The following types of measurements are identified for accounting and control systems:
depending on the objects measured: measurements of bulk forms (bulk samples) to determine the weight and volume and measure material samples (to find out isotopic and/or element composition);
depending on how the measured material is handled: destructive measurements (involve normally destruction of either the chemical or physical form of the material sample) and nondestructive measurements (quantitative or qualitative determination of the type and/or quantity of NM in a sample without producing significant changes to or getting into the sample);
depending on the technique employed to determine the measured quantity: direct and indirect measurements.
Any measurement, as mentioned earlier, involves a certain error. Therefore, formulating the decision to be made based on accounting data needs these errors to be taken into account. What are these and what are their potential sources will be discussed hereinafter.
Direct measurements have the measured and determined quantity compared to the measurement unit directly or using an instrument graduated in respective units.
Indirect measurements have the measured quantity determined (calculated) from direct measurements of other quantities that relate to the measured quantity through a functional dependence.
Measurement error
We will treat error as a deviation of an estimate from the true value of the characteristic estimated for. Then, as applied to measurements, this is a deviation of the measurement result from the true (actual) values of the measured quantity. Note that we do not normally know the true value since uncertainties exist, and properly describing and estimating errors requires that a suitable mathematical model of the error should be selected. Therefore, the actual error is the estimate of the respective uncertainties we have obtained (see p. 4.2.).
The simplest model for a single measurement is
X = T + E, |
(4.40) |
where X is the measurement result; T is the true value and E is the measurement error [5].
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We have already talked about numerous sources of errors as may affect measurement data. All errors are categorized into random errors and systematic errors (also the so-called short-term systematic errors are occasionally introduced).
Indeed, the quantity E can be presented as random and having the mathematical expectation M (E) = S . Then (4.40) will be written as
X = T + S + R, |
(4.41) |
where R is the random quantity with the zero mathematical expectation M (R) = 0 conventionally referred to as random error; and S is the so-called systematic error often treated as an unknown constant shift of the measured quantity about the true value.
Here are several examples to consider for a better conception of this division.
Random error relates to the action of a series of random factors per each individual measurement, e.g. random nature of the nuclear decay process, air motion, sample position and so on.
Consider six UO2 pellets of nominally one composition to be tested for the uranium concentration. Let us denote: хij is the result of j–th uranium concentration measurement in the i– th pellet; μ is the nominal (declared) uranium content in the pellets; ρi is the uranium content deviation from the declared quantity in the i– the pellet; and is the deviation conditioned by the analytical error of the j–th measurement. With the aid of a multiplicative model we will arrive at
х11=μ+ρ1+ε1, х22=μ+ρ2+ε2,
………………
х66=μ+ρ6+ε6.
As ρi and εj differ for each of the six results, these are random errors. According to the classification we have introduced, ρi is the random error of the statistical sampling. If to consider this as a random variable with the average value equal to zero and the variance σρ2, then σρ2 is called the variance of the statistical sampling random error. Similarly, εj in the classification in question is an analytical random error, and σε2 is called the variance of the analytical random error.
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First, since only one measurement was done for each pellet, and ε and ρ have become parts of the equation with one index, then, in the given case, these two errors cannot be separated. In such situation, one can consider them to be a single “random measurement error”. In the event of two or more measurements done for each pellet, errors are separable. Second, a feature of the random error model is that its index varies with each measurement, its influence on the eventual result decreasing with each new measurement. For this reason, random errors are relatively easy to control.
Systematic error is an error caused by the limited accuracy of the instrument, an improper installation of the instrument, the technique used to process data and the effects of external factors. Systematic errors determine the measurement group. Systematic errors are errors caused by the improperly reset scale and the rounding of numerical values.
Let us expand a bit the presented model by giving it one more summand: - a deviation from the nominal value conditioned by the analytical method. It will be the same for all measurements done by the given method. Then
х11=μ+ +ρ1+ε1, х22=μ+ +ρ2+ε2,
………………
х66=μ+ +ρ6+ε6.
The quantity that does not have an index and is common for all measurements is called systematic error or bias. These terms are normally used as equals, still they have a small difference. If measurement data is
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residual bias so occurs, this being equal to the difference between the quantity and its estimate. It is this residual bias that is called systematic error. Not everyone differentiates so between systematic error and bias, still it is important to understand what do you mean exactly by .
If is a random variable with the average value equaling zero and the variance σ 2, then σ 2 will be called the systemic error variance.
Many aspects of accounting and control are more influenced by systematic errors than by random ones. The explanation is that, unlike random error, the systematic error effect does not decrease with each new measurement, thus reducing the efficiency of the NM accounting measures.
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Sources of errors
An error in a given measurement results from several errors overlapping (superimposed). Consider some of the error sources that may affect measurement data.
1. Statistical sampling error. Consider a set of N items, e.g. containers with NM, each of these having a true value of a characteristic (net weight, uranium mass, enrichment, etc.). If several items have been selected randomly from the set (sample), then the average value of a characteristic calculated for these items will differ from the average value of the given characteristic for the whole set (say, from the average enrichment value). Let us determine the statistical sampling error as the difference between the average value of a characteristic for a randomly taken sample and the average value of the characteristic for the entire set.
This error is neglected in some cases, say, inventory taking has the inspector-obtained measurement data for an item compared against the operator data, so the true value of the measured characteristic does not matter and it is reduced in the difference.
In the event of attributive (quality) inventory taking, the true value for each item is equal either to 1, if there is a defect, or to 0, if there is none. And the average value for the group of interest is equal to the relation of the number of defective items in the group to the total number of items: 0<Ndef/Ntot<1. In the given case, therefore, there is a statistical sampling error in each selected item. The influence of this error on the inventory taking data (and so the necessity of taking this into account) will vanish only if all items are checked.
Depending on whether we do measurements for all nuclear material (a universe) or just for a portion of same (a sample), material is accounted for based on three measuring operations:
1)determination of the material net weight or volume (bulk measurements);
2)material sample taking;
3)analysis of the material sampled to find out its element and/or isotopic composition. So, normally, it will be convenient to divide the total measurement error into such components as will match the three measuring operations.
2. Error of measurements in bulk form (bulk measurements) will be defined by us as the difference between the true weight (or volume) of an item and its measured value. With only this definition considered, a bulk measurement error appears to be a directly obtainable simple value.
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However, it may and even is most likely to result from a great deal of bulk measurement errors superimposed, some of these being capable to compensate each other.
3.Material sampling error is found from the relation to characteristic measured. This may be uranium or plutonium concentration, enrichment and so on.
A material sampling error is the difference between the average values of a characteristic for the sample taken and the respective average value of the given characteristic for all material. It is important to understand what “all material” means. If we are interested in the u ranium concentration to be found for the given container, then the sampling error is the difference between the uranium concentration in a small sample taken from this container, and the uranium concentration in the container. This may be referred to as the “sampling error for one containe r”. If the measurement data for a sample taken from the given container with the given uranium concentration needs to be extended to other, nominally the same containers, then the change in the concentration between containers is incorporated in the sampling error together with the “error for one container”.
4.Analysis error. As is the case with the material sampling error, the analytical error (analysis error) is determined for the characteristic of interest. An analytical measurement error is the difference between the true value of the sample characteristic in question and the measured value of this characteristic. Note that this error relates to the sample rather than to all material it characterizes. An error of determining the value of a characteristic for all material will be a combination of the analytical error and the material sampling error.
Where nondestructive measurements are used with no sampling involved, it is the error of measuring a characteristic that will be the analytical error of measurement; with nondestructive measurements done not for the whole of the item but for a sample of it, the sampling error should be also accounted for.
As noted earlier, the given error under determination results from many errors superimposed. For example, a weighing error may depend on how the item is placed on the scale, the type of the scale, the particular scale of the given type, the operator and the environment (air temperature, humidity and so on). The extent to which attention is to be given to the search for and estimation of factors that affect the error depends on a number of circumstances. For instance, if a weighing error at the measurement point of interest does not influence greatly the accounting data quality, there is no special need for identifying and evaluating every source contributing to this
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