Univariate time series in geosciences : theory and examples

738 Pages·2006·11.612 MB·English

by  Hans J Gilgen

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

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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-