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published in: Meteorologicke Zpravy, vol 49 (1996),
p. 129-138
language: Czech + English abstract; includes 18 figures and 1 table with English captions.
This paper is a continuation of Dubrovsky, 1995a.
Martin Dubrovsky
Institute of Atmospheric Physics, Husova 456, 50008 Hradec
Kralove, Czech Republic
ABSTRACT:
The single-site four-variate stochastic weather generator
Met&Roll [4] treats the time series of daily sum of global
solar radiation (SRAD), daily maximum and minimum temperatures (TMAX
and TMIN) and daily precipitation amount (RAIN). Precipitation
occurrence is modelled by first-order Markov chain model,
precipitation amount by Gamma distribution. The standardised
deviations of SRAD, TMAX and TMIN from their means are modelled
by first-order autoregressive model with means and standard
deviations being conditioned on precipitation occurrence. A
capability of the generator to reproduce stochastic structure of
the time series is validated in the present paper with use of
observed 30-y series from 16 Czech stations. The individual tests
are focused on: (a) reproduction of parameters of the generator (figs.
2-5; tab. ), (b) normality of SRAD, TMAX and TMIN (figs. 6-8), (c)
distribution of length of dry and wet periods (fig. 9), (d) the
goodness of Gamma distribution for modelling precipitation
amount, (e) annual cycle of lag-0 and lag-1 correlations among
variables of the autoregressive model (fig.10-12), and (f)
reproduction of the variability of monthly and annual means of
the four generator's variables (fig. 13). The results of the
tests indicate certain deviations (greater in winter) of the
model from reality. Possible modifications to the model are
suggested in the conclusion to achieve better fit to real data.
In particular, the matrices of the autoregressive model should be
allowed to vary during the year to account for the significant
annual cycle of correlations among the variables. Extra tests (figs.
14-18) have confirmed that Met&Roll better reproduces
stochastic structure of real data compared to generators WGEN and
SIMMETEO. Since all three generators use the same model to
generate synthetic series, the result is explained as following:
Met&Roll derives more parameters from the observed data and
uses more detailed representation of their annual cycles.
Fig. 1. Position of stations whose data were used for validation
of the weather generator Met&Roll. Squares (with arrow and
station index) represent reference climatic stations for which
the 30-years series of daily climatic characteristics were
assembled. Horizontal lines denote stations of the actinometric
network, vertical lines denote stations whose sunshine data were
used to localize radiation series. Reference stations: 518 =
Ruzyne, 523 = Hostomice, 561 = Semcice, 563 = Stara Boleslav, 572
= Ondrejov, 627 = Cechtice, 636 = Kostelni Myslova, 649 = Hradec
Kralove, 659 = Pribyslav, 687 = Velke Mezirici, 698 =
Kucharovice, 723 = Brno - Turany, 754 = Stare Mesto, 755 =
Straznice, 774 = Holesov, 779 = Strani.
Fig. 2. Annual cycles of characteristics (average; average +/-
standard deviation) of daily sums of global solar radiation in
Hradec Kralove, 1964-1990. The curves were smoothed by robust
locally weighted regression [2,10]. Heavy and thin lines
represent characteristics for dry and wet days respectively,
derived from the observed series; crosses and circles represent
the same characteristics but derived from the synthetic series
generated by Met&Roll.
Fig. 3. The same as fig.2 but for TMAX.
Fig. 4. The same as fig.2 but for TMIN.
Fig. 5. Annual cycle of parameters of the precipitation model
derived from the observed series (solid lines with filled symbols)
and synthetic series (dashed lines with empty symbols). Circles,
delta-shaped triangles, nabla-shaped triangles and squares
represent parameters PI1 (unconditional probability of
wet day occurrence), PI01 (probability of occurrence
of wet day following the dry day), ALPHA x BETA (the product of
the two parameters of Gamma distribution equals the expectance of
daily precipitation total in wet days), ALPHA (shape parameter of
the Gamma distribution) respectively. Greek characters PI, ALPHA,
BETA are substituted by appropriate latin characters p, a, b in
the figure.
Fig. 6. The annual cycle of coefficients of skewness (squares for
wet days, crosses for dry days) and kurtosis (triangles for wet
days, asterices for dry days) of SRAD for weeks of the
year. The coefficients are standardized in order their sample
values have N(0,1) distribution for normally distributed
variable.
Fig. 7. The same as fig.6 but for TMIN.
Fig. 8. The same as fig.6 but for TMAX.
Fig. 9. Frequency function of the length of dry and wet spells.
Crosses and squares represent observed frequencies of dry and wet
spells of given length, solid and dashed lines denote the model
values (recalculated from the 99y synthetic series).
Fig. 10. Annual cycle of coefficients of correlation (right graph)
and serial lag-1 correlation (left graph) of SRAD with
remaining variables of the autoregressive model (all variables
were standardised). The coefficients were calculated for
individual weeks. The vertical bars at the right-hand portions of
the graphs demarcate intervals for acceptance of the hypothesis (ALPHA
= 5%) that the coefficient of correlation for individual week is
identical to the respective all-year coefficient (see tab.I -
"observed series") used by the weather generator's AR(1)
model.
Fig. 11. The same as fig.10 but for the daily maximum
temperature, TMAX.
Fig. 12. The same as fig.11 but for the daily minimum
temperature, TMIN.
Fig. 13. Reproduction of the variability of monthly and annual
means. The figure displays ratios of standard deviations of
monthly and annual averages of SRAD (squares), TMAX (+), TMIN
(') and RAIN (triangles) derived from 30y observed and
synthetic time series generated by Met&Roll. The horizontal
lines represent square roots of ALPHA/2 and (1 - ALPHA/2)
quantiles (for ALPHA = 0.01 and 0.05) of the F30,30
distribution which are the critical values of the ratio of the
sample standard deviations for rejecting the hypothesis on
equality of variances of the two samples.
Fig. 14. Deviations of the synthetic-data-based annual cycle of
mean daily sums of global solar radiation (SRAD) from the annual
cycle used for generating data. The series were generated by
Met&Roll (solid line), WGEN (dotted line) and SIMMETEO (dashed
line). The thin and heavy lines are used to distinguish between
characteristics valid for dry and wet days respectively.
Fig. 15. The same as in fig.14 but for the standard deviation of
the daily radiation sums.
Fig. 16. The deviations of the synthetic-data-based values of
gamma parameters (parameters ALPHA and BETA are substituted by a
and b in the legend box) from those used for generating the data.
The synthetic data were generated by Met&Roll (solid lines,
M&R), WGEN (dotted lines, WGEN) and SIMMETEO (dashed lines,
SIMM).
Fig. 17. The same as in fig.16 but for the parameters of the
Markov chain model PI1 and PI01.
Fig. 18. Variability (expressed by standard deviation) of the
annual and monthly means of SRAD derived from the observed
data (filled squares) and synthetic data generated by
MET&ROLL (empty square), WGEN (+) and SIMMETEO (x). The
length of all series was 30 years.
Tab. I Covariance matrices, C(X*(t),X*(t)),
and serial covariance matrices, C(X*(t),X*(t-1)),
derived from the standardised values of (a) observed data and (b)
synthetic data generated by Met&Roll, WGEN and SIMMETEO.
==================================================================================================================
| | series generated by
| observed series |--------------------------------------------------------------------------
| | Met&Roll | WGEN | SIMMETEO
------------------------------------------------------------------------------------------------------------------
C[X*(t),X*(t)] =
| | | |
|SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t)
SRADt | 0.98 0.32 -0.17 | 0.98 0.32 -0.18 | 0.98 0.18 -0.17 | 1.00 0.15 -0.14
TMAXt | 0.32 0.97 0.64 | 0.32 0.99 0.65 | 0.18 0.99 0.65 | 0.15 0.99 0.66
TMINt |-0.17 0.64 0.97 |-0.18 0.65 0.99 |-0.17 0.65 0.99 |-0.14 0.66 0.99
-------------------------------------------------------------------------------------------------------------------
C[X*(t),X*(t-1)] =
|SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t) |SRAD(t) TMAX(t) TMIN(t)
SRAD(t-1) | 0.23 0.13 -0.01 | 0.23 0.13 -0.02 | 0.23 0.09 -0.08 | 0.18 0.08 -0.06
TMAX(t-1) | 0.07 0.67 0.68 | 0.07 0.67 0.68 | 0.02 0.62 0.57 | 0.02 0.62 0.58
TMIN(t-1) |-0.04 0.49 0.64 |-0.04 0.49 0.65 |-0.08 0.47 0.67 |-0.07 0.47 0.67
==================================================================================================================
note: Of the listed models only Met&Roll employs matrices derived from the learning series for generation of synthetic series.