Uncertainty example

There are many potential sources of uncertainty. Examples include sampling the nature of the sample matrix sample storage conditions ... 

Like human health assessments of chemical mixtures, ecological assessments usually involve substantial uncertainties. Examples include the assumptions of simple similar action or independent joint action, the extrapolation from laboratory data to field circumstances, and the inclusion of ecological interactions in the assessment. Because of these uncertainties, the assessment should always include a thorough discussion of all assumptions and the identification, when possible, of the major sources of uncertainty. 

Demand profile. Usually, high volume products are associated with comparatively low variety (few skus) and low demand variability and uncertainty. Examples are washing powders in retail, and volume cars like Ford and VW s big sellers. Note that demand variability can be created by means of promotions. Low volume, high variety products - such as most automotive spare parts, and many health and beauty products - tend to have the opposite demand profiles. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

When using annualized capital cost to carry out optimization, the designer should not lose sight of the uncertainties involved in the capital annualization. In particular, changing the annualization period can lead to very different results when, for example, carrying... 

Each of the input parameters has an uncertainty associated with it. This uncertainty arises from the inaccuracy in the measured data, plus the uncertainty as to what the values are for the parts of the field for which there are no measurements. Take for example a field with five appraisal wells, with the following values of average porosity for a particular sand ... 

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. 

The purpose of this exercise is to identify what parameters need to be further investigated if the current range of uncertainty in reserves is too great to commit to a development. In this example, the engineer may recommend more appraisal wells or better definition seismic to reduce the uncertainty in the reservoir area and the net-to-gross ratio, plus a more detailed study of the development mechanism to refine the understanding of the recovery factor. Afluid properties study to reduce uncertainty in (linked to the shrinkage... 

Introduction and Commercial Application JUe objective of performing appraisal activities on discovered accumulations is to reduce the uncertainty in the description of the hydrocarbon reservoir, and to provide information with which to make a decision on the next action. The next action may be, for example, to undertake more appraisal, to commence development, to stop activities, or to sell the prospect. In any case, the appraisal activity should lead to a decision which yields a greater value than the outcome of a decision made in the absence of the information from the appraisal. The improvement in the value of the action, given the appraisal information, should be greater than the cost of the appraisal activities, otherwise the appraisal effort is not worthwhile. 

The general list of factors influencing the uncertainty in the gross rock volume included the shape of structure, dip of flanks, position of bounding faults, position of internal faults, and depth of fluid contacts (in this case the OWC). In the above example, the owe is penetrated by two wells, and the dip of the structure can be determined from the measurements made in the wells which in turn will allow calibration of fhe 3D seismic. 

The example illustrates some important steps in identifying the uncertainties and then beginning to quantify them ... 

Appraisal activity should be based upon the information required. The first step is therefore to determine what uncertainties appraisal is trying to reduce, and then what information is required to tie down those uncertainties. For example, if fluid contacts are a major source of uncertainty, drilling wells to penetrate the contacts is an appropriate tool seismic data or well testing may not be. Other examples of appraisal tools are ... 

The choice of the location for well A should be made on the basis of the position which reduces the range of uncertainty by the most. It may be for example, that a location to the north of the existing wells would actually be more effective in reducing uncertainty. Testing the appraisal well proposal using this method will help to identify where the major source of uncertainty lies. 

Reservoir engineers describe the relationship between the volume of fluids produced, the compressibility of the fluids and the reservoir pressure using material balance techniques. This approach treats the reservoir system like a tank, filled with oil, water, gas, and reservoir rock in the appropriate volumes, but without regard to the distribution of the fluids (i.e. the detailed movement of fluids inside the system). Material balance uses the PVT properties of the fluids described in Section 5.2.6, and accounts for the variations of fluid properties with pressure. The technique is firstly useful in predicting how reservoir pressure will respond to production. Secondly, material balance can be used to reduce uncertainty in volumetries by measuring reservoir pressure and cumulative production during the producing phase of the field life. An example of the simplest material balance equation for an oil reservoir above the bubble point will be shown In the next section. 

Cost estimates can usually be broken into firm items, and items which are more difficult to assess because of associated uncertainties or novelty factor. For example, the construction of a pipeline might be a firm item but its installation may be weather dependent, so an allowance could be included to cover extra lay-barge charges if poor sea conditions are likely. 

As discussed in Section 13.2, the technical, fiscal and economic data gathered to construct a project cashflow carry uncertainty. An economic base case is constructed using, for example, the most likely values of production profile and the 50/50 cost estimates, along with the best estimate of future oil prices and the anticipated production agreement and fiscal system. 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

For some experiments, the solar neutrino flux and the rate of decay of the proton being extreme examples, tire count rate is so small that observation times of months or even years are required to yield rates of sufficiently small relative uncertainty to be significant. For high count rate experiments, the limitation is the speed with which the electronics can process and record the incoming infomiation. 

In general, we know bond lengths to within an uncertainty of 0.00.5 A — 0.5 pm. Bond angles are reliably known only to one or twx) degrees, and there arc many instances of more serious angle enxirs. Tn addition to experimental uncertainties and inaccuracies due to the model (lack of coincidence between model and molecule), some models present special problems unique to their geometry. For example, some force fields calculate the ammonia molecule. Nlln to be planar when there is abundant ex p er i m en ta I evidence th at N H is a 11 i g o n a I pyramid. 

Recording a measurement provides information about both its magnitude and uncertainty. For example, if we weigh a sample on a balance and record its mass as 1.2637 g, we assume that all digits, except the last, are known exactly. We assume that the last digit has an uncertainty of at least 1, giving an absolute uncertainty of ... 

Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. 

A propagation of uncertainty also helps in deciding how to improve the uncertainty in an analysis. In Example 4.7, for instance, we calculated the concentration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty of 2 ppm and a relative uncertainty of 1.6%. How might we improve the analysis so that the absolute uncertainty is only 1 ppm (a relative uncertainty of 0.8%) Looking back on the calculation, we find that the relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank)... 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

Determine the uncertainty for the gravimetric analysis described in Example 8.1. (a) How does your result compare with the expected accuracy of 0.1-0.2% for precipitation gravimetry (b) What sources of error might account for any discrepancy between the most probable measurement error and the expected accuracy ... 

Note that the method of end group analysis is inapplicable to copolymers, since the presence of more than one repeat unit adds extra uncertainty as to the nature of chain ends. The above example included the remark that the molecular weights calculated in the example were average values. In the next section we shall examine this point in greater detail. 

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