A Hidden Markov Model for Rainfall Using Breakpoint Data

Abstract

Pluviographs, which are rainfall accumulation?timeplots, indicate a strong tendency for rainfall intensity to abruptly change from one steady rate of fall to another with these steady rates persisting for some time. Digitizing from pluviographs the times of change from one steady rain rate to another yields breakpoint data, that is, a stream of data pairs consisting of the rainfall rate, which includes zero, and the duration of that rate. Breakpoints provide a complete record of rainfall with information on the rain rates and their durations during periods of continuous steady precipitation and on the durations of dry periods. In a hidden Markov model (HMM), the state of the process at a given time is not known; only the values of the observables, and the range of possible states, are known. For rainfall, there is a hierarchy of states: a precipitation event is either taking place, or not; if one is, then there are episodes when the mechanism is convection (showers) and when it is large-scale uplift (rain); and finally, the current rate of rainfall and its duration will have particular values with periods of zero rate being the dry periods within an episode of a particular mechanism. Thus, there are five states: the time between events when no precipitation is possible, showery times when a shower is taking place, showery times when no shower is taking place, rain times with rain taking place, and dry intervals during a rainy time. Such a model was initially fitted using the expectation maximization (EM) algorithm, but the parameters were reestimated using HMM fitting procedures, which also provided estimated probabilities of the transition matrix. The Viterbi algorithm was used to classify the individual points in the data stream. The rate and duration distributions? parameters, the state transition probabilities, and the classification of the data accord with the view that during widespread rain there may be many changes of rain rate but little dry time, while during showers, shorter periods of steady precipitation tend to be interspersed with longer dry periods. Discrepancies were found between the data and simulations made using the HMM?s estimated parameters. The major of these was that the simulated dwell times within an episode were shorter than in the data, and that the simulated number of episodes per event was greater. Merely restricting certain transitions did not increase the dwell times, but some indications were found that it might be necessary to either change to a hidden semi- Markov model and/or increase the number of states.