Traitement et analyse de séries chronologiques continues de ...

Annexes
Sensor uncertainties. Calibration function is chosen either a straight line, second or third order
polynomial function for different sensor types based on Fischer-Snedecor statistical test. With regards
to the variance of measurements, the ordinary least squares regression for constant variance or the
Williamson regression accounting for uncertainties in both standards and sensor response for non
constant variance are used. (Bertrand-Krajewski, 2004; Bertrand-Krajewski et al., 2007). In both
regression types, uncertainties in function parameters can be calculated either analytically or by Monte
Carlo simulations. The sensor uncertainty is estimated from the variance of the calibration
measurements: the ordinary least squares regression is set a constant sensor uncertainty as the
maximum value of the observed variance, and the Williamson regression is assigned a variable
uncertainty according to the observed variability of the variance along the measurement range.
Field measurement uncertainties. The in situ uncertainty is both sensor and site specific. Its value
should be evaluated from local measurements and expertise. For instance, in the example detailed
hereafter, turbidity is not directly measured in the sewer but in a shelter where a transit flume is
continuously supplied with wastewater from the sewer by means of a peristaltic pump. In this case, the
representativeness of the wastewater in the flume and the varying position of the mobile pump intake
in the sewer should be considered as additional sources of uncertainty. These were estimated to be
approximately equal to 10 % of the measured value. Another example is related to the water level
measurements. The sensor uncertainty of an ultrasonic probe is evaluated to be equal to 3 mm
according to calibration experiments. As (i) the sensor is not perfectly installed and located in the
sewer, and (ii) the free surface is not flat but experiences waves of height ranging from 1.5 cm in dry
weather up to approximately 3 cm in wet weather events, the in situ standard uncertainty in the water
level is set equal to 7.5 mm - 15 mm depending on the water level.
Data correction
This step comprises the correction of raw data at each time step according to the sensor calibration
functions and the calculation of their standard uncertainties from the information provided in the
calibration step.
Corrected values Xˆ i of raw measurements X i at time step i are estimated by the inverse calibration
function, either by direct analytical calculation (in case of calibration functions of degrees 1 and 2) or
by numeric minimization (in less frequent case of calibration function of degree 3), with addition of an
offset value when necessary.
The total standard uncertainty in the corrected values is calculated by:
u ˆ 2 ˆ
tot Xi
um
Xi
2 usiteX
ˆ i 2
where mX i
uncertainties calculated according to the law of propagation of uncertainties (LPU), and
(1)
u ˆ is the measurement standard uncertainty accounting for sensor and calibration
usite Xˆ i is
the in situ standard uncertainty. The standard uncertainty and the LPU approach are applied according
to international standards for the expression of uncertainty in measurements (ENV 1999, ISO, 2009).
Selection of pre-validation tests and setting of test parameters
This step aims to select the tests to be applied (see Figure 1) and to set the parameters required for the
pre-validation tests. The pre-validation method is based on the method initially developed by Mourad
and Bertrand-Krajewski (2002). Three pre-validation marks are used: 1 (valid data), 2 (doubtful data)
or 3 (incorrect data). Eight pre-validation tests are available and selected by the user according to the