Guide to Hydrological Practices, 6th edition, Volume I - Hydrology.nl

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I.2-14 GUIDE TO HYDROLOGICAL PRACTICES Stage Confidence interval of the mean Stage discharge relation Confidence limit of the standard error of the mean S mr Confidence limit of the standard error of estimate S e Figure I.2.4. Explanation of errors in linear regression Discharge Error: The difference between the result of a measurement and the true value of the quantity measured. This term is also used for the difference between the result of a measurement and the best approximation of the true value, rather than the true value itself. The best approximation may be a mean of several or many measurements. Expected value: The best approximation of the true value, which may be a mean of several, or of many measurements. Hysteresis (instrument): That property of an instrument whereby it gives different measures, for the same actual value, according to whether that value has been reached by a continuously increasing or continuously decreasing change of the variable. Measurement: An action intended to assign a number as the value of a physical quantity in stated units. The result of a measurement is complete if it includes an estimate (necessarily in statistical terms) of the probable magnitude of the uncertainty. Normal distribution: A mathematically defined, symmetrical, bell-shaped, continuous distribution, traditionally assumed to represent random errors. Precision: The closeness of agreement between independent measurements of a single quantity obtained by applying a stated measurement procedure several times under prescribed conditions. Accuracy has to do with closeness to the truth, precision has to do only with closeness together. Precision of observation or of reading is the smallest unit of division on a scale of measurement to which a reading is possible either directly or by estimation. Random error: That part of the error that varies in an unpredictable manner, in magnitude and in sign, when measurements of the same variable are made under the same conditions (Figure I.2.3). Range: The interval between the minimum and maximum values of the quantity to be measured, for which the instrument has been constructed, adjusted or set. It can be expressed as a ratio of maximum and minimum measurable values. Reference measurement: A measurement utilizing the most advanced state of science and the latest technologies. The result of the reference measurement is used to estimate a best approximation to the true value. Repeatability: The closeness of agreement, when random errors are present, between measurements of the same value of a quantity obtained under the same conditions, that is, the same observer, the same instrument, the same location, and after intervals of time short enough for real differences to be insignificant. Reproducibility: The closeness of agreement between measurements of the same value of a quantity obtained under different conditions, for example, different observers, instruments, locations, and
CHAPTER 2. METHODS OF OBSERVATION I.2-15 after intervals of time long enough for erroneous differences to be insignificant. Resolution: The smallest change in a physical variable that causes a variation in the response of a measuring system. Sensitivity: The relationship of the change of the response to the corresponding change of the stimulus, or the value of the stimulus required to produce a response exceeding, by a specified amount, the response already present due to other causes. Spurious value: Value known for certain to be in error, for example, due to human mistakes or instrument malfunction (Figure I.2.3). Standard deviation (S y ): This is a measure of the dispersion of values about their mean. It is defined as the positive square root of the sum of the squares of the deviation from the arithmetic mean, divided by (n – 1). It is given by: S y = ⎡ ⎢ ⎢ ⎣ n ∑ 1 ( y i – y ) 2 n –1 ⎤ ⎥ ⎥ ⎦ 1/2 (2.3) where y _ is the arithmetic mean of the sample of n independent measurement of the variable y, and (n – 1) indicates the loss of one degree of freedom. Standard error of estimate (S e ): A measure of the variation or scatter of the observations about a linear regression. It is numerically similar to the standard deviation except that the linear-regression relation replaces the arithmetic mean and (n – 1) is replaced by (n – m): Tolerance: The permissible accuracy in the measurement of a specified variable. Tolerance limit: The limiting lower or upper value specified for a quantitative characteristic. True value: The value that characterizes a quantity in the conditions that exist at the moment when that quantity is observed. It is an ideal value that could be known only if all causes of error were eliminated. Uncertainty: The interval about the measurement within which the true value of a quantity can be expected to lie with a stated probability (Figure I.2.3). The numerical value of uncertainty is a product of the true standard deviation of the errors and a numerical parameter depending on the confidence level: e = ± ασ y ≈ αs y (2.5) The standard deviation, s y , computed from n observations, approaches the true standard deviation, σ y , as n approaches infinity. In the case of normal distribution of error, numerical parameters are as follows: Confidence level 0.50 0.674 0.60 0.842 0.66 0.954 0.80 1.282 0.90 1.645 0.95 1.960 α S e = ⎡ ⎢ ⎣ ∑ (d ) 2 1/2 n − m ⎤ ⎥ ⎦ (2.4) 0.98 2.326 0.99 2.576 0.999 3.291 where d is the deviation of an observation from the computed regression value, m is the number of constants in the regression equation, and (n – m) represent the degrees of freedom in the equation derivation. Systematic error: That part of the error that either: (a) Remains constant in the course of a number of measurements of the same value of a given quantity; or (b) Varies according to a definite law when conditions change (Figure I.2.3). 2.3.3.3 Types of error Spurious errors should be eliminated by discarding the values of measurements concerned. These errors can be identified by a statistical-outlier test, such as the one described in ISO 5168 (ISO, 2005) that gives a rejection criterion. Systematic error originates mainly from instrumentation and cannot be reduced by increasing the number of measurements, if the instruments and conditions remain unchanged. If the systematic
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ANNEX ABBREVIATIONS AND ACRONYMS 2D
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