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Nonhomogeneous
Hidden Markov Models
Allowing Stochastic Downscaling
of Synoptic Atmospheric Patterns
to Local Hydrologic Phenomena
Peter Guttorp' and James P. Hughes2
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Abstract.- A model for multistation precipitation, conditional on synoptic atmospheric patterns, is presented. It postulates the existence of an
unobserved weather state which serves as a link between the large-scale
atmospheric measures and the small-scale spatially discontinuous precipitation field. The weather state process is assumed to be conditionally
Markov, given the atmospheric data. The rainfall process is then conditionally specified given the weather state. Various parameterizations for
the weather state process and the rainfall process are discussed and a
likelihood based estimation procedure is described. As an example the
model is fit to a twenty station network of rain gauge stations in western
Washington. We discuss the NHMM as a method of relating synoptic
atmospheric data, such as the output of a general circulation model, to
rainfall and other local (sub-grid scale) hydrologic processes, and outline
possible extensions to continental scales.
INTRODUCTION
Stochastic downscaling of GCM data involves building a stochastic model
to relate historical measurements of synoptic circulation to historical regional or
local climate data (e.g. precipitation). Then, using GCM circulation data as input,
the model can be used to produce simulations at the regional or local level. The
strength of this procedure lies in its relative simplicity and computational ease
compared to the difficult task of building a nested GCM. However, the procedure
assumes that the relationships (between circulation and regional climate) identified in the historical data will be preserved under the alternative climate scenarios
produced by the GCM.
Initial attempts at stochastic downscaling utilized "weather state" models
( e g . McCabe et al. (1989), Hay et al. (1991), Wilson et a1.(1992), Bardossy and
Plate (1992)). In these models, each day is explicitly classified into one of a few
1
Depurrrnent of Statistics, Box 354322, University of' Wushington, Seuttle, WA 981 95-4322.
3
Dc~parrmentof Biostututm, University of Wushington, Seuttle, WA.
discrete categories (weather states) based on the synoptic circulation pattern for
that day. Then separate rainfall models are fit conditional on the weather state.
Hughes et al. (1993) developed such a model using historical circulation and rainfall and then applied the results to data from a GCM 2 x CO, run to describe rainfall pattems, streamflow and flood frequency under the altered climate.
A difficulty with the weather state models as defined above is that the
weather states must be explicitly defined a priori. A poor choice of the weather
states will lead to a poor model. As an alternative, Zucchini and Guttorp (1991)
proposed a hidden Markov model (HMM) as a model for precipitation. The HMM
presupposes the existence of a few discrete weather states but it assumes that
these states cannot be observed directly. Instead, only rainfall is observed and the
rainfall is assumed to be conditionally temporally independent given the weather
state. In this model the weather states are effectively defined by the rainfall process. The states chosen are those that separate the rainfall process into different
spatial or temporal pattems. Formally, let R, be the measurement (typically multivariate) of the regional or local process ( e g rainfall occurrence) at time t and S ,
be the weather state at time t. Then the hidden Markov model is defined by the
assumptions
where the notation sf means all values of S , from time 1 to T. The term hidden
refers to the fact that the weather state process, S,, is an unobserved quantity and
is not defined a priori (although the weather state for each day may be identified
after the model is fit). Nonetheless, Zucchini and Guttorp (1991) found that the
circulation patterns associated with particular weather states were interpretable. In
effect, the model acts as an automatic classifier of circulation patterns into
weather states that are associated with particular precipitation patterns.
The major drawback of the HMM approach is that it fails to incorporate
synoptic circulation information in the definition of the weather states. The
weather states are defined solely on the basis of rainfall. Thus, the model cannot
be used for downscaling. However, an extension to the HMM which does incorporate atmospheric information and can be used for downscaling was developed by
Hughes and Guttorp (1994a). They termed their model a nonhomogeneous hidden Markov model (NHMM). Letting X, be the measurement (or summary) of
the atmospheric data at time t, the NHMM is defined by the assumptions
The first assumption says that the local or regional process, R,, is conditionally
temporally independent given the current weather state. In other words, the
regional process is driven by the weather state process. The second assumption
states that the (unobserved) weather state process is Markov with transition probabilities depending on the current values of the atmospheric information. Various
parameterizations are possible for P(S,IS,-, ,X,) and P(R,IS,) and maximum likelihood techniques are available to estimate the parameters. Hughes and Guttorp
(1994a) show that both the explicit weather state models mentioned previously
and the hidden Markov model are special cases of the more general NHMM.
Hughes and Guttorp (1994a) applied the NHMM to 24 years of winter data
from a four station rain gauge network in Washington state using the following
model for P(R, IS,):
This model assumes that the probability of rain at each station depends only on
the weather state and is independent of the other stations. For the network analyzed by Hughes and Guttorp (1994a) this assumption was reasonable since the
rain stations were widely separated. Note that such an assumption does not imply
that the rainfall occurrence processes at the stations are uncorrelated. Hughes and
Guttorp (1994a) found strong (unconditional) correlations, on the order of 0.4 to
0.5, between the stations. This correlation is induced by the common weather
state.
In Hughes and Guttorp (1994b), the NHMM was applied to 21 years of
winter data in a densely packed network of 24 stations in the Puget Sound region
of western Washington state. The assumption of spatial independence (even conditional on the weather state) was no longer tenable. Therefore, the following
model was used for P(R, IS,):
where di, is the distance between stations i and j . The parameters P , measure the
spatial dependence between stations. When P , is positive, stations i and j are
positively correlated (within weather states) and the correlation decreases
inversely with distance. A negative value for P, implies negative correlation
between stations i and j (within weather states). When each P , is 0, equation (6)
reduces to the independence model (equation ( 5 ) ) with ai,y
= log(pi,J(l - pi,)).As
before, maximum likelihood may be used for parameter estimation. However, for
large numbers of stations the constant of proportionality in equation (6) is difficult
to evaluate, and Hughes and Guttorp (1994b) describe an alternative estimation
strategy for that situation.
When applied to the 24 station network mentioned above, and based on
mean sea-level pressure and the north-south gradient of 500 mb geopotential
height, the model identified five weather states. State 1 had a southwesterly flow
over the Puget Sound region, both at the surface and aloft, resulting in high precipitation probabilities everywhere. State 2 had westerly surface flow, but southwest flow aloft. There was substantial rainfall in the north and south, but relatively
low probability of rain in the central part of the region. State 3 was characterized
by southwesterly flow at sea level and westerly flow aloft, with fairly weak surface winds. The stations bordering Puget Sound were in a rain shadow, while
more easterly stations had high precipitation probabilities. State 4 was characterized by a high pressure system over the region, with consequent low precipitation
rates. Finally, state 5 was similar to state 2, but with a more westerly 500 mb flow.
The rainfall pattern was similar to that in state 2.
Statistics of the historical rainfall process (i.e. probability of rain, spatial
and temporal correlations, storm duration distribution) were generally well reproduced although stations at the periphery of the network or isolated stations tended
to exhibit a predicted (simulated) duration distribution which was lighter-tailed
than that which was actually observed.
MODELING PRECIPITATION ON LARGER SCALES
The NHMM has not yet been applied on subcontinental or continental
scales. Some modification of the model will be necessary in this case since, in the
present configuration, St is a finite scalar quantity which is meant to describe the
weather state at time r over a (meteorologically) homogeneous region. One possibility would be to identify K distinct precipitation/meteorological regions and
then allow St to be a K-variate vector where the k-th element corresponds to the
weather state in the k-th region. The model for P(S,IS,-l, X,) therefore becomes a
model for a multivariate Markov process. The simplest such model is
which postulates that each regional process evolves independently, conditional on
the observed atmospheric process, X,. Presumably, one would also want to
assume that the rain stations in each region depend only on that region's hidden
state:
where R('} denotes the set of rain stations in region i. Note that this assumption of
spatial independence between regions does not preclude the use of a model for
spatial dependence within each region.
Unfortunately, it seems unlikely that the situation will be as simple as the
(conditional) regional independence model envisioned in (1). It is possible that
some enhancement in skill can be gained by coupling weather states in each
region with weather states in other regions. For instance, the movement of large
scale storm fronts across regions would suggest that yesterday's weather state in
one region could provide information about today's weather state in another
region. On the other hand, if N =
dim(sk)is the total number of states in the
K-variate process St, it is not computationally feasible to attempt to fit the entire
N ( N - 1) transition matrix, P(S,IS,-, ,X,). A reasonable compromise between
these extremes might be to assume that today's state in region i depends only on
n;=,
yesterday's state in the regions surrounding region i. That is,
where Mi) is read "the neighborhood of i". Basic meteorologic principles and
knowledge of the regions under consideration should be used to determine which
regions constitute the neighborhood of region i.
Another approach to modeling continental scale precipitation would be to
divide the atmospheric variables into region-specific measures (call these x:' for
region i) and (sub)continental measures (call these x:) and to postulate the existence of an unobserved continental scale weather state (call this C,). Under these
assumptions, one might write
In other words, the weather state in region i today depends on the weather state in
region i yesterday, the atmospheric data in region i today and the overall continental weather state today. The probability distribution of the continental weather
state is completely defined by the continental atmospheric variables.
SUMMARY
Simulations of precipitation conditional on a set of atmospheric data can be
used to downscale the output of a GCM into local precipitation, and to assess the
regional and local effects of climate change scenarios (by conditioning the simulations on the output of GCM runs assuming altered climate scenarios). This does,
of course, require the untestable assumption that precipitation reacts to the altered
climate in the same fashion as it does to the historical climate. In other words, we
must assume that the climate change affects only the distribution of weather
states, rather than which weather states are appropriate.
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Hughes, J. P., D. P. Lettenmaier, P. Guttorp (1993) A stochastic approach for
assessing the effects of changes in regional circulation patterns on local precipitation. Water Resour. Res., 29, 3303-33 15.
Hughes, J. P. and P. Guttorp (1994a) A Class of Stochastic Models for Relating
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