Trends and Variability in Observations of Winter ... and Variability in Observation… · precipitation cannot be attributed to human activity. Therefore, the cause of observed trends

THE UNIVERSITY OF READING

Trends and Variability in Observations of

Winter Precipitation

Mathew R. P. Sapiano

A thesis submitted for the degree of Doctor of Philosophy

Department of Meteorology

December 2004

Declaration

I confirm that this is my own work and the use of all material from other sources has been

properly and fully acknowledged.

Mathew R. P. Sapiano

i

Abstract

Previous studies have identified an increasing trend in winter mean England and Wales pre-

cipitation of around 10–20% per century but the physical causes for this trend remain un-

clear. Statistical models are developed and applied to identify these trends and relate them to

underlying physical factors. Quasi-geostrophic theory suggests pressure gradient, saturation

specific humidity (qs, which is a function of temperature) and the meridional temperature

gradient as suitable multiplicative factors.

On daily time-scales, the physical relationships can be expressed statistically using a

mixture of Generalized Linear Models: a logistic model for the binary occurrence of precip-

itation and a Gamma distribution model for the wet day precipitation amount. No statistically

significant (at the 5% level) time trend is found in the winter wet day probability over Eng-

land and Wales, although a statistically significant time trend is found in winter wet day

amount. Local sea-level pressure explains much of the variability of both daily and winter

mean England and Wales precipitation, but fails to explain the long-term increasing trend.

Saturation specific humidity qs explains less of the variability, but partly accounts for the

increasing trend in winter precipitation.

The mixture model is also applied globally to October 1979 to March 2002 global grid-

ded pentad (5-day mean) merged satellite/gauge precipitation data. Similar results to those

from the daily analysis are found over the UK. In addition, the North Atlantic Oscillation,

a proxy for the meridional temperature gradient, is also found to have a statistically signif-

icant positive effect on precipitation over much of the Atlantic region. Increased qs caused

by surface warming (a product of climate change) and an increased meridional temperature

gradient have both contributed to the increasing trend in winter precipitation.

ii

Acknowledgements

There are so many people to thank for their input into my life and work during the pursuit

of my doctorate. Firstly, I owe so much of my development to my two supervisors: David

Stephenson and Howard Grubb. I have learned vast amounts from each of them and will be

forever grateful that they chose to take a chance on a young stato with a very limited physical

sciences background. Secondly, I would like to thank my thesis committee, Alan Thorpe and

Giles Harrison, for their input into the project which helped me make several key scientific

advances. A final thanks also goes to my external examiner, Tim Osborn, for taking the time

to carefully read and digest this manuscript. I’d also like to acknowledge useful discussions

with Phil Arkin, Brian Hoskins and Ross Reynolds who all gave me sound advice at some

point to help me make sense of the numbers.

I believe that no thesis from this department is complete without mention of the fantastic

social environment which provides a superb support basis for all in the department. I will

always have fond memories of my time at Reading, and particularly of the people in Meteo-

rology. Most notable of these are my office mates through the years: Ewan, Malc, Tommo,

Keely, Framo, Makund and even Briers.

Finally, I’ve had lots of good times outside of the department and I’d like to acknowledge

the rather distracting influence of my many friends who are not meteorologists. I can’t

honestly say they helped me work harder, but at least we had some good times!

“Many shall run to and fro, and knowledge shall increase”

Daniel, 12-4

iii

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Predicted trends in precipitation . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Scientific questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Plan of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Dynamical Processes Determining Extra-Tropical Precipitation 9

2.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Saturation of rising air . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 The Clausius-Clapeyron equation . . . . . . . . . . . . . . . . . . 10

2.2.2 Humidity changes due to climate change . . . . . . . . . . . . . . 11

2.2.3 The relationship between precipitation and upward motion . . . . . 12

2.3 Vertical motion in the extra-tropics . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Mean flow and the subtropical jets . . . . . . . . . . . . . . . . . . 14

2.3.2 Transients and baroclinic instability . . . . . . . . . . . . . . . . . 15

2.3.3 The omega equation . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.4 The Q-Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.5 Problems with the QG omega equation . . . . . . . . . . . . . . . 20

2.3.6 Scale analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Variability and Trends in Winter Mean England and Wales Precipitation 24

3.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 England and Wales Precipitation . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 The England and Wales Precipitation dataset . . . . . . . . . . . . 25

3.2.2 Methods of calculation of the monthly mean EWP time series . . . 26

3.2.3 Spatial homogeneity of EWP . . . . . . . . . . . . . . . . . . . . . 30

iv

Contents v

3.2.4 Characteristics of seasonal EWP . . . . . . . . . . . . . . . . . . . 30

3.2.5 Differences between seasons . . . . . . . . . . . . . . . . . . . . . 32

3.3 A summary of trends in winter mean EWP . . . . . . . . . . . . . . . . . . 33

3.3.1 Previous studies of trends in EWP . . . . . . . . . . . . . . . . . . 34

3.3.2 Linear time trends . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3 Linear trend model testing . . . . . . . . . . . . . . . . . . . . . . 36

3.3.4 Trends in different-length periods . . . . . . . . . . . . . . . . . . 38

3.3.5 Non-linear trends . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.6 A change in the seasonal pattern? . . . . . . . . . . . . . . . . . . 43

3.3.7 Changes in the variance of winter mean EWP . . . . . . . . . . . . 47

3.4 The role of sea-level pressure . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Previous studies of the pressure and precipitation relationship . . . 49

3.4.2 Mean sea-level pressure series . . . . . . . . . . . . . . . . . . . . 51

3.4.3 Trends in winter mean sea-level pressure . . . . . . . . . . . . . . 52

3.4.4 The relationship between winter mean EWP and SLP . . . . . . . . 53

3.4.5 Residual trends in EWP after accounting for SLP . . . . . . . . . . 56

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Statistical Methodology for Modelling Precipitation 59

4.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 A simple approach for modelling precipitation . . . . . . . . . . . . . . . . 59

4.2.1 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.2 Residual diagnostics from the simple linear regression . . . . . . . 61

4.3 The probability distribution of daily winter EWP . . . . . . . . . . . . . . 63

4.3.1 A mixture of two distributions . . . . . . . . . . . . . . . . . . . . 63

4.3.2 The distribution of the occurrence of precipitation . . . . . . . . . . 65

4.3.3 The distribution of the precipitation amount . . . . . . . . . . . . . 67

4.3.4 The effect of a changing threshold on the Gamma distribution . . . 68

4.4 The Generalized Linear Model (GLM) . . . . . . . . . . . . . . . . . . . . 70

4.4.1 GLMs in climate studies . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.2 The Generalized Linear Model . . . . . . . . . . . . . . . . . . . . 71

Contents vi

4.4.3 Inference for the Generalized Linear Model . . . . . . . . . . . . . 74

4.4.4 The GLM for Binary data . . . . . . . . . . . . . . . . . . . . . . 75

4.4.5 The GLM for Gamma distributed data . . . . . . . . . . . . . . . . 76

4.5 Model residual diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5.1 Residual diagnostics for the Gamma model . . . . . . . . . . . . . 79

4.5.2 Residual diagnostics for the Binary model . . . . . . . . . . . . . . 80

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Modelling of England and Wales Precipitation 83

5.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Scientific questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Modelling strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Explanatory variables for the daily model . . . . . . . . . . . . . . . . . . 87

5.3.1 Sea-level pressure x2 . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.2 Saturation specific humidity x3 . . . . . . . . . . . . . . . . . . . . 88

5.4 The probability of precipitation . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.1 Model fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.2 Statistically significant factors . . . . . . . . . . . . . . . . . . . . 92

5.4.3 Residual diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5 The wet day precipitation amount . . . . . . . . . . . . . . . . . . . . . . 95

5.5.1 Model fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.2 Statistically significant factors . . . . . . . . . . . . . . . . . . . . 97

5.5.3 Residual diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Modelling serially-correlated data . . . . . . . . . . . . . . . . . . . . . . 101

5.6.1 The pentad probability of precipitation . . . . . . . . . . . . . . . . 102

5.6.2 Wet pentad precipitation amounts . . . . . . . . . . . . . . . . . . 104

5.6.3 Residuals for the pentad Gamma model . . . . . . . . . . . . . . . 107

5.7 Time trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7.1 Changes in the distribution of precipitation . . . . . . . . . . . . . 110

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Contents vii

6 Modelling of Global Precipitation 114

6.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.1 GPCP pentad precipitation . . . . . . . . . . . . . . . . . . . . . . 115

6.2.2 Sea-level pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2.3 Saturation specific humidity . . . . . . . . . . . . . . . . . . . . . 119

6.2.4 Meridional temperature gradient . . . . . . . . . . . . . . . . . . . 120

6.3 The GLM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4.1 Mean precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4.2 The annual cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.3 Sea-level pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4.4 Saturation specific humidity . . . . . . . . . . . . . . . . . . . . . 129

6.4.5 The meridional temperature gradient . . . . . . . . . . . . . . . . . 130

6.4.6 Time trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 Conclusions 136

7.1 Time trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A Sample size calculation for linear time trends 142

B Global Precipitation Datasets 144

C Comparison of Time Trends in EWP and GPCP 147

Glossary of Acronyms 151

References 153

Chapter 1

Introduction

1.1 Motivation

Precipitation is an important weather element whose future changes will have a large impact

on society – a decrease in mean precipitation would lead to increased risk of drought, whilst

an increase in mean precipitation would lead to increased risk of flooding. Recent flood

events in the United Kingdom (UK) such as those in October/November 2000 (DEFRA,

2001), the November 2002 and the January 2003 events brought this problem to the fore-

front of the public mind. In 2004, the Environment Agency established an online flood map

allowing the general public to check whether their property is at risk of flooding. Compen-

sation for flood damage is estimated to have cost over £1000 million in 4 of the last 15 years

(Hawes, 2003). An increase in mean precipitation might to be accompanied by an increase

in extreme flood events such as the October 2004 flash flood event in Boscastle (Cornwall,

UK) which received widespread media coverage and led to a loss of property and could po-

tentially have led to a loss of life. Increases in UK precipitation are therefore of more interest

than decreases since the impacts of flooding are a current source of concern to society.

The Intergovernmental Panel on Climate Change concluded that global average surface

temperature had increased by 0.6 ± 0.2◦C over the 20th century (IPCC, 2001). They alsostated that there is evidence that most of the warming observed over the last 50 years is

attributable to human activities. However, there is less certainty associated with trends in

precipitation. IPCC (2001) also concluded that precipitation has increased over much of the

1

Chapter 1. Introduction 2

globe, with a decrease over subtropical areas, but the detection of these trends is hampered by

the fact that they are neither temporally nor spatially uniform (Folland et al., 2001). The DE-

FRA (2001) report on the October/November 2000 flood events and their relation to climate

change found evidence of increasing rainfall (mean and extremes) and river flow extremes in

Britain over the last 30 to 40 years. They conclude that their findings are consistent with pre-

dictions of anthropogenic climate change, however they also stated that observed changes in

precipitation cannot be attributed to human activity. Therefore, the cause of observed trends

in regional precipitation remains unclear.

There are two major methods for investigating precipitation. One is to use a climate

General Circulation Model (GCM) to simulate possible future precipitation, the other is to

study statistical properties of historical observations. Climate GCMs are better suited to the

projection of future climate change and often have a spatial resolution that is too coarse to

accurately model spatially inhomogeneous precipitation. The study of historical data rep-

resents an alternative method for understanding trends and will play an increasingly vital

role in validating the projected scenarios from GCMs. Temporal trends are the focus of this

thesis, therefore statistical models will be applied to historical data to study the variability

and trends in precipitation.

The major difficulty with the statistical modelling of precipitation is that the distribution

of precipitation is difficult to characterise. Precipitation is highly discontinuous - precipita-

tion occurs only during wet events which have non-uniform duration, frequency and inten-

sity. These events aggregate to form the daily and monthly rainfall totals routinely measured

at individual stations. Much of the work on trends in precipitation has concentrated on the

analysis of seasonal mean precipitation since this has a more Gaussian (Normal) distribu-

tion that is amenable to simple analysis techniques. However, seasonal mean precipitation

is far removed from the sub-daily time-scales of the underlying processes. These processes

necessitate careful statistical modelling of sub-seasonal (e.g. daily) rainfall amounts.

1.2 Predicted trends in precipitation

The increasing trend in global mean temperature is now fairly well established. As already

stated, IPCC (2001) showed a warming trend of 0.6 ± 0.2◦C over the 20th century and they


also predicted future warming of between 1.4 and 5.8◦C over the next century. However,

future changes in precipitation are more difficult to predict due to their local or regional

nature and their highly seasonal behaviour (Jones and Conway, 1997).

Hennessy et al. (1997) used the UK Met Office high resolution (UKHI) General Circula-

tion Model (GCM) and the Australian CSIRO9 GCM to look at zonal precipitation changes

and extremes. They found that annual mean global precipitation was 10% higher under a

double CO2 scenario, although this was closer to 20% higher between 50 and 60◦ North.

Hennessy et al. (1997) did not extend their analysis to seasonal changes of mean precipita-

tion and there is strong evidence that the increases in precipitation are not uniform throughout

the year, therefore a seasonal analysis is preferable.

Gregory and Mitchell (1995) used the Hadley Centre climate model to study the effect

of doubled CO2 on the mean and the distribution of temperature and precipitation. They

reported increased winter mean precipitation in mid-latitudes (50◦ to 60◦N) of 0.8–1 mm/day

under a doubled CO2 scenario and decreased winter mean precipitation in lower latitudes of

about of up to 0.8 mm/day, although they also state that these changes were not statistically

significant over England.

Timbal et al. (1995) used the Météo-France atmospheric GCM ARPEGE to study the

effect of doubled CO2 compared to control and found a 6 % increase in winter mean precip-

itation over Northern Europe.

Semenov and Bengtsson (2002) used the Max-Planck Institute coupled atmosphere-

ocean GCM ECHAM4/OPYC3 to study the effect of increased CO2 over the period 1900–

2099. They found an increase of up to 20 % per century in winter mean precipitation over

the UK, with larger (up to 40 %) increases over north Northern Europe. They also studied

changes in the precipitation intensity and the wet day probability. The changes in intensity

were similar to the changes in mean precipitation, but there was a decrease of up to 10% in

the winter wet day probability over much of the Atlantic sector, including the UK.

Finally, Watterson and Dix (2003) used the CSIRO Mark 2 coupled atmosphere-ocean

GCM to study changes in precipitation under rapid increases in CO2 using a five member

ensemble. They compared mean precipitation over 1961–1990 with 2071–2100 and found

an increase in DJF mean precipitation of between 10 and 30 % over the UK.

A major problem for the prediction of precipitation changes using current GCMs is the


relatively coarse resolution available. Many of the hydrological processes required for re-

alistic precipitation estimates take place at sub-grid scales. Various downscaling techniques

exist for estimating local climate data from coarse resolution climate models. Statistical

downscaling can be used to convert the relatively coarse gridded output from GCMs into

local point estimates using statistically based relationships. However, the suitability of sta-

tistical downscaling for predicting point mean precipitation estimates may be of limited ap-

plicability (Jones et al., 1997, Murphy, 2000).

Dynamical downscaling techniques involve using a higher resolution Regional Climate

Model (RCM) on a smaller regional domain. The RCM is generally nested within a GCM,

so that a GCM is used to provide boundary and initial conditions for the RCM. Jones et al.

(1997) used a nested regional climate model (based on the Met Office Unified Model) to

study the effect of a doubling of CO2 by comparing two 10-year equilibrium periods (a

control and a double CO2 run). They compared the results from the RCM with the results

from the forcing GCM. For the GCM, they found an increase in winter mean precipitation

over the UK of between 0.5 and 1 mm/day against a mean of 2–3 mm/day (Jones et al., 1995)

which is equivalent to an increase of between 15 and 50 %. For the RCM, they found slightly

lower values over the UK in winter.

Déqué et al. (1998) used a variable resolution model with high resolution over Eu-

rope and course resolution over the rest of the globe. The model was driven by sea-surface

temperatures from the Hadley Centre HadCM2 model. They ran two 10-year time-slice ex-

periments: a control run and a doubled CO2 run. They found a 10% increase in winter

mean precipitation over Southern parts of the UK, although the north of England, Wales and

Scotland all show a smaller response.

Räisänen and Joelsson (2001) used a regional climate model to study the mean and ex-

tremes of precipitation. They used two different climate models (Hadley Centre HadCM2

and Max-Planck Institute ECHAM4) to force the regional model and ran two 10-year integra-

tions, one with increased greenhouse gas concentrations and the other a control run. They

found that annual mean precipitation over the UK increased by about 10% (a statistically

significant increase). They also looked at extreme precipitation and found an increase in the

maximum precipitation of around 15% over much of Europe, including the areas where the

mean annual precipitation had decreased although these changes in maximum precipitation


were not statistically significant.

Durman et al. (2001) compared the mean precipitation and probability of extreme events

in the Hadley Centre’s HadCM2 global model, a regional version of the same model and

observations. They aggregated the regional model results and the observations to grid boxes

as in the GCM and found that the GCM showed good agreement with the observations,

although the RCM overestimated the grid box mean. They then used a greenhouse gas

forced simulation to study the effect of CO2 on precipitation against an unperturbed control

run. They found that the average European winter precipitation increased by 10-20% and that

the average European summer precipitation had decreased by about 5%. They also compared

the 99th percentile of the control and perturbed runs to examine changes in the probability

of extreme precipitation events. For the GCM, the probability of an extreme event increased

slightly in summer and more than doubled in winter.

Giorgi et al. (2004) used an RCM driven by the Hadley Centre HadAM3H high reso-

lution climate model. They studied the affect of increased CO2 under the A2 and B2 SRES

scenarios and compared the period 1961–1990 with 2071–2100. They found a statistically

significant increase in winter mean precipitation of 0.5 mm/day for the rapid CO2 increase

scenario A2 (a result consistent with Jones et al., 1997) which is approximately equivalent to

an increase of 15 to 25 %. Slightly different results were obtained for the moderate increase

scenario B2, with smaller positive increases in winter mean precipitation.

Räisänen et al. (2004) used the Rossby Centre coupled regional climate model (RCAO)

driven by two different coupled atmosphere-ocean GCMs (Hadley Centre HadCM3H and

Max-Planck Institute ECHAM4/OPYC3) and used two different SRES CO2 scenarios (A2

and B2) to obtain four different climate change scenarios. They compared the period 1961–

1990 with 2071–2100 and found increases of around 20 % in winter mean precipitation over

the UK for the rapid CO2 increase A2 scenario. The results driven by the two models dif-

fered for the more moderate CO2 increase B2 scenario with the ECHAM4/OPYC3 model

giving an increase of 30 % in winter mean precipitation over the UK compared with about 0

% for the HadCM3H run. These differences show that there is still large variation between

estimates of future precipitation changes obtained using different driver models, although

most models seem to show around a 10-20 % increase in winter mean precipitation under

enhanced greenhouse gas conditions. It should also be noted that GCM studies of precip-


itation generally predict an increase in winter precipitation under climate change scenarios

along with an increase in extreme precipitation

1.3 Scientific questions

Previous studies have shown that there has been an increase in UK winter precipitation and a

decrease in UK summer precipitation (e.g. Jones and Conway, 1997), however these studies

have not unambiguously identified the physical causes for these changes. As already stated,

increases in winter precipitation are of greater interest than decreases, due to the substantial

impact of recent flood events on society. An advantage of studying winter precipitation is

that the dynamical mechanisms which lead to precipitation in winter are generally large-scale

(extra-tropical cyclones). This makes modelling of winter precipitation easier as the large-

scale relationships are easier to characterise than small-scale convective processes. The main

question of interest for this thesis is: Why has there been an increasing trend in winter UK

precipitation?

To answer this question requires a number of other questions to be addressed. Firstly,

what is the time-trend in historical UK winter precipitation? This has been addressed in

several previous studies (e.g. Wigley and Jones, 1987, Gregory et al., 1991, Jones and Con-

way, 1997, Osborn et al., 2000, Alexander and Jones, 2001). However, these studies either

failed to supply uncertainty estimates (i.e. standard error or confidence intervals) and those

studies that have considered the uncertainty of trends are based on models where the implicit

assumptions are violated. There is a need for a more detailed statistical modelling of trends

in precipitation.

To address the important question of why time trends in UK winter precipitation have

occured requires consideration of the relevant meteorological factors which affect extra-

tropical winter precipitation. This critical step is generally omitted from the statistical anal-

ysis of precipitation trends despite the fact that consideration of the relevant fundamental

meteorological processes can provide insight to guide statistical modelling. Meteorological

knowledge can be used to identify relevant factors for a statistical model, and can even define

the appropriate form of the model. The second scientific question is therefore: which factors

affect extra-tropical winter precipitation and what is the relationship between these factors


and precipitation?

Many of the previous studies of historical UK winter precipitation have used monthly

and seasonal means. Long historical observational records do not generally exist for smaller

than daily resolution, hence daily represents the smallest time-scale available for analysis.

Therefore, a key question is: how can daily precipitation be accurately modelled statistically?

Once such a model is identified and tested, then this model can be used to address the main

question of why there is an increasing trend in winter England and Wales precipitation.

1.4 Plan of thesis

Chapter 2 of this thesis uses quasi-geostrophic theory identify the key factors affecting extra-

tropical large-scale precipitation in winter. These factors are then used later as the explana-

tory variables in the statistical model.

Chapter 3 introduces the well-known England and Wales Precipitation dataset. Win-

ter means of this time series from 1766–2003 are used to reproduce and extend previously

published analyses of long-term linear time trends in England and Wales rainfall. The pub-

lished trend estimates are revised to include estimates of uncertainty and the possibility of

non-linear trends is assessed. Inhomogeneities arising from different calculation methods

are discussed and tested within a regression framework. Finally, the relationship between

mean precipitation and Sea-Level Pressure is explored using linear regression. Sea-level

pressure helps explain interannual variability and so leads to a more accurate estimate of the

time-trend. Sea-level pressure, however, fails to explain the long-term increasing time trend.

Although seasonal mean precipitation is often approximately Normally distributed, daily

precipitation cannot be described by a single simple probability distribution. Chapter 4

presents a mixture distribution first proposed by Coe and Stern (1982). This technique

involves modelling the probability of the occurrence of precipitation with a Bernoulli (or

Binary) distribution and separately modelling wet day precipitation amount with a Gamma

distribution. The Bernoulli and Gamma distributions can then be made to depend on explana-

tory factors using an extended regression approach known as Generalized Linear Modelling

(GLM) (Nelder and Wedderburn, 1972). The GLM is introduced in Chapter 4 along with

methods for hypothesis testing for model parameters, and residual diagnostics for both com-


ponents of the model (Bernoulli and Gamma).

The mixture GLM is used in Chapter 5 to investigate the effect of sea-level pressure and

saturation specific humidity on daily winter England and Wales precipitation. The statistical

significance of the two explanatory factors is summarised and the effect of the time-trend

assessed. In particular, the residuals of the fitted models are carefully examined to explore

the validity of the model assumptions. Problems arise with serial correlation in the residuals

from the daily model, so pentad (5-day mean) winter England and Wales precipitation is also

investigated. Finally changes in the overall distribution of daily precipitation are considered

and projected forward to the end of the 21st century.

Chapter 6 then applies the mixture GLM to precipitation at each gridpoint of Oct-Mar

winter pentad Global Precipitation Climatology Precipitation (GPCP). Sea-level pressure

and saturation specific humidity are used as explanatory factors. Parameter estimates are

compared with those previously obtained for DJF England and Wales precipitation to vali-

date the global model over the UK. The North Atlantic Oscillation is also used as a proxy for

the meridional temperature gradient that was identified as an important factor in Chapter 2.

Finally conclusions are presented in Chapter 7, along with suggestions for possible fur-

ther work. A summary of time trends in UK winter precipitation is also given, based on the

work carried out in this thesis.

Chapter 2

Dynamical Processes Determining

Extra-Tropical Precipitation

2.1 Aims

To answer the important questions regarding observed trends in precipitation, it is important

to understand the underlying meteorological processes. Trenberth (1999) presented general

arguments for why increased precipitation should be related to climate change. He proposed

that increased concentrations of greenhouse gases lead to increased downwelling infrared

radiation, which increase evaporation at the surface, thus increasing atmospheric moisture

content. He stated that the consequence of this will be an increase in the number of heavy

rainfall events at the expense of more moderate events. Arguments such as this are crucial to

understanding precipitation variability and trends since they inform the statistical model.

The main aim of this chapter is to identify the large-scale factors important for pre-

cipitation on large scales in the extra-tropics and to link these to possible factors, such as

increasing humidity, and the meridional tropospheric temperature gradient that are likely to

change due to global warming.

9

Chapter 2. Dynamical Processes Determining Extra-Tropical Precipitation 10

2.2 Saturation of rising air

As an air parcel rises, it expands and cools and the environmental saturation vapour pressure

decreases. Hence, an air parcel rising adiabatically will eventually become saturated. If the

parcel continues to rise further, its saturation vapour pressure, es, will decrease and the water

vapour contained within it will start to condense out as precipitation. The lifting condensa-

tion level is the point to which a parcel of air has to be raised (adiabatically) for saturation

to occur (Rogers and Yau, 1989). Once the parcel has reached this point, if it is unstable it

will continue to rise (pseudo-adiabatically) releasing latent heat due to condensation of water

vapour, that generally falls as precipitation.

2.2.1 The Clausius-Clapeyron equation

Specific humidity q is defined as

q = εe

p(2.1)

where ε = 0.6213 is the ratio of the molar masses of dry air and water vapour, and p is

pressure and e is vapour pressure. Saturation specific humidity qs can be easily obtained by

replacing e with the saturation vapour pressure es. The saturation vapour pressure is related

to temperature T by the Clausius-Clapeyron equation

desdT

=L

T (α2 − α1)(2.2)

where L is the latent heat of vapourization, and α1 and α2 are the volume occupied by liquid

before and after evaporation, respectively. Since α1 � α2, and assuming that water vapourbehaves as an ideal gas

es ≈ es0exp[

L

Rv

(1

T0− 1

T

)]

(2.3)

where es0 = es(T0), T0 is temperature at sea-level and Rv = 461JK−1Kg−1 is the specific

gas constant for water vapour. The saturation vapour pressure has an exponential dependence

on temperature. Therefore, saturation specific humidity is a function of temperature. Figure


2.1 shows the relationship between saturation specific humidity qs and temperature obtained

from combining (2.1) and (2.3).

Assuming a mean winter temperature of approximately 10◦C (283K), and using (2.3),

a 1◦C increase would yield an increase in saturation specific humidity of approximately

0.5 g/kg (a rise from 9 g/kg to 9.5 g/kg).

qs, g

/Kg

010

2030

40

Temperature, K250 260 270 280 290 300 310

Figure 2.1: Temperature (K) against saturation specific humidity qs (g/Kg).

2.2.2 Humidity changes due to climate change

Trenberth (1999) presents the sequence of processes that could lead to increased rainfall

from increased greenhouse gases. The increased radiative forcing leads to an increase in

surface heating, which in turn will lead to greater evaporation, assuming there is a plentiful

supply of moisture at the surface (e.g. over oceans). Warming will also lead to greater

moisture capacity in the atmosphere, since the saturation specific humidity is an increasing

function of temperature (Section 2.2.1). The extra capacity for water vapour coupled with the

extra evaporation could lead to increased atmospheric moisture content, and hence increased

precipitation rates, assuming that vertical motion does not change.

Ross and Elliott (1996) used radiosonde data for 1973-93, and found increases in the

tropospheric specific humidity, with rates of change as high as 3-7% per decade, south of

∼ 45◦N . Other authors have found evidence of changes in tropospheric moisture, but theattempts to quantify these changes are hampered by por spatial/temporal coverage and ho-

mogeneity problems (Gaffen et al., 1991), however these articles do support the notion of an

increase in tropospheric humidity. Another drawback to these studies is that there has been


very little work on humidity change in the Atlantic and European sector, which is the prime

area of interest in this study.

Allen and Ingram (2002) show results from a number of coupled atmosphere ocean cli-

mate models run with double current CO2 levels which indicate that the Clausius-Clapeyron

relationship may provide an overestimate of the magnitude of the relationship between tem-

perature and atmospheric moisture. They estimate a 3.4% increase in rainfall for each 1K

increase in temperature, which is lower than the 6.5% increase implied by the Clausius-

Clapeyron equation (assuming that the change in qs is exactly balanced by precipitation).

The reasons for this are unclear, although it does suggest that saturated humidity alone does

not explain the trend in precipitation.

2.2.3 The relationship between precipitation and upward motion

The balance equation for water vapour is

dq

dt= s(q) + D (2.4)

where q is specific humidity, t is time, s(q) represents source and sink terms for atmospheric

water vapour and D represents the effect of molecular and turbulent diffusion (Peixoto andOort, 1992). The diffusion term also includes vertical fluxes such as evaporation from the

land surface, which is generally small in the extra-tropics in winter. The main sources and

sinks for water vapour in the atmosphere are evaporation (e) and condensation (c), hence s(q)

can be rewritten as (e − c). The budget of water vapour in the atmosphere can be obtainedby combining (2.4) and the continuity equation

∂q

∂t+ ∇.(qv) + ∂

∂p(qω) = (e − c) + D (2.5)

where v is the horizontal component of vector velocity, p is pressure and ω is vertical velocity

in pressure coordinates.

Assuming the air above the condensation level is saturated, neglecting frictional effects

and integrating this between the condensation level (pc) and the top of the troposphere (pT )

with respect to pressure gives


∫ pT

pc

∂qs∂t

dp +

∫ pT

pc

∇.(qsv)dp +∫ pT

pc

∂(qsω)

∂pdp ≈

∫ pT

pc

(e − c)dp (2.6)

Taking a time average then gives

∫∂∂t

qsdp +∫∇.(qsv)dp + [qsω]ptoppcond ≈

∫(e − c)dp

A B C D

Term A is the time mean of a time derivative and is negligible in comparison to the other

terms, and term B is the time mean humidity horizontal divergence, which is generally small

in comparison to term C over a suitably large area. As humidity is vertically advected, it

will condense and ultimately fall as precipitation. This is the dominant term on the left hand

side. Term C is the mean vertical advection of specific humidity. Term D is the time mean

of vertical mean evaporation minus precipitation. Since evaporation is very small above the

condensation level, the right hand side is approximately the rate of precipitation. Assuming

that the vertical velocity at the top of the troposphere is zero (rigid lid), this then gives:

P ≈∫ pT

pc

cdp ≈ qsω (2.7)

where P is the precipitation rate in hPa s−1 and qs and ω are at the condensation level.

Multiplying (2.7) by the density of air, ρa, and dividing by the density of water, ρw, gives an

estimate of rainfall in ms−1:

P ≈ ρa qswρw

(2.8)

Using this expression, a rough estimate of the precipitation rate during the passage

of an extra tropical cyclone can be obtained. For typical values of ρw ∼ 103kg m−3,ρa ∼ 1kg m−3, qs ∼ 10−2kg/kg, w ∼ 10−2m s−1 for an extra tropical cyclone, gives aprecipitation rate of P ∼ 10−7 ms−1, equivalent to values of 10 mm day−1 typically ob-served.


2.3 Vertical motion in the extra-tropics

There are a number of mechanisms that lead to vertical motion in the atmosphere: surface

heating leading to convection, flow over orography and the uplift along weather fronts associ-

ated with cyclonic conditions. These mechanisms can all lead to saturation and the formation

of cloud (as explained in section 2.2), which can then precipitate.

Much of the vertical motion in winter in the extra-tropics is due to extra-tropical cy-

clones caused by large scale atmospheric instability in the form of unstable Rossby waves.

These large scale wave instabilities are generally associated with zonally asymmetric per-

turbations on the zonal mean flow. Section 2.3.1 deals with the general pattern of the mean

zonal flow, whilst section 2.3.2 gives a brief outline of the Eady model for baroclinic in-

stability, which is the main mechanism for the formation and enhancement of extra-tropical

cyclones.

2.3.1 Mean flow and the subtropical jets

Extra-tropical winter time flow is dominated by large amplitude, synoptic scale transient

disturbances that develop on the background flow. Figure 2.2 shows the mean 500 hpa

geopotential height for winter 2002/3. This is a typical example of how the seasonal mean

pattern appears smooth, although it is actually the average of many transient disturbances,

all moving approximately horizontally, and generally westerly in mid-latitudes.

Figure 2.3a shows the time-mean, zonal-mean zonal wind velocity maxima that occur

near the tropopause in the the extra-tropics. The subtropical jet is not equally as strong at

all latitudes, with variations in strength due to the uneven distribution of land and sea in the

Northern Hemisphere. The Northern Hemisphere subtropical jet is strongest downstream of

the Rocky mountains and the Tibetan Plateau, over the Atlantic and Pacific Oceans, respec-

tively.

The geostrophic wind is related to the geopotential height, Φ, by:

vg =1

fk×∇Φ (2.9)

where f is the Coriolis parameter. Assuming hydrostatic balance, this equation can be used


520

520

520

520

520

540

540

540

540

540

540

560

560

560

560

560

560

560

580

580

580

580580

580

580

580

Figure 2.2: Plot of mean 500 hpa geopotential height for winter (DJF) 2002/3 in decametres.Data from NCEP/NCAR reanalysis.

to obtain the thermal wind equation:

∂vg∂p

= − Rfp

k ×∇T (2.10)

where R is the universal gas constant (R = 287JK−1Kg−1). This shows that changes in the

vertical profile of the geostrophic wind are balanced by horizontal gradients in temperature.

This is clearly evident in Figure 2.3, where the zonal wind is strongest where the meridional

temperature gradients are largest.

2.3.2 Transients and baroclinic instability

It is generally accepted that the principal source of transient eddies in the the extra-tropics

is baroclinic instability. The theory of baroclinic instability was substantially developed

by Charney (1947) and Eady (1949); a review of these theories can be found in numerous

textbooks such as James (1994). Baroclinic instabilities grow on the mean background flow

with vertical wind shear.


Figure 2.3: Plot of time-mean zonal-mean zonal wind, [u] (upper panel, m s−1) and time-mean zonal-mean temperature, [T ], (lower panel, K) both plotted on pressure height (y-axis)and latitude (x-axis) for DJF (Source: Holton (1992)).

Figure 2.4 shows the structure of the most unstable Eady wave. The pressure field tilts

westwards with height. The vertical velocity is greatest ahead of the area of low geopo-

tential, and smallest ahead of the area of low geopotential. There is a correlation between

the pressure field and the temperature field that causes a net poleward temperature transport.

There is also a correlation between vertical motion and temperature, that leads to net vertical

temperature flux.


Figure 2.4: Structure of the most unstable Eady wave: (a) Contours of perturbation geopoten-tial height (H and L represent high and low pressure zones), (b) Contours of vertical velocity(arrows indicate direction of vertical velocity) and (c) Contours of perturbation temperature(W and C indicate warm and cold air masses). All plots are against height (y-axis) and thezonal direction (x-axis) (Source: Holton (1992)).

2.3.3 The omega equation

Quasi-Geostrophic (QG) theory can be used in the extra-tropics to diagnose the vertical

velocity, ω = dpdt

, in terms of the geostrophic wind. This diagnostic equation is known as the

omega equation (Hoskins et al., 1978, Durran and Snellman, 1987).

The omega equation can be obtained by eliminating time derivatives from the thermo-

dynamic energy equation and the vorticity equation, and by ignoring diabatic heating and

friction. Following Holton (1992), the quasi-geostrophic omega equation is given by the

elliptic differential equation:


[

σ∇2 + f 20

∂2

∂p2

]

ω = f0∂∂p

[

vg.∇(

1f0∇2Φ + f

)]

+∇2[

vg.∇(

−∂Φ∂p

)]

A B C

(2.11)

where f0 is the constant component of the coriolis parameter and the static stability parameter

σ is given by

σ ≡ −RT0p

d ln θ0dp

where θ0 is the potential temperature corresponding to the basic state temperature in

the mid-troposphere, T0. Term A attempts to balance the sum of terms B and C. Term B

represents the horizontal advection of vorticity. This term is associated with ascent above the

surface low and subsidence above the surface high. Low pressure deviations, for example,

are associated with positive vorticity, which implies a falling geopotential and a decrease in

the thickness (temperature), which is balanced by the vertical motion. Term C represents

thermal advection. If there is warm (cold) advection, term C will be positive (negative).

There is cold advection to the east of the surface low, which leads to ascent in the same

region.

2.3.4 The Q-Vector

Terms B and C of the omega equation are often of similar magnitude, and opposite sign

and often cancel each other out (Hoskins et al., 1978, Trenberth, 1978). Hence, a method

is needed to combine these two terms without cancellation. Trenberth (1978) showed that

the right hand side of the omega equation could be written as the advection of vorticity by

the thermal wind plus terms involving the deformation of the wind field. This approach

provided an easy method for calculating ω from existing charts of vg and T . Hoskins et al.

(1978) introduced the Q-vector approach, where the Q-vector is equal to the rate of change of

the horizontal potential temperature gradient, which would develop in a fluid parcel moving

with the geostrophic wind if the vertical velocity was exactly zero Durran and Snellman

(1987).


The omega equation can be written in Q-vector form as:

[

σ∇2h + f 20∂2

∂p2

]

ω = −2∇. Q (2.12)

where Q represents the Q-vector and

Q =

[∂vg∂x

.∇(

∂Φ

∂p

)

,∂vg∂y

.∇(

∂Φ

∂p

)]

(2.13)

When Q is convergent, upward motion is expected and when it is divergent, downward

motion is expected (Hoskins and Pedder, 1980). This can be seen in Figure 2.5, which shows

a schematic of the Q-vectors for an idealised pattern of wave train pressure anomalies.

Figure 2.5: Q-vectors (bold arrows) for idealised pattern of isobars (solid lines) withisotherms (dashed lines) (Source: Sanders and Hoskins (1990)).

Following Sanders and Hoskins (1990), (2.13) can be rewritten by differentiating (2.9)

and using hydrostatic balance:

∂Φ

∂p= −RT

P(2.14)

Taking the gradient of (2.14) and assuming ∂T∂x

� ∂T∂y

, gives:

∇(

∂Φ

∂p

)

= −RP∇T ≈ −R

P

∂T

∂y~̂y (2.15)

and so

Q = −RP

∂T

∂y∇vg (2.16)


where vg is the meridional geostrophic wind component. Rewriting in terms of time mean

and transient terms gives

Q = −RP

∂T

∂y∇v − R

P

∂T

∂y∇v′ − R

P

∂T ′

∂y∇v − R

P

∂T ′

∂y∇v′ (2.17)

For the justifiable assumptions that v � v′ and ∂T ′∂y

� ∂T∂y

this then leads to

Q ≈ −RP

∂T

∂y∇v′ (2.18)

and so, using v′g =1fρ

∂p′

∂x,

ω ≈ L−1[

2R

fρp

∂T

∂y∇2∂p

′

∂x

]

(2.19)

where the elliptic Laplacian-like operator is defined as

L =[

σ∇2h + f 20∂2

∂p2

]

(2.19) shows that ω is a function of the time mean meridional temperature gradient, ∂T∂y

,

the transient pressure anomaly, p′ and the static stability σ. Hence, a stronger meridional

temperature gradient will increase vertical velocities for a given storm pressure p′. Static sta-

bility is contained within the L operator, and hence has an inverse effect on vertical motion.A reduction in static stability will allow for increased vertical velocities.

2.3.5 Problems with the QG omega equation

In the quasi-geostrophic omega equation, vertical motion is caused by geostrophic vorticity

and temperature advection (Hoskins et al., 1978). The dry quasi-geostrophic equations ne-

glect certain terms contained in the primitive equations, such as enhanced vertical motion

due to diabatic processes (the release of latent heat) and frictional effects.

Krishnamurti (1968) analysed vertical motion produced by a 5-level general balance

model, which includes some of the non-geostrophic effects. Krishnamurti (1968) found, by

partitioning the various terms contributing to the vertical velocities, that latent heat increases

the rising motion and that the deformation of divergence effects cut down the intensity of


sinking motion in the more complete balance model.

Pauley and Nieman (1992) studied the large-scale departures from quasi-geostrophic

vertical motions for a model simulation of a particular storm using a mesoscale model. Ver-

tical motion was computed using both the QG omega equation and a hydrostatic generalized

omega equation in order to assess the importance of non-quasi-geostrophic effects. It was

found that the forcing due to diabatic processes yielded the greatest departure from QG the-

ory in regions of ascent with a negative effect as large as 4pa s−1 (but the mean value is closer

to 0.5pa s−1 difference between the QG model and the full model), and ageostrophic advec-

tion caused the largest departures in the regions of descent, which supported the findings

of Krishnamurti (1968). However, Pauley and Nieman (1992) chose a particularly intense

simulated storm, and hence the results may not be applicable to more typical systems. In

addition, their results are based on only one specific case study.

Räisänen (1995) also pointed out that these case studies focus on single storms that are

stronger than usually observed. Räisänen (1995) used a similar generalized omega equa-

tion to Pauley and Nieman (1992), and compared this to the QG omega equation. For

mid-latitudes, a correlation of 0.7 was found between the ω estimated using the QG and

generalized omega equations. Räisänen (1995) concluded that the dominant cause of verti-

cal motion in the mid-latitudes are vorticity and thermal advection, but that the effect from

diabatic heating is far from negligible.

These studies suggest that the quasi-geostrophic omega equation provides a good first

order approximation for estimating synoptic scale vertical velocities, but that a more detailed

treatment that includes diabatic effects may be required for smaller scales.

2.3.6 Scale analysis

Section 2.2 showed that saturation specific humidity is related to temperature via the Clausius

Clapeyron equation. Section 2.3.3 showed that vertical velocity can be diagnosed using the

omega equation. This section will perform a simple scale analysis to check whether these

relationships are reasonable. Table 2.1 gives some typical values for an extra-tropical cyclone

in mid-latitudes. Substituting some of these values into (2.19) gives


ω ≈ L−1 102

5 × 1045

106

(1

106

)2

10

with L ∼ 10−18Pa−2s−2, and hence ω ∼ 0.1Pas−1 equivalent to w ∼ 10−2ms−1

Parameter Value Units DescriptionU 10 m s−1 Horizontal velocityw 0.05 m s−1 Vertical velocityω 0.5 Pa s−1 Vertical velocity in pressure coordinatesL 106 m Horizontal length scaleH 104 m Vertical length scaleρa 1 kg m

−3 Density of dry airρw 10

3 kg m−3 Density of waterqs 10

−2 kg kg−1 Saturation specific humidityp′ 5 × 10−4 Pa Typical pressure anomaly∂T∂y

5 × 10−6 Km−1 Mean meridional temperature gradientf0 10

−4 s−1 Coriolis parameterσ 2 × 10−6 m2Pa−2s−2 Static stability parameterP 10−5 m s−1 Precipitation rate

Table 2.1: Synoptic values typical for extra-tropical cyclones.

Using (2.8) which relates the precipitation rate to the humidity and the vertical veloci-

ties, and taking qs = 10−2 kg/kg gives

P ≈ ρaρw

wqs ≈ 10−5w

Substituting the value of w from above then gives a typical precipitation rate of about

10 mm day−1. This is in good agreement with observed precipitation rates in extra-tropical

cyclones and justifies the use of our simplified equations.

2.4 Summary

Combining (2.8), (2.19) and writing qs as a function of T gives:

P ≈ 2Rpρwf

qs(T )∂T

∂y

[

σ∇2 + f 2 ∂2

∂p2

]−1∇2 ∂

∂xp′ (2.20)

From this equation it can be seen that the key factors affecting extra-tropical precipitation

rates are:


1. Saturation specific humidity, qs, which depends strongly on lower tropospheric tem-

perature, T .

2. The zonal transient pressure gradient, ∂p′

∂x.

3. The time-mean meridional temperature gradient, ∂T∂y

.

4. Static stability σ

The precipitation rate is dependent on the saturation specific humidity (qs) and the mag-

nitude of vertical velocity in an extra-tropical cyclone (w′). The Clausius-Clapeyron equation

dictates that qs is exponentially related to inverse temperature and hence an increase in mean

temperature will lead to increases in saturation specific humidity. This will generally lead to

greater moisture content in the atmosphere, and increased precipitation rates in the absence

of any other changes.

Equation (2.20) shows that vertical motion is a function of the time-mean meridional

temperature gradient, the zonal transient pressure gradient and the static stability. Atmo-

spheric pressure is known to be an important factor for vertical motion, with ascent around

the surface low. As a result of this, many studies have used atmospheric pressure to explain

variability in precipitation. Eqn. (2.20) shows that the main area of ascent is ahead of the

surface low where the pressure tendency is decreasing as shown in Fig. 2.4.

In contrast to the many studies which have used atmospheric pressure to explain pre-

cipitation variations, the meridional temperature gradient has received scant attention. This

analysis demonstrates that this is an important factor for vertical motion in the extra-tropics.

The quasi-geostrophic omega equation provides a good approximation for diagnosing

vertical velocities in the extra-tropics. It represents the synoptic scale features but not the

finer details such as mesoscale features. Räisänen (1995) concluded that the most impor-

tant omission from the quasi-geostrophic omega equation is the diabatic heating feedback

caused by the release of latent heat as the water condenses, which serves to locally warm the

atmosphere.

Chapter 3

Variability and Trends in Winter Mean

England and Wales Precipitation

3.1 Aims

Much of the published work on identifying and estimating trends in England and Wales

precipitation has focussed on trends in the seasonal winter mean. This is perhaps the most

obvious approach for detection of long-term climate change because such changes should be

most evident in the seasonal mean, due to the sub-seasonal variations being averaged out of

the data, leaving only year-to-year changes. The other major advantage of a seasonal mean

analysis is the long record which is not available below the monthly level.

The first aim of this chapter is to summarise and supplement previous studies that esti-

mated the long-term trends in seasonal mean precipitation. Several studies have attempted

to use variables related to circulation (such as geopotential height at various levels, or flow

indices) to explain trends and variability in precipitation. Whilst this approach can partly ex-

plain short-term variations in precipitation (e.g. Conway et al., 1996, Wibig, 1999, Osborn

and Jones, 2000), it has not been demonstrated to explain the long-term trend in England and

Wales precipitation. The second aim of this chapter is to summarise the relationship between

winter mean sea-level pressure and winter mean precipitation, and to demonstrate that this

relationship can be used to reduce the year-to-year variation in precipitation, thus making

long-term trends easier to detect. This is done by fitting statistical models and assessing the

24

Chapter 3. Variability and Trends in Winter Mean England and Wales Precipitation 25

goodness of fit using residual diagnostics.

In Section 3.2.5 the area averaged England and Wales Precipitation (EWP) dataset is

introduced. After the calculation method has been summarised, some of the problems with

the analysis are discussed and then the annual cycle is briefly summarised. Long-term linear

time trends in winter mean EWP and their statistical significance are calculated in Section

3.3.7 and the assumptions of the statistical model are challenged. Changes in the annual

cycle and the variance of the season over time are also explored. Winter mean sea-level

pressure is included as an explanatory factor in the model in Section 3.4.5. The statistical

significance of this relationship and other model diagnostics are presented. Finally, residual

linear time trends in winter mean EWP from this model are carefully considered.

3.2 England and Wales Precipitation

This section introduces the main precipitation dataset for England and Wales used in this

thesis. A short history of UK precipitation data is given, followed by a summary of some

potential problems with the data.

3.2.1 The England and Wales Precipitation dataset

Burton (1993) described the motivation for the collection of precipitation records in Britain

that was initiated in 1859 by the Royal Meteorological Society president G. J. Symons.

Symons founded the British Rainfall Organization, a network of volunteer precipitation ob-

servers to record and publish precipitation data in the annual volume British Rainfall, first

published in 1860 and last published in 1968, when electronic storage methods became avail-

able. These early records contained monthly total precipitation from UK stations (the 1900

issue, for example, contained monthly records from 231 stations, and this number increased

throughout the lifetime of the publication).

Glasspoole (1952) described the motivation for the synthesis of precipitation records in

Britain “because of the exceptional run of dry years, 1854 to 1858, at a time when the urban

population was increasing and arrangements had to be made for piped supplies from storage

reservoirs”. The British Rainfall data formed the basis of the spatially averaged, monthly


mean version of the EWP time series compiled by Nicholas and Glasspoole (1931), in con-

nection with the economic and statistical investigation of the History of English Prices by Sir

William Beveridge, who “wished to obtain general values of rainfall over England and Wales

for each month for as long a period as possible” (Nicholas and Glasspoole, 1931). Precipi-

tation is an important factor on the quality and yield of wheat, hence precipitation is related

to the price of wheat, and was a factor of interest in the History of English Prices. Nicholas

and Glasspoole (1931) compiled the measurements collected by the the British Rainfall Or-

ganization and annual values published in the Meteorological Magazine (Glasspoole, 1928)

as well as other, unpublished, records.

3.2.2 Methods of calculation of the monthly mean EWP time series

There have been several different calculation methods employed for the monthly mean EWP

time series, due to changes in the observing network and, more recently, the invention of

the computer. Precipitation data for Wales was not available before 1870, and so there was

a major change in the calculation method used prior to 1870. The series was updated by

Wigley et al. (1984), with another substantial change in the calculation, although the data

calculated by Wigley et al. (1984) were reported to be in good agreement with those from

Nicholas and Glasspoole (1931). Since 1997, the EWP time series has been updated using

the recent method of Alexander and Jones (2001).

Let yijk denote mean precipitation for month i, year j and station k and denote the mean

over one of the subscripts by a dot (i.e. yij• is the mean monthly precipitation for each year

averaged over all stations).

The first calculation method of Nicholas and Glasspoole (1931) was used between 1727-

1869, and was necessary due to the lack of available data for Wales. Monthly station values

were expressed as a fraction of annual totals at each station, and the mean over all stations

obtained. This quantity was then expressed relative to the England only monthly-to-annual

fraction for the reference period, 1881-1915:

pij =

(

1

m

m∑

k=1

yijky•jk

)

/y

(E)i•

y(E)••


where y(E)j and y(E) are the monthly and annual area-averaged precipitation for England only

for 1881-1915 calculated by Glasspoole (1924) from isohyetal (lines of constant precipita-

tion) maps, m is the number of stations and pij is the monthly proportion of precipitation

in England compared to the reference period. This was then scaled to be consistent with

the monthly totals obtained by Glasspoole (1928). Nicholas and Glasspoole (1931) quoted

the monthly England only precipitation as a proportion of the England only reference pe-

riod, so the actual value of precipitation for each month between 1776 and 1869 was ob-

tained as p∗ijy(EW )j , where p

∗ij is pij standardized to agree with the monthly totals obtained by

Glasspoole (1928), and y(EW )j is the monthly mean England and Wales precipitation for the

reference period 1881-1915. This assumes that the England only precipitation totals are the

same fraction of the reference period values as the England and Wales precipitation totals.

The second calculation method of Nicholas and Glasspoole (1931) was used between

1870 and 1930, once data for Wales became available, and is described by Glasspoole (1928).

For each station, the monthly precipitation total was expressed as a fraction of the monthly

mean for that station over the reference period (1881-1915):

pij =1

m

m∑

k=1

yijkyi•k

where yi•k is the monthly mean precipitation for the reference period 1885-1915 at station k,

m is the number of stations and pij is the area-averaged England and Wales precipitation for

month i in year j expressed as a fraction of the reference period. Nicholas and Glasspoole

(1931) quoted the monthly England and Wales precipitation as a proportion of the England

and Wales reference period, so the actual value of precipitation for each month between 1870

and 1930 was obtained as pijy(EW )j . Annual totals were calculated in the same way, but not

necessarily using the same network. The monthly values were then scaled to be consistent

with the annual values, although for some years the sum of the monthly values may have

been used as the annual value (Wigley et al., 1984).

The method of Wigley et al. (1984) (with updates by Wigley and Jones (1987), Gregory

et al. (1991) and Jones and Conway (1997)) was used to calculate the monthly EWP time

series between 1931 and 1996. Wigley et al. (1984) used 7 stations from each of the 5 coher-

ent precipitation regions as defined by the principal component analysis of 55 precipitation


Figure 3.1: Map showing the 9 rainfall regions defined by Wigley et al. (1984), using thenaming convention of Gregory et al. (1991). Source: Osborn et al. (2000)

SEE South-East EnglandSWE South-West England and South WalesCE Central and East EnglandNEE North-East EnglandNWE North-West England and North Wales

Table 3.1: The five coherent precipitation sub-regions of England and Wales as defined byWigley et al. (1984) and Gregory et al. (1991).

station records from 1861–1970. These regions are shown in Table 3.1 and Figure 3.1. The

mean of the 7 stations was calculated for each of the sub-regions, and a weighted average of

these regions was then used to construct a series that was consistent with that compiled by

Nicholas and Glasspoole (1931). The weights were obtained using multiple linear regression

with the previously published monthly EWP time series as the response and the un-weighted

spatial means of the five regions as the explanatory factors.


Alexander and Jones (2001) introduced a new method of calculation for the EWP time

series in 1997 which allowed for automation of the calculation process for real time updating.

UK stations which send National Climatological Messages to the United Kingdom Meteo-

rological Office (UKMO) are used in the dataset. The daily value for each station is scaled

by the reference period ratio of the station’s mean monthly total and the sub-region’s mean

monthly total, using the sub-regions defined by Wigley et al. (1984). This correction ensures

that the data agree with those obtained using the technique of Wigley et al. (1984). The

monthly sub-region value is then obtained by taking the monthly mean of the scaled daily

station values for that sub-region. The monthly EWP value is then obtained as the weighted

mean of the 5 sub-regions, following the calculation method of Wigley et al. (1984). Values

calculated using the method of Alexander and Jones (2001) are reported to be consistent with

those obtained using the method of Wigley et al. (1984), and allow for the stations used in

the calculation to vary as the network changes, as long as the station has a mean monthly

total available for the reference period, 1961–1990. The data are quality controlled before

they are assimilated into the UKMO database, and finally, the monthly EWP value is checked

on the 5th of each month, with final quality control taking place up to several months later

(Alexander and Jones, 2001). The Alexander and Jones (2001) estimates are used from 1998.

Num

ber

of s

tatio

ns

Year

015

3045

60

1780 1800 1820 1840 1860

Figure 3.2: Plot of the number of stations used in the early calculations of the EWP timeseries, 1766-1869.

The number of records used over different periods varied greatly, particularly in the

original compilation of Nicholas and Glasspoole (1931). Figure 3.2 shows the number of

stations used by Nicholas and Glasspoole (1931) (note that Nicholas and Glasspoole (1931)

provided data from 1727, which does not appear in the current compilation of the EWP time

series, due to the small number of stations used during that time). From 1767, at least 5


stations were available for each year, with this total rising steadily to 58 by 1869. The exact

number of stations used after 1869 was not given by Nicholas and Glasspoole (1931), nor

were the station names, but the number of stations dropped to 26 between 1881 and 1915

(Wigley et al., 1984). From 1931 to 1996, 35 stations were used (7 in each of the 5 spatially

coherent sub-regions) and approximately 50 are used in the current automatic compilation,

depending on which stations report values in time (Alexander and Jones, 2001).

3.2.3 Spatial homogeneity of EWP

Wigley and Jones (1987) and Gregory et al. (1991) presented regional precipitation totals

to give some insight into the spatial inhomogeneities in England and Wales precipitation.

Wigley and Jones (1987) used the regions defined by the principal component analysis of

Wigley et al. (1984), and constructed series for geographical sub-regions of England and

Wales with each sub-region being the average of 7 stations (these stations were consistent

with those used by Wigley et al. (1984), although many changes to the network were needed

to update the series due to the unavailability of data). Gregory et al. (1991) updated these

series, added series for Scotland and Northern Ireland and presented seasonal and annual

means for each of the series. Figure 3.1 shows the 9 different regions. The updated analysis

by Gregory et al. (1991) (which includes Scotland and Northern Ireland) showed that the

south-west of England and south-west of Wales (SWE) were the wettest regions in England

and Wales over the period 1931 to 1989 (with an annual precipitation value of 2.83 mm/day).

The north-west of England and north-west of Wales (NWE) was the second wettest region

(2.76 mm/day). The central and eastern parts of England (CEE) were the driest of the regions

with annual precipitation of 1.77 mm/day. This summary shows that there is large east-west

variation in UK precipitation amounts. However, these differences are small when compared

with those observed in Scotland (north-western parts (NS) received around 4.33 mm/day

compared with eastern parts (ES) which experienced 2.12 mm/day).

3.2.4 Characteristics of seasonal EWP

EWP is the spatial average of several stations, which means that it will generally have re-

duced temporal variance to data from any individual station, as much of the single site vari-


0 5 10 15

050

010

0015

00

Daily DJF EWP

Cou

nt

a)

0 5 10 15

050

010

0015

00

Daily DJF precip, Reading, UK

Cou

nt

b)

0 1 2 3 4 5

02

46

810

12

Mean DJF EWP

Cou

nt

c)

0 1 2 3 4 50

24

68

1012

Mean DJF precip, Reading, UK

Cou

nt

d)

Figure 3.3: Histograms of (a) daily DJF EWP, (b) daily DJF precipitation at Reading, UK, (c)mean DJF EWP and (d) mean DJF precipitation at Reading, UK. All data is from 1971–2002and is in units of mm/day.

ability is averaged out. This means that the EWP time series is not representative of the

values observed at a single station. Figure 3.3a shows a histogram of the daily EWP val-

ues and Fig. 3.3b shows the distribution of daily precipitation at a single station at Reading

(UK), for winter (DJF) days from 1971 to 2002. The mean DJF EWP value over this period

is 2.92 mm/day, whilst the mean DJF precipitation at Reading was 1.85 mm/day – so precip-

itation at Reading is approximately 60% of that for the EWP series. The spatially averaged

daily EWP has more days with precipitation larger than 1 mm/day compared with the single

site data from Reading so that a wet day is more likely with EWP. It is also likely that the

daily EWP has fewer extreme precipitation events, which can be locally extreme rather than

homogeneous across the whole of the UK. The maximum value of daily EWP over 1971 to

2002 was 28.2 mm/day compared with a maximum value at Reading of 35.8 mm/day, which

is a large difference given that mean precipitation at Reading is, on average, 60% of that for


the EWP time series.

Figure 3.3a shows that the probability distribution of daily precipitation in England and

Wales is positively skewed. Temporal averaging makes the distribution of precipitation more

Normal, as revealed by the histogram of the DJF mean of the EWP data (Fig. 3.3c). This

enables methods based on the Normal distribution to be used to model the seasonal mean

of the EWP series (e.g. ordinary least squares linear regression). One drawback of using

temporally averaged data is that the storm-related meteorological processes governing winter

precipitation (identified in Chapter 2) act on daily and sub-daily time-scales so that these

processes are neglected.

3.2.5 Differences between seasons

Figure 3.4 shows boxplots for 30-year periods of each season together with linear trend

estimates obtained using ordinary least squares linear regression of seasonal means. The

mean annual precipitation (of 2.51 mm/day) over the period 1766 to 2003 is included as a

dotted horizontal line. The mean precipitation was 2.59 mm/day for winter, 2.00 mm/day for

spring, 2.47 mm/day for summer and 2.97 mm/day for autumn. Hence, winter and summer

have similar mean precipitation, spring has the least precipitation of the four seasons over

England and Wales, and autumn has the most. The linear trend estimates confirm that there

is an increasing trend in winter and a decreasing trend in summer with very little trend in

autumn and spring (as shown by Jones and Conway, 1997).

The different sign for the observed trend in winter and summer suggests that different

mechanisms might be responsible for the changes in winter and summer. Since the mecha-

nisms governing winter precipitation are related to large-scale atmospheric dynamics in the

extra-tropics (and therefore more compatible with the spatially-averaged EWP time series)

and because winter has a larger trend than summer, the rest of this thesis will focus on winter

precipitation.


01

23

45

••

1766

-1795

1796

-1825

1826

-1855

1856

-1885

1886

-1915

1916

-1945

1946

-1975

1976

-2002

Win

ter

EW

P, m

m/d

ay a)

01

23

45

•• •

1766

-1795

1796

-1825

1826

-1855

1856

-1885

1886

-1915

1916

-1945

1946

-1975

1976

-2003

Spr

ing

EW

P, m

m/d

ay b)

01

23

45

•

•

1766

-1795

1796

-1825

1826

-1855

1856

-1885

1886

-1915

1916

-1945

1946

-1975

1976

-2003

Sum

mer

EW

P, m

m/d

ay c)

01

23

45 •

•

•

1766

-1795

1796

-1825

1826

-1855

1856

-1885

1886

-1915

1916

-1945

1946

-1975

1976

-2003

Aut

umn

EW

P, m

m/d

ay d)

Figure 3.4: Boxplots of EWP split into approximate 30 year periods for (a) winter (DJF),(b) spring (MAM), (c) summer (JJA) and (d) autumn (SON). The bold straight line is a trendline obtained using ordinary least squares linear regression of seasonal means (followingJones and Conway, 1997). The dashed horizontal reference line denotes the mean annualprecipitation for the period (2.51 mm/day). For each boxplot the centre line represents themedian and the notched section represents an approximate 95% confidence interval for themedian. The top and bottom of the box represent the upper and lower quartiles. The whiskersextend to the furthest value that is not beyond 1.5 times the inter-quartile range (the differencebetween the upper and lower quartiles); any values beyond this are considered outliers andare individually represented by a solid circle. All values are mm/day over the period 1766 to2003.

3.3 A summary of trends in winter mean EWP

This section reviews the literature concerning trends in winter mean EWP. The possible non-

linearity of these trends is explored, and the length of series required to detect a statistically

significant trend is also considered. Finally, the possibility of the data having non-constant

variance over time is investigated.


3.3.1 Previous studies of trends in EWP

Since the EWP data series was revised by Wigley et al. (1984), there have been a number

of attempts to summarise long- and short-term trends in the data. Wigley and Jones (1987)

used a Gaussian filter1 to remove variability on time-scales of less than (approximately) 10

years for each season of EWP from 1931 to 1985. From the resulting smoothed time series

of precipitation they concluded that there was no long-term trend in the mean of precipitation

over the period. Wigley and Jones (1987) also modelled extremes for each of the seasons as

a hypergeometric distribution, and observed a tendency towards drier summers and wetter

springs. They speculated that the extreme dry summers and extreme wet springs would be

statistically significant, although no formal statistical test was carried out. On the other hand,

Gregory et al. (1991) found no strong evidence of drier summers and wetter springs although

they only used the shorter period of 1931 to 1989.

Jones and Conway (1997) investigated the mean and standard deviation of the monthly

mean EWP series between 1766 to 1995, and summarised the linear trends in the mean

rainfall for each three month season. They found a statistically significant trend (at the 5%

level of significance) in DJF winter means of EWP with an overall increase in the mean

equivalent to 0.33 mm/day per century over the period 1766 to 1995. They also reported a

decreasing trend in JJA summer mean EWP, a slight increase in SON autumn means and a

slight decrease in MAM spring means, although none of these were found to be statistically

significant at the 5% level. These trends were confirmed in the more recent study by Alexan-

der and Jones (2001), where the trend in summer precipitation was reported to be statistically

significant at the 5% level of significance.

3.3.2 Linear time trends

Jones and Conway (1997) used ordinary least squares (OLS) linear regression with year as

the explanatory variable to model long term trends in seasonal EWP totals. The model can

be written as:

1Gaussian smoothing weights, with the same length as the desired smoothing period and with unit sum, areconvolved with the raw data series to produce the smoothed series


yi = β0 + β1xi + εi i = 1, . . . , n; εi ∼ N(0, σ2) (3.1)

where yi is i’th observation of mean DJF EWP, xi is the year, n is the number of observations,

β0 and β1 are the intercept and the slope parameters respectively and εi is the i’th residual

from the linear fit, which are assumed to be independently Normally distributed with zero

mean and constant variance σ2.

Year

Win

ter

EW

P (

mm

/day

)

1800 1850 1900 1950 2000

12

34

95% Prediction interval95% Confidence interval

Figure 3.5: Winter mean EWP from 1766 to 2003. The solid line is the linear trend estimatedusing least-squares linear regression with year. The intercept for the solid line at year 2003is 2.95 mm/day and the slope is 0.30 mm/day per century (with a standard error of 0.065mm/day per century). Dashed lines represent the 95% confidence interval for the line, anddotted lines represent the 95% prediction interval for the points.

Figure 3.5 shows a time series plot of winter mean England and Wales Precipitation from

1776 to 2002, along with the fitted OLS linear regression with time, and the 95% confidence

and prediction intervals. The confidence interval is 95% certain to contain the true value of

the fitted line and the prediction interval is 95% certain to contain any single datum. Table

3.2 gives estimates of β0 (the EWP intercept at 2003) and β1 from the regression, along with

the standard error of β̂1 (and the corresponding p-value) as well as the trend expressed as a

percentage of the mean (taken as the intercept at 2003). This provides an update to the trend

estimates given by Jones and Conway (1997) based on data for the period 1766 to 2003 and

gives uncertainty estimates on the trend not provided by Jones and Conway (1997). Table 3.2

also gives the trend over the more recent period 1931-2001. Table 3.2 shows that different


Data Period β̂0 Trend β̂1 Std Err (P-value) % trendJones and Conway (1997) 1766-1995 0.33 (


-2 -1 0 1 2

020

4060

Residuals (mm/day)

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quen

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Documents

Trends and Variability in Observations of Winter ... and Variability in Observation… · precipitation cannot be attributed to human activity. Therefore, the cause of observed trends