Section 11.4: Stochastic Precipitation Models
207
fall oeeurrenees, and to model preeipitation amounts explieitly. The weather
state classifieation proeedure was the same as in Bardossy and Plate (1991),
and they retained the assumption of eonditional independenee under ehanges
in the weather state. Rather than modeling the oeeurrenee proeess explicitly,
they modeled a multivariate normal random variable W, negative values of
whieh eorresponded to the dry state, and (a transform of) positive values are
the precipitation amount. Within a run of a weather state, W was assumed
to be lag-one Markov. Spatial eorrelation in the oeeurrenee proeess, and in
the precipitation amounts, is modeled via the first two moments ofW, whieh
were weather state-dependent. The model was applied to 44 stations in the
Ruhr River eatehment. The model was able to reproduee the first two uneonditional moments of rainfall amounts, and preeipitation probabilities, as
weIl as the dry day durations, reasonably weIl at one of the stations (Essen,
also used in the 1991 paper) selected for more detailed analysis.
Wilson et al. (1991) developed a weather classifieation seheme for the
Pacifie Northwest based on cluster analysis of surfaee pressure and 850 mb
temperature over a 10 degree by 10 degree grid mesh loeated over the North
Paeifie and the westem eoast of North Ameriea. A ten-year sequenee of
weather states (1975-84) was formed, and was furt her classified aeeording
to whether or not preeipitation oeeurred at astation of interest. The partitioned weather state vector was then modeled as a semi-Markov proeess.
For wet states, preeipitation amounts were simulated using a mixed exponential model. The wet and dry period lengths were simulated quite weIl,
although some of the weather state frequencies were mis-estimated, espeeially
in summer. The authors noted that the semi-Markov model used a geometrie
distribution for the lengths-of-stay, they suggested that a heavier tailed distribution might be neeessary. The above model is somewhat limited in that
its generalization to multiple stations results in rapid growth in the number
of parameters.
Wilson et al. (1992) explored a slightly different multiple station model,
based on a Polya um structure. Rather than explieitly ineorporating the
wet-dry state with the weather state, they developed a hierarehieal modified model for the rainfall state eonditioned on the weather state and the
wet-dry state of the higher order station(s). In a Polya um, the wet-dry
state is obtained by drawing from a sampie, initially of size N + M, astate,
of whieh N are initially wet, and Mare initially dry. For eaeh wet state
drawn, the sampie of wet states is inereased by n, likewise for eaeh dry state
drawn, the sam pie of dry states is inereased by m, and the original state
drawn is "replaeed". Thus, the Polya um has more persistenee than a binomial proeess, in whieh the state drawn would simply be replaeed, and the
probability of the wet or dry state is independent of the dry or wet period
length. The modifieation to the Polya um (employed by others as weIl, e.g.,
Wiser, 1965) is to replaee the persistent proeess with a binomial proeess onee
a given run (wet or dry period) length w has been reaehed. In addition, the
207
fall oeeurrenees, and to model preeipitation amounts explieitly. The weather
state classifieation proeedure was the same as in Bardossy and Plate (1991),
and they retained the assumption of eonditional independenee under ehanges
in the weather state. Rather than modeling the oeeurrenee proeess explicitly,
they modeled a multivariate normal random variable W, negative values of
whieh eorresponded to the dry state, and (a transform of) positive values are
the precipitation amount. Within a run of a weather state, W was assumed
to be lag-one Markov. Spatial eorrelation in the oeeurrenee proeess, and in
the precipitation amounts, is modeled via the first two moments ofW, whieh
were weather state-dependent. The model was applied to 44 stations in the
Ruhr River eatehment. The model was able to reproduee the first two uneonditional moments of rainfall amounts, and preeipitation probabilities, as
weIl as the dry day durations, reasonably weIl at one of the stations (Essen,
also used in the 1991 paper) selected for more detailed analysis.
Wilson et al. (1991) developed a weather classifieation seheme for the
Pacifie Northwest based on cluster analysis of surfaee pressure and 850 mb
temperature over a 10 degree by 10 degree grid mesh loeated over the North
Paeifie and the westem eoast of North Ameriea. A ten-year sequenee of
weather states (1975-84) was formed, and was furt her classified aeeording
to whether or not preeipitation oeeurred at astation of interest. The partitioned weather state vector was then modeled as a semi-Markov proeess.
For wet states, preeipitation amounts were simulated using a mixed exponential model. The wet and dry period lengths were simulated quite weIl,
although some of the weather state frequencies were mis-estimated, espeeially
in summer. The authors noted that the semi-Markov model used a geometrie
distribution for the lengths-of-stay, they suggested that a heavier tailed distribution might be neeessary. The above model is somewhat limited in that
its generalization to multiple stations results in rapid growth in the number
of parameters.
Wilson et al. (1992) explored a slightly different multiple station model,
based on a Polya um structure. Rather than explieitly ineorporating the
wet-dry state with the weather state, they developed a hierarehieal modified model for the rainfall state eonditioned on the weather state and the
wet-dry state of the higher order station(s). In a Polya um, the wet-dry
state is obtained by drawing from a sampie, initially of size N + M, astate,
of whieh N are initially wet, and Mare initially dry. For eaeh wet state
drawn, the sampie of wet states is inereased by n, likewise for eaeh dry state
drawn, the sam pie of dry states is inereased by m, and the original state
drawn is "replaeed". Thus, the Polya um has more persistenee than a binomial proeess, in whieh the state drawn would simply be replaeed, and the
probability of the wet or dry state is independent of the dry or wet period
length. The modifieation to the Polya um (employed by others as weIl, e.g.,
Wiser, 1965) is to replaee the persistent proeess with a binomial proeess onee
a given run (wet or dry period) length w has been reaehed. In addition, the