Martin Dubrovsky
Institute of Atmospheric Physics, Hradec Kralove‚
Czech Republic
Dubrovsky M., (1997): Creating Daily Weather Series With Use of the Weather Generator. Environmetrics 8, 409-424.
This HTML version of my paper has many bugs.. I will try to eliminate them soon...
2 DOWNSCALING GENERAL CIRCULATION MODEL OUTPUT
2.1 Nested mesoscale model
2.2 Analogues
2.2.1 Historical analogues
2.2.2 Spatial analogues
2.3 Modification of existing time series
2.4 Weather generator
2.5 Downscaling upper-air circulation pattern to surface weather
characteristics
3 MET&ROLL-1: FOUR-VARIATE WEATHER GENERATOR
FOR THE CROP GROWTH MODEL
3.1 Model
3.2 Stochastic generation of the weather series
3.3 Validation of the weather
generator
3.3.1 Reproduction of the parameters of the generator
3.3.2 Normality of variables of autoregressive model
3.3.3 Daily precipitation totals
3.3.4 Length of dry and wet periods
3.3.5 Lag-0 and lag-1 correlations between-quantities of the
autoregressive model
3.3.6 Variability of monthly and annual means
4 MET&ROLL-2: WEATHER GENERATOR LINKING THE
SURFACE WEATHER CHARACTERISTICS WITH UPPER-AIR CIRCULATION
SUMMARY
Estimation of quantitative impacts of potential climate change on
environment and various aspects of human being requires high-resolution
surface weather data. Since the direct output from General
Circulation Models (GCMs) is unreliable at the local scale,
alternative approaches - most frequently based on statistical
techniques - should be used to downscale coarsely resolved GCM
output patterns to finer spatial and/or temporal resolution. The
downscaling techniques are briefly reviewed in the paper. Two of
the approaches were followed in developing two versions of the
stochastic weather generator (WG) called Met&Roll: (i)
generation of synthetic weather series by generator with
parameters derived from the local observed series and then
modified according to the climate change scenario that prescribe
monthly increments of the individual variables, (ii) downscaling
the GCM-simulated daily circulation pattern, using statistical
linkage between the circulation patterns and the surface weather
characteristics.
Met&Roll-1 is a four-variate surface weather generator
which employs a Markov chain approach to model precipitation
occurrence and an autoregressive model to simulate the solar
radiation and the diurnal extreme temperatures. The validation of
the generator is performed by comparison of the stochastic
structure of observed and synthetic series. Uncertainties in
projecting the climate change scenario into the parameters of the
WG are discussed. Met&Roll-2 is a generator which links the
four surface weather variables with upper-air circulation
patterns (CPs). CPs are characterised by principal components
derived from 500 hPa geopotential field. The series of CPs is
either generated by autoregressive model or taken from the GCM
output. The first test of this generator is focused on the
correlation between CPs and surface weather characteristics.
key words: climate change, stochastic model, time series analysis, Markov chain, autoregressive model
It is generally assumed that the increasing concentration of
greenhouse gases in the atmosphere will significantly contribute
to the change of climate in near future. The question stands:
What would be the impact of the potential climate change on
environment and various aspects of the human being? The potential
subjects of the climate change impacts include, e.g.,
hydrological cycle and water resource management, agriculture,
forestry, tourism and human health.
To estimate the impacts, we need:
In crop growth modelling,[1,2,3,4] 4 daily
characteristics for single location are usually required:
precipitation amount (RAIN), extreme temperatures (TMAX
and TMIN), and global solar radiation (SRAD). In
modelling hydrological cycle, rainfall-runoff models are supplied
with precipitation data from more locations simultaneously or
continuously in space inside the given area. The data
representing present-climate conditions may be taken from the
archived records of observational data. Various techniques may be
used to synthesise weather data for changed climate conditions.
Most of them are based on General Circulation Model (GCM)
simulations.
General circulation models are the primary tools today for
simulating present and changed climate conditions. They simulate
physical processes occurring in the atmosphere with use of set of
partial differential equations which represent conservation laws
for atmospheric mass, momentum, total energy, and water vapour.
GCMs include representations of surface hydrology, sea ice,
cloudiness, convection, atmospheric radiation and other relevant
processes.[5] Although GCMs are run with short
time steps (commonly less than one hour) their spatial resolution
(typically of the order of 300 km) is too coarse to account for
the local effects and capture the mesoscale phenomena which play
a great role in formation of site-specific weather conditions.
These phenomena may generally distort even the largest scale
patterns of meteorological variables, due to the pronounced non-linearity
of the atmospheric processes.
In result, application of the site-specific surface weather
characteristics in daily or shorter time steps as provided by
direct GCM output for changed climate conditions is unreliable.[6] It is therefore recommended to use daily
larger-scale upper-air characteristics or monthly characteristics
and downscale them into finer spatial and/or temporal resolution
by alternative approaches. Selected downscaling techniques are
schematically displayed in Figure 1] and
discussed in Chapter 2. Regarding the fact that GCMs cannot
satisfactorily reproduce even the present climate conditions[7] (often characterised by atmospheric CO2
concentration), climate change scenarios are often given in terms
of monthly increments which are based on comparison of GCM
simulations for present and doubled CO2 concentrations. The
increments may be used to modify either surface weather series (Section
2.3) or parameters of the weather generator (Section 2.4), or to
identify analogues.
The present paper consists of two parts. In the first part (Chapter 2), the approaches to development of
high resolution weather data from GCM output are reviewed with a
special emphasis on weather generators. In the second part (Chapters
3 and 4, the weather
generator Met&Roll coded by the author, with the aim to
provide an input to the crop growth model, is described and
tested. The first version of the weather generator (Chapter 3) is a surface weather generator, which
generates series of surface weather characteristics based on
previous analysis of observed surface weather series. The second
version of the generator (Chapter 4) relates
the surface weather characteristics with upper-air circulation
and may be run in two modes: in an independent mode in which the
series of circulation patterns is stochastically generated, or in
a dependent mode in which the surface weather characteristics are
generated conditionally on circulation patterns derived from GCM
output or historical observations.
Several approaches to downscaling the GCM output are schematically displayed in Figure 1.
In this approach, the low-resolution information provided by a
GCM is used to initialise the nested limited-area mesoscale model
and then to provide time dependent lateral and possibly upper
boundary conditions for the mesoscale simulation.[8,9,10,11]
Although the validation experiments show considerable skill of
mesoscale models to simulate interdiurnal variability of surface
weather characteristics, more straightforward statistical methods
are mostly preferred because of their simplicity and ability to
better simulate stochastic structure of the weather series.
Anyway, provided the GCM well reproduces variability of the
atmospheric circulation in changed climate conditions on the
synoptic-scale, the nested mesoscale model have greater potential
to realistically simulate site-specific climatic conditions
compared to the statistical approaches (sections 2.2 - 2.4) which
have problems to account for the changes in circulation
conditions.
Historical climatic data are searched for the periods (years/seasons)
with climate conditions similar to those prescribed by the
climate change scenario. For example, daily series from extremely
warm winters are used to represent the changed climate
characterised by increased winter temperatures.
Spatial analogues use monthly or seasonal means to identify
one region's current climate with the perturbed GCM climate for
another selected region. For example, if the changed climate
monthly characteristics projected for site A correspond to the
present climate conditions in site B, then the observed daily
series from B are used to represent changed climate series in
site A. This approach is most suitable for temperature, but may
be of limited value for discontinuous fields such as
precipitation.[12]
Let x(t) is the observed series of a single variable (e.g.,
temperature or precipitation amount). The values of the series
may be modified by one of the two following equations:
x'(t) = x(t) * Di(d)
x'(t) = <x>(d) + Di(d)[x(t)
- <x>(d)] (2)
where <x>(d) is the mean annual cycle of
the original series with d representing Julian day, Di(d)
is a modification function which may be given in terms of GCM-based
monthly increments, symbol * stands for either additive or
multiplicative operator and x'(t) is the new series.
Additive modification is typically used for temperature
characteristics, multiplicative modification is used for
precipitation and solar radiation.[13] The
method allows to change only mean (1) or variance (2) of the
series without changing other features of the stochastic
structure of the series (shape of the distribution,
autocorrelation function, correlations between variables, etc.).
Clearly, equations (1) and (2) may be combined to change both
mean and variance simultaneously. Operators + and - in (2) may be
substituted by multiplication and division, as well.
Two shortcomings of the method may be named: (i) the method
cannot be used for locations without historical observations, and
(ii) the length of the series is limited by length of the
observed series (this is, anyway, the shortcoming of the method
of analogues, too) which need not be long enough to make a
conclusive sensitivity analysis. In addition, there arises the
problem of ambiguity how to project the climate change scenario
into the series - this item is discussed at the end of the
forthcoming section, since this appears to be a problem in using
weather generators, too.
In developing the weather generator, stochastic structure of
the series is described by a statistical model. The model is
often based on Markov chains and autoregressive models of first
or higher (less frequently) order.[14,15,16,17,18]
Having derived parameters of the generator from the observed
series, we may generate arbitrarily long series with stochastic
structure similar to the real data. To generate data for changed
climate conditions, parameters of the generator are modified
according to the climate change scenario. For example, to
generate series of daily mean temperatures, we analyse the
observed series to get characteristics of the mean annual cycle
and interdiurnal variability. The annual cycle may be
characterised by the mean and variance for each day of the year,
the interdiurnal variability may be characterised by lag-correlations
(only lag-1 correlations are usually assumed). Having the set of
characteristics estimated from the observed series, we may
generate synthetic series for the present climate conditions
using random number generator. To generate series for changed
climate conditions, the annual cycles of the mean and/or variance
may be modified in accordance with the GCM-based climate change
scenario. The modification equations may have a form:
v'(d) = v(d) * Di(d)
(3)
where v'(d) and v(d), respectively, are
modified and original annual cycles of any characteristic.
Since interpolation of monthly characteristics, which are often
included in the set of parameters of the weather generator, is
considered more correct than interpolation of daily weather
series,[19] the weather generator may be used
even for sites without observational data. The disadvantage of
the model is that the reproduction of the stochastic structure of
the series depends on the choice of the model Ä not all
important characteristics of the stochastic structure of the data
are satisfactorily reproduced. Normally, preservation of the mean
is not the problem. The problems arise with reproduction of
variability (e.g., variability of monthly means, cf. Section 3.3.6)
and occurrence of extreme events. These characteristics, which
are sensitive to discrepancies in selecting appropriate
statistical model, may have a crucial impact on a stability of
the systems being studied: greater variability of the
hydrological cycle and crop production implies a requirement on
greater flexibility in water resource or food planning and, of
course, more extreme events or higher frequency of extreme events
(extremely low/high temperatures, long-lasting droughts or heavy
showers) may calamitously affect both hydrological cycle and crop
production.
In the case of a direct modification of a weather series [equations
(1) and (2)] as well as in case of modifying parameters of the
weather generator (3), the additive and multiplicative factors
for the changed climate may be taken from the GCM-based scenario.
The question stands, how to project climate change scenario into
the weather series or the parameters of the weather generator.
For example, consider the four-variate (RAIN, SRAD, TMAX,
TMIN) weather generator described in Section 3.1, which is a
typical representative used by the crop growth models. The set of
generator's parameters (Table I]) consists of
16 characteristics (4 characteristics per variable) which are
allowed to have an annual cycle and two 3x3 matrices of the
autoregressive model which are constant in course of the year.
The typical climate change scenario used for the four variables
consists of only 3 parameters per month: multiplicative
increments for solar radiation and precipitation amount and
additive increment for temperature. Therefore, we face the task,
how to project the three increments into the parameters of the
generator. Specifically, following questions must be answered:
Number of the uncertainties may be decreased (but not eliminated!)
with a help of the Wilks'[13] approach: the
links between the monthly statistics (means and variances) and
the parameters of the daily series can be used to adjust the
generator's parameters in a manner consistent with imposed
changes in monthly statistics.
It is accepted that GCMs are becoming increasingly reliable in
modelling upper-air circulation on a daily scale.[20]
This assumption is followed in many recent studies in which the
surface weather characteristics are downscaled from upper-air
circulation patterns based on relations obtained by statistical
analysis of observational data. The circulation patterns are
typically classified based on assessment of fields of
geopotential height at either 700 or 500 hPa pressure level. The
set of circulation types either coincides with some commonly used
synoptic classification (Lamb Weather Types[21]
are used by Wilby,[22] `Grosswetterlagen'
catalogue[23] is used by Bardossy and Plate[24]) or is increasingly frequently based on
`objective' classification techniques.[25,26]
Regression equations are used either to estimate surface weather
characteristics from the GCM-based upper-level characteristics on
a day-by-day basis, or they relate parameters of the surface
weather generator with upper-air circulation. In the latter
method, the series of surface weather characteristics is
synthesised by the weather generator with parameters conditioned
on upper-air characteristics.
In addition to using the circulation patterns derived from the
GCM output, the series of circulation patterns may be generated
by a stochastic generator (Markov chains or semi-Markov chains)
or taken from historical records with climatic conditions similar
to the projected climate.
The method is often employed to generate multi-site precipitation
data for hydrological studies: Wilson et al.[25]
use hierarchical modified Polya urn model to simulate multiple-station
precipitation conditioned on the regional weather type.
Precipitation amount distributions are assumed to be drawn from
spatially correlated mixed exponential distributions, whose
parameters vary by season and weather class. Weather types are
alternatively determined by several statistical methods,
including k-means algorithm and principal component analysis.
Weather types series is modelled by semi-Markov chain in which
the length of stay in a single state (weather type) is modelled
alternatively by geometric and mixed geometric distributions.
Bardossy and Plate[24] and Bogardi et al.[26] model multisite precipitation by truncated
multiple autoregressive model with parameters conditioned on
circulation types determined by subjective and `objective'
classification techniques, respectively. Wilby[22]
relates daily rainfall at two sites in southern England to the
Lamb weather types by using conditional probabilities, the time
series of weather types is generated by Markov chain.
Three crucial assumptions condition reliability of this
downscaling method:
Met&Roll is a user-friendly graphical PC program, which is
driven either by interactive menus or run in a fully automatic
regime. In the latter case, all required procedures are specified
in an initialisation file. The formulation of the weather
generator implemented in Met&Roll was adopted from Wilks.[13] The model variables are 4 daily weather
characteristics: total solar radiation (SRAD), maximum
temperature (TMAX), minimum temperature (TMIN) and
precipitation amount (RAIN). The precipitation occurrence
is modelled by a non-stationary first-order Markov chain with
transition probabilities varying throughout the year,
P01(d(t))
if RAIN(t-1) = 0
Pr(RAIN(t) > 0) = { (4)
P11(d(t))
if RAIN(t-1) > 0
where P01(*) and P11(*)
are probabilities of occurrence of a wet day (defined as the day
with precipitation amount exceeding 0.1 mm) following the dry or
wet day, respectively, d is a Julian day (d = 1
for January 1) and t is a rank of the day in the series.
Precipitation amount on wet days is modelled by gamma
distribution, GAMMA(ALPHA,BETA). The
standardised deviations of SRAD, TMAX and TMIN
from their mean annual cycles are modelled by a first-order
autoregressive model:
x*(t)
= Ax*(t-1) + Be(t) (5)
where A and B are [3x3] matrices,
e(t) is a random vector of three mutually
independent normally distributed variables (white noise) and x*
= (x*1,x*2,x*3)
is a vector of standardised values of SRAD, TMAX
and TMIN:
x*i(t) = [xi(t) - Mij(d)] / Sij(d) (6)
where x*i, i = 1,2,3,
stands for SRAD, TMAX and TMIN,
respectively, and Mij(d) and Sij(d)
are means and standard deviations of xi
for given day d of the year without (j=0) and
with (j=1) precipitation.
The set of parameters of the model (Table I)
includes 16 parameters which are allowed to vary during the year
and two matrices of the autoregressive model which are considered
constant. Annual cycles of the parameters are smoothed by robust
locally weighted regression,[27] which was
found superior to Fourier series commonly used for this purpose.[17,28,29]
The main procedures available in Met&Roll are:
Each day of the synthetic series is generated according to the following algorithm:
The purpose of the stochastic generator is to produce data
which are statistically similar to the observed series. In other
words, the statistics (including means, variances, frequency of
occurrence of extremes, correlations and lag-correlations between
variables) derived from the synthetic data should be
statistically insignificantly different from those derived from
the observed data. In validating Met&Roll, sufficiently long
(30 years for testing variability of monthly and annual means, 99
years in all other cases) synthetic time series were generated
for the tests to be well resolute. The synthetic series were
analysed and resultant statistics were compared with those
derived from the observed series. The 30-year (1961-1990, i.e.
WMO normal period) daily weather series from 16 Czech
meteorological stations were used for that purpose. The results
obtained for Hradec Kralove‚ are briefly discussed below.
Reproduction of annual cycles of parameters, except for the
shape parameter of the Gamma distribution, is within the frame of
random errors. The discrepancies in the case of the shape
parameter are due to the rounding errors, which cause the
information on the distribution of RAIN to be partly
`lost'. Reproduction of the parameters of the weather generator,
however, cannot be considered to be a generator's quality
certificate but rather a proof of numerical correctness of the
analysis and generation procedures.
The variables generated by the autoregressive model, equation (5), are normally distributed, which implies that the standardised sample coefficients of skewness and kurtosis have normal distribution, N(0,1). In present tests, these coefficients were calculated for individual weeks of the year from the observed 30y series. The results of the tests applied to observational data indicate:
Sample distribution of daily precipitation amount calculated
for individual months differs from the fitted Gamma distribution
in 3 (10) months of the year at the level of significance ALPHA
= 0.01 (0.05), based on CHI2 test. Considering the
whole set of 16 stations, the greatest deviations from the Gamma
distribution occur within the cold part of the year (October -
March): the number of stations with deviation statistically
significant at ALPHA= 0.05 is 14, 13, 15, 12, 14 and 14
in individual months. On the contrary, the best fit is indicated
within June - August: statistically significant deviation at ALPHA=
0.05 occurs at 4, 5 and 4 stations in individual months.
The tests comparing the distribution functions of the length of dry and wet spells derived from the observed and synthetic series indicate:
Although it was stated above that the generator well preserves
all its parameters (including matrices of the autoregressive
model), it does not automatically imply that the AR(1) model with
matrices constant in the course of the year satisfies the
structure of the data. On the contrary, the graph in Figure 3] indicates that the correlations
between variables may dramatically vary - even change the sign -
during the year, which is best pronounced in case of correlations
between TMAX and SRAD.
Reproduction of the variability of the longer-period (monthly,
seasonal, annual) characteristics represents a crucial test of a
stochastic weather generator.[18] Figure 4 gives a comparison of the synthetic-series-derived
and observed-series-derived standard deviations of monthly and
annual means of the four variables treated by Met&Roll. It is
seen that the model reduces the variability - especially for the
cold months (by about 30%). The values of the F-test statistic
indicate that the differences between observed and model
variances are statistically significant at the 5% level for 16 (out
of 48 for all 12 months and 4 variables) variances of monthly
averages and for 3 (out of 4, i.e. all except for TMIN)
variances of annual averages.
It is hypothesised that the underestimation of the variances may
result from
The second version of Met&Roll treats the series of daily
weather characteristics conditionally on upper-air circulation
patterns. Contrary to the models cited in Section 2.5 which use a
finite set of circulation types, circulation patterns in
Met&Roll are characterised by a small number of mutually
independent continuous variables - principal components (PCs).[30] The PC analysis was applied to the vector of
63 grid-related geopotential heights of 500 hPa air pressure
level in 00 GMT inside the area bordered by 40 W, 40 E, 35 N and
65 N (9x7 grid points). The resultant 5 (7) PCs extracted to
characterise winter (summer) circulation pattern explain 70 (73)
% of the variance in geopotential height field. This approach has
some apparent advantages:
The weather generator Met&Roll-2 may be alternatively used in
two modes:
Let x = (SRAD*, TMAX*, TMIN*) be a vector of
the standardised [equation (6)] daily weather characteristics,
and y = (PC1, PC2, ..., PCp)
is a vector of PC scores characterising the circulation pattern.
The full model (the word `full' refers to the fact that the
circulation patterns are stochastically generated, not taken from
observational data or GCM output) of the generator is then
expressed by following equations:
1. time series of the circulation patterns is modelled by
autoregressive model of the first order:
y(t) = W.y(t-1)
+ Z.e(t)
(7)
where W and Z are matrices to be
determined from the observed series and e is a p-dimensional
white noise.
2. The four surface daily weather characteristics (RAIN,
SRAD, TMAX and TMIN) are linked with the
circulation pattern by multiple linear regression:
Pr(RAIN(t)) = p0
+ qR.y(t)
(8)
x(t) = Q.y(t)
where p0 (scalar), qR
(vector) and Q (matrix) are to be
determined from the observed series.
The above equations imply that the surface weather variables also
follow the AR(1) model. Met&Roll-2 run in an independent mode
thus follows the same model as Met&Roll-1. However, matrices
A and B in equation (5) determined solely from the surface
weather data will better fit structure of surface weather series
compared to Met&Roll-2 whose matrices are affected by both
structure of the circulation pattern series and by relationship
between surface weather and upper air circulation. It is
therefore advisable that the generator Met&Roll-2 is used
only in a dependent mode. In this case, the three requirements
listed in section 2.5 should be fulfilled in order the synthetic
series of the surface weather characteristics to have a realistic
stochastic structure.
Having available only limited sample of data which do not include GCM output (required for validation of item (i) in section 2.5), only a preliminary test concerning item (ii) was performed. The results given in Table II show that correlations between upper-air circulation and surface weather characteristics are not high enough for the downscaling of the GCM output by this approach to be sufficiently effective.
In the present paper, approaches to creating high resolution surface weather data for changed climate conditions were discussed. Two of them were considered in developing the two versions of the Met&Roll weather generator.
Met&Roll-1 is a surface weather generator which may stochastically generate daily series of 4 weather characteristics. In validating Met&Roll-1, the tests were done to examine its ability to reproduce the stochastic structure of daily weather series. It was found that the generator well preserves some features of the stochastic structure of the series. On the other hand, some discrepancies were found in reproducing the shape of the distributions of SRAD and RAIN and the length of dry spells. Unfortunately, these characteristics may have a great effect on reproducing occurrence of the extreme events: For example, the frequency of occurrence of extremely high or low temperatures, extremely long droughts or long-lasting rain may be under- or over-estimated by the model. Furthermore, the correlations and lag-correlations among SRAD, TMIN and TMAX have a significant annual cycle, which contradicts the use of invariable matrices in the autoregressive model [equation (5)]. Evidently, the model of the generator may be modified to account for the above discrepancies. The modifications might include:
In addition to the problem with reproducing the observational
series, it appears that projection of the climate change scenario
into parameters of the generator requires some amount of
speculations. To overcome the uncertainties, further research was
focused on development of Met&Roll-2.
Weather generator Met&Roll-2 relates the surface weather
characteristics with upper-air circulation and may be run in two
modes: in an independent mode in which the series of circulation
patterns is stochastically generated or in a dependent mode in
which the surface weather characteristics are generated
conditionally on circulation patterns derived from a GCM output
or historical observations. Circulation patterns are
characterised by principal components obtained from 63 grid-related
geopotential heights of 500 hPa pressure level over Europe, and
the four surface weather characteristics are related to the
circulation patterns by multiple linear regression. In order the
Met&Roll-2 may produce reliable daily series of weather data
for changed climate, following conditions are required to be
fulfilled:
Since the generator is still under development and only
limited amount of data was available at the time of writing this
paper, only a preliminary test is referred here. The test is
focused on the potential of the upper-air circulation to explain
the surface weather conditions. The results (Table
II) show that the four weather characteristics are only
weakly correlated with the circulation. This means, that even if
the variability of the circulation pattern would be perfectly
simulated by GCM, the stochastic structure of the synthetic
surface weather series cannot satisfactorily fit the structure of
the observed series. The task for the future may be thus
formulated as following: optimisation of the method of
circulation pattern characterisation, which would be
satisfactorily reproduced by GCM and which would well explain the
surface weather conditions.
Acknowledgement:
I would like to thank my colleague Radan Huth
for providing circulation data and giving necessary explanation
to them, Jaroslava Kalvova for her reminders
concerning General Circulation Models, and the Czech
Hydrometeorological Institute for surface weather data
used in validating the weather generator. The study was performed
within the frame of grant projects 205/96/1669 and 205/96/1670
supported by the Grant Agency of the Czech Republic.
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Figure I. Five approaches to creating high-resolution weather data from the GCM output. References relate to sections of the paper.
Figure 2. The weekly series of coefficients
of skewness (a) and kurtosis (b) derived from the 30y observed
series of daily sum of global solar radiation (SRAD).
Triangles relate to dry days, squares are for wet days. The
coefficients are standardised in order their sample values would
have normal distribution, N(0,1), for normally distributed
variable.
Figure 3. Annual cycle of lag-1 correlations (a) and lag-0 correlations (b) between standardised values of TMAX and the other variables of the autoregressive model. The coefficients for individual weeks are calculated from the 30y observed series. The vertical bars at the right part of the graphs mark the 95% confidence interval around the all-year correlations used by the weather generator's AR(1) model.
Figure 4. Reproduction of the variability of monthly and annual means by Met&Roll-1. The figure displays ratios of sample standard deviations of monthly and annual averages of SRAD, TMAX, TMIN and RAIN as derived from 30y observed and synthetic time series. The horizontal heavy dashed and dotted lines delineate 99% and 95% regions for accepting hypothesis that the variances derived from the both series differ statistically insignificantly.
Table I. Parameters of the model of the
weather generator Met&Roll-1.
| Parameter | Description |
Number of stored values - representation of the annual cycle |
| P1, P01 | parameters of the first-order Markov chain for precipitation occurrence | one per day |
| ALPHA, BETA | parameters of the Gamma distribution for daily precipitation amount | one per month |
| M(x|dry), S(x|dry), M(x|wet), S(x|wet), where x = SRAD, TMAX, TMIN respectively | smoothed annual cycles of averages and standard deviations of SRAD, TMAX and TMIN; separately for dry and wet days | one per day |
| A and B | Matrices of the first-order autoregressive model for interdiurnal variability of standardised values of SRAD, TMAX and TMIN | one per year |
Table II. The variance of the four surface weather characteristics explained by circulation on a given day and surface weather characteristics on the previous day. The circulation is characterised by vector of principal components, PC(t), the vector of surface weather characteristics is defined as SW*(t-1) = (RAIN, SRAD*, TMAX*, TMIN*). Asterices indicate that the variables were standardised.
Dependent ¦ Independent variables
variables +------------------------------------
¦ PC(t) ¦ SW*(t-1)
(time = t) +------------------+-----------------
¦ summer winter ¦ summer winter
---------------------+------------------+-----------------
SRAD* ¦ 0.209 0.036 ¦ 0.213 0.109
TMAX* ¦ 0.533 0.304 ¦ 0.568 0.626
TMIN* ¦ 0.311 0.235 ¦ 0.615 0.657
(1) (RAIN)1/4 ¦ 0.079 0.172 ¦ 0.352 0.233
(2) (RAIN)1/4 (RAIN>0)¦ 0.022 0.084 ¦ 0.108 0.109
(3) RAINOCC (lin) ¦ 0.098 0.155 ¦ 0.077 0.133
(3) RAINOCC (log) ¦ 0.101 0.156 ¦ 0.077 0.133
(1) days with zero precipitation amount are included
(2) days with zero precipitation amount are not included
(3) the regression function between the dependent binary variable (RAINOCC = occurrence of precipitation) and explanatory variables were determined by both linear (lin) and logistic (log) regression.