198
Chapter 11: Stochastic Modeling oE Precipitation
probability that the growing season precipitation will be less than the threshold required for crop survival. Likewise, the design of small flood protection
structures, particularly in urban areas, requires knowledge of what engineers
term a probability intensity duration relationship. A probability intensity
duration relationship is simply the family of prob ability distributions of the
annual maximum precipitation for duration D where D might take on, for
example, values of 1, 2, 3, 6, 12, and 24 hours.
Stochastic models represent the joint probability density function of precipitation, where (discrete) time ti, i = 1 ... n is the index, that is
(11.2)
(11.3)
Stochastic models of precipitation were originally developed to address
practical problems of data simulation, particularly for water resource systems
design and management in data-scarce situations, and to aid in understanding
the probabilistic structure of precipitation. However, they have important
applications for local interpretation of large area climate simulations, such as
are produced by General Circulation Models. Some of these applications are
discussed in Section 11.5. First, however, some important probabilistic and
stochastic characteristics of precipitation are described.
11.2 Probabilistic Characteristics
of Precipitation
The probabilistic characteristics of precipitation are strongly dependent on
time and space scales. At short time scales (e.g., hourly to daily), precipitation is intermittent, that is, precipitation amounts do not have a continuous
prob ability distribution. Instead, there is a finite probability of zero precipitation, and a continuous probability distribution for non-zero amounts (in
practice, there is a minimum measurement threshold, typically 0.25 mm, and
larger measurements are in increments of the threshold; nonetheless, non-zero
amounts are usually approximated with a continuous distribution). At longer
time scales (for instance, monthly or annual), precipitation is usually not intermittent, except in some arid areas. In addition to the tendency away from
intermittency, the probability distribution of precipitation amounts tends to
become more symmetrie as the time scale increases. For instance, Figure
11.1 shows empirical prob ability distributions of daily, monthly, and annual
precipitation for West Glacier, Montana, USA. In Figure 11.1, the precipitation amounts have been normalized by the mean for each time scale, and the
abscissa has anormal prob ability scale, so that a straight line corresponds
to a normal distribution. In addition, the slope of the plot is related to the
Chapter 11: Stochastic Modeling oE Precipitation
probability that the growing season precipitation will be less than the threshold required for crop survival. Likewise, the design of small flood protection
structures, particularly in urban areas, requires knowledge of what engineers
term a probability intensity duration relationship. A probability intensity
duration relationship is simply the family of prob ability distributions of the
annual maximum precipitation for duration D where D might take on, for
example, values of 1, 2, 3, 6, 12, and 24 hours.
Stochastic models represent the joint probability density function of precipitation, where (discrete) time ti, i = 1 ... n is the index, that is
(11.2)
(11.3)
Stochastic models of precipitation were originally developed to address
practical problems of data simulation, particularly for water resource systems
design and management in data-scarce situations, and to aid in understanding
the probabilistic structure of precipitation. However, they have important
applications for local interpretation of large area climate simulations, such as
are produced by General Circulation Models. Some of these applications are
discussed in Section 11.5. First, however, some important probabilistic and
stochastic characteristics of precipitation are described.
11.2 Probabilistic Characteristics
of Precipitation
The probabilistic characteristics of precipitation are strongly dependent on
time and space scales. At short time scales (e.g., hourly to daily), precipitation is intermittent, that is, precipitation amounts do not have a continuous
prob ability distribution. Instead, there is a finite probability of zero precipitation, and a continuous probability distribution for non-zero amounts (in
practice, there is a minimum measurement threshold, typically 0.25 mm, and
larger measurements are in increments of the threshold; nonetheless, non-zero
amounts are usually approximated with a continuous distribution). At longer
time scales (for instance, monthly or annual), precipitation is usually not intermittent, except in some arid areas. In addition to the tendency away from
intermittency, the probability distribution of precipitation amounts tends to
become more symmetrie as the time scale increases. For instance, Figure
11.1 shows empirical prob ability distributions of daily, monthly, and annual
precipitation for West Glacier, Montana, USA. In Figure 11.1, the precipitation amounts have been normalized by the mean for each time scale, and the
abscissa has anormal prob ability scale, so that a straight line corresponds
to a normal distribution. In addition, the slope of the plot is related to the