Chapter 7: Automated fuzzy rule based classification
of circulation patterns: basis for multivariate
stochastic generating of precipitaion series
jií stehlík, andrás bárdossy
1. Introduction
Much attention has been given to the problem of downscaling surface climate behavior from large scale atmospheric circulation because of the need to deal with future
climate scenarios. Due to the high nonlinearity and influence of local factors present
in the atmospheric system, it is difficult to describe the relationship between atmospheric circulation and surface climate using a deterministic approach. However, from
a statistical point of view there is evidence that surface climate variables depend on
large scale atmospheric circulation.
Several studies have examined the relationships between circulation patterns and
climatic variables. Bürger (1958) studied the relationship between the atmospheric
circulation patterns and mean, maximum and minimum daily temperatures, precipitation amounts and cloudiness using a time series from 1890 to 1950 measured
at four German cities (Berlin, Bremen, Karlsruhe and Munich). He found a good correspondence between climatic variables and atmospheric circulation. Lamb (1972)
stated that even highly varying precipitation is strongly linked to atmospheric circulation. Trenberth (1990) analyzed the importance of atmospheric dynamics in climate
change. He also stated that atmospheric circulation is the major link between regional
change in climate variables.
In CP classification techniques, two main groups of methods can be distinguished
(Yarnal, 1984). The first type of method is the subjective classification. The advantage
of this method is that the knowledge and experience of meteorologists is fully used
in the classification. A major disadvantage is that the results cannot be reproduced,
and the method can only be applied for certain geographical regions. Many subjective classifications have been developed for different regions with different scales: e.g.
Baur et al. (1944) and Hess and Brezowsky (1969) for Central Europe, Lamb (1972) for
the British Isles, Dzerdzeevskii (1968) for the extratropical latitudes of the Northern
Hemisphere.
The second type of CP-classification method is an objective technique. This method
is based on automatized algorithms and allows fast classification, which is especially
necessary for climate change scenarios. Objective classification includes the following
methods: k-means clustering (Wilson et al., 1992), those based on physical quantities (Jenkinson and Collison, 1977), fuzzy classification based on subjectively defined
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jií stehlík — andrás bárdossy
rules (Bárdossy et al., 1995), principal component clustering (Goodess and Palutikof,
1998), principal component analysis coupled with k-means (Bogárdi et al., 1994), and
neural networks methods (Cawley and Dorling, 1996).
The precipitation characteristics of CPs from all of the above mentioned CP classification methods are studied “ex post“. The mean precipitation behavior conditioned
on the CPs is determined after the classification. The objective of the classification
method presented in this paper is to define CPs so that they explain the variability
of precipitation in a locally specific functional form. Therefore, the CPs explain the
dependence between the large scale atmospheric circulation and the surface climate.
The classified CPs are often used for simulating surface variables at smaller scales.
For this purpose it is also possible to use the regression among daily precipitation
characteristics and daily airflow indices, such as vorticity and components of geostrophic air-flow (Wilby et al., 1998 a, b). However, the advantage of the presented CP
classification method is that it is valid for large scale regions and enables meteorological interpretation.
The method presented in this paper is an automated circulation pattern classification
based on fuzzy rules. It is an objective method which takes precipitation for the classification procedure into account. The data used are daily 700 hPa and 500 hPa elevations
and daily precipitation totals for 9 stations in Germany and 21 stations in Greece.
The existence of highly nonlinear relationships among variables governing the
climate system was proved by many studies dealing with chaotic behavior of rainfall
data (e.g. Rodriguez-Iturbe et al., 1987, Sivakumar et al., 1998, Stehlík, 2000). These
nonlinear relationships cause the effect known as sensitive dependence of initial
conditions, which makes long term predictions impossible. Therefore, the problem of
long term rainfall modeling can be solved only in a stochastic way.
The major problem of mathematical modeling of daily precipitation series is the
spatial and temporal variability and clustering effects both in time and space. These
effects have natural causes, such as the movement of atmospheric fronts. Dealing
with the simulation of precipitation time series is important not only from the pure
scientific point of view, but also because of practical consequences in hydrology. The
generated series can be used for making predictions of recurrence intervals of various extremes, such as dry periods or extreme precipitation events. Using generated
extreme precipitation events as input in hydrological models enables one to asses the
hydrological influence. The observed series are often too short or not enough spatially
resoluted for the purposes of hydrological modeling. The precipitation generators
make it possible for one to derive parameters from the observed series and then generate long series with high spatial resolution. Stochastic rainfall simulators can also
provide an input for hydrological models in order to estimate the peak discharges for
design purposes on catchments with only limited available data.
The advantage of the presented model is that it takes into account the spatial correlation among precipitation series. Therefore, it is also suitable for generating areal
precipitation.
The first attempts for precipitation modeling were done by simple ARMA models
for annual precipitation totals. Also, the monthly totals can be modeled with the help
automated fuzzy rule based classification of circulation patterns
of appropriate transformations. In the case of daily rainfall the problem of time intermittence occurs. This cannot be solved by classical time series models because they
assume continuous and mostly normal distribution of variables. However, the daily
precipitation series are discrete-continuous. The same problem also occurs in space.
Since rainfall does not always cover the entire area, spatial intermittence must also be
considered.
Markov models and Markov renewal models are often applied for modeling of precipitation occurence and duration. Chang et al. (1984) used a discrete ARMA model
for modeling sequences of dry and wet days. Wilks (1989) developed a model for
precipitation occurence and amounts with parameters depending on monthly totals.
Review of models for continuous precipitation modeling was done by RodriguezIturbe et al. (1987). A stochastic space-time model for simultaneous modeling of precipitation on several locations was developed by Binark (1979). Wilson et al. (1992)
introduced a precipitation model which takes large scale atmospheric circulation into
account. Bárdossy and Platte (1992) developed a multidimensional stochastic model
for space-time distribution of daily precipitation. The rainfall is linked to atmospheric
circulation patterns using conditional distributions and conditional spatial covariance
functions. The model is a transformed, conditional, multivariate autoregressive model
with parameters depending on the atmospheric circulation pattern. The model was
applied in the catchment of the river Ruhr using the classification scheme of the German Weather Service.
Another problem is downscaling of precipitation under climate change conditions.
Global circulation models (GCM) do not provide realistic precipitation series even
for present climate scenarios. Even if this would be the case, it would not be possible
to generate precipitation time series for smaller regions or selected locations because
of coarse space resolutions of GCM. However, it is possible to downscale precipitation behavior from the GCM air pressure or geopotential height outputs. A lot of
attention has been paid in the last years to the problem of statistical downscaling
from GCM. A review of methods for GCM precipitation downscaling was done by
Giorgi and Mearns (1991). Wilby et al. (1998b) proposed a procedure for comparing
the performance of various precipitation downscaling models.
This paper is organized into three sections. Section 2 describes the models for the
classification of circulation patterns and downscaling of precipitation. Section 3 contains case studies for Germany and Greece together with a brief description of results
for Northern America. The final section consists of a discussion and conclusions.
2. Methodology
2.1 Classification of circulation patterns
Usually the classification of circulation patterns is done on a statistical basis or with
meteorological considerations. Here it was done to achieve optimal precipitation
downscaling.
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jií stehlík — andrás bárdossy
The daily circulation patterns were defined using geopotential pressure heights of
500 hPa or 700 hPa and precipitation amounts from several climate stations. A classification based on fuzzy rules was applied. The advantage of this classification is that it
is objective, automated and takes into account the precipitation behavior in a certain
region. The task of the classification is to obtain circulation patterns which differ as
much as possible from the average precipitation behavior. This means that the objective is to get some wet and dry circulation patterns in the frame of one classification.
The position of the highs and lows is given in the form of fuzzy rules, which provide
a framework for dealing with vague and/or linguistic information in modeling. Fuzzy
logic accepts overlapping boundaries of sets. Instead of elements belonging or not
belonging to a given set, partial membership in any set is possible.
Here the classification approach uses the concept of fuzzy sets, which enables one
to deal with imprecise statements as “high pressure” or “above normal”. Each circulation pattern is described by a set of fuzzy rules. The classification is performed by
selecting the pattern for which the fulfillment grade of the corresponding rules is the
highest. For more details refer to Bárdossy et al. (2002).
Classification Method
The classification consists of three steps:
a) Data transformation
b) Definition of the fuzzy rules
c) Classification of observed data
In order to get normalized pressure anomalies, g(i,t), the daily geopotential height
data for each grid point i has been transformed by subtracting the mean value and
dividing by the standard deviation for a certain period t. The pressure data used are
the NMC grid data for different windows over Europe on a grid resolution of 5° × 5°.
The procedure for fuzzy rule based classification is described in Bárdossy et
al. (1995). Every CP is described with a fuzzy rule k represented by a vector v(k) =
(v(1)⁽k⁾, …, v(n) ⁽k⁾), where n is the number of gridpoints for which the air pressure data
are available. The v(i)⁽k⁾-s are the indices of the membership function corresponding
to the selected locations i. Five possible classes of membership functions, v, of the
rule premises appear to be adequate according to the normalized geopotential height
values. The membership functions are defined as triangular fuzzy numbers:
v=1
v=2
v=3
v=4
v=5
Very low: (–∞, –1, –0.2)T
Medium low: (–1.4, –0.6, 0)T
Medium high: (0, 0.6, 1.4)T
Very high: (0.2, 1, +∞)T
The membership function is the constant 1.
The fifth membership function was introduced to allow any possible geopotential
height anomalies for those locations which have no influence on the circulation pattern. The location and number of such gridpoints depend on the class to be described.
Usually most gridpoints belong to this class; only characteristic ones are assigned to
other classes. The location of these gridpoints might vary for different CPs.
automated fuzzy rule based classification of circulation patterns
For the data classification, the membership grades of the normal height anomalies
were computed. For a given time, t, and location, i, the membership grade corresponding to rule k is defined as: μ (i, k) = μv(i)(k) (g (i, t)).
These membership values are combined to calculate the degree of fullfilment DOF
of the rule:
DOF (k , t ) =
4
Ȇ
l =1
§
1
¨
μ (i, k ) Pl
¨ N (ν (i)( k ) = l ) ¦
(k )
ν ( i ) =l
©
1
· Pl
¸
¸
¹
(1)
where N is the number of gridpoints classified by class l and Pl ≥ 1 is the parameter
which allows one to emphasize the influence of selected classes on the DOF. The k for
which the DOF(k,t) is maximal is selected as CP for day t.
Using the classification scheme for CP definitions obtained from the optimization
procedure enables to classify outputs from GCM.
Optimization Criteria
Two types of objective functions were introduced in order to measure the performance of the classification. The task of the optimization is to achieve a set of CPs which
explain the variability of precipitation behavior as much as possible. The optimization
is carried out for all selected stations simultaneously, whereas the validation is done
for each individual station.
Two types of objective functions are defined. The first deals with the probability
of precipitation on a given day. Considering the threshold ϑ for daily precipitation
amount enables the general definition of the first objective function:
S
O1 (ϑ ) = ¦
i =1
1
T
¦ ( p(CP (t ))
T
t =1
i
− pi )2
(2)
Here S is the number of stations, T is the number of days, p(CP(t))i is the probability of precipitation exceeding the threshold ϑ on a day with given CP on station i,
pi is the probability of a day with precipitation exceeding the threshold ϑ for all days
without classification and within the time period T.
For the precipitation amounts the following objective function is defined:
1 T §¨ z(CP (t ))i ·¸
¦ ln¨ z ¸
i =1 T t =1
i
¹
©
S
O2 = ¦
(3)
Here z(CP(t))i is the mean precipitation amount on a day with a given CP on station i, and zi is the mean daily precipitation without classification at the same station.
The higher the values of O1(ϑ) and O2, are, the better is the optimization. It is possible
to optimize more objective functions at the same time. This can be done defining an
overall objective function:
O = a1O1 (ϑ1 ) + ... + anO1 (ϑ n ) + an +1O2
(4)
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where a1 … an+1 are weights. They are fixed so that all objective functions have the
same range.
Optimization Algorithm
The objective of the optimization procedure is to find such fuzzy rules for which the
optimization criterion is maximal. This means that the problem is combinatorial. For
a given number of fuzzy rules (CPs), it is necessary to consider that each rule has a
given number of terms (gridpoints) depending on the pressure window, and that one
of the five states (fuzzy numbers) can be defined for each gridpoint. Because of the
enormous number of possible combinations, it is not possible to compute the objective function for each combination systematically. The number of all combinations
depends on the size of the pressure window and number of CPs to be optimized. For
example, in the case studies presented in this paper the number of combonations
are approximately 10¹,⁰⁰⁰. Therefore, a Simulated Annealing algorithm was used as an
optimization procedure. For the fixed number of rules, the algorithm can be briefly
described as follows:
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Initialize the rules randomly and evaluate the performance O
Set the initial annealing temperature to qo
Select a rule k randomly
Select a location i randomly
Select class v* randomly
If v(i)(k) = v* return to step 2
Set v(i)(k) = v* and perform the classification
Calculate performance O* for the new rules
If O* > O then accept the change
If O* ≤ O then with probability exp((O – O*)/qs) accept the change
If the change is accepted, replace O by O*
Repeat steps 2–10 M times
Decrease the annealing temperature by setting qs+1 = α qs, <1
Repeat steps 2–12 S times, or until no more changes are performed.
The algorithm makes it possible to accept negative changes. The willingness for accepting these changes depends on the annealing temperature, which decreases during
the computation. It is worth mentioning that the initial classification does not influence the appearance of the resulting optimized CPs at all. The optimization process
adjusts any initial classification to the optimum solution.
Validation of CP Classifications
A split sampling approach was used for the process of CP optimization and validation. The formulae used as optimization criteria can also be applied as measures of
the classification quality. However, despite the optimization already carried out, it
is still inevitable to evaluate the overall quality of the optimized classification, e.g in
comparison to completely different classification techniques. Parameters to evaluate
automated fuzzy rule based classification of circulation patterns
the performance of precipitation-oriented classifications are based on the following
quantities:
HH [%]
Relative occurence frequency of a CP for a given time period
p(CP(t)) [%] Probability of precipitation on a day with a given CP
A/HH
Wetness index: ratio of the percentage of annual precipitation total
and the precipitation total for a given CP to its appearance rate (high
values indicate ‘wet’ CP’s)
m [mm]
Mean precipitation total on a wet day for a given CP
s [mm]
Standard deviation of precipitation total on a wet day for a given CP
2.2 Precipitation downscaling model
Mathematical Basis
Both spatial and temporal intermittence of precipitation is a major problem in its
mathematical description because of a high probability of dry day occurrences and
continuous distribution describing the rainfall amounts at a selected locations. Therefore random variables with mixed distributions are required to describe daily precipitation. Another problem occurs in the clustering of wet and dry day occurrence. This
clustering has natural causes and was usually modeled with the help of appropriate
distributions, for example using Neymann-Scott models. In the time-space model,
clustering is the consequence of the persistence of atmospheric circulation patterns.
The space time model is described in detail in Stehlík and Bárdossy (2001). It is a
modified version of the model presented in Bárdossy and Plate (1992). Let A = {α₁, … αn}
be a set of possible atmospheric circulation patterns. Let Ãt be the random variable
describing the actual atmospheric circulation, taking its values from A. Let the daily
precipitation amount at time t and point u in the region U be modeled as the random
function Z (t, u), u in U. The distribution of rainfall amounts at a selected location is
skewed. In order to relate it to a simple normally distributed random function, W (t, u)
(for any locations u1,...., un the vector (W (t, u1), …, W (t, un)) is a multivariate normal
random vector) the following power transformation relationship is introduced:
0
if W (t, u) ≤ 0
Z (t, u) =
(5)
Wβ (t, u)
if W (t, u) > 0
Here β is an appropriate positive exponent. Therefore, the mixed (discrete-continuous) distribution of Z (t,u) is related to a normal distribution. As the process Z (t, u)
depends on the atmospheric circulation pattern, the same applies to W (t, u). The reason for this transformation is that multivariate processes can be modeled much easier
if the process is normal. The problem of intermittence can also be handled this way,
as the negative values of W are declared as dry days and dry locations. The exponent
β is needed because the distribution of precipitation amounts is usually much more
skewed than the truncated normal distribution.
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jií stehlík — andrás bárdossy
The relationship between W (t, u) and the circulation pattern Ãt is obtained through
the rainfall process Z (t, u) using (5).
The following notation is introduced for the subsequent development:
W(t) = (W(t,u1), ... , W(t,un))
(6)
Z(t) = (Z(t,u1), ... , Z(t,un))
(7)
The expectation of W (t, u) for a given CP is:
~
wα i (t,u) = E[W(t, u) | At = i]
(8)
To simplify the following development the vector of the expectations of W (t,u) is
introduced:
w Įi (t ) = ( wα i (t , u1 ) , ... , wα i (t,un))
(9)
i
The random process describing W(t) is described
by using the following equation:
W (t ) = r (t ) ( W (t − 1) − w ~ (t − 1)) + C ~ Ψ (t ) + w ~ (t )
where
At − 1
Ȍ (t ) = (ψ (t ,u1 ), ..., ψ (t ,un ))
At
At
(10)
(11)
is a random vector of independent N(0,1) random variables.
The function r(t) is an autocorrelation of one day time lag. It is independent of the
circulation pattern, but dependent on an annual cycle which is approximated by using
the Fourier series:
K
a
(12)
r (t ) = o + ¦ (a k cos(kωt ) + bk sin(kωt ))
2 k =1
where the frequency ω is (2π/365). The advantage of Fourier series approximation
(instead of a simple polynomial fit) is that adding new ak and bk parameters does not
change the values of the former ones. For K = n/2 (where n is number of points in the
series), the Fourier approximation is identical with the observed series. Usually the
first three Fourier parameters are enough to simulate the annual cycle of autocorrelation.
The proportion of variance accounted for by each harmonic can be computed as
follows:
n / 2 (a k2 + bk2 )
2
(13)
Rk =
(n − 1) s 2
where s² is the sample variance of the series. Since each harmonic provides independent information about the data series, the joint proportion of variance exhibited
by a regression equation with only harmonic predictors is the sum of the Rk² values for
each of the harmonics (for maximum number (n/2) of harmonics the sum of Rk² is 1).
The matrix CÃt takes the process of spatial variability into account. This n × n matrix is related to W(t) through (Bras and Rodriguez-Iturbe, 1985):
automated fuzzy rule based classification of circulation patterns
Γ
~
0 At
= E ª( W (t ) − w ~ (t )) ( W T (t ) − w ~ (t ))º
At
At
«¬
»¼
(14)
T
ΓT1 A~t = E ª«( W(t − 1) − w A~ (t )) ( W (t ) − w A~ (t ))º»
t
t
¬
¼
(15)
C A~t CTA~t = Γ1 A~t − Γ0−1A~t ΓT1 A~t
(16)
Here Γ0Ãt is spatial covariance matrix and Γ1Ãt is space-time covariance matrix for
the time lag of one day. Assuming that these two matrices are related to each other,
through:
r (t ) Γ ~ = Γ ~
(17)
0 At
1 At
leads to
C ~ C ~ T = (1 − r 2 (t ))Γ
At
At
~
0 At
(18)
Equation (5) establishes the link between W(t) and the rainfall Z(t). Thus, the multisite process is fully described.
Parameter Estimation
The temporal parameters are computed by means of a maximum likelihood method.
Assuming that the parameters of the precipitation annual cycle depend on the CP, i,
and time of the year t, one has:
w i (t , u) =
σ i (t , u) =
a 0 (w i , u)
2
c 0 (σ i , u)
2
K
+ ¦ (a k (w i , u)cos(kωt ) + bk (w i , u) sin (kωt ))
(19)
k =1
K
+ ¦ (c k (σ i , u)cos(kωt ) + d k (σ i , u) sin (kωt ))
(20)
k =1
The log-likelihood function can be written following the derivation of Henze and
Klar (1993):
§ - w (t , u) ·
¸+
log Φ¨ i
¨ σ (t , u) ¸
Z ( t ,u ) = 0
© i
¹
1
1
ª
·º
−1 §
¨ Z (t , u) β - w i (t , u) ¸»
« Z (t , u) β
+ ¦ log «
φ¨
¸»
βσ i (t , u) ¨
σ i (t , u)
Z ( t ,u ) > 0
¸»
«
©
¹¼
¬
L(w i (t , u), σ i (t , u); Z (t , u), t = 1,..., T ) =
¦
(21)
where φ and Φ denote the density function and the cumulative distribution function of the standard normal distribution respectively.
This equation can be simplified to:
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jií stehlík — andrás bárdossy
L(.) =
§ - w (t , u) ·
¸+
log Φ¨ i
¨ σ (t , u) ¸
Z ( t ,u ) = 0
© i
¹
¦
¦ logσ
Z ( t ,u ) > 0
i
(t , u) -
§1
·
logβ + ¨¨ − 1 ¸¸ ¦ log Z (t , u) β
Z ( t ,u ) > 0
©
¹ Z ( t ,u ) > 0
¦
1
§
·
¨ Z (t , u) β - w i (t , u) ¸
1
1
−
¨
¸
¦ log(2π) + 2 Z (¦
2 Z ( t ,u ) > 0
σ i (t , u)
t ,u ) > 0 ¨
¸
©
¹
2
(22)
Summing the terms which do not depend on the Fourier coefficients to K, one has:
2
1
§
·
§ - w i (t , u) ·
¨ Z (t , u) β - w i (t , u) ¸
1
¸+
L(.) = K + ¦ log Φ¨
logσ i (t , u) +
¸ (23)
¦ ¨ σ (t , u)
¨ σ (t , u) ¸ Z (¦
2 Z ( t ,u ) > 0 ¨
Z ( t ,u ) = 0
¸
i
¹ t ,u ) > 0
© i
©
¹
The above function must be maximized as a function of a₀(wi, u) …, aK(wi, u),
b₁(wi, u) …, bK(wi, u), c₀(σi, u), …, cK(σi, u), d₀(σi, u), …, dK(σi, u). This can be done
by numerical optimization using appropriate algorithms. The convergence can be
substantially increased by using the first approximation for wi(t, u) and σi(t, u) for a
selected set of points (for example, 12 mid-days of the 12 months) and then fitting
initial a₀(wi, u) …, aK(wi, u), b₁(wi, u) …, bK(wi, u), c₀(σi, u), …, cK(σi, u), d₀(σi, u), …,
dK(σi, u). Starting from this point insures a quick optimization.
The spatial structure of the rainfall is described using a circulation pattern dependent covariance structure. The covariance structure is assumed to be translation
invariant, but time dependent supposing:
cov[ Z x , Z y ]i (t ) = pi (t ) e − h( x , y )qi (t )
(24)
where h is distance. The parameters pi and qi depend on the circulation pattern and
the time of the year, and are also modeled by the Fourier series. Anisotropy is taken
into acccount by introducing a coordinate transformation:
x' = λ ( x cos φ + y sin φ )
(25)
y' = − x sin φ + y sin φ
(26)
where (x, y) are the original coordinates, and (x’, y’) in the transformed system.
φ is the rotation angle and λ is the ratio of two orthogonal ranges representing the
highest and the lowest variability. It is supposed that the anisotropy changes with the
circulation pattern, but for a given circulation pattern remains constant during the
year (due to flow direction).
The present model is described in Stehlík and Bárdossy (2001). It is an advancement of the precipitation model developed by Bárdossy and Plate (1992). This model
included winter and summer seasons, where the new model includes the annual cycle
and generates precipitation time series for several sites. Now it is possible to simulate
precipitation for a set of points simultaneously. Further, the annual cycle of spatial
covariances was introduced and the parameter estimation is improved.
automated fuzzy rule based classification of circulation patterns
3. Application
3.1 CP classification
The above described methodology was applied in many regions at different scales: at
the local scale in Central Europe (Germany, Danube basin and Otava basin) and in
Eastern Mediterranian (Greece); at the continental scale the classification has been
done for Northern America.
Because the classifications carried out for Germany and Greece have been used
as a basis for precipitation downscaling, results of these classifications are discussed
in detail. The locations of Greek and German stations are shown in Figure 1 and Figure 2.
For the CP optimization, the calibration period 1980–1989 and validation period
1970–1979 were selected. Many CP optimizations, subsequent classifications and
validations were performed. The goal was to assess the influence of geopotential surfaces (700, 500 hPa and sea level pressure) as well as the window size from which to
take the pressure values.
Another question investigated was which number of CPs should be generated to
best describe rainfall characteristics in an area. Using too few CPs leads to a loss of
information since their behavior only represents averaged weather conditions. Too
many CPs result in very low occurrence frequencies, which make statistical investigations impossible. Also, it was of interest to see to what extent the form of the objective
functions influence the validation results.
Results for Germany/Central Europe
For CPs classification which explains the precipitation and temperature variability respectively, it was found that the 500 hPa data in the window 35° N – 65° N,
15° W – 40° E provided the best results. The best number of CPs was found to be 12,
both for precipitation and temperature based optimizations. Daily precipitation totals
Fig. 1 Location of German precipitation
stations used for CPs optimization and subsequent downscaling.
Fig. 2 Location of Greek precipitation stations
used for CPs optimization and subsequent
downscaling.
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jií stehlík — andrás bárdossy
Table 1 CP and precipitation characteristics at precipitation station Stuttgart. Values in the table
are averaged over the validation period 1970–79.
Mean wet-day
amount [mm]
CP occurrence
frequency [%]
CP wetness index
Precip.
contribution /
occurrence
frequency
CP
Precipitation
probability [%]
106
1
2
3
4
5
6
7
8
9
10
11
12
Year
0.63
1.12
1.65
0.88
0.86
0.85
0.93
1.58
1.61
0.88
1.27
1.23
Winter
0.42
0.63
2.04
1.20
1.14
0.97
1.16
1.63
2.50
1.64
0.34
1.13
Fall
0.41
0.93
2.07
1.31
0.83
0.67
0.53
2.07
1.28
1.10
1.70
1.23
Summer
0.84
1.71
1.56
0.26
0.81
0.51
1.22
1.34
1.46
0.67
1.24
1.30
Spring
0.56
1.42
1.52
1.15
0.89
1.36
0.45
1.21
1.51
0.54
1.25
0.88
Year
40.2
9.0
7.9
2.8
3.8
4.6
1.4
10.7
6.0
5.6
3.7
2.7
Winter
39.7
12.1
11.3
2.7
5.1
3.3
1.4
8.2
5.7
4.3
2.8
1.7
Fall
40.5
7.0
8.0
3.1
3.4
5.1
1.8
13.4
4.2
5.4
3.0
3.3
Summer
45.2
5.4
4.6
2.8
3.0
5.1
1.7
10.4
6.0
7.1
4.1
3.3
Spring
35.2
11.3
7.9
2.5
3.6
4.9
0.8
10.9
8.0
5.8
4.9
2.7
Year
1.1
2.0
2.9
1.5
1.5
1.5
1.6
2.8
2.8
1.5
2.2
2.2
Winter
0.5
0.7
2.3
1.3
1.3
1.1
1.3
1.8
2.8
1.8
0.4
1.3
Fall
0.7
1.5
3.4
2.2
1.4
1.1
0.9
3.4
2.1
1.8
2.8
2.0
Summer
2.1
4.3
3.9
0.7
2.0
1.3
3.0
3.3
3.6
1.7
3.1
3.2
Spring
1.0
2.5
2.6
2.0
1.5
2.4
0.8
2.1
2.6
0.9
2.2
1.5
Year
25.6
41.5
66.9
51.5
41.3
26.2
61.5
73.2
66.5
41.7
58.5
62.0
Winter
26.3
30.3
62.7
54.2
41.3
33.3
46.2
82.4
70.6
69.2
32.0
73.3
Fall
18.4
42.2
74.0
53.6
48.4
26.1
56.3
76.2
68.4
34.7
66.7
73.3
Summer
30.5
56.0
64.3
38.5
39.3
14.9
75.0
67.7
65.5
29.2
71.1
56.7
Spring
26.9
46.2
67.1
60.9
36.4
33.3
71.4
68.0
63.5
43.4
57.8
48.0
or daily mean temperatures from nine Germany-wide distributed stations were used
for both the calibration and validation procedure.
For the optimization, a threshold of mm in the objective functions (2) and (3) were
used. Table 1 presents CPs and precipitation characteristics for the Stuttgart precipitation station. The average occurence frequency of most CPs is approximately the
same in each season. An exception is, for example, CP 3 with 4.6 % of occurence in
summer and 11.3 % in winter. Generally, it holds that the wettest (considering annual
average) CPs are wet and the driest CPs are dry in every season. If the annual wetness
index is close to one, the wet/dry character of the CP may vary from season to season,
as for example CP 2. Whereas it is wet in summer (wetness index 1.42), it contributes
relatively little to precipitation in winter (wetness index 0.63).
Figure 3 shows the distributions of the mean (1970–1979) normalized 500 hPa
geopotential height anomalies for the wettest and the driest CP. CP 1 is the most frequent CP and dominates during the whole year having the average annual frequency
of about 40 %. At the same time it is the driest CP with the lowest precipitation probability (25.6 %), lowest mean wet-day amount (1.1 mm), and lowest wetness index
(0.63). The map shows that CP 1 is characterized by a pronounced high pressure
anomaly east of British Isles which causes a weak air movement and transport of dry
air masses from northeastern Europe to central Europe. CP 8 is a typical wet CP with
automated fuzzy rule based classification of circulation patterns
Fig. 3 Mean normalized distributions of 500 hPa geopotential height anomalies for CP 1 and
CP 8, averaged over 1970–79 (classification for Germany).
107
108
jií stehlík — andrás bárdossy
Fig. 4 Mean normalized distributions of 700 hPa geopotential height anomalies for CP 2 and
CP 5, averaged over 1970–79 (classification for Northern America).
automated fuzzy rule based classification of circulation patterns
2
3
4
5
6
7
8
9
10
11
12
CP wetness index
Precip.
contribution /
occurrence
frequency
CP
3.62
1.32
0.72
0.75
1.42
0.89
0.82
0.53
0.22
2.04
0.34
0.93
Winter
3.42
1.23
0.67
0.76
0.36
1.28
1.63
0.43
0.29
2.12
0.34
1.37
Fall
3.77
1.13
0.44
1.04
2.73
0.32
0.58
0.49
0.18
2.59
0.39
0.52
Summer
3.50
1.84
0.22
0.09
2.93
0.00
0.00
0.25
0.12
2.16
0.61
1.76
Spring
3.71
1.26
1.53
0.54
1.42
0.80
0.38
0.80
0.16
2.10
0.18
0.87
Year
8.95
8.43
7.80
7.64
4.96
1.75
4.90
3.81 31.05 7.64
7.89
3.48
Winter
10.11 10.33 8.22
6.67
4.44
2.00
3.78
4.44 30.22 6.22
9.33
3.00
Fall
7.47
8.13 10.77 7.91
4.73
1.21
5.05
3.63 32.31 6.26
6.04
3.41
Summer
8.70
7.07
6.74
8.15
5.98
1.63
5.33
3.48 30.33 9.13
8.26
4.35
Spring
9.57
8.26
5.54
7.72
4.67
2.17
5.43
3.70 31.30 8.91
7.93
3.15
Year
3.84
1.40
0.77
0.80
1.51
0.95
0.87
0.57
0.24
2.16
0.36
0.99
Winter
6.38
2.29
1.25
1.43
0.68
2.39
3.04
0.81
0.55
3.97
0.64
2.56
Fall
5.30
1.59
0.62
1.46
3.85
0.45
0.82
0.69
0.26
3.65
0.55
0.73
Summer
0.69
0.36
0.04
0.02
0.58
0.00
0.00
0.05
0.02
0.43
0.12
0.35
Spring
2.96
1.01
1.22
0.43
1.13
0.63
0.30
0.64
0.13
1.68
0.15
0.70
Year
44.34 31.17 18.60 19.00 22.10 18.75 17.32 17.99 6.08 35.13 10.07 24.41
Winter
61.54 44.09 24.32 33.33 32.50 50.00 35.29 27.50 14.34 55.36 19.05 48.15
Fall
48.53 21.62 22.45 23.61 25.58 9.09 26.09 15.15 2.72 52.63 7.27 29.03
Precipitation
probability [%]
Mean wet-day
amount [mm]
1
Year
CP occurrence
frequency [%]
Table 2 CP and precipitation characteristics at precipitation station Athens. Values in the table are
averaged over the validation period 1970–79.
Summer
17.50 12.31 3.23
Spring
47.73 40.79 21.57 18.31 27.91 10.00 14.00 23.53 6.60 32.93 9.59 24.14
4.00
7.27
0.00
0.00
3.13
1.08 11.90 2.63
5.00
a high precipitation probability (73.2 %), mean wet-day amount (2.8 mm) and wetness index (1.58). The map shows that the eastern position of low pressure anomaly
and western position of higher anomaly result in transport of wet air from the North
sea. The maps of pressure anomalies show that the automated classification method
produces physically realistic results.
Results for Greece/Eastern Mediterranian
A characteristic of the Greek climate is the extreme difference in precipitation between summer and winter, and also the great differences of precipitation amounts
at different stations on the same day. While one station may measure high rainfall
amounts, another may measure no rain at all. The advantage of the automated classification optimization takes each individual station behavior into account.
Precipitation data from 21 climate stations evenly spread over the whole of Greece
were taken into account for calibration and validation purposes. For CP classification the 700 hPa pressure data in the window 20° N – 65° N, 20° W – 50° E yielded the
best results. It was found that twelve is the optimal number of CPs. The result of
the validation of many different optimizations with different weightings is that the
precipitation-based optimization, with emphasis on extreme precipitation (five times
higher weight for 5 mm threshold than for 0 mm), yielded very good results.
109
110
jií stehlík — andrás bárdossy
CP and precipitation statistics for the Athens precipitation station, which is located
approximately in the middle of the investigated area and has a typical annual cycle
of precipitation with very little precipitation in summer and higher precipitation in
winter, is shown in Table 2. The table shows that CP 6 and CP 7 do not contribute to
summer precipitation at all due to the precipitation probability being 0. As in the case
of the German study, the occurence frequency of most CPs remains approximately
the same in each season.
Whereas other CPs have frequency between 2 and 10 %, CP 9 is exceptionally
frequent, occuring in approximately 30 % of the days in each season. For Athens, very
wet and very dry CPs have a wet or dry character in every season. CPs which are
not pronounced from the annual wetness index point of view vary their wet and dry
character seasonally, e.g. CP 7 is wet in winter (wetness index 1.63) and absolutely dry
in summer.
Examples of CP classification for Northern America
Two outputs of CPs classification done for Northern America are shown in Figure 4.
For example, whereas the CP 2 causes wet weather in British Columbia, the CP 5
brings very dry conditions to Ontario.
3.2 Precipitation Downscaling
For the German stations, the average monthly amounts are mostly more or less
comparable during the year or they reach a maximum during the summer months.
However, for most Greek stations a pronounced precipitation cycle with almost no
rain in summer is typical. Also, there are great differences of precipitation measurements at different stations at the same day. It happens that one station might measure
high rainfall amounts while no rainfall is recorded in some others.
Results for Germany/Central Europe
For validation of the downscaling model, the observed and simulated daily precipitation amounts in the time period 1970–1990 were compared.
Fig. 5 Average annual cycles of precipitation for observed and simulated series in Germany
(1970–1990).
automated fuzzy rule based classification of circulation patterns
Graphs of observed and modeled
annual cycles show that these cycles are
reproduced very well (Figure 5). For comparison between observed and simulated
precipitation series the standard diagnostics according to Wilby et al. (1998b)
was applied. In Table 3 the diagnostics is
presented for three stations with different elevations. It shows that in general
the simulations slightly overestimate the
Fig. 6 Annual cycle of precipitation conditiamount of rainfall on rainy days. The oned to occurence of CP 1 and CP 8: compareason for this is that the model was de- rison between observed data and Fourier apveloped with a special emphasis on the proximation by 3 harmonics for the Stuttgart
reproduction of extreme rainfall events station (1970–1990).
and therefore tends to overestimate rainfall amounts. The standard deviations of
0.6
wet day amounts are reproduced very
well. The unconditional probability of a
0.5
wet day and the conditional probabilities P₀₀ and P₁₁ yield comparable values
0.4
both for observed and simulated series.
Also, the reproduction of the mean spell
lengths is quite well; only the long dry0.3
day periods during the vegetation period
are somewhat overestimated.
Table 3 also shows that standard devi- Fig. 7 Average annual cycle of lag 1 autocorations of monthly precipitation totals are relation of daily precipitation for eight German
stations (1970–1990).
comparable. Even such values like maximum daily precipitation are reproduced
very well. There is no systematic over- or
underestimation. Thus applicability of
the model is proved.
CP-conditioned annual cycles of
precipitation for both above described
CPs are presented in Figure 6. The cycles differ substantially from each other.
Whereas CP 1 – conditioned cycle has
distinct maximum in summer, the cycle Fig. 8 Spatial covariance functions for precipibelonging to CP 8 is more smooth. In tation conditioned to occurence of CP 1 and
Figure 6 the Fourier series of three har- CP 8 (CPs optimized for Germany) both in
monics for each precipitation cycle is summer and in winter.
also presented.
The agreement between each pair of lines proves that the Fourier series yields very
good approximation of the observed data.
111
[mm]
96.90
32.97
21.00
3.34
2.74
0.45
0.29
0.38
2.00
5.89
4.01
observed
59.00
39.15
38.00
3.85
2.41
0.39
0.23
0.46
2.60
6.00
4.73
simulated
Stuttgart
(315 m a.s.l.)
57.90
34.27
26.00
3.16
3.54
0.53
0.38
0.32
2.20
5.34
4.10
observed
51.00
38.31
30.00
3.31
2.75
0.45
0.29
0.38
2.70
5.54
4.60
simulated
Köln
(92 m a.s.l.)
62.90
67.82
10.00
2.67
4.90
0.65
0.52
0.22
3.30
7.83
6.26
observed
66.30
57.93
15.00
2.78
3.46
0.55
0.39
0.29
4.40
7.64
6.91
simulated
Kahler Asten
(839 m a.s.l.)
[mm]
[mm]
Rx maximum daily precipitation
[ ]
Standard deviation of monthly precipitation total
Number of dry spells > 10 days in vegetation period (Apr.–Sep.)
[day]
Ld mean dry-spell length
[ ]
[day]
πW unconditional probability of a wet-day
Lw mean wet-spell length
[ ]
[mm]
Median wet-day amount
P11 wet-day probability conditional on previous day being wet
[mm]
Standard deviation of wet-day amount
[ ]
[mm]
Mean wet-day amount (>0,05 mm)
P00 dry-day probability conditional on previous day being dry
Unit
Diagnostic
171.70
72.37
68.00
6.03
2.26
0.27
0.15
0.61
4.90
13.05
9.28
observed
87.30
75.87
83.00
5.88
1.96
0.25
0.12
0.62
5.40
12.17
9.88
simulated
Agrinio
(47 m a.s.l.)
129.10
86.16
79.00
6.00
2.52
0.30
0.18
0.59
4.70
14.02
10.11
observed
113.90
85.92
90.00
5.79
2.10
0.27
0.14
0.61
6.00
13.00
10.58
simulated
Corfu
(4 m a.s.l.)
95.80
31.53
80.00
5.67
1.96
0.26
0.13
0.61
2.80
7.49
5.37
observed
112.90
36.31
76.00
5.77
1.76
0.23
0.10
0.63
3.10
8.41
6.03
simulated
Kozani
(627 m a.s.l.)
Table 5 Precipitation diagnostics according to Wilby et al. (1998b) for the stations Agrinio, Corfu and Kozani for the validation period 1970–1987.
Comparison of observed and simulated precipitation series.
[mm]
Rx maximum daily precipitation
[ ]
Standard deviation of monthly precipitation total
Number of dry spells > 10 days in vegetation period (Apr.–Sep.)
[day]
Ld mean dry-spell length
[ ]
[day]
πW unconditional probability of a wet-day
Lw mean wet-spell length
[ ]
[mm]
Median wet-day amount
[ ]
[mm]
Standard deviation of wet-day amount
P11 wet-day probability conditional on previous day being wet
[mm]
Mean wet-day amount (>0,05 mm)
P00 dry-day probability conditional on previous day being dry
Unit
Diagnostic
Table 3 Precipitation diagnostics according to Wilby et al. (1998b) for the stations Stuttgart, Köln and Kahler Asten for the validation period
1970–1990. Comparison of observed and simulated precipitation series.
112
jií stehlík — andrás bárdossy
automated fuzzy rule based classification of circulation patterns
Table 4 Explained variance for precipitation and precipitation probability by each from the first
three Fourier harmonics and their sum.
number of
harmonics
German stations
precipitation
precipitation
probability
64.45
62.78
20.01
20.78
6.98
6.91
91.44
90.56
1
2
3
sum
Greek stations
precipitation
precipitation
probability
58.09
87.28
21.69
7.00
8.61
2.45
88.39
96.73
According to Eq. 13 the proportion of variance of CP-conditioned annual cycle
of precipitation and precipitation probability for each station and each CP was computed. The results are shown in Table 4, where a mean value for each from three
computed harmonics is presented. Table 4 shows that most of the variation in the
precipitation data is described by the first harmonics, for which the R² is 64.45 %. The
first three harmonics account for (in average) 91.44 % of variance of CP-conditioned
precipitation annual cycle. This fact proves the suitability of the Fourier transform
approximation. Similar results were obtained for precipitation probability.
As presented in Figure 7, the annual cycle of autocorrelation is quite well pronounced. From this figure one can see that the temporal persistence in the precipitation is higher in winter than in summer. It proves the fact that winter rainfall is mostly
caused by cyclonic atmospheric conditions at the large scale whereas summer rains
are often produced by local convective systems.
Spatial covariance functions for both CP 1 and CP 8 for the summer and winter
season respectively are shown in Figure 8. For both CPs (with the exception of CP 8 in
summer for short distances) the covariance functions in the same distance are lower
for summer than for winter. This is due to the lower spatial persistence of summer
rainfall events in comparison to the winter ones. This is probably because the summer
rainfall is often produced by small-scale convective events, whereas winter rainfall is
associated with low pressure anomalies at larger scale, as discussed above, in case of
annual cycle of autocorrelation.
Agrinio
Korfu
200
160
Observed
180
Observed
140
Simulated
160
Simulated
Precipitation totals [ mm ]
Precipitation totals [ mm ]
180
120
100
80
60
40
20
0
140
120
100
80
60
40
20
0
Jan Feb Mar
Apr
Mai
June July Aug Sept Oct
Nov
Dez
Jan Feb Mar
Apr
Mai
June July Aug Sept Oct
Nov
Dez
Fig. 9 Average annual cycles of precipitation for observed and simulated series in Greece
(1970–1987).
113
114
jií stehlík — andrás bárdossy
Results for Greece/Eastern Mediterranian
For the comparison between observed and simulated precipitation series, the time period 1970–1987 was considered. In regards to the seasonal cycle of precipitation, the
simulation results are very good, especially for stations having a typical pronounced
cycle with almost no precipitation in winter (Figure 9). The precipitation diagnostics was done for three stations with different elevations ranging from 4 meters a.s.l.
(Korfu) to 627 meters a.s.l. (Kozani) (see Table 5). Whereas Agrinio and Korfu have
typical seasonal precipitation cycles, the average monthly precipitation amounts for
Kozani are approximately the same during the year. The results show a good agreement
between observed and simulated precipitation series for all three stations. Similar to
the case of German stations, the mean wet day amounts are slightly overestimated,
however only by 5 to 12 %. The unconditional (πΩ) and both conditional (P₀₀ and
P₁₁) probabilities are in very good agreement. Also other statistical tests, like mean
wet and dry spell length, number of dry spells with more then ten days in vegetation
period and standard deviation of monthly precipitation totals, showed that the model
is able to reproduce all features of daily precipitation series under Greek climate conditions very well.
Examples of annual cycles of precipitation and their Fourier approximations
are presented in Figure 10. Results of
normalized spectral density analyses are
again shown in Table 4.
Because of the highly pronounced
cycle of precipitation probability, the
first harmonic explains almost all the
variability of the annual cycle.
Annual cycle of precipitation autoFig. 10 Annual cycle of precipitation condicorrelation reveals similar features, like tioned to occurence of CP 1 and CP 8: comin the German case (Figure 11), with a parison between observed data and Fourier
winter maximum and summer mini- approximation by 3 harmonics for the station
mum. Because the distances among the Agrinio (1970–1987).
selected precipitation stations are quite
0.6
high, there is no possibility to take the
0.5
spatial persistence of rainfall into ac0.4
count.
0.3
0.2
4. Discussion and Conclusions
In this paper an automated and objective method of CP classification and a
downscaling model for generating daily
precipitation series has been presented.
The classification method is based on
0.1
Fig. 11 Average annual cycle of lag 1 autocorrelation of daily precipitation for 21 Greek
stations (1970–1987).
automated fuzzy rule based classification of circulation patterns
a fuzzy rules approach, which enables one to deal with verbal descriptions of geopotential heights and pressure data anomalies in a mathematical way. The precipitation generator uses a CP-conditioned concept, where the condititional precipitation
characteristics for each circulation pattern are computed. It is generally possible to
take an arbitrary number of precipitation stations into account.
The seasonal cycle of precipitation is simulated by using the Fourier series. The
generated precipitation time series can be used as input data in various types of hydrological models, especially in regions where only relative short time precipitation
series are available. Two case studies in different climatological regions are presented
in the paper.
The results of this paper can be summarized as follows:
1. The CP classification method and precipitation downscaling model have
been applied in two European regions with different climate conditions:
Central Europe (Germany) and Eastearn Mediterranian (Greece).
2. Some outputs of the classification at the continental scale are presented for
Northern America.
3. To establish the rules for CP definition, the simulated annealing algorithm has
been applied. This algorithm makes it possible to solve a complicated optimization problem with reasonable computational effort.
4. Changing the form of the objective function enables one to incorporate various criteria into the CP definition.
5. When looking at the maps of pressure anomalies, it is evident that the method
produces physically realistic CP defintions.
6. By taking precipitation series from a set of stations into account, the method
serves as a basis for downscaling the large scale atmospheric behavior represented by pressure data into smaller geographical regions.
7. Using the classification scheme for CP definitions obtained from the optimization procedure would enable one to classify outputs from Global Circulation
Models.
8. A model for downscaling daily precipitation has been developed. The rainfall
is modeled as process coupled with atmospheric circulation. It is linked to the
circulation patterns using conditional probabilities and conditional distribution functions.
9. In the downscaling model an annual cycle of model parameters, including
autocorrelation and spatial correlation, is taken into account and modeled by
means of Fourier series.
10. Several tests, such as comparison of monthly precipitation averaged over the
validation period, comparison of mean values and deviations of yearly totals
and other statistics diagnostics, show that simulated values agree fairly well
with historical data.
11. Simulated precipitation time series can be used as input for hydrological
models.
12. Both presented methods are universal; their applications are not restricted to
a given area, but can be applied in every region with available data.
115
116
jií stehlík — andrás bárdossy
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