Rainfall Simulation

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Using GLMs to simulate daily rainfall under scenarios of

climate change

Nadja Leith

December 8, 2005


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Contents

1 Introduction 2

2 Generalised Linear Models for daily rainfall 2

3 Data 3

3.1 Rainfall data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 Atmospheric data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2.1 Climate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.2 Reanalysis data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Atmospheric predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Model building 11

5 Simulation 16

5.1 1961-1990 period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 2071-2100 period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Conclusion 27

7 Appendix A 28

8 Appendix B 31

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1 Introduction

There is a pressing need to understand the possible effects of climate change. One areawhich requires attention is the effect of a changing global climate on local scale rainfall. Forexample, in order to run hydrological rainfall-runoff models for future flood risk assessmentwe need to be able to simulate realistic sequences of daily and sub-daily rainfall that areconsistent with projected changes in climate.

This document outlines work undertaken for DEFRA project FD2113, Work Package1.2. The aim of this work is to establish a methodology for simulating realistic sequencesof future daily rainfall conditional on atmospheric information from deterministic climatemodels. We employ Generalised Linear Models (GLMs), introduced in Section 2, to relatedaily rainfall to atmospheric predictors. The GLMs are fitted to observed daily rainfall and

the corresponding atmospheric reanalysis data, with this relationship then being applied toClimate Model output. The data and climate model output are discussed in Section 3, modelfitting is addressed in Section 4 and simulation results are presented in Section 5. Section 6presents the conclusions of this work.

2 Generalised Linear Models for daily rainfall

The work presented here makes use of the GLIMCLIM software (Chandler, 2002) in whichdaily rainfall, denoted Yt, is modelled in two stages, both of which use a GLM. Firstly, as

in Chandler and Wheater (2002), rainfall occurrence is modelled using logistic regression. Letpi denote the probability of rain at a single site on day i, and let xi denote a correspondingpredictor vector for that site which includes atmospheric measures. Then the occurrencemodel is

ln(pi/(1 pi)) = xTi .

Wet day rainfall amounts are modelled using a gamma distribution with a common dispersionparameter. Conditional on predictor vector zi, the mean rainfall for the site on the ith wetday is mi with

ln(mi) = zTi .

Such models can be fitted simultaneously to data from a number of sites. In order to simulaterealistic multi-site rainfall sequences GLIMCLIM allows for spatial dependence, resultingfrom large scale weather systems, by constructing a joint distribution of rainfall at all siteswhich respects the marginal distributions from the GLMs (Yang et al., 2006b). However,in the work reported here different sites are modelled separately, as this work package isprimarily focused on single site rainfall. Modelling the sites separately allows different sitesto have different dispersion parameters, ensuring that even if different sites have the samemean rainfall on a particular day, they may have different values for rainfall variance.

Predictors considered for both the occurrence and amounts models include periodic func-tions along with specific month effects to represent seasonality, and transformations of rainfall

values on previous days to account for persistence. Atmospheric predictors must also be in-cluded if the models are to simulate realistic daily rainfall sequences under future climate

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scenarios (the types of atmospheric predictors used will be discussed in Section 3). Finally,interactions between the predictors are considered. This allows for some covariates to mod-

ulate the effect of others. For example, it will be seen in Section 4 that in the model forrainfall amounts the effect of the rainfall on the previous two days varies with season. Thisis appropriate because convective and frontal rainfall, which dominate in summer and winterrespectively, result in different degrees of autocorrelation.

In the work reported below, the choice of covariates was based on the Likelihood RatioStatistic (see Dobson (1990) for further details). Standard residual analyses were also usedto check the models and suggest improvements. These resulted, for example, in the inclusionof a February effect beyond the sine and cosine waves in the occurrence model of Section 4.Once a final model has been selected its properties can be obtained via simulation. This isrequired because the nonlinear structure of the models does not allow the properties to be

obtained analytically.

3 Data

3.1 Rainfall data

Models have been fitted separately to daily rainfall data from three gauges (at Heathrow,Birmingham and Manchester airports). Data from the 1961-1990 period were used to fit themodels (this corresponds to the control period of the climate models). Data were missing at

Heathrow for January to August of 1988 and February 1989. Birmingham airport had missingobservations for December 1983 and August 1989. The records for Manchester airport werecomplete. It should be noted that during the mid-1970s there was a conversion from imperialto metric measurement units (all rainfall values in this work are in millimetres), resultingin a change of measurement resolution. To minimise the effect of this, it is appropriateto threshold the rainfall data in such a way that small rainfall amounts are set to zerothroughout the period of record (Yang et al., 2006a). Here, a threshold of 0.5mm has beenused. This threshold is subtracted from all rainfall values with Yt > 0.5 before fitting, sothat the GLMs are fit to rainfall data

Y

t = max(0, Yt 0.5).Data simulated using the GLMs then have the threshold added back to any non-zero values.The simulated rainfall will therefore contain no values between zero and 0.5mm.

3.2 Atmospheric data

The aim of this work is to simulate future rainfall conditional on projections of future climate.The GLMs are fit using observed atmospheric data and are then simulated conditional onclimate model output for the 2071-2100 period. We therefore need to identify atmospheric

variables that are closely related to rainfall, and are also reproduced reasonably by climatemodels.

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Model type Institution Model nameGCM Canadian Centre for Climate Modelling and Analysis cgcm2

GCM Commonwealth Scientific and Industrial Research Organisation csiromk2GCM Max-Planck-Institute echam4GCM Hadley Centre hadcm3RCM Danish Meteorological Institute hirhamRCM Swedish Meteorological and Hydrological Institute rcaoRCM Hadley Centre hadrm3p

Table 1: Details of employed climate models.

3.2.1 Climate Models

Deterministic climate models attempt to represent explicitly the physical processes under-lying climate. Both General Circulation Models (GCMs) and Regional Climate Models(RCMs) are used in this work. The specific models used are summarised in Table 1.

RCM data were obtained from the PRUDENCE (Prediction of Regional scenarios andUncertainties for Defining EuropeaN Climate change risks and Effects) project(http://www.prudence.dmi.dk). This was an investigation which produced high-resolution(grid distances of between 0.2 and 0.5) climate simulations for the 1961-1990 (control) and2071-2100 periods, using RCMs from a number of climatological institutions across Europe.At boundaries all models were forced with an atmosphere only global model which was inturn forced by sea surface temperatures (SSTs). For the future 2071-2100 period, SSTs weresimulated using a General Circulation Model (GCM) without flux adjustment (HadCM3).All results presented here are based on the A2 transient green-house gas scenario as specifiedby the Intergovernmental Panel on Climate Change (IPCC), in which total annual CarbonDioxide emissions in 2100 are approximately 3.5 times greater than in 2000 (IPCC, 2001,Appendix II SRES Tables). For reasons of convenience, RCM years have 360 days only (30days per month). RCMs developed by three institutions were considered (under the advice ofDavid Stephenson): the Danish Meteorological Institute (DMI), the UK Hadley centre (HC)and the Swedish Meteorological and Hydrological Institute (SMHI). The DMI and SMHIRCMs were forced using a different atmosphere only model to the HC RCM.

As previously mentioned, all three RCMs considered here are forced using SSTs fromthe same GCM. Yet due to the extremely complex nature of the climate system and theresulting uncertainties in model formulation and parameterisation, different GCMs can givevery different projections for future climate. Rowell (2004) observes that the uncertaintyin projected surface air temperature over the UK due to RCM formulation is relativelysmall in comparison to the uncertainty resulting from the formulation of the driving GCM.Therefore, in order that this source of uncertainty be addressed, we consider output fromGCMs developed by four different institutions: the Canadian Centre for Climate Modellingand Analysis (CCCma; cgcm2), Commonwealth Scientific and Industrial Research Organ-isation (CSIRO; csiromk2), Max-Planck-Institute (MPI; echam4) and the Hadley Centre(HC; hadcm3). The data were obtained from the Statistical DownScaling Model (Wilby and

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Dawson, 2001) (SDSM) website (http://www-staff.lboro.ac.uk/ cocwd/SDSM/archive.html)on a grid of 2.5 latitude by 3.75 longitude. The CCCma and CSIRO models both have

365 days per year and run from 1961 to 2100, the MPI and Hadley Centre models have 360days per year and run from 1961 to 2100 and 1961 to 2099 respectively. Again, all outputused is conditional on the A2 transient green-house gas scenario, so emission scenario un-certainty is not addressed. It is noted that the relative humidity values from CSIRO, MPIand particularly the CCCma model should be treated with caution having been derived al-gorithmically from pressure, temperature and specific humidity (Harris, 2004). Harris (2004)observes that calculation of relative humidity from other CCCma outputs did not result inrealistic values, suggesting it may be inappropriate to attempt to derive relative humidityfrom existing variables. One possible explanation offered for this is the inappropriateness ofthe calculated saturation vapour pressure.

Given that the aim of this work is to simulate daily rainfall it would be legitimate tosuggest using the climate model rainfall output directly. We do not take this approachfor a number of reasons. Firstly, climate model rainfall output is a spatial average butsingle site values are often required. Secondly, impact assessment studies ideally requirea number of possible rainfall sequences and this cannot be easily achieved using climatemodels due to computational costs. Finally, climate models cannot explicitly resolve theprocesses controlling rainfall. Instead, they build empirical relationships between rainfalland other atmospheric variables, a technique called parameterisation. It has been foundthat different climate models give very different projections for rainfall, even at large spatialand temporal scales, giving a lower level of confidence in the rainfall realisations. Stefan

Hagemann and Daniela Jacob (personal communication, 2004) found that the different RCMsin the PRUDENCE project gave both negative and positive proportional changes in rainfallbetween the control and future periods over the Elbe and Rhine catchments. Frei et al. (2003)report that the PRUDENCE RCMs underestimated the average intensity over the Alps by16-42%. Nevertheless, it is sensible to compare future rainfall properties obtained via theGLM rainfall simulations with those based on climate model rainfall outputs directly. If thetwo different approaches give similar results, this may improve confidence in our projectionsof future rainfall. In Section 5.2 we will compare the values of mean daily rainfall obtainedvia simulation of the GLMs with the values based on the climate model rainfall outputs.

We have discussed the limitations of climate model rainfall output. There are also lim-

itations of climate model output for other variables, although in general these are not soserious as for rainfall. Due to the chaotic nature of the climate system climate models can-not generate actual climate sequences, but aim to simulate realisations with climate likeproperties (von Storch and Zwiers, 1999; Smith, 2002). In other words, the best any cli-mate model can hope to do is to produce a sequence which is, in terms of its properties,indistinguishable from the true trajectory (at least within the period of interest). In realitythere are two further issues to consider when using climate model output, and while previ-ous discussion in the literature has focused on GCMs the same arguments can be applied toRCMs.

Firstly, it is likely that properties (e.g.mean and variance) of model outputs may match

those of observations only when aggregated over large temporal scales. The IPCC (2001,

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p. 473) suggest that GCMs provide credible simulations of climate, at least down to sub-continental scales and over temporal scales from seasonal to decadal. Prudhomme et al.

(2002) note that despite the belief that GCMs model atmospheric variables (such as the sea-level pressure) better at a daily time step than they model rainfall, there is still no consensusabout the reliability at that time scale.

Secondly, the IPCC (2001, p. 750) note that studies rarely use climate model outputsdirectly because biases are too great. Therefore, the output from any climate model runto investigate the effect of anthropogenic forcings is always compared to the output froma control climate model run (after both have had spin-up time). Changes in mean climatebetween the baseline period and the future are often calculated either as the differenceor as the ratio between the simulated forced climate and the simulated control climate(IPCC, 2001, p. 751). The IPCC note that ratios are often used with those surface variables

that are either positive or zero (e.g. precipitation and pressure), while for other variables(e.g. temperature) the difference is usually used. These relative or absolute changes (oftenstratified by season) are then used in conjuction with the observed data for the baselineperiod to create a future climate scenario. Both these forms of manipulation are consistentwith the assumption that a simulated climatological mean is linearly biased. Furthermore,it seems reasonable to extend this idea and to assume that the observed or projected valueof some atmospheric variable, Xot (where t indexes at an appropriate temporal scale), hasthe same distribution as + Xst , where X

st is the corresponding climate model output for

time t.

3.2.2 Reanalysis data

The NCEP (Kalnay et al., 1996) and ERA40 (at http://data.ecmwf.int/data/d/era40 mnth/)reanalysis data sets are derived by feeding quality controlled observations into a physicalmodel, which then outputs gridded values of many different variables. While a number ofatmospheric variables (e.g. temperature) are closely related to input observations, othervariables (e.g. precipitation) are not constrained by actual observations for that variable.This means that the reanalysis data sets are more reliable for some variables than for others.1

The reanalysis data play the role of observed atmospheric data in the work presented here,and are used to fit the GLMs relating observed rainfall to observed atmospheric variables.

3.3 Atmospheric predictors

In this section we discuss how the reanalysis data and climate model output were manipu-lated in order to provide atmospheric predictors for the GLMs. The GLMs are fitted usingboth the NCEP and ERA40 data in order to assess whether the model parameter estimatesare robust to differences in the reanalysis data. As the fitted models will then be simu-lated conditional on the GCM/RCM output, atmospheric predictors were limited to thosethat are reasonably represented by the climate models. Therefore temperature, sea level

1

NCEP rainfall data were not included in the set of potential atmospheric predictors due to qualityissues (Chandler, 2000).

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2070 2075 2080 2085 2090 2095 2100

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CCCma

Year

Sea

levelpressure

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levelpressure

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levelpressure

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0.0

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Hadley

Year

Sea

levelpressure

SB

W

EE

SE

Figure 1: Annual mean sea-level pressure standardised with respect to the 1961-1990 controlperiod, for grid squares with centre points 50N, 0 (Southern England, SE), 52.5N, 0

(Eastern England, EE), 52.5N, 3.75W (Wales, W), 55N, 3.75W (Scottish Borders, SB).

pressure, relative humidity were considered (specific humidity and temperature being too

highly correlated in the reanalysis data to both be used). All reanalysis data and climatemodel output for these variables were regridded onto the SDSM NCEP grid, measuring 2.5

latitude by 3.75 longitude. As measures of humidity were only available for the HC RCM,this realisation was used in rainfall simulation from all RCMs. However, humidity outputfor DMI and SMHI may be available in the future.

In Section 2 it was discussed that different models will be fit for different sites. Thisallows the use of atmospheric predictors corresponding to an area local to the rainfall site.This is particularly important as the projected changes in atmospheric values are differentfor different grid squares, as shown in Figures 1 to 3. For example, Figure 2 shows that theCSIRO model projects greater increases in temperature for the more southerly grid squares.

Atmospheric predictors corresponding to a particular site were calculated using a weightedaverage of nearby grid squares to allow for the non-continuous nature of the data. Theweighting was based on the assumption that the atmospheric values for a particular gridsquare corresponded to its centre point. Let (xi, yi) be the coordinates for the centre ofgrid square i, let (xs, ys) be the coordinates of the site, and let d(xs, xi) and d(ys, yi) be theeast-west and north-south distances (in kilometres) respectively. Then the weight for gridsquare i is proportional to

w(xs, xi) + w(ys, yi)

where

w(xs, xi) = 275 d(xi, xs) if d(xi, xs) 2750 if d(xi, xs) > 275

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2070 2075 2080 2085 2090 2095 2100

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ture

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Year

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ture

SB

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SE

Figure 2: Annual mean temperature standardised with respect to the 1961-1990 controlperiod, for grid squares with centre points 50N, 0 (Southern England, SE), 52.5N, 0


2070 2075 2080 2085 2090 2095 2100

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Year

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SE

Figure 3: Annual mean relative humidity standardised with respect to the 1961-1990 controlperiod, for grid squares with centre points 50N, 0 (Southern England, SE), 52.5N, 0


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1960 1980 2000 2020 2040 2060 2080 2100

0.4

0.2

0.0

0.2

0.4

Year

Yearlyaveragesea

levelpressure

CCCmaCSIRO

MPI

HC

NCEP

Figure 4: Yearly average sea-level pressure at Heathrow, standardised with respect to the1961-1990 mean and standard deviation.

and equivalently for w(ys, yi). The cut-off of 275 was chosen as this is approximately themaximum distance between the centre points of two grid squares. Hence, if a site is located

exactly at the centre of a grid square, the values of atmospheric variables are obtained solelyfrom that square. To illustrate the resulting atmospheric sequences, Figures 4, 5 and 6 showyearly averages from the different GCMs for sea-level pressure, temperature and relativehumidity respectively at Heathrow.

Given that climate model years do not have the same number of days as calendar years itwould be inappropriate to use daily atmospheric variables in our GLMs. Therefore, monthlymean values of the spatially averaged data were obtained for the 1961-1990 period. Sucha degree of temporal smoothing will still allow for simulation of rainfall typical of weathersystems that occur on smaller temporal scales, as the structure of such rainfall events willbe incorporated into the variance and autocorrelation structure of the GLMs.

The monthly atmospheric values were then standardised by month (i.e. with respect tothe monthly means and standard deviations for the 1961-1990 period). This standardisationwas done for two reasons. Firstly, it was desired to deseasonalize the predictors as seasonalitywill be included explicitly elsewhere in the GLMs. Secondly, it accounts for the climatemodel bias. To explain this, let Xoym be the NCEP or ERA40 monthly mean value ofa particular atmospheric variable in year y and month m; the values of Xoym have beenobserved for the control period but, obviously, not for the future. Similarly let Xsym bethe GCM/RCM simulated monthly mean value of the atmospheric variable in year y andmonth m; here, values are available for both control and future periods. Let Xom and S

om

be the month dependent sample mean and standard deviation for the Xoym in the control

period, and similarly for Xs

m and Ss

m. Thus the GLMs are fitted using atmospheric predictors

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1960 1980 2000 2020 2040 2060 2080 2100

0.4

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0.0

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1.0

Year

Yearlyaveragetemperature

CCCma

CSIRO

MPI

HC

NCEP

Figure 5: Yearly average temperature at Heathrow, standardised with respect to the 1961-1990 mean and standard deviation.

1960 1980 2000 2020 2040 2060 2080 2100

1.0

0.5

0.0

Year

Yearlyav

eragerelativehumidity

CCCma

CSIRO

MPI

HC

NCEP

Figure 6: Yearly average relative humidity at Heathrow, standardised with respect to the1961-1990 mean and standard deviation.

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(Xoym Xom)/S

om. Now let us assume, as discussed in Section 3.2.1, that the GCM/RCM

outputs are biased so that + Xsym follows the same distribution as Xoym. It then follows

that

(Xoym Xoym)/S

oym ( + X

sym X

sym)/S

sym.

where denotes equality of distribution. Clearly,

( + Xsym Xsym)/S

sym = (X

sym X

sym)/S

sym

and so

(Xoym Xoym)/S

oym (X

sym X

sym)/S

sym.

Therefore, to simulate rainfall using based on the GCM/RCM output the fitted GLMs areconditioned on Zsym = {(X

sym X

sym)/S

sym}.

4 Model building

In this section we describe the construction of GLMs for rainfall at the three selected sites(see Section 3.1). This work aims to provide a methodology for using GLMs to simulatedaily rainfall, and the predictors included in the models presented here should provide aguide to suitable predictors for other sites. Therefore in this work, if a predictor is foundto be significant at any site, it is included in the GLMs for all sites. The predictors forHeathrow, and the corresponding parameter estimates and standard errors obtained whenfitting using the NCEP and ERA40 reanalysis data, are listed in Tables 2 to 5. The modelsfor other sites can be found in the appendix. We let Yt denote the rainfall value on dayt, I(Y[t k] > 0) be an indicator of rainfall occurrence k days previously at the site andI(Y[t k] > 0 : k = 1 to n) be an indicator of rain on each of the previous k days.

Both the NCEP and ERA40 fitted models are fairly similar. The interactions betweenI(Y[t k] > 0 : k = 1 to 2) and the sine and cosine components in both the fitted amountsmodels imply a seasonally varying autocorrelation structure: the dependence on the previoustwo days rainfall occurrence is greater in winter than in summer. This agrees with the results

of Chandler and Wheater (2002) and is expected because there are fewer long-lasting frontalweather systems in summer in mid-latitudes. Interestingly, the opposite effect is observedin the occurrence model: here, the dependence on the previous days rainfall is greater insummer. Such a surprising result is not observed in either the Manchester or Birminghamoccurrence models. The Heathrow models show that autocorrelation for both occurrence andamounts varies with sea level pressure, and the parameter estimates suggest that dependencebetween rainfall on successive days is greater in the more stable weather systems associatedwith higher pressure. Although the amounts model explains only 3.2 percent of the variancein daily rainfall, it will be illustrated in Section 5.1 that this apparently small degree ofpredictive power captures the seasonal and inter-annual variation in rainfall properties well.

The fitted GLMs for the three sites are examples of models that could also be fitted todata from other sites. We note that when fitting and then extrapolating a rainfall amounts

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Effect Parameter Estimate Standard ErrorConstant 1.33 0.03

Sea-level pressure -0.08 0.03Temperature -0.002 0.02ln(1 + Y[t 1]) 0.08 0.03ln(1 + Y[t 2]) 0.06 0.03I(Y[t k] > 0 : k = 1 to 2) -0.15 0.07I(Y[t 1] > 0) 0.15 0.07Daily seasonal effect, cosine component -0.22 0.03Daily seasonal effect, sine component -0.14 0.03InteractionsI(Y[t k] > 0 : k = 1 to 2) with Dailyseasonal effect, cosine component

0.18 0.07

I(Y[t k] > 0 : k = 1 to 2) with Dailyseasonal effect, sine component

-0.11 0.06

Daily seasonal effect, cosine component withsea-level pressure

-0.12 0.03

Daily seasonal effect, sine component withsea-level pressure

0.02 0.03

I(Y[t k] > 0 : k = 1 to 2) with sea-levelpressure

0.12 0.05

Table 3: Gamma amounts model for Heathrow, fitted using the NCEP reanalysis data.

This model accounts for 3.2% of the variance. The dispersion parameter for this model wasestimated to be 0.73.

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Effect Parameter Estimate Standard ErrorConstant 1.33 0.03

Sea-level pressure -0.08 0.03Temperature -0.02 0.02ln(1 + Y[t 1]) 0.08 0.03ln(1 + Y[t 2]) 0.06 0.03I(Y[t k] > 0 : k = 1 to 2) -0.15 0.07I(Y[t 1] > 0) 0.15 0.07Daily seasonal effect, cosine component -0.22 0.03Daily seasonal effect, sine component -0.14 0.03InteractionsI(Y[t k] > 0 : k = 1 to 2) with Dailyseasonal effect, cosine component

0.18 0.07

I(Y[t k] > 0 : k = 1 to 2) with Dailyseasonal effect, sine component

-0.11 0.06

Daily seasonal effect, cosine component withsea-level pressure

-0.13 0.03

Daily seasonal effect, sine component withsea-level pressure

0.02 0.03

I(Y[t k] > 0 : k = 1 to 2) with sea-levelpressure

0.12 0.05

Table 5: Gamma amounts model for Heathrow, fitted using the ERA40 reanalysis data.

This model accounts for 3.2% of the variance. The dispersion parameter for this model wasestimated to be 0.73.

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model, constraints are required on the coefficients corresponding to previous days rainfallif the simulated rainfall is to remain within reasonable bounds. For example, a three way

interaction between temperature, ln(1 + Y[t 1]) and seasonality was also found to besignificant when included in the amounts model presented here. However, future simulationsfrom a model including this interaction escalated, at times, to unrealistic values of more thana meter of rainfall in a day. This was a result of the projected increase in temperature, asthis in turn increased the effect of Yt1 on t. Given this problem the interaction was notincluded in the final model. This highlights the need to consider carefully the structure ofmodels that are to be used for extrapolation purposes. Given that extrapolation of fittedmodels is fundamental to this project, we suggest models should be robust and capture themain features of rainfall at the possible expense of detail.

5 Simulation

5.1 1961-1990 period

To investigate the performance of the fitted models, rainfall is simulated for the 1961-1990period and the properties of these sequences compared with those of the observations. Suchmodel checking is required since small errors in model specification may cumulate over along period of time (Yang et al., 2006b). Furthermore, simulation allows us to assess theperformance of the occurrence and amounts models when used together. Figures 7 through

to 15 are checks on the fit of the NCEP fitted models presented in Section 4. Given thesimilarity between the NCEP and ERA40 fitted models, the latter will not be consideredfurther here. 200 simulations of the GLMs were run for the control period, conditional onthe NCEP data.

Figures 7 to 9 investigate how well the GLMs simulate total annual rainfall at eachsite. The first plot in each Figure is a Probability Integral Transform (PIT) (Dawid, 1984)which can be used to assess the calibration of modelled distributions. The PITs show thenumber of years the observed total annual rainfall fell in between certain percentiles of thecorresponding simulated distributions. If the observed values had actually come from thesimulated distributions the results should look like samples from a uniform distribution. The

second plots in Figures 7 to 9 show the simulated distributions of total rainfall in each year,along with the corresponding observed values. The bands correspond to the 5th, 10th, 25th,50th, 75th, 90th, and 95th percentiles and the black line shows the observed value (thickerlines indicate missing data that have been simulated ten times conditional on the observeddata as described in Yang et al. (2006b)). If the simulations are realistic the observed valuesshould look as if they have been sampled from the distributions. The notable inter-annualvariability in the simulated distributions (e.g. the shift upwards in 1974) is due to the NCEPexplanatory variables. Based on these plots the models appear to perform well; the PITsare convincing as 30 independent realizations from the uniform distribution.

Figures 10 to 12 show the simulated distributions of total summer and winter rainfall

at the different sites and the corresponding observed values (black line). Again, if the

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Probability integral

Percentiles

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

6

7

1960 1965 1970 1975 1980 1985 1990

0

40

0

800

1200

Year

mm

Total annual rainfall

Figure 7: i) Probability Integral Transform for annual rainfall at Manchester: observedvalues as a sample from simulated distributions. ii) Simulated distributions of total annualrainfall in Manchester. The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and95th percentiles and the thick line shows the observed values.

simulations are realistic the observed values should look as if they have been sampled fromthe seasonal distributions. All three Figures show that the simulated distributions of seasonaltotals are consistent with the observed values. Finally, Figures 13 to 15 show the simulateddistributions of various monthly summary statistics and the corresponding observed values(black line). There is generally a good match to the summary statistics, particularly tothe mean and the proportion of wet days. The fact that the simulated standard deviation,standard deviation when wet and maximum are generally too small at all sites in summershows that the use of a constant dispersion parameter estimate throughout the year is notideal.

5.2 2071-2100 period

The ultimate aim of this work is to simulate a range of future rainfall sequences that areconsistent with the future climates projected by the climate models. Therefore, 200 simu-lations were run for the 2071-2100 period for each GCM and RCM, using the NCEP fittedGLMs presented in Section 4.

To condense the results, monthly rainfall summary statistics were computed for eachsimulation. Figures 16, 17 and 18 show, for each GCM, the simulated distributions of De-cember daily rainfall mean, 5-day maximum and variance at Heathrow for each year between2071-2100. The bands in the figures represent the 5th, 10th, 25th, 50th, 75th, 90th, and95th percentiles of the simulated distributions. The irregular nature of the distributions is

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Percentiles

Frequency

0.0 0.2 0.4 0.6 0.8

0

2

4

6

8

1960 1965 1970 1975 1980 1985 1990

0

400

800

1200

Year

mm


Figure 8: i) Probability Integral Transform for annual rainfall at Birmingham: observedvalues as a sample from simulated distributions. ii) Simulated distributions of total annualrainfall in Birmingham. The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and95th percentiles and the thick line shows the observed values.


Percentiles

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

6

1960 1965 1970 1975 1980 1985 1990

0

200

600

1000

Year

mm


Figure 9: i) Probability Integral Transform for annual rainfall at Heathrow: observed valuesas a sample from simulated distributions. ii) Simulated distributions of total annual rainfallin Heathrow. The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th

percentiles and the thick line shows the observed values.

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1960 1965 1970 1975 1980 1985 1990

0

100

300

500

Year

mm

Total summer (JJA) mean areal rainfall

1960 1965 1970 1975 1980 1985 1990

0

100

30

0

500

Year

mm

Total winter (DJF) mean areal rainfall

Figure 10: Simulated distributions of total seasonal rainfall at Manchester (top:summer,bottom:winter). The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95thpercentiles and the thick line shows the observed values.

1960 1965 1970 1975 1980 1985 1990

0

100

200

300

400

Year

mm


1960 1965 1970 1975 1980 1985 1990

0

100

200

300

400

Year

mm


Figure 11: Simulated distributions of total seasonal rainfall at Birmingham (top:summer,bottom:winter). The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th

percentiles and the thick line shows the observed values.

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1960 1965 1970 1975 1980 1985 1990

0

100

200

300

400

Year

mm


1960 1965 1970 1975 1980 1985 1990

0

100

3

00

500

Year

mm


Figure 12: Simulated distributions of total seasonal rainfall at Heathrow (top:summer, bot-tom:winter). The bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95thpercentiles and the thick line shows the observed values.

2 4 6 8 10 12

1.0

1.5

2.0

2.5

3.0

3.5

Month

mm

Mean

2 4 6 8 10 12

2

3

4

5

6

7

Month

mm

Std Dev

2 4 6 8 10 12

0.3

0

0.3

5

0.4

0

0.4

5

0.5

0

0.5

5

Month

Proportion

P(wet)

2 4 6 8 10 12

3

4

5

6

7

8

Month

mm

Mean when wet

2 4 6 8 10 12

3

4

5

6

7

8

9

Month

mm

SD when wet

2 4 6 8 10 12

0

20

40

60

80

100

12

0

Month

mm

Maximum

2 4 6 8 10 12

0.0

0.1

0.2

0.3

0.4

Month

Correlation

Lag 1 ACF

2 4 6 8 10 12

0.0

5

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

0.25

Month

Correlation

Lag 2 ACF

Figure 13: Simulated distributions of monthly summary statistics at Manchester. The bandscorrespond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles and the thick line

shows the observed values.

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2 4 6 8 10 12

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Month

mm

Mean

2 4 6 8 10 12

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Month

mm

Std Dev

2 4 6 8 10 12

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

0.5

0

Month

Proportion

P(wet)

2 4 6 8 10 12

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

Month

mm

Mean when wet

2 4 6 8 10 12

3

4

5

6

7

8

9

Month

mm

SD when wet

2 4 6 8 10 12

0

20

40

60

80

100

Month

mm

Maximum

2 4 6 8 10 12

0.0

0.1

0.2

0.3

0.4

Month

Correlatio

n

Lag 1 ACF

2 4 6 8 10 12

0.0

5

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

Month

Correlatio

n

Lag 2 ACF

Figure 14: Simulated distributions of monthly summary statistics at Birmingham. Thebands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles and thethick line shows the observed values.

2 4 6 8 10 12

1.0

1.5

2.0

Month

mm

Mean

2 4 6 8 10 12

2

3

4

5

6

Month

mm

Std Dev

2 4 6 8 10 12

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

Month

Proportion

P(wet)

2 4 6 8 10 12

3

4

5

6

7

8

Month

mm

Mean when wet

2 4 6 8 10 12

3

4

5

6

7

8

9

1

0

Month

mm

SD when wet

2 4 6 8 10 12

0

20

40

60

80

100

Month

mm

Maximum

2 4 6 8 10 12

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.

35

Month

Correlation

Lag 1 ACF

2 4 6 8 10 12

0.0

5

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

0.

25

Month

Correlation

Lag 2 ACF

Figure 15: Simulated distributions of monthly summary statistics at Heathrow. The bandscorrespond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles and the thick line

shows the observed values.

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2070 2075 2080 2085 2090 2095 2100

0

2

4

6

8

10

Year

mm

Simulated mean daily rainfall,conditional on CCCma output.

2070 2075 2080 2085 2090 2095 2100

0

2

4

6

8

10

Year

mm

Simulated mean daily rainfall,conditional on HC output.

2070 2075 2080 2085 2090 2095 2100

0

2

4

6

8

10

Year

mm

Simulated mean daily rainfall,conditional on CSIRO output.

2070 2075 2080 2085 2090 2095 2100

0

2

4

6

8

10

Year

mm

Simulated mean daily rainfall,conditional on MPI output.

Figure 16: Simulated distributions of December mean daily rainfall at Heathrow, 2071-2100

a result of the GLMs being simulated conditional on only one realization of each climate

model. These plots show that conditioning a rainfall simulation on a single climate modelrealization will underestimate the variability in rainfall properties at any particular timepoint. This is because, even if a climate model simulates sequences with the correct proper-ties, no individual realisation provides the atmospheric values that will actually be observedin the future.

However, there is also a second issue to consider. Figures 19 and 20 show the meanannual rainfall and proportion of wet days at Heathrow, calculated from the 200 simulationsfor each GCM. Visual inspection of these plots suggests that the annual rainfall statisticssimulated using the different GCMs have different distributions, rather than just being differ-ent realizations from the same distribution. For example, the mean rainfall and proportion

of wet days are generally higher for rainfall simulated using the CSIRO GCM atmosphericpredictors, as compared to the rainfall simulated using the other three GCMs. This is notsurprising given the differences in the GCM projections shown in Figures 4 to 6.

Tables 6 to 9 summarise the GLM simulated rainfall in December and June at Heathrowand Manchester respectively for the 2071-2100 period. They also show the rainfall meanscorresponding to the rainfall output generated by the climate models. As for the atmosphericvariables, GCM/RCM precipitation corresponding to a particular site has been calculatedusing a weighted average of grid squares (see Section 3.3). As is standard in the literature,we have assumed climate model rainfall output to have a multiplicative bias (i.e. Xsymfollows the same distribution as Xoym) and so proportional changes were calculated from

climate model outputs, with these changes then being applied to the observed 1961-1990

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2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated five day maximum daily rainfall,conditional on CCCma output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated five day maximum daily rainfall,conditional on HC output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated five day maximum daily rainfall,conditional on CSIRO output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated five day maximum daily rainfall,conditional on MPI output.

Figure 17: Simulated distributions of December 5-day maximum rainfall at Heathrow, 2071-2100

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated variance of daily rainfall,conditional on CCCma output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated variance of daily rainfall,conditional on HC output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated variance of daily rainfall,

conditional on CSIRO output.

2070 2075 2080 2085 2090 2095 2100

0

50

100

150

Year

mm

Simulated variance of daily rainfall,

conditional on MPI output.

Figure 18: Simulated distributions of December daily rainfall variance at Heathrow, 2071-

2100

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2070 2075 2080 2085 2090 2095 2100

1.0

1.5

2.0

2

.5

Year

MeanRainfall

CCCma

CSIRO

MPI

HC

Figure 19: Simulated annual mean daily rainfall at Heathrow, 2071-2100

daily rainfall mean for each site. We cannot compare the GLM simulated variances and

five day maxima with those generated directly by the climate models, as climate models aregenerally considered to be area averages while we are looking a single site rainfall.

Tables 6 and 8 show that both the GLM simulations and the climate models projectwetter winters at Heathrow and Manchester airports in the 2071-2100 period with respect tothe 1961-1990 period. Tables 7 and 9 show that both the GLM simulations and the climatemodels generally suggest drier summers in the future period, although the GLM approachalmost always projects a greater decrease in mean rainfall. The slight differences between therainfall means from the two different approaches reflect the differences in the methodologyused to relate rainfall to atmospheric variables. In favour of the GLM approach, we notethat the climate model parameterisations are not constructed separately for each grid square;

also, the climate model rainfall values are area averages whereas point rainfall is of morerelevance for hydrological applications.

Tables 6 to 9 also show that there are significant differences in the rainfall statistics sim-ulated using the GLMs when conditioned on different GCMs/RCMs. The standard errorsin Tables 6 to 9 were calculated using the result presented in Appendix 8, assuming station-arity within months for the 2071-2100 period. Although this is technically incorrect, sincethe atmospheric predictors from the climate models are not stationary in this period (see,for example, Figure 5), resulting simulated trends in the rainfall statistics appear minimalso the standard errors will be reasonably accurate. On the basis of these tables we concludethat no single GCM or RCM can be assumed to reproduce the properties of future climate.

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2070 2075 2080 2085 2090 2095 2100

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

0.5

0

Year

P

roportionofwetdays

CCCma

CSIRO

MPI

HC

Figure 20: Simulated annual proportion wet days at Heathrow, 2071-2100

Climate Model Daily mean Daily variance 5-day maxGLM mean (s.e.) CM mean GLM mean (s.e.) GLM mean (s.e.)

CCCma (GCM) 2.18 (0.14) 2.15 16.27 (1.28) 28.74 (1.37)CSIRO (GCM) 2.76 (0.14) 2.41 20.82 (1.29) 34.22 (1.25)MPI (GCM) 2.18 (0.16) 2.36 16.85 (1.61) 28.69 (1.74)HC (GCM) 2.21 (0.18) 2.05 18.36 (1.92) 29.39 (1.94)

DMI (RCM) 2.08 (0.12) 2.21 16.11 (1.23) 28.25 (1.23)SMHI (RCM) 2.12 (0.12) 2.30 16.60 (1.22) 28.75 (1.26)HC (RCM) 2.25 (0.15) 2.19 17.78 (1.56) 29.77 (1.61)

Observed 1961-1990 1.81 (0.16) 13.76 (2.24) 27.61 (2.53)

Table 6: Summary of the rainfall distribution in December at Heathrow airport. The climatemodel statistics are for the 2071-2100 period, with the observed 1961-1990 values shown forcomparative purposes. The GLM mean values are based on the GLM simulations and theCM mean values correspond to the rainfall output produced directly by the climate models.

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Climate Model Daily mean Daily variance 5-day maxGLM mean (s.e.) CM mean GLM mean (s.e.) GLM mean (s.e.)

CCCma 1.05 (0.10) 2.09 10.11 (0.88) 18.36 (1.28)CSIRO 1.60 (0.10) 2.23 14.39 (0.87) 27.44 (0.92)MPI 1.33 (0.13) 1.40 12.34 (1.09) 21.33 (1.44)HC (GCM) 1.38 (0.08) 1.88 12.99 (0.72) 24.70 (0.85)DMI (RCM) 1.01 (0.07) 1.60 9.62 (0.67) 18.26 (0.97)SMHI (RCM) 0.99 (0.07) 1.74 9.49 (0.70) 17.66 (1.02)HC (RCM) 1.28 (0.12) 1.59 12.16 (1.05) 20.96 (1.41)

Observed 1961-1990 2.20 (0.22) 17.97 (3.13) 31.69 (2.83)

Table 9: Summary of the rainfall distribution in June at Manchester airport. The climatemodel statistics are for the 2071-2100 period, with the observed 1961-1990 values shown forcomparative purposes. The GLM mean values are based on the GLM simulations and theCM mean values correspond to the rainfall output produced directly by the climate models.

6 Conclusion

In the work presented here we have fitted GLMs relating observed single site rainfall to ob-served sea-level pressure, temperature and relative humidity. The observed atmosphericvariables were obtained from either NCEP or ERA40 reanalysis data, and results were insen-

sitive to which data set was used. It is noted that interactions involving autocorrelation andtrending terms should be avoided if the models are to simulate realistic rainfall under futureatmospheric conditions. The GLMs performed well when simulating rainfall for the 1961-1990 model fitting period; interannual rainfall variability and monthly summary statisticswere generally successfully reproduced. However, the assumption of a constant dispersionparameter throughout the year was not ideal as it resulted in the models underestimatingrainfall variability in summer. Sequences of future rainfall were simulated by conditioningthe models on GCM or RCM atmospheric outputs. The GLM simulations of future rainfallsuggest drier summers and wetter winters in the 2071-2100 period with respect to the 1961-1990 period. These results agree with the rainfall output directly by the climate models,although the GLMs lead to drier summer projections.

It was shown that more than one realisation of atmospheric predictors is needed fromeach climate model in order to obtain a comprehensive range of plausible rainfall simulations.Furthermore, conditioning the GLMs on different climate models gave significantly differentprojections of future rainfall properties. These issues will be addressed in further work to beundertaken for FD2113, which will provide a methodology for simulating a comprehensiverange of future daily rainfall sequences that accommodate the uncertainty in projectionsfrom deterministic climate models.

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7 Appendix A

Logistic occurrence model for Birmingham airport fit using the NCEP reanalysis data.

Main effect: Coefficient Std Err

------------ ----------- -------

Constant -1.122993 0.0375

slp -0.466950 0.0335

temperature 0.004546 0.0249

relative humidity 0.079479 0.0269

I(Y[t-1]>0) 0.912695 0.0794

I(Y[t-2]>0) 0.444489 0.0629

I(Y[t-3]>0) 0.108103 0.0456I(Y[t-k]>0: k=1 to 2) -0.497323 0.0899

Daily seasonal effect, cosine component 0.163514 0.0415

Daily seasonal effect, sine component 0.052204 0.0414

Smooth February effect -0.569312 0.1796

Ln(1+Y[t-1]) 0.222263 0.0404

2-way interactions: Coefficient Std Err

------------------- ----------- -------


with I(Y[t-1]>0)


with I(Y[t-1]>0)

Smooth February effect 0.290003 0.2726

with I(Y[t-1]>0)

relative humidity -0.051803 0.0355

with Daily seasonal effect, cosine component


with Daily seasonal effect, sine component


with Smooth February effect

temperature 0.032619 0.0352with Daily seasonal effect, cosine component

temperature -0.009333 0.0328




slp 0.282116 0.0468

with I(Y[t-1]>0)

Gamma model for Birmingham airport fit using the NCEP reanalysis data.


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------------ ----------- -------

Constant 1.385830 0.0306

slp -0.070117 0.0241temperature 0.035137 0.0198

Ln(1+Y[t-1]) 0.098816 0.0307

Ln(1+Y[t-2]) 0.054616 0.0271

I(Y[t-k]>0: k=1 to 2) -0.136968 0.0664

Daily seasonal effect, cosine component -0.136556 0.0320

Daily seasonal effect, sine component -0.129932 0.0322

I(Y[t-1]>0) -0.017867 0.0645


------------------- ----------- -------

slp 0.007374 0.0443

with I(Y[t-k]>0: k=1 to 2)

slp -0.073739 0.0285


slp -0.031641 0.0289


I(Y[t-k]>0: k=1 to 2) 0.041937 0.0584


I(Y[t-k]>0: k=1 to 2) -0.019790 0.0580


Logistic occurrence model for Manchester airport fit using the NCEP reanalysis data.


------------ ----------- -------

Constant -1.087429 0.0390

slp -0.314110 0.0343

temperature -0.009573 0.0265


I(Y[t-1]>0) 1.123724 0.0792

I(Y[t-2]>0) 0.618078 0.0632I(Y[t-3]>0) 0.147290 0.0450

I(Y[t-k]>0: k=1 to 2) -0.730473 0.0886



Smooth February effect -0.680694 0.1885

Ln(1+Y[t-1]) 0.271478 0.0390


------------------- ----------- -------

Daily seasonal effect, cosine component 0.094655 0.0617with I(Y[t-1]>0)

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with I(Y[t-1]>0)

Smooth February effect 0.362890 0.2770with I(Y[t-1]>0)













slp 0.115326 0.0451

with I(Y[t-1]>0)

Gamma model for Manchester airport fit using the NCEP reanalysis data.


------------ ----------- -------

Constant 1.380710 0.0301slp -0.059167 0.0289


Ln(1+Y[t-1]) 0.078862 0.0265

Ln(1+Y[t-2]) 0.022942 0.0242

I(Y[t-k]>0: k=1 to 2) 0.004166 0.0564

Daily seasonal effect, cosine component -0.158346 0.0384


I(Y[t-1]>0) 0.048001 0.0592

2-way interactions: Coefficient Std Err------------------- ----------- -------

slp 0.030049 0.0366

with I(Y[t-1]>0)

slp -0.033750 0.0254


slp -0.042745 0.0256


I(Y[t-1]>0) 0.106037 0.0494


I(Y[t-1]>0) 0.006836 0.0494


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8 Appendix B

This method calculates standard errors for Y for a non-independent sample of random vari-ables Yi. Let there exist a distribution for X, in this case the atmospheric predictors, and adistribution for Y|X, in this case a rainfall statistic. Assume we sample nx independent Xsand for each X we sample ny conditionally independent Ys. Let Yi be the mean of the Yscorresponding to the ith X and let Y be the overall mean of the Ys. Then

V ar(Y) = V ar(Y1 + Y2 + ... + Ynx

nx)

=1

n2xV ar(Y1 + Y2 + ... + Ynx)

=1

n2x(V ar(Y1) + V ar(Y2) + ... + V ar(Ynx))

and

V ar(Yi) = E[V ar(Yi|Xi)] + V ar[E(Yi|Xi)]

=1

nyE[V ar(Y|Xi)] + V ar[E(Y|Xi)]

where Y is generated conditional on Xi. Then

V ar(Y) =1

n2

x

[nx(1

nyE[V ar(Y|Xi)] + V ar[E(Y|Xi)])]

=E[V ar(Y|Xi)]

nynx+

V ar[E(Y|Xi)]

nx

Thus the standard errors were estimated by calculating the sample mean and variance of therainfall summary statistics in a particular month of each year, and then taking the samplevariance and mean of these values over the years.

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