Time Series Analysis in Meteorology and Climatology

Series foreword vii 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