Generalized Gaussian Error Calculus
For the first time in 200 years Generalized Gaussian Error Calculus addresses a rigorous, complete and self-consistent revision of the Gaussian error calculus. Since experimentalists realized that measurements in general are burdened by unknown syste
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Michael Grabe
Generalized Gaussian Error Calculus
With 47 Figures
123
Dr. rer. nat. Michael Grabe Am Hasselteich 5 38104 Braunschweig, Germany [email protected]
ISBN 978-3-642-03304-9 e-ISBN 978-3-642-03305-6 DOI 10.1007/978-3-642-03305-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009940174 c Springer-Verlag Berlin Heidelberg 2010 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, broadcasting, 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. Violations are liable to prosecution under the German Copyright Law. 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 protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Lucy, Niklas, Finley, and Rafael
Preface
The book of nature is written in the language of mathematics Galileo Galilei, 1623
Metrology strives to supervise the flow of the measurand’s true values through consecutive, arbitrarily interlocking series of measurements. To highlight this feature the term traceability has been coined. Traceability is said to be achieved, given the true values of each of the physical quantities entering and leaving the measurement are localized by specified measurement uncertainties. The classical Gaussian error calculus is known to be confined to the treatment of random errors. Hence, there is no distinction between the true value of a measurand on the one side and the expectation of the respective estimator on the other. This became apparent not until metrologists considered the effect of so-called unknown systematic errors. Unknown systematic errors are time-constant quantities unknown with respect to magnitude and sign. While random errors are treated via distribution densities, unknown systematic errors can only be assessed via intervals of estimated lengths. Unknown systematic errors were, in fact, addressed and discussed by Gauss himself. Gauss, however, argued that it were up to the experimenter to eliminate their causes and free the measured values from their influence. Unfortunately, this is not possible. Considering the present state of measurement technique, unknown systematic errors are of an order of magnitude comparable to that of random errors and this causes the Gaussian error calculus to break down. Consequently, the metrological community needs to consider how the error calculus to-be should address the coexistence of r
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