Propagation of imprecision

In the next two sections we encountered the problem of propagation of experimental imprecision through a calculation. When the calculation involves only one parameter, taking its first derivative will provide the relation between the imprecision in the derived function and that in the measured parameter. In general, when the final result depends on more than one independent experimental parameter, use of partial derivatives is required, and the variance in the result is the sum of the variances of the individual parameters, each multiplied by the square of the corresponding partial derivative. In practice, the spreadsheet lets us find the required answers in a numerical way that does not require calculus, as illustrated in the exercises. While we still need to understand the principle of partial differentiation, i.e., whatit does, at least in this case we need not know how to do it, because the spreadsheet (and, specifically, the macro PROPAGATION, see section 10.3) can simulate it numerically. 

The estimated excess activity A and associated uncertainty for each slice of core are listed in Table II. The error, associated with each count is estimated using the propagation-of-errors method (Bevington, 1969). Counting statistics account for most of the imprecision, although uncertainties associated with counter background and efficiency also contribute. 

The software allows the user to customize the Monte Carlo-DSTE method implemented in MUSTADEPT for the representation and propagation of the uncertainty associated to the imprecise information provided by the experts. In particular, the user has to set two parameters the length of the time steps partitioning the time horizon of interest and the number of Monte Carlo simulations. These variables determine the level of precision of the final results and the computational efforts required to carry out the analysis the... 

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

These equations identify the dominant source and loss processes for HO and H02 when NMHC reactions are unimportant. Imprecisions inherent in the laboratory measured rate coefficients used in atmospheric mechanisms (for instance, the rate constants in Equation E6) can, themselves, add considerable uncertainty to computed concentrations of atmospheric constituents. A Monte-Carlo technique was used to propagate rate coefficient uncertainties to calculated concentrations (179,180). For hydroxyl radical, uncertainties in published rate constants propagate to modelled [HO ] uncertainties that range from 25% under low-latitude marine conditions to 72% under urban mid-latitude conditions. A large part of this uncertainty is due to the uncertainty (la=40%) in the photolysis rate of 0(3) to form O D, /j. 

For variables vAiose errors are not Independent, a general form of Equation 11 Incorporates covarleince as well as variance. Also, another form of the equation has been derived for non-random error (20). As will be seen below, certain ccxnputatlons, well known In SBC, cure now being found to be Intrinsically Imprecise because of error propagation. 

As our second example, we will estimate the standard deviation in [H+] when the concentration of hydrogen ions is calculated from a pH reading p with a corresponding imprecision Ap. In order to see how the imprecision propagates, we nowuse the relation [H+] = 10-pH = 1 (T(p Ap) = io-pd- p/p) = 10 pX 10 ldp/p. There does not exist a closed-form expression for 10 4p/p analogous to that which we used for (d + Ad)3 in our earlier example, but instead we can use the series expansion... 

We will now carry the above to its logical spreadsheet conclusion. The spreadsheet is there to make life easy for us in terms of mathematical manipulations, and three-quarters of a page of instructions to describe how to do it may not quite be your idea of making life easy. Touche. But this was only the introduction once we know how to make the spreadsheet propagate imprecision for us, we can encode this knowledge in a macro. That is what we have done, and have described in detail in chapter 10. The macro is called Propagation, and if you have downloaded the macros from the website (as described in section 1.13) you can nowuse that macro. Below we illustrate howto use Propagation. 

After you have entered the function, and pushed the OK button, you will see the propagated imprecision appear in cell CIO, in italics. Compare it with your earlier results. That s it, no mathematics, no manipulations, just enter the data and push the OK button the macro does the rest. Figure 2.4-1 shows the result, and the entire region of the spreadsheet used. 

The fact that basic thermodynamic data are imprecisely known means that model results such as p values, SI values, and so on, will all be to some extent imprecise as well. The imprecision of the input data is propagated through the calculation procedure and appears in the results. The nature of this propagation has not been extensively investigated, but it depends not only on uncertainties in the thermodynamic and analytical data, but also on the nature of the geochemical system involved. See Anderson (1976, 1977) and Criscenti el al. (1996) for discussions. 

Even more than spectrometer designers fear photodiode fading, computer programmers suffer from a neurotic fear of bugs, because they know that the consequences of even an apparently harmless imprecision in a computer code can propagate itself by unpredictable waves of ruin at any unpredictable moment. 

Reliability engineering addresses these questions by structured and formal methods of analysis, which entail the representation and modeling of the structure, equipment, or system based on the available knowledge and information, the quantification of the model based on the available data, and the representation, propagation, and quantification of the uncertainty in the behavior of the structure, equipment, or system as inferred from the knowledge, information, and data available, which is always incomplete, imprecise, and often scarce. 

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