Naldmann et al. 
unstable conditions are usually available from environmental 
measurement series. Therefore, a time interval must be defined, 
in which the measurement conditions can roughly be considered 
as being approximately stable. This means, the standard 
deviation should not exceed the change of the parameter in 
that interval. An estimate for the change could be the difference 
of the moving average at the beginning and the end of the 
interval!. Then, assuming a normal distribution, fluctuation 
ancertainty can be estimated by the standard deviation of the 
data within this interval (equation 3). The interval should 
however be large enough to have a minimum number of 
values included (at least 10) to be statistically meaningful. 
Otherwise, a factor, a, has to be applied that is given by the 
student-£ distribution (GUM, 2008). Thus, the fluctuation 
uncertainty of the #” temperature value T; is estimated by all m 
values T;; in the chosen interval around the value T;: 
„ (Ty-T)* 
üfuc (Ti) =a zu, Ca 0 =a - STD; (egq. 3) 
For instance, a 2 min interval in our measurement series 
would include 12 data points for sensors 4, 5, and 6 (having a 
sample rate of 6/min), which involves a student-£ factor of 
4=1.05 (=1) for a 68% probability range (see Table G2 in 
/GUM, 2008)). 
Obviously, the sampling rate of a measurement series must 
be sufficiently large so that a suitable interval can be defined. If 
che sampling rate is too small to catch the fluctuation of the 
measurement signal other ways must be found to estimate 
fuctuation uncertainty. In this case it may be quantified by 
independent experiments in the lab or simply based on the 
experience of the scientist evaluating the data [so called type B 
uncertainty (GUM, 2008)]. 
It must be emphasized that the time interval mentioned in 
:his subsection is used to calculate an estimate for the fluctuation 
uncertainty of a single (raw) data point, which reflects the 
temperature variability seen in the insets of Figure 5. Likewise, 
:he panel in the middle of Figure 4 shows the fluctuation 
uncertainty of single temperature points in the measurement 
period Pl (based on a 5 min interval). However, fluctuation 
uncertainty of single points must be distinguished from that of 
mean values. Fluctuation uncertainty of mean values, using a 
5 min time interval as an example, will be discussed in the 
next subsection. 
{i) Temperature estimated by the arithmetic mean of values 
measured in a 5 min interval (“5 min means”) 
„A more sophisticated criteria would include a trend analysis. To this 
and, a linear regression would be applied in the defined interval. If the 
;lope of the regression line is smaller than the expanded uncertainty of 
'he slope, the parameter can be considered stable. 
Zrontiers in Marine Science 
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10.3389/fmars.2022.1002153 
An estimate for fluctuation uncertainty can be calculated 
with the standard deviation of the mean of n values T;; within the 
* 5_minute period: 
(su (Ti-T) 
Ufac (Ti)smin= a 1/ X; AR n-D a-SEM (egq. 4) 
Here T; is the arithmetic mean over 5 min in the 7” time 
interval, with nı = 30 for the sensors having a sample rate of 10 s, 
and nı = 300 for sensors having a 1 s sample rate. a = 1 as n > 10. 
The fluctuation uncertainty of the mean is obviously smaller 
than that of a single result (see i) due to the additional factor 
1/\/n. The fluctuation uncertainty of the mean is reflected by the 
smoother behavior seen in the main parts of Figure 5 and the 
smaller values illustrated in the lower panel of Figure 4. 
„igure 6 on the left-hand side shows the combined, 
expanded uncertainties of four sensors during a period with 
low variability (P3). Their values are a few mK, they are largely 
constant over the complete measurement time and are 
dominated by calibration uncertainty as the temperature has 
low variability. The sensor corresponding to the orange results 
has a slightly larger calibration uncertainty. The dashed line 
ndicates the expanded combined uncertainties of individual 
‚esults, meaning representation (i) for comparison. Note that it 
corresponds only to those sensors with smaller calibration 
uncertainties. The uncertainty of the raw data is somewhat 
larger, so that averaging is also advantageous under low 
variability conditions. 
The figure on right shows expanded uncertainties of the 
5 min means during the period with high variability (P2). There, 
che dashed line indicates the calibration uncertainty of the 
sensors. The fluctuation uncertainty increases the expanded 
uncertainty by about a few hundred’s Kelvin. It can also be 
seen that those sensors measuring with higher time resolution 
(sensor 1 & 3), have smaller fluctuation uncertainties compared 
to those with lower sample rates (sensor 4 & 5) because of the 
averaging, despite smaller fluctuations of the raw data of the 
latter (see inset of Figure 5). Hence, it seems that larger numbers 
of samples lead to less uncertainty in comparison to longer 
integration times of the other two sensors. The uncertainty of the 
'ndividual results is not shown, since it is too large to be shown 
on that scale. 
4.3 Data analysis based on multiple 
sensor measurements 
rigure 7 shows exemplary spreads of the results of four 
sensors. Each result is the mean of a 5 min interval and is shown 
as a colored dot. The uncertainty bar of each result indicates its 
combined uncertainty as described in the previous section. The 
ihree groups seen in Figure 7 belong to 3 different, but 
subsequent 5 min intervals. Note that the results of each 
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