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A comparative assessment of different methods for detecting inhomogeneities in Turkish temperature data set

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Int. J. Climatol. 18: 523–539 (1998)
Department of Ci6il Engineering, Cassie Building, Claremont Rd, Uni6ersity of Newcastle, Newcastle upon Tyne, NEI 7RU, UK
Climatic Research Unit, Uni6ersity of East Anglia, NR4 7TJ, UK
Recei6ed 3 August 1997
Re6ised 12 No6ember 1997
Accepted 13 No6ember 1997
Regression models are developed to predict point rainfall statistics, with potential application to downscaling general
circulation model (GCM) output for future climates. The models can be used to predict the mean daily rainfall
amount and the proportion of dry days for each calendar month at any site in England and Wales, and use the
following explanatory variables: (i) geographical (altitude, geographic coordinates, and distance from nearest coast);
and (ii) atmospheric circulation variables (mean values of air-flow indices derived from mean sea-level pressure grids).
Values predicted by the models, for 10-km grid squares covering the whole of England and Wales, are in reasonable
agreement with the 1961–1990 climatology of Barrow et al. (1993). The potential use of the models in hydrological
climate change impact studies is discussed. © 1998 Royal Meteorological Society.
KEY WORDS: England
and Wales; climate change; hydrological modelling; general circulation models; downscaling; rainfall;
atmospheric circulation; vorticity; regression models
The impact of changes in climate on hydrological systems is a problem of crucial and growing importance
to society. A capability for producing realistic rainfall statistics or rainfall time series for future climate
conditions would be of great utility for a range of environmental impact studies. Hydrological impact
assessments often require a suitable rainfall time series to be generated for input to a hydrological model
of a system. For example, at the catchment-scale, a daily rainfall time series for a future climate may be
generated and used with a rainfall-runoff model to simulate future river flow scenarios. In urban drainage
studies, where an even finer time resolution is required, an hourly time series may be generated and used
with a hydraulic flow simulation model to predict the frequency of sewer overflow discharges to receiving
water courses. At the monthly scale, a rainfall series may be simulated to provide predictions for future
reservoir levels.
The usual tool for simulating future climates is a general circulation model (GCM) of the atmosphere,
assuming prescribed changes in atmospheric composition and forcing (Houghton et al., 1996). However,
GCMs are not appropriate for simulating rainfall series at the catchment-scale, because they operate at
very coarse spatial scales (typically some 300 km × 300 km). Hence, a key requirement for the production
of rainfall scenarios from GCMs is the development of methods to ‘downscale’ GCM output to scales
appropriate for catchment models of hydrological systems. In particular, methods are required to relate
* Correspondence to: Department of Civil Engineering, Cassie Building, Claremont Rd, University of Newcastle, Newcastle upon
Tyne, NE1 7RU, UK. e-mail:
Department of Statistics, Massey University, Albany, Private Bag 102 904, North Shore, Auckland, New Zealand
Contract grant sponsor: EC Environment Research Programme under the POPSICLE project; Contract grant number: EV5V-CT940150, Climatology and Natural Hazards
CCC 0899–8418/98/050523 – 17$17.50
© 1998 Royal Meteorological Society
GCM output to single-site rainfall data. Before these methods may be applied to the future climate, they
must first be developed and validated for the present climate. This work sets out to develop a simple
downscaling tool for relating point rainfall statistics to atmospheric circulation variables, such as those
that are output from GCMs. The downscaling tool is based on two simple regression models. The point
rainfall statistics can then be used as the basis for parameterizing a stochastic rainfall model for
generating time series of rainfall.
There is an extensive literature on simulating rainfall scenarios for future climates: recent examples are
Bardossy and Plate (1992), Buishand and Brandsma (1997), Conway and Jones (1997) and Hughes and
Guttorp (1994). The models described here differ from those in previous work because they enable the
prediction of rainfall statistics for any site in England and Wales, and are therefore applicable at sites
where data are limited or do not exist.
The paper is divided into sections as follows. In Section 2 the observational data used in the regression
analysis are described. Appropriate models are then proposed and fitted in Section 3, and their predicted
values compared with the observed values. In Section 4, the models are implemented within a geographical information system (GIS) and are used to predict values over 10-km grid squares covering the whole
of England and Wales. A validation compares the predicted values with those of the 1961–1990
climatology of Barrow et al. (1993). The paper concludes with a discussion of possible applications to
studies of the impacts of climate change on hydrology.
2.1. Explanatory 6ariables
The data used comprise daily time-series records (1961–1990) of atmospheric circulation variables (see
Jones et al., 1993 for details) and rainfall at 67 sites (Figure 1). The rainfall sites were distributed
throughout England and Wales, with two located in the adjacent borders region of Scotland, and were
obtained from the UK Meteorological Office. The rainfall sites were chosen for uniform spatial coverage,
and as complete temporal coverage as possible. Ten sites at higher elevations (in the Welsh mountains,
Pennines and Lake District) were specially selected in order to reduce the bias towards low elevations
which is inevitable in operational rainfall measurement networks.
Four atmospheric circulation variables are used: mean sea-level pressure (p), total shear vorticity (z),
and the zonal and meridional components of geostrophic air-flow, (u and 6). A positive value of u
corresponds to air flowing from west to east, whilst a positive value of 6 corresponds to air flowing from
south to north. The vorticity (z) is a measure of atmospheric rotation, with positive values corresponding
to low pressure (cyclonic weather) and negative values corresponding to high pressure (anti-cyclonic
weather). The data are derived from a relatively coarse grid of mean sea-level pressure (5° latitude by 10°
longitude) shown in Figure 2. The gridded data set was initially developed by Jenkinson and Collison
(1977), and later used by Jones et al. (1993).
The vorticity and flow-velocity units are geostrophic and expressed as hPa per 10 ° latitude at 55 ° North
(see Jones et al., 1993) and when multiplied by 1.2 are speeds in ms − 1. These variables differ from those
used in previous work (Conway and Jones, 1997): they used flow strength and direction instead of u and
6, and considered z alone as a measure of cyclonicity. Monthly mean values of the circulation variables
are shown in Figure 3. A seasonal variation is evident in the two air-flow velocities, reflecting the
dominance of westerly frontal weather systems during the winter in the UK.
Each site is characterised by the variables: altitude (a), Ordnance Survey east grid reference (e), north
grid reference (n), and distance from the nearest coast (c). Thus, there are a total of eight explanatory
variables available for the analysis: p, u, 6, z, a, e, n, and c.
For each year-month in each record of data, the mean daily value was found for each variable to give
monthly mean time-series of u, 6, z, p and rainfall amount (Y) at each site (Table I). The proportion of
dry days was also found for each year-month to give monthly proportion dry (P) time series at each site.
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
A day is deemed dry if less than 0.2 mm of rainfall is recorded. Excluding missing data results in a total
of 17 567 site-months available for model fitting. The term ‘point rainfall statistic’ will be used here to
refer to either the mean daily rainfall or the proportion of dry days at a single rain gauge or site.
2.2. Temporal aggregation
A sampling interval, or aggregation level, of one month was adopted. The use of a daily interval (or
n-day mean where n is less than five or so) would be preferable, in order to resolve individual synoptic
weather systems. However, we could not obtain a reasonable degree of explained variance at aggregation
levels below monthly with the present data set. If a daily level is desired for downscaling, then other
approaches may be more suitable (e.g. Conway and Jones, 1997). A monthly time level does have a
number of advantages:
(i) the work presented here provides point rainfall statistics which can be used in the estimation of the
parameters of a stochastic rainfall model. Statistics are required for each calendar month (discussed
in Section 5). It is therefore appropriate to use this aggregation level throughout (rather than
aggregating predicted daily values, for example). The stochastic model will then perform the
sub-monthly disaggregation process, and is able to produce daily (and hourly) rainfall amounts with
the correct statistical properties.
Figure 1. Location of raingauges used in the regression analysis
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 2. Grid points used for calculations of circulation indices
(ii) each observed time series can be treated as approximately stationary over one month (a longer
sampling interval can result in a non-stationary time series due to seasonal variations or seasonal
(iii) the sample autocorrelation for monthly rainfall series tends to be small so that the data may be
treated as approximately independent in time.
We have analysed all of the months of the year together rather than performing individual analyses on
the seasonal or monthly timescale. This ensures that only the explanatory variables are used for
producing seasonal variations, rather than the time of year. Otherwise regression relations would be
produced for each fixed calendar month, with the risk that the relationship between climate and time
of year may not be maintained in future climates. The seasonal variation can then also be used in
validation of the method.
2.3. Spatial aggregation
We have analysed the whole of England and Wales together rather than performing regional
analyses, for example following the sub-division into five regions used by Wigley et al. (1984) or four
areas identified by Mayes (1991). Such a regional analysis would be valuable in better resolving
synoptic scale weather systems. However, the coarse resolution of the circulation variables used here
does not support such an analysis, which could perhaps be performed using Mayes (1991) as a basis.
However, for similar reasons as outlined above for the seasonal case, any given regionalization may
not be appropriate in a changed future climate, and a better solution is to use circulation indices at
higher spatial resolution.
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
3.1. Formulation of models
The mean daily rainfall at site i, year j, and month k, denoted Yijk, always takes values greater than
zero. Hence, it is appropriate to formulate the following regression model:
Yijk =exp{a0 +aaai +aeei +anni +acci +auujk + a66jk + azzjk + appjk + oijk }
where the a’s are parameters to be estimated and oijk is random or modelling error. When oijk has a normal
distribution with mean zero and standard deviation s, the expected value is
E(Yijk )= ry × exp{a0 +aaai +aeei +anni +acci + auujk + a66jk + azzjk + appjk },
where ry is a correction ratio to allow for the bias resulting from re-transformation from ln(Y) to Y (see
Section 3.2). If the oijk come from a normal distribution, ry = exp{s 2/2}. In practice, however, the oijk come
from a skewed distribution; provided the sample is large enough, which it is here, an estimate for ry can
be found empirically to give an approximately unbiased estimate of E(Yijk ) in Equation (2).
The random variable Pijk, representing the proportion of dry days for site i, year j, month k, always
takes values between 0 and 1. Hence, it is appropriate to postulate the following regression model:
Figure 3. Mean values of z, u, 6, p over the period 1961– 1990 plotted against month. Units of z, u, 6 are hPa per 10° latitude at
55° North
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Table I. Variables used in the regression analysis*
Variable name
Mean daily rainfall (mm)
Proportion of days with less than 0.2 mm of rain
Altitude (10 m)
Easting (100 km)
Northing (100 km)
Distance from nearest coast (km)
West–east component of flow velocity
South–north component of flow velocity
Pressure (hPa)
Error in predicting Yijk
Error in predicting Pijk
* Indices i, j and k refer to site, year, and month, respectively.
Pijk = 1/[1 +exp{ − (b0 +baai +beei +bnni + bcci + buujk + b66jk + bzzjk + bppjk + jijk )}],
where the b’s are parameters to be estimated and jijk is random or modelling error. After taking the
expectation of the Taylor expansion of Pijk =f(jijk ),
E(Pijk )=rp /[1 + exp{ − (b0 +baai +beei + bnni + bcci + buujk + b66jk + bzzjk + bppjk )}],
where rp is a correction ratio for the retransformation bias (greater than 0 and less than 1). Again,
provided the sample size is large enough, this ratio can be found empirically to ensure that an
approximately unbiased estimate is obtained for E(Pijk ) in Equation (4).
3.2. Fitted models
The transformations ln{Yijk } and ln{(Pijk /(1 − Pijk )} were made to give linear equations in the model
parameters. The a’s and b’s were then estimated by linear least squares regression (forced entry method);
the resulting estimates are shown in Table II, with the percentage of explained variance (r 2), the standard
deviation (S.D.) of the residual errors (oijk and jijk ) and empirical estimates of the correction ratios (r).
Three model variants were investigated. The first variant (denoted model ‘Z’) used independent
variables u, 6, z, a, e, and c. It was found that northing (n) had negligible effect. The second variant
(model ‘P’) replaced vorticity (z) with pressure (p) to characterise the cyclonicity of the circulation.
Coefficients from both of these models are shown in Table II, and it can be seen that they perform
similarly: differences between the two models are discussed further in Section 3.3. The third variant was
to construct seasonal models for mean and proportion dry, i.e. four separate regressions for samples
separated according to season; winter (December, January, February (DJF)), spring (MAM), summer
(JJA) and autumn (SON). Although this seasonal model produced marginally higher percentages of
explained variance, it is not considered a practical model, since it introduces a seasonal variable which
cannot be guaranteed to behave in a similar fashion under future climate change.
The model coefficients appear to be in general agreement with the observed trends in England and
Wales rainfall climatology (see for example Hulme and Barrow, 1997). Both models predict more rain at
high altitudes and in the west of England and Wales, and less rain on the coast (Table II). Lower rainfall
is predicted in the east of England, because of the rain shadow in the lee of the Welsh and Pennine
mountains. Moreover, greater rainfall is predicted for low values of pressure or high values of vorticity,
as well as for stronger zonal flow (u), corresponding to westerly-type frontal weather systems.
Both models explain approximately half the variability in the data, which is reasonable in view of their
simple form. More complex models were also considered and found to slightly improve the r 2 values. For
example, models using powers, cross-products, and more detailed sample distributional properties (e.g. the
variance and autocorrelation of the daily vorticity) were considered. However, it was decided to retain the
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
© 1998 Royal Meteorological Society
Table II. Estimated parameters
Model P
Model Z
a (altitude)
c (distance
from coast)
e (easting)
p (pressure)
u (zonal flow)
6 (meridional
z (vorticity)
r (correction
Mean daily rainfall (ln Y)
Proportion of dry days (ln{P/(1−p)})
S.E. (b. )
parameter (b. )
S.E. (â)
(b. )
S.E. (b. )
Int. J. Climatol. 18: 523 – 539 (1998)
Residual S.D.
S.E. (â)
Proportion of dry days (ln{P/(1−p)})
Mean daily rainfall (ln Y)
simpler models formulated above because the parameter estimates are easier to interpret, and the small
improvements to the r 2 values do not justify the additional complexity.
The percentage of explained variance for each month for both models P and Z was calculated and is
shown in Table III. This shows a poorer performance in the months April–July inclusive, due to a higher
relative error caused by the lower mean rainfall in these months. The model performance could again be
improved by formulating a seasonal model; this has not been done for the reasons given above.
The use of temperature as an explanatory variable was also considered. Monthly time series of Central
England temperature were taken from Manley (1974) and Parker et al. (1992) and the regression models
re-fitted with a further variable for temperature included. However, the residual standard deviation of the
re-fitted models had a value very close to the fitted models of Table II (the differences were less than
0.001). It was therefore decided to omit the temperature variable for this analysis, because it appears to
be of no practical value in the present day case.
Clearly, for some geographical regions, with climates different from England and Wales, temperature
is likely to be an important explanatory variable and should then be included in the model (e.g. see
Brandsma and Buishand, 1997). Also, for future impact studies, there is a strong case for the inclusion of
temperature, as a rise in temperature may be the major observed signal of climate change. There is clearly
potential for more detailed analyses using other variables, including relative humidity as well as a range
of synoptic variables at different heights in the atmosphere (e.g. 500 hPa geopotential height). This is
discussed further in Section 6.
3.3. Analysis of model performance
The residual errors (observed−predicted values) were plotted against the explanatory variables and
against predicted values of the point rainfall statistics. These plots revealed no systematic departures in the
model assumptions; for example, the linear relationship for ln(Y) and ln{P/(1− P)} does not appear to
be violated.
A more stringent test involves plotting the residual errors against month as a test of the models’
representation of seasonal variation. This is particularly important as predictions for each month are
likely to be required in applications. Consequently, summary statistics of the predicted and observed
values were evaluated by pooling the data for each calendar month and are shown in Figures 4 and 5.
Plots of all the residuals against month were also considered, rather than summary statistics, but it was
difficult to detect discrepancies in these plots because of the large number of data.
Table III. r 2 values for regression models for all England and Wales 10 km grid values
validated against climatology
Mean daily rainfall
Proportion of dry days
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 4. Summary statistics of mean daily rainfall (Y) plotted against month for 67 sites used in regression; (a) mean; (b) standard
deviation; (c) skewness
The sample means are in reasonable agreement; the mean of the predicted values generally follow
the seasonal variation evident from the mean of the observed values (Figure 4(a) and 5 (a)). The
major discrepancy to be noted is the overprediction of mean rainfall (and corresponding underprediction of proportion dry) by model ‘Z’ in May. However, model ‘P’ performs adequately throughout
the year, and for this reason has been preferred for further use in this analysis despite theoretical
support for the use of vorticity rather than pressure. The reasons for the overprediction are not clear,
but may be found in the seasonal variation in land-sea temperature differences. Sea surface temperature was not included in these models, but can be important for some wind directions and warrants
further study.
Some minor discrepancies may also be noted, for example a slight tendency to underpredict from
August to December (Figure 4(a)). Greater discrepancies are evident in the plots for the sample
standard deviations (Figure 4(b)), where it is clear that the models under-predict these values. The
skewness plots (Figure 4(c) and 5(c)) show a reasonable agreement between the predicted and observed values for most months; however, a clear over-prediction is evident for some months (e.g.
August in Figure 5(c)). The practical implication of these discrepancies, which clearly depends on the
intended application, is discussed in Section 5.
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
4.1. GIS implementation
Barrow et al. (1993) derived a baseline climatology for the period 1961–1990 for mean monthly
precipitation and number of rain days. They fitted thin-plate splines to monthly data taken from a large
number of sites (2376 for mean, 168 for rain days) scattered throughout Great Britain, interpolated onto
a 10-km grid using elevation as an independent variable. Although some of the sites correspond with
those in the regression analysis (13 for mean rainfall, and eight for rain days), the climatology is the most
comprehensive and accurate available and forms a good basis for a spatial validation of the regression
models over England and Wales.
The regression models and climatology were implemented in a geographical information system (GIS)
for 10-km grid squares covering England and Wales, and identical to that of Barrow et al. (1993). The
GIS was first used to calculate mean values of the regression site-variables over each grid square. The
regression models and the GIS information were then used to predict the mean monthly rainfall and the
proportion of dry days for each month over all the grid squares. Time series of predicted Y and P were
calculated for each grid square using the 1961–1990 time series of circulation variables and the grid
square geographical information. The values for each calendar month were then averaged to give
1961–1990 mean-monthly rainfall statistics.
Figure 5. Summary statistics of proportion of dry days (P) plotted against month for 67 sites used in regression; (a) mean; (b)
standard deviation; (c) skewness
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 6. Summary statistics for mean of all England and Wales 10 km grid square values for (a) mean daily rainfall; (b) proportion
of dry days
The spatial mean for England and Wales was calculated for each calendar month for both model ‘P’
and model ‘Z’, and are plotted in Figure 6 for comparison with the climatological values. This confirms
the poor performance of the vorticity model (model ‘Z’) in May. The annual averages are mapped in
Figures 7 and 8 for comparison with the climatology, and an example monthly average (for January) is
shown in Figure 9.
Figure 7. Maps of mean annual rainfall on a 10 km grid. (a) Climatology of Barrow et al. (1993); (b) predicted values given by
regression model P
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 8. Maps of mean annual proportion of dry days. (a) Climatology of Barrow et al. (1993); (b) predicted values given by
regression model P
4.2. Ele6ation dependence
The predicted values are found to be rather sensitive to the elevation value taken for the grid square.
It is not obvious what the correct elevation estimates are for use at a 10-km scale; e.g. Barrow et al. (1993)
discuss whether the grid-square mean or maximum elevation should be used in interpolation. Elevation
data taken from the Digital Chart of the World (Defense Mapping Agency, 1992) at approximately 1 km
resolution were used to investigate this problem. The use of the mean of the 1-km elevations (over the
same 10-km grid used in Barrow et al., 1993) results in a general under-estimation of mean rainfall, whilst
use of the highest 1-km grid square value results in an over-estimation.
It was found that the best fit to the climatology was obtained using mean elevation plus one standard
deviation of the 1-km grid squares contained in the 10-km grid square. One standard deviation ranges
from a few metres in a flat region such as East Anglia, to some 100 m in mountainous areas.
Figure 9. Maps of mean monthly rainfall for January. (a) Climatology of Barrow et al. (1993); (b) predicted values given by
regression model P
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
4.3. Validation
The general reproduction of spatial patterns and amounts is good, given that single regression models
are used to predict the point rainfall statistics for the whole of England and Wales. The primary gradients,
lee and coastal effects are well reproduced. Discrepancies are apparent, particularly at high altitudes
where an underprediction occurs (Figure 7), probably because of insufficient data at these altitudes when
fitting the regression models.
Note that a more detailed map is produced by the regression model than is available for the climatology
for the proportion of dry days (Figure 8). This is due to fewer data being used in the climatology for rain
days than for the mean rainfall: only 168 of 7201 sites used for the mean daily rainfall climatology had
this information available.
Caution must be exercised in the use of both climatologies for high elevation sites, and particularly so
for the rain day case in view of the small number and low elevation of the sites employed.
Some discrepancies are evident for individual months, the most noticeable being an under prediction in
the mean monthly rainfall along the south coast in winter (Figure 9 illustrates this for January). Failure
to reproduce this more detailed aspect of spatial patterns is unsurprising given the regional nature of the
circulation variables and regression models and the relative sparsity of rain gauges used in the regression
The distribution of mean monthly rainfall and proportion of dry days for the climatology, regression
models and gauge records were plotted using values for all the grid squares and for all months, averaged
over the period 1961 – 1990 (Figures 8 and 9). Satisfactory agreement was obtained, although an
under-prediction of high rainfalls is again evident in the tail of the sample distributions (Figure 8). This
corresponds with the under-estimation of rainfall at high altitudes evident in the GIS maps (Figure 5).
4.4. Assessment of model stability
If the regression models derived above are to be applicable to future climates, the model parameters
must be shown to be robust with respect to climate changes. In other words, the relationship between
rainfall statistics and circulation indices must be time invariant (or stationary). Wilby (1997) examined the
problem of ‘stationarity’ in UK precipitation time series and concluded that it may be necessary to use
other atmospheric variables to reliably reproduce observed time series attributes in downscaling models.
Other variables suggested include sea surface temperatures and larger scale atmospheric indices (e.g. the
North Atlantic Oscillation Index).
As this increased complexity is beyond the scope of the present models, it is necessary to assess how
stable the present simple models are under the observed range of climate variability. To test the models’
performance the mean annual England and Wales precipitation was calculated using models P and Z for
parts of the period for which circulation indices were available (1881–1993). These were then compared
with the observed England and Wales precipitation series constructed by Gregory et al. (1991) and
updated by Jones and Conway (1997). To allow for differences in the series averages (mainly due to the
elevations of the sites used) the model series have been standardized to the observed 1961–1990 average
and the 1961–1990 annual rainfall totals are shown for comparison in Figures 10–12.
The period 1940 – 1960 was not used for validation because a different procedure was used for gridding
mean sea level pressure (MSLP) from surface analyses in the UK region in the period 1940–1965.
Although weather type analysis is generally unaffected, quantitatively different circulation indices are
produced (Hulme and Jones, 1991). The procedure involved extra smoothing which has the effect of
increasing the MSLP of depressions more than the corresponding decrease for anticyclones, which are
generally less intense than depressions. This results in higher average MSLP, smaller values of positive
vorticity with generally unchanged values of negative vorticity for this period. The smoothing in the
period 1961–1965 will have influenced the regression results, but removal of this period from the
regression would not allow the direct use of 1961–1990 climatologies in model validation. The effects of
smoothing and the possibility of re-gridding the MSLP fields will be addressed in future work.
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 10. Sample distributions of monthly mean rainfalls (mean of period 1961 – 1990)
The mean rainfall and correlation coefficients for the calibration period and a validation period from
1881 to 1939 are shown in Table IV. The models generally perform as well in the validation period as in
the calibration period, and the mean rainfall for both models is within 5% of the observed value. Note also
that, at least at the aggregated annual level shown here, model Z performs better than model P. This
validation provides some confidence in the stability of the models under climate variability experienced to
date. However, there can be no guarantee that the model parameters are stable to future climate changes.
Improvement to the models’ physical basis is therefore desirable by the inclusion of other local variables,
and is discussed further in Section 6.
A potential application, which has motivated the work here, is to use the predicted point rainfall statistics
to modify the parameters of a Neyman – Scott (N–S) stochastic model of precipitation (e.g. see
Cowpertwait et al., 1996) used to produce daily and hourly time series. The proposed method is as follows.
Figure 11. Sample distributions of monthly mean proportion of dry days (mean of period 1961 – 1990)
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
Figure 12. Annual time series of precipitation totals for England and Wales for the period 1961 – 1990
(i) The N–S model is first fitted to data taken from the present climate. If no site data are available at
the catchment of interest, and the application is in the UK, then the regionalised Neyman–Scott
models developed by Cowpertwait et al. (1996) or Cowpertwait and O’Connell (1997), can be used.
(ii) Future values of the mean daily rainfall and the proportion of dry days are then predicted using the
regression models together with future values of circulation variables derived from output from
perturbed GCM integrations.
(iii) Two N–S parameters are then modified using the predicted point rainfall statistics for future
(iv) Hourly rainfall time series are then simulated and used as input to a hydrological catchment model
to assess the impact of the future climate on river flows. For larger catchments, or cases where there
is significant spatial variability (for example due to orographic effects) a spatial-temporal N-S model
(Cowpertwait, 1995) may be used to simulate multi-site hourly rainfall time series for input to the
hydrological catchment model.
Clearly, the method described above involves some broad assumptions. For example, it is necessary to
assume that some of the N – S model parameter estimates remain constant in the future. Consideration of
the rainfall model structure indicates that two parameters are directly related to changes in the mean
rainfall and proportion of dry days: the rate of storm arrivals and the mean cell intensity. These
parameters are therefore the most appropriate for refitting the model. Furthermore, if only a small change
occurs in a N–S model parameter, when estimated at sites taken over a large geographical region
containing significant climate variation, it is reasonable to suggest that this parameter estimate can be
treated as approximately constant in the future as long as no major changes in the dominant precipitation
mechanism occur. The regionalised N – S model (Cowpertwait et al., 1996) can be used to provide this
evidence, supporting the choice of the parameters identified previously.
Table IV. Mean annual rainfall and correlation coefficient (r) for England and Wales
model and observed annual rainfall. The period 1961 – 1990 was used for development of
the regression models; 1881–1939 is used as a validation period. Figures in parentheses
have been standardized by the ratio of observed to model mean annual rainfall for the
1961 – 1990 period
1881 – 1939
Mean (mm)
Mean (mm)
977 (881)
931 (876)
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 523 – 539 (1998)
A further point is that the method presented here is flexible enough to allow its use with transient GCM
simulation output. Previous work in this area has used equilibrium GCM integrations, and the N–S
model would have been fitted to statistics derived from the whole simulation period. The latest GCM
integrations however are transient (Mitchell et al., 1995), and a number of complications arise.
The N–S model may be re-parameterized and a simulation of a time series performed at any multiple
of 12 months. At one extreme therefore, this allows the fitting of the model to circulation statistics taken
from 1-year time windows of a transient GCM simulation, so that the inter-annual variability is derived
from the GCM and direct correspondence with other GCM variables is maintained. At the other extreme,
a longer time window may be used, which of course may not be stationary, and will result in inter-annual
variability produced by the N – S model itself, and with no direct temporal correspondence with the GCM
The regression models provide a simple and parsimonious method of predicting rainfall statistics across
England and Wales, accounting for the effects of large scale circulation, and thus allowing their use in
downscaling. The use of rather coarse resolution circulation data means that some important regional
climatic effects may be ignored (e.g. Mayes, 1991). Given this limitation, and that very simple models have
been formulated, it is not surprising that some discrepancies are evident in the validations.
However, in some applications, these discrepancies may be unimportant. For example, if the models are
used to predict expected values for each month (obtained by averaging all available data for each month
in a long record), then the discrepancies in the sample standard deviation plots (Figure 4(b) and 5(b)) are
not important. Even if the models are used to predict monthly time series, with under-prediction of
rainfall variability by the models, the model variability could be increased by simulating monthly time
series of point rainfall statistics. This could be achieved by adding the random variables o and j to the
predicted values (i.e. using Equations (1) and (3) instead of the expected values in Equations (2) and (4)).
It is obvious that other explanatory variables may be used in regression analyses such as presented here.
As longer duration atmospheric data sets from observation and model re-analysis become available to
replace the MSLP data set, it will be possible to improve the physical basis of these models. Such data
sets include upper air circulation (e.g. at 750 hPa and 500 hPa), humidity and atmospheric stability. The
use of spatial time series of sea-surface temperatures can also provide information on land-sea temperature contrasts which have important effects on rainfall generation in England and Wales. An increased
physical basis will in turn allow more confidence in the simulated future rainfall statistics, particularly in
regard to possible changes in the relationship between rainfall statistics and circulation.
It is hoped that the use of such comprehensive and higher spatial resolution observed data sets will
allow improvement on the regression models presented here. These models would then afford significant
benefits for downscaling, with better resolution of both the temporal sequence of synoptic events and the
spatial patterns which produce regional climates.
Useful discussions with Adri Buishand (KNMI), and Andrew Metcalfe (University of Newcastle), and
comments from two anonymous referees are gratefully acknowledged. This research was supported by the
EC Environment Research Programme under the POPSICLE project (Contract: EV5V-CT94-0510,
Climatology and Natural Hazards).
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