Environmental Engineering Reference
In-Depth Information
data eliminated or replaced by interpolation techniques. Random errors are filtered
by observers.
The quality of a measured value is characterized by its accuracy,
i.e.
, the absence
of bias, and by its reliability, quantified by random error variances. In the following,
the statistical properties of any measurement error
e
will be defined by
e
∼
N
(
b
,
ν
),
(2.23)
where
N
stands for the normal law,
b
for the bias and νfor the variance. For multiple
unbiased measurements (vector
e
), the statistical properties are
e
∼
N
(
0
,
V
),
(2.24)
where, most frequently,
V
is a diagonal matrix. However, it may happen that the
measurement errors of different process variables are correlated. In these conditions
V
contains non-zero off-diagonal terms. This may arise when a common sample
or a common measuring device is used to measure different process variables. The
following situations are examples of such correlated errors:
•
Particle size analysis by sieving. Particles that are not on the right sieve are nec-
essarily present on another one, thus creating a negative correlation between the
errors of fractions belonging to different size intervals [55].
•
Measurement systems involving a common sensor, such as an X-ray fluorescence
analyzer used at different sampling locations, create correlation between the er-
rors.
•
Matrix effects in X-ray fluorescence analysis might correlate concentration mea-
surement errors of different metals.
•
Synchronous incremental sampling of different streams may induce error corre-
lation due to intercorrelation of the streams dynamics, which is created by the
process itself (see integration error in the following section).
It is important to point out common misinterpretation of error correlation. Even
if two process variables are correlated,
e.g.
, concentrations of two different metals in
an ore, their measurement errors are not necessarily correlated. If the sampling and
analysis steps are uncorrelated, the measurement errors are usually uncorrelated,
even when the process variables are correlated.
2.4.2 Measurement Errors for Particulate Materials
Since they are made of randomly organized grains of various minerals, ores are
heterogeneous materials. Also, as these grains take random shapes and sizes, lo-
cal properties of ores may not be representative of their overall average properties.
Breakage of ores into smaller particles may help ore homogenization. However, if
the particles are not perfectly mixed, local properties of a batch of particles may not
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