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Simple stochastic models Rt to time series of daily precipitation amount have a marked tendency to underestimate the observed (or interannual) variance of monthly (or seasonal) total precipitation. By considering extensions of one particular class of stochastic model known as a chain-dependent process, the extent to which this "overdispersion" phenomenon is attributable to an inadequate model for high-frequency variation of precipitation is examined. For daily precipitation amount in January at Chico, California, fitting more complex stochastic models greatly reduces the underestimation of the variance of monthly total precipitation. One source of overdispersion, the number of wet days, can be completely eliminated through the use of a higher-order Markov chain for daily precipitation occurrence. Nevertheless, some of the observed variance remains unexplained and could possibly be attributed to low-frequency variation (sometimes termed "potential predictability"). Of special interest is the fact that these more complex stochastic models still underestimate the monthly variance, more so than does an alternative approach, in which the simplest form of chain-dependent process is conditioned on an index of large-scale atmospheric circulation.