Time Series Analysis in Meteorology and Climatology: An Introduction

Preface ix

1. Fourier analysis 1

1.1 Overview and terminology 2

1.2 Analysis and synthesis 6

1.3 Example data sets 14

1.4 Statistical properties of the periodogram 23

1.5 Further important topics in Fourier analysis 47

Appendix 1.A Subroutine foranx 83

Appendix 1.B Sum of complex exponentials 86

Appendix 1.C Distribution of harmonic variances 86

Appendix 1.D Derivation of Equation 1.42 92

Problems 93

References 99

2. Linear systems 101

2.1 Input–output relationships 102

2.2 Evaluation of the convolution integral 104

2.3 Fourier transforms for analog data 110

2.4 The delta function 113

2.5 Special input functions 118

2.6 The frequency response function 122

2.7 Fourier transform of the convolution integral 128

2.8 Linear systems in series 130

2.9 Ideal interpolation formula 132

Problems 137

References 142

3. Filtering data 143

3.1 Recursive and nonrecursive filtering 144

3.2 Commonly used digital nonrecursive filters 150

3.3 Filter design 159

3.4 Lanczos filtering 161

Appendix 3.A Convolution of two running mean filters 173

Appendix 3.B Derivation of Equation 3.20 176

Appendix 3.C Subroutine sigma 177

Problems 180

References 182

4. Autocorrelation 183

4.1 Definition and properties 184

4.2 Formulas for the acvf and acf 188

4.3 The acvf and acf for stationary digital processes 192

4.4 The acvf and acf for selected processes 195

4.5 Statistical formulas 201

4.6 Confidence limits for the population mean 206

4.7 Variance of the acvf and acf estimators 211

Appendix 4.A Generating a normal random variable 215

Problems 216

References 221

5. Lagged-product spectrum analysis 223

5.1 The variance density spectrum 223

5.2 Relationship between the variance density spectrum and the acvf 226

5.3 Spectra of random processes 230

5.4 Spectra of selected processes 232

5.5 Smoothing the spectrum 236

Appendix 5.A Proof of Equation 5.11 239

Appendix 5.B Proof of Equation 5.12 240

Problems 241

References 243

Index 245