Uncertainties in calculated results

Relevant hydrological fundamentals are utilized (21) to take account of the complex interaction of physical and chemical processes involving sod or rock, water, and contaminant. Attention is paid to uncertainties in calculated results. 

Error analysis. The mathematical analysis done to show quantitatively how uncertainties in data produce uncertainty in calculated results, and to find the sizes of the uncertainty in the results. [In mathematics the word analysis is synonymous with calculus, or a method for mathematical... 

One significant feature of the Parametric Method is that it indicates, through the (1 + K 2) value, the relative contribution of each variable to the uncertainty in the result. Subscript i refers to any individual variable. (1 + K ) will be greater than 1.0 the higher the value, the more the variable contributes to the uncertainty in the result. In the following example, we can rank the variables in terms of their impact on the uncertainty in UR. We could also calculate the relative contribution to uncertainty. 

It is these contrasting effects of aerosol particles, combined with uncertainties in the contribution of absorption due to 03, that provide the largest uncertainties in calculations of actinic fluxes and photolysis rates in the boundary layer (e.g., Schwander et al., 1997). As a result, it is important to use the appropriate input... 

The RMS error of Eq. 1.13 is related to the 2-norm of the vector of differences Pi — P°, while that of Eq. 1.14 is related to the oo-norm [1], We are not asserting that the latter should replace the former as an estimate of uncertainties in calculations, but that both error estimates should be considered if the uncertainties in predicted values for which there are no standards are to be realistic. We should note that a more elaborate approach to comparing computed results to a set of standard values has been devised by Maroulis and co-workers using principles of information theory [2]. Their work has been little used in practice, but it provides a convenient way of comparing many values. 

What is the net sample rate and its uncertainty This raises the general question of calculating the uncertainty in the result of some mathematical operations on an uncertain number. If we consider two independently determined numbers and their uncertainties (standard deviations), A + [Pg.572]

Only in situations where numerical uncertainty is small compared to modeling uncertainty we can successfully validate a calculation. After minimizing numerical errors there will still be other uncertainties in calculations due to for example variations in inlet conditions or due to inherent uncertainty in tabulated material properties, etc. These can be best handled by repeating the calculations with appropriate variations in the uncertain input quantities, thus resulting in say nc calculations with seemingly n, random outcomes the mean and variance of which are donated by Xc and. S 2, Similarly there would be ne repeated experiments of the same phenomenon with a, random outcomes with the corresponding mean and variance, Xe and S2e, respectively. The estimated modeling error is by definition the difference between the experimental mean and calculation mean, i.e. 

Absolute uncertainty. The uncertainty in a measured quantity is due to inherent variations in the measurement process itself. The uncertainty in a result is due to the combined and accumulated effects of these measurement uncertainties which were used in the calculation of that result. When these uncertainties are expressed in the same units as the quantity itself they are called... 

Experimental error. The uncertainty in the value of a quantity. This may be found from (1) statistical analysis of the scatter of data, or (2) mathematical analysis showing how data uncertainties affect the uncertainty of calculated results. 

Uncertainty. Synonym error. A measure of the the inherent variability of repeated measurements of a quantity. A prediction of the probable variability of a result, based on the inherent uncertainties in the data, found from a mathematical calculation of how the data uncertainties would, in combination, lead to uncertainty in the result. This calculation or process by which one predicts the size of the uncertainty in results from the uncertainties in data and procedure is called error analysis. 

At the same time, calling this a fit to the bands is very much understating the accomplishment. The set of four parameters in Table 2-1 and the term values in Table 2-2 (all in the Solid State Table) allow calculation of energy bands for any of the homopolar semiconductors or any of the zincblcndc-structure compounds, as simply for one as for the other, without computers, with consistent accuracy, and without need for a previous accurate calculation for that compound. Only in first-row compounds is there indication of significant uncertainty in the results. Furthermore, as we noted in Table 2-1, the theoretical matrix elements are very nearly equal to the ones obtained by fitting bands thus, if we had plotted bands in Fig. 3-8,a that were based upon purely theoretical parameters, the curves would have been hardly distinguishable. 

Values are means standard errors for 2 years of data. Numbers of observations range from 15 (HNO3) to 26 (particles) to 128 (precipitation) to 730(802). In comparing these deposition rates it must be recalled that any such estimates are subject to considerable uncertainty. The standard errors given provide only a measure of uncertainty in the calculated sample means relative to the population means hence additional uncertainties in analytical results, hydrologic measurements, scaling factors, and deposition velocities must be included. The overall uncertainty for wet deposition fluxes is about 20% and that for dry deposition fluxes is approximately 50% for SOj", Ca ", K", and approximately 75% for NOj" and... 

In the case of a nonlinear or unknown dependence y = f x), the application of simple linear interpolation leads to some uncertainty in the results. However, if the equation for nonlinear dependence is known, a transformation of Eq. (1) into Eq. (2) can be applied to remove the nonlinearity. This approach gives the results of the same precision as that of direct calculation with the equation y = f x) ... 

The combined uncertainty in the result at a concentration y is calculated as follows ... 

Values for F and f are compiled in tables (11, 13). The calculation becomes more tedious (and uncertainty in the result increases) for more complex structures. A computer program has been developed to aid in the calculation (C. Hansch and A. Leo, personal communication). 

Regression analysis is often employed to fit experimental data to a mathematical model. The purpose may be to determine physical properties or constants (e.g., rate constants, transport coefficients), to discriminate between proposed models, to interpolate or extrapolate data, etc. The model should provide estimates of the uncertainty in calculations from the resulting model and, if possible, make use of available error in the data. An initial model (or models) may be empirical, but with advanced knowledge of reactors, distillation columns, other separation devices, heat exchangers, etc., more sophisticated and fundamental models can be employed. As a starting point, a linear equation with a single independent variable may be initially chosen. Of importance, is the mathematical model linear In general, a function,/, of a set of adjustable parameters, 3y, is linear if a derivative of that function with respect to any adjustable parameter is not itself a function of any other adjustable parameter, that is. 

With this value we can estimate that the kilogram of beans contains about 1000 g -i- 0.2024 g bean = 4940 beans. This estimate, however, was obtained from the observation of only 140 beans, or 3% of the total assuming the 1-kg package contains about 5000 beans. We cannot expect this estimate to be equal to the exact value, which remains unknown. Our calculation yields the sample average, not the population mean. We will see later how to estimate the uncertainty in this result. 

---