Naldmann et al. 
approach must consider the correlation between 7% ; and Tr; 
and correlations between the results of the individual sensors, 
ie., that all sensors have been calibrated by the same institute in 
che same way. Using other estimators, e.g., the weighted mean, as 
ı representative for the average of several sensors, also effects the 
form of eq. 6. However, eq. 6 can be considered as an acceptable 
approximation for practical use. Intuitively, the uncertainty bars 
of an individual sensor result must overlap with that of the mean 
to some reasonable extent. For the sake of simplicity, we will 
discuss the representative character of a sensor result rather in 
terms of the overlap of uncertainties considering Figures 6-8. 
Basically, three cases should be distinguished which have 
different implications on the representativeness of an 
individual result. 
a) The combined uncertainty value of a result is in the order 
af the calibration uncertainty. Hence, fluctuation uncertainty is 
small compared to calibration uncertainty, as can be seen in the 
left-hand side of Figure 6. Temperature can be assumed stable 
and homogenous in the vicinity of the sensors. In this case, the 
uncertainties of all sensors will overlap quite well, as can be seen 
in the cloud on the left-hand side of Figure 7 (around 80 min). 
Bach sensor measures a good estimate for the mean temperature 
and the combined uncertainty of the sensor result is a good 
estimate for the uncertainty of the temperature in the considered 
dume window and measurement volume. The deviation of an 
individual sensor (e.g., that of sensor 5) from the mean cannot 
necessarily be considered as systematic measurement bias that 
should be compensated. In fact, the deviation lies in the range of 
che calibration uncertainty. Correcting results for deviations that 
are smaller than calibration uncertainty, cannot be justified. 
b) Both calibration and fluctuation contribute to the 
zombined uncertainty of a sensor result in roughly the same 
order of magnitude. Temperature variation with respect to 
measurement time and special distribution must be assumed 
cO some extent. The combined uncertainties of the individual 
sensors do still overlap fairly well with the uncertainty of the 
mean, as illustrated by the cloud on the right-hand side of 
%igure 7 (90 min). However, the overlap of some individual 
sensors with each other is marginal (see sensors 3, 4 and 5). 
Hence, the uncertainty u, of an individual sensor does not well 
represent the actual mean temperature and its uncertainty. An 
uncertainty factor a; reflecting the spread of several sensor 
results should therefore be included in the uncertainty of an 
individual sensor if only one sensor had been deployed (and 
temperature variability cannot be neglected): 
Us = u (eq. 7) 
u; is calculated according to eq. 2 from the available 
measurement data. a, is an estimated value reflecting the 
spread of the results of several sensors and u, denotes the 
Zrontiers in Marine Science 
L. 
10.3389/fmars.2022.1002153 
enlarged uncertainty of an individual sensor. Obviously, 
assigning a number to a, is somewbhat arbitrary if only results 
of a single sensor are available, as is typical for oceanographic 
practice. However, GUM (section 4.3 of GUM, 2008) suggests 
evaluation of a so-called type B standard uncertainty that is 
based on the available information if repeated observations (here 
ın the sense of several sensors) are not possible. Thus, looking at 
the results shown in Figure 7, the uncertainty bars of all sensors 
would reasonably overlap if they were about 50% larger. 
Therefore, setting a,=1.5 is an arbitrary, but reasonable choice. 
If uque is smaller than 0.5 uca1 its contribution to uc becomes less 
televant. The relative difference between u-21 and u. is then less 
than 11%. Hence, it is also reasonable to set u Umquc< 0.5 Ucay AS a 
limit, below which it is reasonable to assume stable temporal and 
spatial conditions and, consequently, to set a,=1 in that case. 
c) Figure 8 compares the uncertainties assigned to the means 
of the sensor results with those of the individual sensors. There 
are measurement intervals in which the combined uncertainties 
of the individual sensors are in the order of several tens of mK. 
Hence, the corresponding fluctuation uncertainties are 
significantly larger than calibration uncertainties. Due to 
ongoing mixing processes significant instability in temporal 
and spatial temperature distribution must be assumed. While 
temporal averaging still provides a reasonable estimate of 
temperature and its uncertainty due to temporal variability at 
the exact position of a sensor, it cannot readily be assumed that 
the values are also adequate representatives for the entire time 
‚ange of the measurement. Additional information is needed for 
.nstance by averaging the results of several sensors, potentially 
by weighing the individual results with their uncertainties 
\Maronna et al., 2006) and assigning uncertainties to the 
averages as mentioned above. 
Cases a) and b) apply to measurement results where the 
uncertainties indicate no or moderate temperature variability. 
We propose, as a rule of thumb, that the combined uncertainty 
of a single sensor measurement can be considered as an adequate 
rtepresentative of the mean temperature in the specified 
measurement time (here, 5 min) and for the ambient water 
body next to the sensor within reasonable limits, if the 
Auctuation uncertainty of an individual sensor is not larger 
than two times the calibration uncertainty. If fluctuation 
uncertainty is larger, the uncertainty must be estimated using 
additional means. For instance, a multi sensor approach could 
provide reasonable uncertainties, also accounting for spatial 
inhomogeneity. If no further experimental data is available, 
the factor a; can only be quantified based on the experience of 
the scientist evaluating the data. 
A proposed flowchart for processing uncertainty 
information is presented in the Supplementary Materials, 
Appendix 3. 
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