Time Series Analysis in Meteorology and Climatology

Claude Edward Duchon, Professor Emeritus, School of Meteorology, University of OklahomaRobert C. Hale, Research Scientist, Cooperative Institute for Research in the Atmosphere, Colorado State University

• Series foreword vii Preface ix1. Fourier analysis 11.1 Overview and terminology 21.2 Analysis and synthesis 61.3 Example data sets 141.4 Statistical properties of the periodogram 231.5 Further important topics in Fourier analysis 47Appendix 1.A Subroutine foranx 83Appendix 1.B Sum of complex exponentials 86Appendix 1.C Distribution of harmonic variances 86Appendix 1.D Derivation of Equation 1.42 92Problems 93References 992. Linear systems 1012.1 Input–output relationships 1022.2 Evaluation of the convolution integral 1042.3 Fourier transforms for analog data 1102.4 The delta function 1132.5 Special input functions 1182.6 The frequency response function 1222.7 Fourier transform of the convolution integral 1282.8 Linear systems in series 1302.9 Ideal interpolation formula 132Problems 137References 1423. Filtering data 1433.1 Recursive and nonrecursive filtering 1443.2 Commonly used digital nonrecursive filters 1503.3 Filter design 1593.4 Lanczos filtering 161Appendix 3.A Convolution of two running mean filters 173Appendix 3.B Derivation of Equation 3.20 176Appendix 3.C Subroutine sigma 177Problems 180References 1824. Autocorrelation 1834.1 Definition and properties 1844.2 Formulas for the acvf and acf 1884.3 The acvf and acf for stationary digital processes 1924.4 The acvf and acf for selected processes 1954.5 Statistical formulas 2014.6 Confidence limits for the population mean 2064.7 Variance of the acvf and acf estimators 211Appendix 4.A Generating a normal random variable 215Problems 216References 2215. Lagged-product spectrum analysis 2235.1 The variance density spectrum 2235.2 Relationship between the variance density spectrum and the acvf 2265.3 Spectra of random processes 2305.4 Spectra of selected processes 2325.5 Smoothing the spectrum 236Appendix 5.A Proof of Equation 5.11 239Appendix 5.B Proof of Equation 5.12 240Problems 241References 243Index 245

“In summary, I unequivocally endorse this book as a valuable contribution to the literature of time series analysis in the geosciences. It is clear and includes examples that make it accessible for students; knowledgeable practitioners will also gain new insights from this book.”  (Bulletin of the American Meteorological Society,1 September 2012)