Univariate time series in geosciences : theory and examples PDF
Preview Univariate time series in geosciences : theory and examples
Hans Gilgen Univariate Time Series in Geosciences Theory and Examples Hans Gilgen Univariate Time Series in Geosciences Theory and Examples With 220 Figures AUTHOR: Dr. Hans Gilgen Institute for Atmospheric and Climate Science Swiss Federal Institute of Technology (ETH) Zurich Universitätsstr. 16 8092 Zurich Switzerland E-MAIL: [email protected] ISBN 10 3-540-23810-7 Springer Berlin Heidelberg New York ISBN 13 978-3-540-23810-2 Springer Berlin Heidelberg New York Library of Congress Control Number: 2005933719 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Cover design: E. Kirchner, Heidelberg Production: A. Oelschläger Typesetting: Camera-ready by the Author Printed on acid-free paper 30/2132/AO 543210 in memory of JoJo and MaiMai to Lu¨sai Preface La th´eorie des probabilit´es n’est au fond que le bon sens r´eduit au calcul. Probability theory is, basically, nothing but common sense reduced to calculation. Laplace, Essai Philosophique sur les Probabilit´es, 1814. In Geosciences, variables depending on space and time have been mea- sured for decades or even centuries. Temperature for example has been ob- served worldwide since approximately 1860 under international standards (those of the World Meteorological Organisation (WMO)). A much shorter instrumental (i.e., measured with instruments) record of the global back- ground concentration of atmospheric carbon dioxide is available for Hawaii (Mauna Loa Observatory) only dating back to 1958, owing to difficulties in- herent in the routine measurement of atmospheric carbon dioxide. Further examplesoflong-termrecordsarethoseobtainedfrommeasurementsofriver discharge. Incontrasttostandardisedroutinemeasurements,variablesarealsomea- sured in periods and regions confined in time and space. For example, (i) groundwaterpermeabilityinagraveldepositcanbeapproximatedfromgrain size distributions of a few probes taken (owing to limited financial resources for exploring the aquifer), (ii) solar radiation at the top of the atmosphere has been measured by NASA in the Earth Radiation Budget Experiment for the period from November 1984 through to February 1990 using instru- ments mounted on satellites (since the lifetime of radiation instruments in space is limited), (iii) three-dimensional velocities of a turbulent flow in the atmospheric boundary layer can be measured during an experiment (seeing thatameasurementcampaignistoocostlytomaintainfordecades),andlast butnotleast(iv),measurementshavebeenperformedunderoftenextremely adverse conditions on expeditions. Many variables analysed in Geosciences depend not only on space and time but also on chance. Depending on chance means that (i) all records observed are reconcilable with a probabilistic model and (ii) no determinis- tic model is available that better fits the observations or is better suited for practical applications, e.g., allows for better predictions. Deterministic mod- VIII Preface els have become more and more sophisticated with the increasing amount of computing power available, an example being the generations of climate models developed in the last two decades. Nevertheless, tests, diagnostics and predictions based on probabilistic models are applied with increasing frequency in Geosciences. For example, using probabilistic models (i) the North Atlantic Oscillation (NAO) index has been found to be stationary in its mean, i.e., its mean neither increases nor decreases systematically within the observational period, (ii) decadal changes in solar radiation incident at theEarth’ssurfacehavebeenestimatedformostregionswithlong-termsolar radiation records, (iii) Geostatistical methods for the optimal interpolation of spatial random functions, developed and applied by mining engineers for exploringandexploitingoredeposits,arenowusedwithincreasingfrequency inmanydisciplines,e.g.,inwaterresourcesmanagement,forestry,agriculture or meteorology, and (iv) turbulent flows in the atmospheric boundary layer are described statistically in most cases. If a variable depending on space and/or time is assumed to be in agree- ment with a probabilistic model then it is treated as a stochastic process or random function. Under this assumption, observations stem from a realisa- tion of a random function and are not independent, precisely because the variable being observed depends on space and time. Consequently, standard statistical methods can only be applied under precautions since they assume independent and identically distributed observations, the assumptions made in an introduction to Statistics. Often, geophysical observations of at least one variable are performed at a fixed location (a site or station) using a constant sampling interval, and time is recorded together with the measured values. A record thus obtained is a time series. A univariate time series is a record of observations of only one variable: a multivariate one of simultaneous observations of at least two variables.Univariatetimeseriesareanalysedinthisbookundertheassump- tion that they stem from discrete-time stochastic processes. The restriction to univariate series prevents this book from becoming too long. In contrast to the other examples given, the Mauna Loa atmospheric carbon dioxide record grows exponentially, a property often found in socio- economic data.Forexample,thepowerconsumed inthecityofZurich grows exponentially, as demonstrated in this book. SubsequenttointroducingtimeseriesandstochasticprocessesinChaps.1 and 2, probabilistic models for time series are estimated in the time domain. Anestimationofsuchmodelsisfeasibleonconditionthatthetimeseriesob- served are reconcilable withsuitable assumptions. Amongthese, stationarity playsaprominentrole.InChap.3,anon-constantexpectationfunctionofthe stochastic process underanalysis is capturedby means of estimatingalinear model using regression methods. Chap. 4 introduces the estimation of mod- els for the covariance function of a spatial random function using techniques developed in Geostatistics. These models are thereafter used to compute op-
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