statistical downscaling relationships for precipitation in the netherlands and north germany

INTERNATIONAL JOURNAL OF CLIMATOLOGY

Int. J. Climatol. 22: 15–32 (2002)

DOI: 10.1002/joc.718

STATISTICAL DOWNSCALING RELATIONSHIPS FOR PRECIPITATION INTHE NETHERLANDS AND NORTH GERMANY

BJORN-R. BECKMANN† and T. ADRI BUISHAND*Royal Netherlands Meteorological Institute (KNMI), De Bilt, The Netherlands

Received 8 March 2001Revised 26 July 2001

Accepted 30 July 2001

ABSTRACT

The statistical linkage of daily precipitation to the National Centers for Environment Prediction (NCEP) reanalysis datais described for De Bilt and Maastricht (Netherlands), and for Hamburg, Hanover and Berlin (Germany), using daily datafor the period 1968–97. Two separate models were used to describe the daily precipitation at a particular site: an additivelogistic model for rainfall occurrence and a generalized additive model for wet-day rainfall. Several dynamical variablesand atmospheric moisture were included as predictor variables. The relative humidity at 700 hPa was considered as themoisture variable for rainfall occurrence modelling. For rainfall amount modelling, two options were compared: (i) theuse of the specific humidity at 700 hPa, and (ii) the use of both the relative humidity at 700 hPa and precipitable water.

An application is given with data from a time-dependent greenhouse gas forcing experiment using the coupledECHAM4/OPYC3 atmosphere–ocean general circulation model for the periods 1968–97 and 2070–99. The fittedstatistical relationships were used to estimate the changes in the mean number of wet days and the mean rainfall amountsfor the winter and summer halves of the year at De Bilt, Hanover and Berlin. A decrease in the mean number of wet dayswas found. Despite this decrease, an increase in the mean seasonal rainfall amounts is predicted if specific humidity isused in the model for wet-day rainfall. This is caused by the larger atmospheric water content in the future climate. Theeffect of the increased atmospheric moisture is smaller if the alternative wet-day rainfall amount model with precipitablewater and relative humidity is applied. Except for an anomalous change in mean winter rainfall at Hanover, the estimatedchanges from the latter model correspond quite well with those from the ECHAM4/OPYC3 model.

Despite the flexibility of generalized additive models, the rainfall amount model systematically overpredicts the meanrainfall amounts in situations where extreme rainfall could be expected. Interaction between predictor effects has to beincorporated to reduce this bias. Copyright 2002 Royal Meteorological Society.

KEY WORDS: The Netherlands; Germany; precipitation; NCEP reanalysis; general circulation models (GCMs); statistical downscaling;generalized additive models; climate change

1. INTRODUCTION

A major problem associated with future global warming, because of its impacts on hydrology and waterresources, is the potential change in the precipitation intensities and seasonal amounts. Simulations withgeneral circulation models (GCMs) form the primary source of information about future climate change.These models provide time series of many climate variables on a rather coarse grid, about 300 km × 300 km,which is not suitable for direct use in climate-change impact studies. A downscaling technique is needed toobtain the required climate variables at the local scale. The term ‘statistical downscaling’ refers to the use ofa statistical model for this purpose.

Statistical downscaling of precipitation has often focused on the link between local precipitation andatmospheric flow characteristics. Several techniques have been used, ranging from multiple regression (e.g.

* Correspondence to: T. Adri Buishand, Royal Netherlands Meteorological Institute (KNMI), PO Box 201, 3730 AE De Bilt, TheNetherlands; e-mail: [email protected]† Present address: Deutscher Wetterdienst (DWD), Offenbach, Germany.

Copyright 2002 Royal Meteorological Society

16 B.-R. BECKMANN AND T. A. BUISHAND

Kilsby et al., 1998; Wilby et al., 1998) and canonical correlation analysis (e.g. von Storch et al., 1993) tomultivariate adaptive regression splines (Corte-Real et al., 1995), kriging (Biau et al., 1999) and artificialneural networks (e.g. Hewitson and Crane, 1996), and from parametric time series modelling (e.g. Bardossyand Plate, 1992; Corte-Real et al., 1999) to nonparametric resampling (e.g. Zorita et al., 1995; Conway andJones, 1998). Although long-term variations in precipitation during the past may be explained by fluctuations inthe atmospheric circulation, various GCM experiments suggest that the changes in precipitation resulting fromthe global warming can not be derived from the changes in the atmospheric circulation alone (Matyasovszkyet al., 1993; Jones et al., 1997; Wilby and Wigley, 1997).

Besides the atmospheric circulation, there is also a temperature effect on precipitation, because warm air canhold more moisture than cold air. Shower intensity also depends on temperature. For De Bilt (Netherlands)these phenomena can easily be identified by plotting the mean wet-day precipitation amounts versus the dailymean or daily maximum 2 m temperature (Konnen, 1983; Buishand and Klein Tank, 1996). A similar analysisfor a number of sites in England, Switzerland, Portugal and Italy showed, however, that the temperature effecton precipitation is often obscured by other meteorological factors (Brandsma and Buishand, 1996). Becausethe correlation between daily precipitation and temperature is generally weak in observed data, it is quitedifficult to use temperature as a covariate in stochastic models for daily precipitation.

With the exception of a paper by Karl et al. (1990), a measure of the upper-air humidity has only recentlybeen included in statistical downscaling models for daily precipitation. The correlation between precipitationand atmospheric moisture is generally stronger than that between precipitation and temperature. Crane andHewitson (1998) and Cavazos (1999) considered the specific humidity at different heights as a measure of theatmospheric water content. Charles et al. (1999) used the 850 hPa dew point temperature depression in theirmodel for daily rainfall occurrence, because the probability of rain is better related to the degree of saturationof the atmosphere than to the total amount of water.

New downscaling relationships are developed for five lowland stations in the Netherlands and NorthGermany. Separate models are used to describe the probability of rain and the wet-day rainfall amounts.Several circulation variables are included in these models. The probability of rain is further linked to therelative humidity at 700 hPa. Two measures of absolute humidity are compared for rainfall amount modelling.Results from a time-dependent greenhouse gas (GHG) forcing experiment with the ECHAM4/OPYC3 climatemodel are used to estimate the changes in precipitation for the period 2070–99 for three of the five stations.These estimates are compared with the changes in precipitation in the climate model output.

The statistical models for the probability of rain and the wet-day rainfall amounts, and the main resultsof their fit to the daily precipitation data are discussed in Section 2. Section 3 deals with the estimation ofthe changes in the seasonal mean rainfall amounts. The reproduction of daily rainfall in extreme situations isinvestigated in Section 4. A concluding discussion is given in Section 5.

2. RAINFALL MODELLING

Observed daily rainfall for the period 1968–97 has been analysed for De Bilt, Maastricht, Hamburg, Hanoverand Berlin (Figure 1). Generalized additive models have been used to link these data to circulation variablesand atmospheric moisture. These models are an extension of the standard linear regression model for normallydistributed data, covering both nonlinearity and a variety of distributions (Hastie and Tibshirani, 1990). Theywere applied successfully to describe the dependence of the wet-day precipitation amounts on 2 m temperatureand circulation variables in earlier work (Brandsma and Buishand, 1997). Here, generalized additive modelsare also used to describe rainfall occurrence.

The predictor variables were derived from the National Centers for Environmental Prediction (NCEP)reanalysis (Kalnay et al., 1996). Six-hourly values of numerous weather variables were available ona 2.5° × 2.5° grid. The grid point data nearest to the precipitation measurement stations were usedfor the statistical analysis. Daily averages of the predictor variables were obtained by taking thefour six-hourly values in the reanalysis data that were within the sampling interval of daily rainfallmeasurements.

Copyright 2002 Royal Meteorological Society Int. J. Climatol. 22: 15–32 (2002)

DOWNSCALING PRECIPITATION 17

De Bilt (2)

Maastricht (114)

GERMANY

2° 39' E

Climatological Station

National Boundary

15° 18' E

2° 39' E 15° 18' E

46° 25' N

55° 10' N

46° 25' N

55° 10' N

Hanover (55)

Hamburg (11)

Berlin (51)NETHERLANDS

Figure 1. Locations of stations with their altitude (metres above m.s.l.) in parentheses

Section 2.1 describes rainfall occurrence modelling with an additive logistic model. The generalizedadditive model for the wet-day rainfall amounts is discussed in Section 2.2. The statistical relationshipsare assumed to be constant over the year in these sections. The question of seasonal variation is addressedin Section 2.3.

2.1. Rainfall occurrence modelling

The occurrence of wet and dry days can be represented as a binary sequence {yt }, where yt = 1 if day t

is wet and yt = 0 if day t is dry. A wet day is defined here as a day with a precipitation amount of 0.1 mmor more. The key parameter in rainfall occurrence modelling is the probability P of a day being wet. In theadditive logistic model P is given by:

logit(P ) = ln(

P

1 − P

)= a0 +

m∑i=1

fi(xi) (1)

where the fi(xi) are arbitrary smooth functions of the predictor variables xi . The logit transformation ensuresthat P lies in the interval between zero and one. The predictor functions have been iteratively estimatedby a locally weighted running-line (loess) smoother with a span of 0.5 (proportion of the data entered inthe local fit), using the S-Plus statistical software package (Chambers and Hastie, 1993). To exclude freeconstants, the fitted values for each function fi(xi) are adjusted to average zero. In the case of a linearfunction, this implies that fi(xi) = ai(xi − xi), where ai is a regression coefficient and xi is the average ofthe variable xi .

Atmospheric variables having relatively strong correlation with rainfall occurrence were considered aspredictor variables. Different sets of predictor variables were compared using Akaike’s information criterion,and the significance of every predictor variable and the nonlinearity of the predictor functions in the selectedset were checked by an approximate χ2-test (see Appendix A). A predictor variable was excluded if its effecton rainfall occurrence at one or more stations was opposite to that expected from physical grounds as a resultof correlation with other predictor variables. This led to a rainfall occurrence model with seven predictor



variables: the relative humidity rh at 700 hPa, the sea-level pressure slp, the u-velocity (westerly flow), thev-velocity (southerly flow), the relative vorticity ζ , all derived from sea-level pressure, the 1000–500 hPathickness (thick ) and the baroclinicity b. For the sake of consistency with the climate-change application inSection 3, the relative humidity was derived from the specific humidity and air temperature at 700 hPa. Thebaroclinicity was taken as the absolute value of the air temperature gradient at 700 hPa.

As an illustration, Figure 2 shows the estimates of the functions fi(xi) in Equation (1) for all seven predictorvariables for Hanover. In this example these functions are linear for the relative vorticity, the v-velocity, thebaroclinicity and the sea-level pressure, and are nonlinear for the u-velocity, the relative humidity and the1000–500 hPa thickness. The rainfall probability on the right-hand side of each panel is given by:

P = 1

1 + exp[−a0 − fi(xi)](2)

−20

−20 −10 0 10 20 5100 5400 5700 0.00 0.01 0.02

−2

0

2

4

−2

0

2

4

−2

1000 1020 1040

0

2

4

98.5

89.9

54.7

14.1

98.5

89.9

54.7

14.1

98.5

89.9

54.7

14.1

−10 0

u in m/s

f i(x i

)f i(

x i)

f i(x i

)

P in

%

P in

%P

in %

rh in % ζ in khz

v in m/s

Rainfall occurrence

slp in hPa

thick in m b in K/km

0 20 40 60 80 100 −0.05 0.00 0.05 0.1010 20 30

Figure 2. Estimates of the functions fi (xi ) for the seven predictor variables in the additive logistic model for rainfall occurrence atHanover. The dashed lines mark pointwise 2× standard-error bands



The results in Figure 2 reflect the influences of the predominant precipitation mechanisms in the Netherlandsand North Germany. Rain is often due to cyclonic disturbances moving from west to east over the areaand strong westerly winds over central Europe are usually associated with cyclonic conditions. Hence,the probability of rain is high on days with strong westerly flow and is also strongly influenced byvorticity. Figure 2 further shows that the probability of rain increases with increasing wind speed fromnortherly directions. The occurrence of rain showers in the deep layer of unstable cold air at the rear ofcyclones contributes to the relatively high rainfall probability in situations with a strong northerly flow.In agreement with meteorological principles, the probability of rain increases with increasing vorticity(or cyclonality), relative humidity and baroclinicity, and decreases with increasing sea-level pressure andincreasing 1000–500 hPa thickness.

For all precipitation stations, additive logistic regression models were fitted to the rainfall occurrence data.Table I summarizes the results. The predictor variables are ordered with respect to their statistical significancein terms of the AIC statistic in Appendix A. Deletion of the most significant variable gives the largest increasein AIC. The proportion of explained variance in the third column of Table I was calculated as:

EV = 1 −

N∑t=1

(yt − Pt)2

N∑t=1

(yt − y)2

(3)

where yt is the observed rainfall occurrence at day t , y is the average of the yt values (fraction of wetdays), Pt is the estimated rainfall probability for day t and N is the number of days in the record. Thisis allowed because the average of the Pt values is almost equal to that of the yt values. The residualautocorrelation in the last column of Table I refers to the lag 1 autocorrelation coefficient of the residuals{yt − Pt }.

About half of the predictor functions fi(xi) are nonlinear. The order of the statistical significance of thepredictor variables is almost identical for the five stations. The u-velocity and the relative humidity are alwaysthe two most significant predictors, quite often followed by vorticity. Moreover, for each predictor variablethe function fi(xi) has the same shape for all stations.

The sequence of wet and dry days exhibits moderate persistence. For instance, the lag 1 autocorrelationcoefficient for Hanover is 0.37. The residual autocorrelation coefficients are much smaller (see Table I), butare statistically significant at the 5% level. The persistence of daily rainfall occurrence is thus not fullycaptured by the autocorrelation in the predictor variables. Inclusion of the wet–dry status of the previous daymay improve the reproduction of the persistence of daily rainfall occurrence. Another possibility is to develop

Table I. Predictor variables for rainfall occurrencea

Station Predictor variables ExplainedVariance

(%)

Residualautocorrelation

De Bilt u, rh, ζ , v, thick, b, slp 47.8 0.111Maastricht u, rh, v, ζ , thick, b, slp 47.5 0.117Hamburg u, rh, ζ , thick, v, b, slp 49.1 0.138Hanover u, rh, ζ , v, thick, b, slp 46.2 0.138Berlin rh, u, v, ζ , thick, b, slp 43.1 0.125

a The first variable in the row of predictors is the most significant predictor variableand the last variable the least significant one. The predictors in bold face denote fi(xi )

is nonlinear.



separate models for the probabilities of a wet day following a wet day and that of a wet day following a dryday (Conway et al., 1996; Wilby et al., 1998).

2.2. Rainfall amount modelling

The generalized additive model with constant coefficient of variation and log link function has been adoptedto explore the relationship between the wet-day precipitation amounts and the predictor variables. The loglink function specifies the expected wet-day amount as:

R = exp

[a0 +

m∑i=1

fi(xi)

](4)

Note that R cannot become negative with this representation. A constant coefficient of variation implies thatthe variance of the wet-day precipitation amounts increases with the expected value, which is for wet-dayprecipitation much more realistic than the more familiar assumption of a constant variance. As in Equation (1),the fi(xi) are arbitrary smooth functions of the predictor variables xi . Model selection was done in the sameway as for rainfall occurrence. However, an approximate F -test (Appendix A) was used instead of a χ2-testto assess the significance of predictor variables and nonlinearity of predictor functions.

Two different models were created for every site: one model with specific humidity q at 700 hPa asatmospheric moisture variable and another one with precipitable water pw and relative humidity rh at 700 hPa.It was not possible to include both specific humidity and relative humidity into one model because of thecorrelation between these two variables (0.51 for wet days at Berlin up to 0.59 for wet days at De Bilt). Thecorrelation between relative humidity and precipitable water is at every site weaker than that between relativehumidity and specific humidity. Because of its strong correlation with specific humidity and precipitablewater, the 1000–500 hPa thickness cannot be entered into the statistical models for the wet-day rainfallamounts.

Figure 3 shows the estimates of the functions fi(xi) in Equation (4) for all six predictor variables inthe model with specific humidity for Hanover. The rainfall amount on the right-hand side of each panel isgiven by:

R = exp[a0 + fi(xi)] (5)

The expected wet-day rainfall amount increases with increasing specific humidity. In the alternative rainfallamount model for Hanover, the expected rainfall amount increases with increasing precipitable water andrelative humidity. The remaining predictor functions for wet-day rainfall show qualitatively the same behaviouras those in rainfall occurrence modelling (compare Figure 2 with Figure 3).

For all stations Table II gives some details about the fitted models. The model with precipitable water andrelative humidity generally explains a somewhat larger percentage of the variance of wet-day rainfall thanthat with specific humidity. For the Netherlands, Hamburg and Hanover the models explain about 30% of thevariance of the wet-day rainfall amounts. The percentage of explained variance is slightly less for the morecontinental station Berlin. For all stations the moisture variables are powerful predictors for wet-day rainfall.The significance of the various dynamical variables shows strong similarities with that for rainfall occurrencemodelling. The u-velocity is a powerful predictor variable for wet-day rainfall at De Bilt, Maastricht, Hamburgand Hanover. It is, however, marginally significant for wet-day rainfall at Berlin.

2.3. Reproduction of the seasonal cycle

A quick test of the assumption of a constant statistical relationship over the year is possible by comparingthe observed rainfall probabilities and mean wet-day rainfall amounts for each calendar month with thoseexpected from the fitted models. Figure 4 shows the reproduction of the seasonal cycles for the probabilityof rain and mean wet-day rainfall amount at Hanover. The models describe the observed seasonal cycle



1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

1.5

11.9

7.2

4.4

2.7

1.6

1.0

0.6

11.9

7.2

4.4

2.7

1.6

1.0

0.6

1.0

0.5

0.0

−0.5

−1.0

−1.5

−0.10 −0.05 0.00

ζ in kHz q in g/kg

Wet-day rainfall

f i(x i

)f i(

x i)

R in

mm

/day

R in

mm

/day

u in m/s

slp in hPa v in m/s b in K/km975 0.00 0.01 0.021000 1025 −20 −10 0 10

0.05 0.10 0 1 2 3 4 5 6 −20 −10 0 10 20 30

Figure 3. Estimates of the functions fi (xi ) for the six predictor variables in a generalized additive model for wet-day rainfall at Hanover.The dashed lines mark pointwise 2× standard-error bands

Table II. Predictor variables for wet-day rainfalla

Station Predictor variables Explained Variance (%)

De Bilt q, ζ , u, slp, b, v 29.7ζ , u, pw, slp, rh, b, v 32.0

Maastricht q, u, v ,ζ , slp, b 30.2u, pw, v, rh, ζ , b, slp 33.3

Hamburg u, q, ζ , slp, v, b 28.3u, ζ , pw, rh, v, slp, b 29.6

Hanover ζ , q, u, slp, v, b 29.4ζ , rh, u, pw, b, v, slp 31.3

Berlin q, ζ , v, slp, b, u 25.1pw, rh, v, ζ slp, b, u 28.3

a The first variable in the row of the predictors is the most significant predictor variableand the last variable the least significant one. The predictors in boldface denote fi(xi ) isnonlinear.

reasonably well. The errors are generally not more than twice the standard error of the observed monthlymeans. The observed monthly mean wet-day probabilities in Figure 4 explain only 1.0% of the variance ofdaily rainfall occurrence. For the monthly mean wet-day precipitation amounts, the proportion of explainedvariance is 1.7%. These values are small compared with those for the fitted generalized additive models.

Figure 4 shows that the rainfall probability is slightly overestimated in the first half of the year andunderestimated in the second half of the year. This discrepancy was also found in the fits for the otherstations. A standard one-sample t-test on the differences between the observed and modelled monthly valuesshows that several of these discrepancies are statistically significant at the 5% level.



Month

70

65

60

55

50

45

40

35

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5Feb Apr Jun Aug Oct

observedmodelled

observedmodel 1 (q)model 2 (pw)

Dec Feb Apr Jun Aug Oct Dec

Rai

nfal

l pro

babi

lity

in %

Rai

nfal

l am

ount

in m

m/d

ay

Month

Figure 4. Observed and predicted seasonal cycles of the daily rainfall probability (left panel) and mean wet-day rainfall amount (rightpanel) at Hanover for the period 1968–97. Atmospheric moisture is represented as specific humidity in model 1, and as relative humidity

and precipitable water in model 2. The error bars indicate twice the standard error of the observed monthly means

−0.05−3

−2

−1

0

1

2

−1.0

−0.5

0.0

0.5

1.0

1.5

3

4 Winter

Summer

Rainfall occurrence Wet-day rainfall

ζ in kHz

f(ζ)

f(ζ)

ζ in kHz0.00 0.05 0.10 −0.05 0.00 0.05 0.10

Figure 5. Seasonal estimates of the effects of vorticity ζ on rainfall occurrence (left panel) and wet-day rainfall (right panel) at Hanover.Specific humidity is used as the moisture variable in the model for wet-day rainfall

The two rainfall amount models describe the seasonal cycles at Hanover equally well. This holds alsofor Berlin. For De Bilt, Maastricht and Hamburg, however, the model with precipitable water and relativehumidity performs somewhat better than that with specific humidity. The differences between the observed andpredicted monthly mean wet-day precipitation amounts are sometimes larger than twice the standard errorof the observed monthly means.1 The mean wet-day rainfall amount in May and June is underestimatedat all stations. The largest underestimations (up to 0.9 mm per wet day) were found for De Bilt andHamburg.

The discrepancies between the observed and modelled seasonal cycles suggest a seasonally varyingrelationship. This was investigated for Hanover by modelling the data for the winter (October–March) andsummer (April–September) halves of the year separately. Small, but statistically significant (Appendix A)differences emerged. For instance, in winter vorticity has a somewhat stronger effect on rainfall occurrence



and wet-day rainfall than in summer (Figure 5). The impact of relative humidity on rainfall occurrence isrelatively strong in summer.

3. CHANGES IN SEASONAL MEANS

The changes in the seasonal mean number of wet days and the seasonal mean precipitation amountswere estimated for De Bilt, Hanover and Berlin for the end of the 21st century using monthly meanvalues of the predictor variables from the time-dependent GHG forcing experiment with the coupledglobal atmosphere–ocean model ECHAM4/OPYC3 of the Max-Planck Institute for Meteorology and theDeutsche Klimarechenzentrum (both in Hamburg). A steadily growing concentration of GHGs, as observedbetween 1860 and 1990 and according to IPCC scenario IS92a (‘business as usual’) from 1990 onward,was prescribed in that experiment (Arpe and Roeckner, 1999; Roeckner et al., 1999). The estimation of thechanges in the seasonal mean values is explained first in Section 3.1 and the results are then discussed inSection 3.2.

3.1. Method of calculation

Given the values of the predictor variables, the mean number of wet days Nw per season (winter or summerhalf of the year) and the mean rainfall amount Rtot per season can be calculated as:

Nw = 1

J

∑t

Pt (6)

Rtot = 1

J

∑t

PtRt (7)

where Pt is the modelled probability of rain for day t , Rt is the expected rainfall amount for day t given thatit is wet, and J is the number of years. The summation is over all days in the period concerned. The wet-dayrainfall amount model is thus applied for every day t and not only for the wet days. The differences betweenthe estimated and observed seasonal means are not more than 3%.

To estimate the seasonal mean number of wet days N∗w and the seasonal mean rainfall amount R∗

tot for theend of the 21st century, the daily values of the predictor variables were perturbed with the change in theirmean according to the climate model experiment:

x∗ti = xti + �xi (8)

where xti is the value of the ith predictor for day t in the period 1968–97 (from the NCEP reanalysis data) and�xi is the change in the mean of xi between the periods 1968–97 and 2070–99 in the climate model output.The daily values for the relative humidity in the future climate were, however, derived from the perturbeddaily values of the specific humidity and air temperature at 700 hPa. Equations (1) and (4) were then appliedwith the x∗

ti as predictor variables, and the resulting rainfall probabilities and wet-day rainfall amounts weresubsequently substituted into Equations (6) and (7). Separate climate-change signals were determined for thewinter and summer halves of the year.

The wind velocity components, the relative vorticity and the baroclinicity for the climate model data wereobtained in the same way as those for the reanalysis data. The precipitable water pw was not available in theclimate model output. The seasonal means of precipitable water were therefore derived from those of specifichumidity at 700 hPa using linear regression relationships fitted to the reanalysis data. Separate relations wereused for the winter and summer halves of the year.

For the grid point closest to Hanover, Tables III and IV compare the seasonal means of the predictorvariables in the climate model output with those in the NCEP reanalysis for the winter and summerrespectively. The changes for the 2070–99 period are also presented. There is almost always a significant



Table III. Winter means (1968–97) of NCEP predictor variables, bias in the ECHAM4/OPYC3winter mean relative to the NCEP mean, and changes for the period 2070–99 for Hanovera

Variable NCEP mean1968–97

GHG bias1968–97

GHG change2070–99

q (g/kg) 1.59 +0.18 +0.57slp (hPa) 1015.5 +1.7 +0.3u (m/s) 4.59 +4.18 +0.42v (m/s) 0.83 +0.43 −0.01ζ (kHz) −0.0083 +0.0138 −0.0046thick (m) 5369.9 +37.9 +89.7b (K/km) 0.0076 −0.0030 −0.0003

a Biases and changes printed in italics are significant at the 5% level (t-test).

Table IV. Summer means (1968–97) of NCEP predictor variables, bias in the ECHAM4/OPYC3summer mean relative to the NCEP mean, and changes for the period 2070–99 for Hanovera

Variable NCEP mean1968–97

GHG bias1968–97

GHG change2070–99

q (g/kg) 2.62 +0.40 +0.87slp (hPa) 1015.3 −1.0 +0.2u (m/s) 1.60 +1.97 +0.03v (m/s) −0.77 +2.04 −0.01ζ (kHz) 0.0021 +0.0006 −0.0014thick (m) 5505.9 +54.6 +95.6b K/km 0.0066 −0.0029 +0.0003

a Biases or changes printed in italics are significant at the 5% level (t-test).

difference between the ECHAM4/OPYC3 and NCEP mean values. The most notable biases are found for u

and ζ in the winter season caused by a strong north–south gradient in sea-level pressure over North Germanyin the climate model output (Beckmann and Buishand, 2001). A discussion of the differences between thepatterns of sea-level pressure in the NCEP reanalysis and the ECHAM4/OPYC3 data can be found in Arpeand Roeckner (1999) and Knippertz et al. (2000).

For the 2070–99 period the changes in specific humidity and thickness are statistically significant andexceed the bias in both summer and winter. The increase in thickness of about 90 m is in agreementwith that expected from the hypsometric equation for the simulated temperature increase of about 4.5 °Cby the climate model. The relative increase in specific humidity amounts to 7.5%/ °C in winter and6.4%/ °C in summer, which is close to the change in the saturated vapour pressure per degree centrigrade(Clausius–Clapeyron relation). Significant changes are also found for the vorticity, but for the winter halfof the year the change is smaller than the bias. The changes in the other variables are small compared withtheir bias.

The representative grid points of the other stations show similar changes and biases in q and thick. ForDe Bilt and Berlin there is less bias in the u-velocity in the winter season. The change in vorticity atthe chosen grid points for these stations for the winter season is about 60% of that at the grid point nearHanover. For the grid point near Berlin there is a considerable decrease in the average v-velocity in thefuture climate, both in the winter (0.30 m/s) and the summer half of the year (0.35 m/s). There is also arelatively strong increase in the u-velocity at that grid point (0.62 m/s in winter and 0.49 m/s in summer).The changes and biases in the baroclinicity for De Bilt and Berlin are more or less similar to those forHanover.



In the case of climate change it is obvious that the range of the predictor variables is somewhat differentfrom that used in model construction. Unfortunately, the S-Plus software only provides the smoothed functionswithin the range of the observed data. Therefore, the smoothed estimates of nonlinear predictor functions werereplaced by piecewise linear functions. As an example, Figure 6 shows the piecewise linear approximationto the predictor function f (v) in the rainfall amount model with specific humidity for Hanover. The dashedline in that figure is given by:

f (v) ={av0 if v > cv2

av0 + av1 (cv2 − v) if cv1 < v ≤ cv2

av0 + av1 (cv2 − cv1) + av2 (cv1 − v) if v ≤ cv1

(9)

where the knots cv1 and cv2 are located at −1 m/s and 5 m/s respectively. The coefficient av0 is chosen suchthat the average of the fitted values of f (v) equals zero.

Such piecewise linear approximations to nonlinear predictor functions were developed for the rainfalloccurrence and amount models of De Bilt, Hanover and Berlin. Although the resulting approximationsare continuous, this does not hold for their derivatives. Brandsma and Buishand (1997) used naturalcubic splines. This leads to smoother approximations, but it makes the comparison between stations moredifficult.

3.2. Changes for De Bilt, Hanover and Berlin

The estimated changes in the seasonal mean number of wet days and rainfall amounts are presented inTable V. These changes were obtained by applying Equations (6) and (7) both for the period 1968–97 andthe period 2070–99.

The decrease in the number of wet days at all stations is predicted mainly by the increased 1000–500 hPathickness. This change in thickness has the strongest effect in summer because the slope of f (thick ) is steeperat large values of thick (see Figure 2). A decrease in vorticity over the Netherlands and North Germany inthe climate model data during winter (Table III) also contributes to the decrease in the number of wet days,in particular at Hanover. The relatively small decrease in the number of wet days at Berlin is partly due tochanges in the velocity components at the chosen grid point for that site. The effects of the changes in theother predictor variables are negligible.

Despite the decrease in the mean number of wet days, a larger seasonal mean rainfall amount isobtained if the wet-day rainfall model with specific humidity is applied. This is mainly caused by

v in m/s

cv1 = −1 m/s

0.8

0.6

0.4

0.2

0.0

−0.2

−0.4

−0.6

−20 −15 −10 −5 0 5 10 15

cv2 = 5 m/s

f(v

)

Figure 6. Smooth estimate of the function fi (xi ) for the southerly flow v in a generalized additive model for wet-day rainfall at Hanover(solid line). The dotted lines mark the accompanied 2× standard-error bands. The dashed line marks the piecewise linear approximation



Table V. Observed seasonal mean number of wet days and seasonal mean rainfall amounts (1968–97)at De Bilt, Hanover and Berlin, and their estimated changes for the end of the 21st century (2070–99)a

De Bilt Hanover Berlin

Winter Summer Winter Summer Winter Summer

Wet daysObserved number 107 86 101 97 94 81Change (%) −9 −17 −14 −18 −6 −13

Rainfall amountObserved (mm) 416 385 299 349 258 324Change (%)

Model 1 +13 +6 0 +1 +19 +3Model 2 +3 −9 −9 −10 +7 −9ECHAM4/OPYC3 +10 −11 +12 −7 +12 −13

a Atmospheric moisture is represented as specific humidity in the relationship for the wet-day precipitation amountsin model 1, and as relative humidity and precipitable water in model 2.

Table VI. Estimated coefficients of the linear and piecewise linear functions forthe relative vorticity in the rainfall occurrence models and the rainfall amount

models with specific humidity for De Bilt, Hanover and Berlina

De Bilt Hanover Berlin

Rainfall occurrenceaζ 37.81 40.82 30.31

Rainfall amountaζ1 19.92 23.38 18.78aζ2 7.56 10.67 10.07

a For rainfall occurrence f (ζ ) is a linear function with slope aζ , whereas for therainfall amounts f (ζ ) is piecewise linear with slope aζ1 for ζ < 0.02 kHz and aζ2 forζ ≥ 0.02 kHz.

the increased specific humidity in the future climate, which leads to an increase of 20–25% in themean wet-day precipitation amounts. For De Bilt and Berlin the increase in mean summer rainfall issmaller than that in mean winter rainfall because of the stronger decrease in the number of wet daysin summer. The changes in precipitation are quite different if the alternative wet-day rainfall modelwith precipitable water and relative humidity is applied. Then, there is no longer a general increasein seasonal mean rainfall amounts. Precipitable water and relative humidity are positively correlated.Each individual moisture variable then contributes less to the wet-day precipitation amounts than specifichumidity in the other model. The smaller effect of the change in precipitable water than that in specifichumidity and the slight change in relative humidity result in a relatively small change in the mean wet-day precipitation amounts. The decrease in vorticity has a negative impact on the mean wet-day rainfallamounts, but this impact is much smaller than that of the increase in specific humidity, in particular forDe Bilt and Berlin. The perturbations of the other predictor variables were not large enough to cause asignificant change in seasonal mean precipitation. In particular, the expected changes in the seasonal meanrainfall amounts from the statistical model with precipitable water and relative humidity are comparableto those from the ECHAM4/OPYC3 output. The only exception is the change in mean winter rainfall atHanover. This can be attributed to the strong vorticity effect on the estimated change from the statisticalmodel.



There are two reasons for the strong vorticity effect at Hanover. First, the grid point closest to this siteshows the largest change in vorticity in the climate model output. Second, the estimated regression coefficientsfor vorticity are relatively large in the statistical models for Hanover (Table VI). The latter is partly due tothe correlation between vorticity and sea-level pressure (correlation coefficient about −0.4). Removing sea-level pressure from the statistical models leads to larger regression coefficients for vorticity. The discrepancybetween the estimated change in mean winter rainfall from the statistical models and that in the direct climatemodel output would then even be larger.

4. MODEL DEFICIENCIES

Similar statistical models, as those discussed in Section 2, were used for time series generation of dailyprecipitation for Berne in Switzerland (Buishand and Beckmann, 2000). Difficulties were met with thereproduction of extremes. For this reason the rainfall amount model is re-examined in this section. In Figure 7the observed wet-day rainfall amounts at De Bilt, Hanover and Berlin are plotted against their expected valuesfrom the fitted model with specific humidity as moisture variable and piecewise linear approximations tononlinear predictor functions. There is a good correspondence between the observed and predicted valuesexcept for the last class, in which the models strongly overpredict the true wet-day rainfall amounts. Thelargest bias of 5.7 mm is found for Hanover. This bias also appears if the loess smoother is used to estimatethe nonlinear functions fi(xi) instead of piecewise linear approximations. The bias could not be removed byfitting separate models to the data for the winter and summer halves of the year.

The reproduction of extremes is sensitive to the choice of the link function in the model for the wet-dayrainfall amounts. The use of a log link function implies that η = ln R is additive in the fi(xi). A more generallink function is the power link function:

η = Rλ − 1

λ(10)

The log link corresponds to the limit case λ = 0. A simple test of whether the log link is acceptable can beobtained from a Taylor series expansion of η about λ = 0 (Pregibon, 1980; McCullagh and Nelder, 1989):

η ≈ lnR + 1

2λ (lnR)2 (11)

00

De Bilt

Hanover

Berlin

4

8

12

16

20

4 8

Expected rainfall amount in mm

Obs

erve

d ra

infa

ll am

ount

in m

m

12 16 20

Figure 7. Observed mean wet-day rainfall amounts versus the expected wet-day rainfall amounts from fitted relationships for De Bilt,Hanover and Berlin. The fitted relationships contain specific humidity as moisture variable. Each mean value is based on about 350

values, except for the last class, where only about 50 values were considered



Table VII. Estimated regression coefficients au and abfor the u-velocity and baroclinicity in the rainfall amountmodel with specific humidity for Hanover. Separate fits

for the three different vorticity classes

Vorticity class au ab

1 0.038 36.132 0.041 41.163 0.023 52.81

Equation (11) can be rewritten as:

lnR ≈ η − 1

2λ (lnR)2 = a0 +

m∑i=1

fi(xi) − 1

2λ(lnR)2 (12)

Given the first estimate R1 using the log link function, the analysis is repeated with the extra predictor variable(lnR1)

2/2. For Hanover, it turned out that the contribution of this extra predictor is not significant, whichsupports the use of the log link function.

In the generalized additive model with log link function, the contribution of the ith predictor to lnR isfi(xi), independent of the values of other predictors. Interaction occurs if the value of a predictor determinesthe effect of other predictors. The fact that possible interactions are ignored seems to be the main reason forthe overestimation of the mean wet-day precipitation amounts in extreme situations. For Hanover, significantinteractions were discovered by dividing the data into three vorticity categories:

1. ζ <–0.01 kHz2. −0.01 ≤ ζ < +0.01 kHz3. ζ ≥ +0.01 kHz

Each class contains about the same amount of data. Table VII shows that the regression coefficients for theu-velocity and baroclinicity in the third vorticity class strongly deviate from those in the other two classes.The use of separate models for the three vorticity classes reduces the bias in extreme wet-day rainfall atHanover from 5.7 mm to 2.4 mm.

Incorporation of interaction between predictor effects is not straightforward. For Hanover, the interactionterms relate to vorticity that was poorly reproduced by the ECHAM4/OPYC3 model. Deletion of vorticitycould therefore be an option, even though it is one of the most significant predictors. The bias in the upperwet-day rainfall amount class is then not more than 0.5 mm.

5. DISCUSSION AND CONCLUSION

Generalized additive models are flexible tools to explore the dependence of daily precipitation on othermeteorological variables. Within this modelling framework it is convenient to analyse rainfall occurrence andthe wet-day rainfall amounts separately. The relative humidity at 700 hPa is consistently one of the mostsignificant predictors for describing the rainfall occurrence for the Netherlands and North Germany, whereasthe specific humidity at 700 hPa and the precipitable water are powerful predictors for the wet-day rainfallamounts. This was also the case in a similar analysis of daily rainfall for Vienna and sites in the Swisslowland and central Spain (Beckmann and Buishand, 2001). These results are, however, different from thoseof Murphy (2000), who fitted linear regression equations to monthly rainfall amounts at 976 sites in Europe.Atmospheric moisture was rarely selected in that work.



Various circulation variables were considered. Vorticity is one of the most significant predictors for dailyprecipitation in the Netherlands and North Germany. Except for the wet-day rainfall amounts at Berlin,the u-velocity is also an important predictor variable. The order of the statistical significance of circulationvariables depends strongly on the region of interest. Different results were obtained for Vienna, the Swisslowland and central Spain (Beckmann and Buishand, 2001). Another notable feature of the application toother areas is that the percentage of explained variance for the wet-day rainfall amounts can be much lowerthan the figures in Table II (e.g. about 10% for central Spain).

Small discrepancies in the modelled seasonal cycles of rainfall occurrence and the mean wet-day rainfallamounts were observed. For De Bilt, Hanover and Berlin the changes in the seasonal mean rainfall amountswere calculated for the end of the 21st century using the fitted statistical models with perturbed predictorvariables. The perturbations were derived from the output of the time-dependent GHG experiment with theECHAM4/OPYC3 model. The estimated change in seasonal mean rainfall from the statistical downscalingrelationships is influenced strongly by two factors: (i) the increase in the 1000–500 hPa thickness, leadingto a decrease in the number of wet days; (ii) the increase in the specific humidity and precipitable water,leading to larger wet-day rainfall amounts. The changes in seasonal mean rainfall are sensitive to the moisturevariables entered into the model for the wet-day precipitation amounts. The use of precipitable water andrelative humidity especially leads to a reasonable agreement between the expected changes from the statisticalmodels and those in the simulated rainfall of the ECHAM4/OPYC3 model. More comparisons with GCMsimulations are needed to get a better insight into which moisture variables to use.

A decrease in the number of wet days at mid latitudes has also been observed in other GCM simulationswith enhanced GHG concentrations (Cubasch et al., 1995; Hennessy et al., 1997). This has been attributed toa larger fractional contribution of convective precipitation to the total seasonal amounts. However, decreasesin the number of wet days without an increase in the convective fraction have also been observed (Jones et al.,1997). Furthermore, it is not clear whether the change in the number of wet days is adequately described bythe increased 1000–500 hPa thickness in the future climate.

A large decrease in the relative vorticity at the grid point considered for Hanover strongly influences thechange in mean winter rainfall at this site. However, little confidence could be put on this decrease because ofthe large bias in the relative vorticity in the ECHAM4/OPYC3 simulation, resulting from a poor reproductionof the gradients in the mean sea-level pressure at neighbouring grid points. Circulation variables shouldbe calculated over a larger domain to cope with such biases. It is, however, not obvious how large thisdomain must be. First, a larger domain may reduce the performance of the regression models, and second,the biases in the simulated circulation variables are model dependent. For Berlin, the changes in the velocitycomponents contribute substantially to the change in the mean number of wet days. The other changes incirculation variables have little influence on the mean number of wet days and rainfall amounts. Moreover,these changes are often not statistically significant and small compared with the bias in the climate modeloutput.

A limitation of the downscaling relationships presented is that they are unable to take the effects ofpotential changes in the vertical atmospheric stability into account. A humid unstable vertical temperaturegradient is a necessary condition for the development of atmospheric convection and convective precipitation.For De Bilt, the use of a number of stability indices was examined, among which Hanssen’s (1965) thicknessindex and the K-index (Karl et al., 1990; Murphy, 2000). The latter shows the strongest correlation withdaily precipitation. However, the K-index is correlated with the moisture variables in this study, because itcontains the temperature and the dew point temperature at 700 hPa. It was therefore not possible to enterboth the K-index and relative humidity into the rainfall occurrence model. The relative humidity appears tobe the better predictor. Also unsuccessful was the use of the K-index together with the specific humidity inthe rainfall amount model. The Hanssen index is only marginally significant. One difficulty is that in veryunstable situations there can be no or little precipitation at the target point and much precipitation somewhereelse. The Hanssen index has a much better skill if one is interested in the largest rainfall amount in a regionrather than local rainfall.

Besides the choice of predictor variables, the structure of the statistical models can be questioned, inparticular that for the wet-day rainfall amounts. For Hanover, it was shown that the assumption of additivity



of predictor effects leads to an overprediction of extreme daily rainfall. This overprediction is reduced byincorporating an interaction between the vorticity and the u-velocity and an interaction between the vorticityand the baroclinicity. The reproduction of extremes needs more attention in the literature on statisticaldownscaling.

ACKNOWLEDGEMENTS

The authors thank the reviewers for their comments on an earlier version of the paper. They are also gratefulto H. Thiemann and M. Lautenschlager of the Deutsche Klimarechenzentrum (Hamburg) for their effortsto make the ECHAM4/OPYC3 data available and to the Climatic Research Unit, University of East Anglia(Norwich) UK, for extracting a subset of the NCEP reanalysis data for a European window. Daily rainfall datafor Hamburg, Hanover and Berlin were kindly provided by the Deutsche Wetterdienst (Offenbach). Figure 1was produced by F.J.M. van der Wel. This research was in part supported by the EU Environment and Cli-mate Research Programme under the WRINCLE project (contract: ENV5-CT97-0452, Climate and NaturalHazards).

APPENDIX A: DEVIANCE STATISTICS

The deviance is a measure of the discrepancy of a fit. It is based on the likelihood function. For the standardlinear regression model the deviance reduces to the residual sum of squares. This appendix summarises theuse of the deviance in model selection. A detailed discussion of the deviance is given in McCullagh andNelder (1989). The approximate χ2- and F -tests for generalized additive models are presented in Hastie andTibshirani (1990).

Using the same notation as in Section 2.1, the deviance for the rainfall occurrence model reads:

D = 2N∑t=1

[yt ln

(yt

Pt

)+ (1 − yt ) ln

(1 − yt

1 − Pt

)](A1)

For the rainfall amount model, the deviance is given by:

D = 2Nwo∑τ=1

[Roτ − Rτ

Rτ

− ln(Roτ

Rτ

)](A2)

where Roτ and Rτ are the observed and expected rainfall amounts respectively for the τ th wet day and Nwo

is the observed number of wet days. Note that D = 0 for a perfect fit, i.e. Rτ = Roτ for all τ . The better thefit, the smaller the deviance will be.

Akaike’s selection criterion is based on the statistic:

AIC = D + 2νφ (A3)

where φ is the dispersion parameter (φ = 1 for the rainfall occurrence model and φ = CV 2 for the rainfallamount model with constant coefficient of variation CV ) and ν is the number of degrees of freedom used inthe fit. Inclusion of a linear predictor results in one extra degree of freedom and using a running-line smootherwith a span of 0.5 corresponds to about three degrees of freedom for each predictor (Hastie and Tibshirani,1990). The best model is the one with the lowest AIC statistic rather than that with the smallest deviance.The penalty term 2νφ penalises models that overfit the data.

For the inclusion of an extra predictor variable or the use of a nonlinear predictor, the approximate χ2- andF -tests are more severe criteria than the AIC statistic (Hastie and Tibshirani, 1990). These tests are basedon the decrease �D in the deviance:

�D = D1 − D2 (A4)



where D1 is the deviance for the smaller model (with ν1 degrees of freedom) and D2 the deviance for thelarger model (with ν2 degrees of freedom). Under the null hypothesis that the smaller model is correct, thedistribution of �D/φ can be approximated by the χ2-distribution with ν2 − ν1 degrees of freedom. This testassumes that the dispersion parameter φ is known (rainfall occurrence model). The F -distribution has to beused if φ is replaced by an estimate (model for wet-day rainfall). Because of the large sample sizes in thisstudy, the approximate F -test is almost identical to the χ2-test. The significance of a predictor variable andthe nonlinearity of the predictor functions was tested at the 1% level.

Seasonal variation of relationships can also be tested with the approximate χ2- and F -tests. For Hanover, thedeviance D1 for the constant model was compared with the deviance D2 for the case of two separate models forthe winter and summer halves of the year. The latter is the sum of the deviances for the two seasons. Both forthe rainfall occurrence and the rainfall amount model, the change in the deviance is significant at the 1% level.

NOTE

1. The standard error of the observed monthly mean wet-day precipitation amount was derived here from the sample standard deviationof the wet-day precipitation amounts, assuming independence between the daily values. The use of the mean wet-day rainfall amountsfor each month meets difficulties because of the random number of wet days in a month.

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