Section 11.3,' Stochastic Models 0/ Precipitation
201
11.3 Stochastic Models of Precipitation
11.3.1 Background
As noted above, the main characteristic that distinguishes precipitation time
scales in the context of stochastic modeling is intermittency. Our interest
here is primarily in short time scales, at which distinct wet and dry periods
are apparent in the record. Early attempts to model the stochastic structure
of the precipitation arrival process (wetj dry occurrences) were based on first
order homogeneous Markov chains (e.g., Gabriel and Neumann 1957; 1962),
which essentially state that the precipitation state (wet or dry) prob ability
on day t depends only on the state on days t - i, i = 1 ... k, where k is the
order of the Markov chain (often one). The probabilities of state p, p = 1,2
(wet or dry) occurring on day t given the state on day t - i, i = 1 ... kare
the transition probabilities.
Various extensions of Markov models have been explored to accommodate inhomogeneity (such as seasonality) ofthe transition probabilities (e.g.,
Weiss, 1964; Woolhiser and Pegram, 1979; Stern and Coe, 1984) and to incorporate and discrete amounts intervals (e.g., Khanal and Hamrick, 1974;
Haan, 1976). Markov models have generally fallen from favor, however, because they are unable to reproduce the persistence ofwet and dry speIls (that
is, they underestimate the prob ability of long runs of wet or dry sequences,
also known as clustering) that is observed in rainfall occurrence series at daily
or shorter time intervals (Foufoula-Georgiou, 1985).
Much of the recent activity in stochastic precipitation modeling, starting
with the work of Kavvas and Delleur (1981), has sought to develop andjor
apply more advanced point process models that represent clustering [see, for
example, Rodriguez-Iturbe et al. (1987), Foufoula-Georgiou and Lettenmaier
(1987), Smith and Karr (1985), and others). Most of this re cent work, which
is reviewed in Georgakakos and Kavvas (1987), is based on a branch of probability known as point process theory (e.g., LeCam, 1961). Markov chain
and point process models are similar to the extent that they are restricted
to single-station applications, and are not easily generalizable to multiple
station applications, at least without (in the case of Markov chain models)
explosive growth in the number of parameters. In addition, all of the above
models describe the precipitation process unconditionally, that is, they do
not incorporate cause-effect information, such as descriptors of large-scale
meteorological conditions that might give rise to wet, or dry, conditions.
11.3.2 Applications to Global Change
Recent interest in assessments of the hydrologie effects of climate change has
placed different demands on stochastic precipitation models. Much of the
concern about global warming has been based on simulations of climate produced by global general circulation models of the atmosphere (GCMs). These
201
11.3 Stochastic Models of Precipitation
11.3.1 Background
As noted above, the main characteristic that distinguishes precipitation time
scales in the context of stochastic modeling is intermittency. Our interest
here is primarily in short time scales, at which distinct wet and dry periods
are apparent in the record. Early attempts to model the stochastic structure
of the precipitation arrival process (wetj dry occurrences) were based on first
order homogeneous Markov chains (e.g., Gabriel and Neumann 1957; 1962),
which essentially state that the precipitation state (wet or dry) prob ability
on day t depends only on the state on days t - i, i = 1 ... k, where k is the
order of the Markov chain (often one). The probabilities of state p, p = 1,2
(wet or dry) occurring on day t given the state on day t - i, i = 1 ... kare
the transition probabilities.
Various extensions of Markov models have been explored to accommodate inhomogeneity (such as seasonality) ofthe transition probabilities (e.g.,
Weiss, 1964; Woolhiser and Pegram, 1979; Stern and Coe, 1984) and to incorporate and discrete amounts intervals (e.g., Khanal and Hamrick, 1974;
Haan, 1976). Markov models have generally fallen from favor, however, because they are unable to reproduce the persistence ofwet and dry speIls (that
is, they underestimate the prob ability of long runs of wet or dry sequences,
also known as clustering) that is observed in rainfall occurrence series at daily
or shorter time intervals (Foufoula-Georgiou, 1985).
Much of the recent activity in stochastic precipitation modeling, starting
with the work of Kavvas and Delleur (1981), has sought to develop andjor
apply more advanced point process models that represent clustering [see, for
example, Rodriguez-Iturbe et al. (1987), Foufoula-Georgiou and Lettenmaier
(1987), Smith and Karr (1985), and others). Most of this re cent work, which
is reviewed in Georgakakos and Kavvas (1987), is based on a branch of probability known as point process theory (e.g., LeCam, 1961). Markov chain
and point process models are similar to the extent that they are restricted
to single-station applications, and are not easily generalizable to multiple
station applications, at least without (in the case of Markov chain models)
explosive growth in the number of parameters. In addition, all of the above
models describe the precipitation process unconditionally, that is, they do
not incorporate cause-effect information, such as descriptors of large-scale
meteorological conditions that might give rise to wet, or dry, conditions.
11.3.2 Applications to Global Change
Recent interest in assessments of the hydrologie effects of climate change has
placed different demands on stochastic precipitation models. Much of the
concern about global warming has been based on simulations of climate produced by global general circulation models of the atmosphere (GCMs). These