumption that log gage rainfall and log radar rainfall are jointly multivariate normal, the estimation problem is equivalent to lognormal co-kriging with uncertain mean and variance of the gage rainfall field. The posterior distribution is obtained under the assumption that the prior for the mean and inverse of the variance of log gage rainfall is normal-gamma 2. Estimate and estimation variance do not have closed-form expressions, but can be easily evaluated by numerically integrating two single integrals. To reduce computational burden associated with evaluating sufficient statistics for the likelihood function, an approximate form of parameter updating is given. Also, as a further approximation, the parameters are updated using raingage measurements only, yielding closed-form expressions for estimate and estimation variance in the Gaussian domain. With a reduction in the number of radar rainfall data in constructing covariance matrices, computational requirements for the estimation procedures are not significantly greater than those for simple co-kriging. Given their generality, the estimation procedures constructed in this work are considered to be applicable in various estimation problems involving an undersampled main variable and a densely sampled auxiliary variable.