Open AccessArticle

Assessment of the Representativeness and Uncertainties of CTD Temperature Profiles

Marc Le Menn

Franck Dumas

and

Baptiste Calvez

Service Hydrographique et Océanographique de la Marine (Shom), CS 92803, 29228 Brest Cédex 2, France

Author to whom correspondence should be addressed.

J. Mar. Sci. Eng. 2025, 13(2), 213; https://doi.org/10.3390/jmse13020213

Submission received: 29 November 2024 / Revised: 8 January 2025 / Accepted: 10 January 2025 / Published: 23 January 2025

(This article belongs to the Special Issue Progress in Sensor Technology for Ocean Sciences)

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Figure 1
Carousel water sampler used for the measurements at sea. The SBE 9+ CTD profiler is located at the bottom with the first pair of CT sensors. The second pair (not visible on this photo) is located at the top. The pump and the pipe connecting it to the conductivity sensor are visible in the foreground. "> Figure 2
Drift over time of SBE 3 sensors used during the campaigns at sea. During acquisitions, data are corrected for these drifts. "> Figure 3
(a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The combined expanded uncertainty per water layer is superimposed on the error signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (6). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and, in green, the error signal between the model and the measurement. (d) Amplitudes of the different terms of Equation (6). The green colour represents the transient part of the sensor temperature rise of the equation. The orange colour represents the sensor’s response time error, τ v. The blue colour represents the temporal thermal variations obtained with the v t term. "> Figure 3 Cont.
(a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The combined expanded uncertainty per water layer is superimposed on the error signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (6). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and, in green, the error signal between the model and the measurement. (d) Amplitudes of the different terms of Equation (6). The green colour represents the transient part of the sensor temperature rise of the equation. The orange colour represents the sensor’s response time error, τ v. The blue colour represents the temporal thermal variations obtained with the v t term. "> Figure 4
(a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The total expanded uncertainty per water layer is superimposed on the deviation signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (10). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and the instantaneous errors between the model and the measured signal. (d) Zoom of the pressure signal between the descent phase and the ascent phase to illustrate the oscillations due to the movement of the boat. "> Figure 4 Cont.
(a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The total expanded uncertainty per water layer is superimposed on the deviation signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (10). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and the instantaneous errors between the model and the measured signal. (d) Zoom of the pressure signal between the descent phase and the ascent phase to illustrate the oscillations due to the movement of the boat. "> Figure 5
On the left, enlargement of one part of the profile no. 0006 descent (orange line) and visualisation of the numerical model obtained with Equation (8) (green line). Large and rapid variations in temperature appear that do not follow the detected temperature gradients (in red). On the right, visualisation of the corresponding recorded pressure variations. ">

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Abstract

CTD profilers are used as reference instruments to qualify temperature and salinity data. Their metrological specifications can be controlled in a calibration bath, and calibration coefficients can be applied to correct the linearity of sensors and the trueness of measured data with a given uncertainty. However, in ocean areas with thermal gradients, the uncertainty of the measured data is questionable due to the thermal inertia of sensors and the movements of the CTD in relation to the roll or pitch of the boat. In order to evaluate these measurement uncertainties and in order to be able to use the upcast profiles, a double C–T sensor SBE 9 profiler was fixed under a carousel water sampler, the second C–T couple being at the top of the carousel frame. This configuration allows the evaluation of the temperature measurement deviations of recorded profiles. In order to quantify the different sources of instrumental uncertainties, the temperature signal has been modelled accounting for the movements induced by the boat. The result allows one to quantify what can be called the representativeness of CTD’s temperature measurements. This notion is very useful in the data assimilation process. A table quantifying the various sources of uncertainty has been created from profiles obtained during four offshore campaigns. In the future, it could be used to find the representativeness of similar profiles obtained with a single pair of sensors. Ship-based CTD profiles are generally considered as perfect or without uncertainty in data assimilation and in the qualification per comparison of other instruments (XBT, Argo profiles, etc.). Our findings imply that this hypothesis will have to be reconsidered.

Keywords:

CTD profiler; representativeness; uncertainty; response time; temperature gradient

1. Introduction

CTD profilers were designed in response to the World Ocean Circulation Experiments (WOCE) programme. The programme suggested that the quantities temperature and conductivity should be measured, respectively, to 0.002 °C and 0.002 mS cm⁻¹, resulting in a salinity measurement accuracy of ±0.002 [1]. Le Menn [2] showed that if the uncertainty of ±0.002 °C could be kept in the calibration bath, the expanded uncertainty of calculated practical salinity was close to ±0.003.

Therefore, ship-based CTD profiles are used as a reference to qualify temperature and salinity data from other instruments, like XBT [3], Argo floats [4], or marine mammal data loggers [5], using collocated profiles. The CTD profiler’s metrological specifications can be controlled in a calibration bath, and calibration coefficients can be applied to correct the linearity of sensors and the trueness of measured data, but no methodology exists to quantify the uncertainties obtained during in situ measurement profiles. In 2018, Raiteri et al. [6] applied the formula of uncertainties propagations to salinity data acquired during profiles made in the Gulf of La Spezzia, but the temperature profiles were not evaluated and the errors in relation to the dynamic effects were excluded. In 2022, Waldmann et al. [7] proposed a methodology to assess the uncertainties of CDTs used on moorings, but this configuration excluded the uncertainties in relation to the dynamic effects. In 2023, Wong et al. [8] published an article to describe the Argo delayed-mode process and to validate it to quantify residual errors and regional variations in uncertainty, but this procedure cannot be applied to ship-based CTD profiles. The goal of this publication is to propose a methodology to assess temperature profile measurement uncertainties that include static and dynamic effects.

When used at sea, CTDs are generally fixed onto the bottom of a carousel water sampler frame (See Figure 1). This equipment is lowered at different depths, depending on the depth of the seafloor, via a trawl. This configuration makes it possible to exploit the conductivity–temperature (C–T) profiles obtained during downcasts. In order to exploit the upcast profiles by avoiding the water mixing in the carousel water sampler, the idea came to fix another C–T pair at the top of the carousel. This innovative configuration also opens up the possibility to assess the natural variability of the medium in terms of temperature between the downcast and the upcast in relation to a variable time scale of measurements (10 min to several hours). With properly calibrated sensors, in a quiet and homogeneous medium, this difference might be close to the calibration uncertainty or, in the best cases, to the resolution of the instrument. However, the profiles acquired during different campaigns at sea showed that this case is rarely met; several factors in relation to the natural variability and the instrument ensure that deviations are observed.

If the measurement uncertainty of the instrument in relation to its calibration, its sensors specifications, and its drift in time can be evaluated, the remaining differences can be used to quantify the natural variability during the time of the down–up profile. This quantification can be used to describe the representativeness of temperature measurements at different depths, at the location of the profile. This is another goal of this publication.

Representativeness has been the subject of several different definitions. In 1981, Nappo et al. [9] defined it as “the extent to which a set of measurements taken in a given space–time domain reflects the actual conditions in the same or different space–time domain”. This definition is similar to the definition of reproducibility given in the International Vocabulary of Metrology, or VIM [10], and corresponds fairly well with the notion of representativeness of the CTD profiles as we try to assess it in this publication. It is also close to the one given by R. Cooley et al. in 2020 [11], for whom representativeness can be defined purely in terms of the ability of the observational sampling to resolve the spatiotemporal scales of interest, which is entirely independent of measurement/instrument error. According to them, it is also in relation to the error component associated with the representativeness of a single observation for a certain application.

Therefore, the notion and the definition of representativeness has prompted great interest in data assimilation with optimal estimation [12,13,14]. Since the 1980s, the literature on this subject has been abundant, and we will only focus on recent publications that have tried to explain the use of this concept. In 2015, Hodyss et al. [15] defined it as the inability of a forecast model to accurately simulate the climatology of the truth. This very general definition illustrates the fact that it is difficult to discern the sources of errors in forecast models. In this representation, “the truth” refers to the observation point, which is considered to have a neglectable uncertainty and which is considered to see the small-scale process, so that the model achieves relatively coarser states. In fact, the representativeness error, also called the representation error (RE), includes the measurement errors and the representation errors obtained by numerical models, and it describes the uncertainty of using a single measurement to represent the gridded averages for a certain spatial and temporal resolution.

In 2008, Oke and Sakov [16] defined representativity as the component of observation error due to unresolved scales and processes, and they considered that the main source of RE is due to the limited (spatial and temporal) resolution of available observations. Their results suggest that the values of REs are typically greater than or at least comparable to measurement errors, particularly in regions of strong mesoscale variability [17]. This description was taken up by Janji’c et al. in 2018 [13]. However, according to them, the observation error has two components, the representation error, which depends on how the measurements are used, for example in a data assimilation process, and the measurement error, which is associated with the measurement device alone. In this description, the natural variability around the device during the data acquisition is basically ignored.

According to Schiller et al. [18], RE is also dependent on eddy activity. Efforts are being made to improve forecasts in ocean eddies because, according to Rykova [19], the misfits between observations and analyses from high-resolution ocean forecast systems are large. She quantified this mis-fit between observed and analysed fields to be 0.4–0.9 °C for subsurface temperature, 0.06–0.16 for subsurface salinity, and 0.2–0.6 °C for sea surface temperature, so that measurement errors are shown to be 0.004 °C for subsurface temperature and 0.01 for subsurface salinity for Argo profilers.

With the double CTD sensor configuration described in Section 2, we have the possibility to quantify the representativity or the degree of trueness of single profiles and of groups of profiles, in order to make a better assessment of the measurement error and of the natural variability during the measurements, as per oceanic areas, and to determine if this error is really neglectable or is of the same order as the observation error, as defined by Oke and Sakov. It is generally assumed that REs are horizontally uniform and only depth dependent [19]. This hypothesis could also be tested by considering the measurement uncertainties of measured temperatures. An analysis of measurement uncertainties is developed in Section 3 to determine the origin of deviations between downcast and upcast. This uncertainty budget accounts for the location in depth of temperature measurements and the effects of sensor response time. We hope that this approach will allow a better understanding of the existing deviations between observed and analysed fields. For that, and to evaluate the uncertainties in relation to the response time, we developed a numerical model of the measured temperature that was used to simulate profiles obtained during different campaigns at sea. This model includes the effects of the boat’s movements. It is developed in Section 4 and Section 5, and the results are presented in Section 6, based on two examples of profiles. Section 7 presents a short description of the campaign locations used in this study. A discussion of these results is presented in Section 8, along with the perspectives they offer.

2. The Experimental Apparatus

Measurements at sea are made with a carousel water sampler with a double frame under it (see Figure 1). The frame at the bottom contains a SBE 9⁺ CTD profiler fixed in a horizontal position. This CTD contains the first CT pair (SBE 3 and SBE 4 sensors from Sea-Bird Scientific) used to measure temperature–conductivity data during the descent of the carousel. The water is pumped from a hose, the end of which is fixed to the lowest part of the frame. The second CT pair is fixed to the frame of the sample bottle rosette, along with a second pump. The water is drawn in at the same level as the top of the frame.

The SBE 3 and SBE 4 sensors were calibrated before and after the campaigns with the SBE 9⁺ that was used during the campaign to ±0.002 °C in the range 0–32 °C and ±0.004 mS/cm in the range 0 mS/cm to 60 mS/cm. For the SBE 4 sensors, the calibration uncertainty can be higher according to the linearity and the reproducibility of its measurements in the bath, but it is never higher than ±0.006 mS/cm. This study is focused mostly on temperature profiles. The drift over time of each sensor is known and can be quantified using the calibration correction history.

3. The Measurement Uncertainties of Temperature Measurements

The sensitive elements of SBE 3 sensors are thermistors. Thermistors have a non-linear response corrected with the Bennett formula and the coefficients given in Sea-Bird Scientific calibration reports. After using these coefficients, the remaining linearity uncertainty, u_l, is less than 0.1 mK [20]. This uncertainty appears like a residual Gaussian noise.

As thermistors are resistive sensors, they are fed by a current; thus, another source of uncertainty is the self-heating created by the Joule effect. To reduce this, manufacturers use very small currents to feed the thermistor. Sea-Bird Scientific warrants a self-heating error inferior to 0.1 mK in still water for the SBE 3 sensor [20]. Since the only knowledge we have of this uncertainty, u_sh, is its maximum value, we can assign it a rectangular distribution.

Sea-Bird Scientific warrants an initial temperature accuracy of 0.002 °C. This can be considered as an expanded calibration uncertainty. It can be reached between −1 °C and 32 °C with a very stable calibration bath and a reference sensor regularly calibrated in fixed point cells of the International Temperature Scale of 1990 (ITS-90), regularly linked to the references of National Metrology Institutes. Considering these constraints, in this assessment the standard calibration uncertainty, u_c, will be considered to be 1 mK.

At depth, pressure can compress the needle which shelters the thermistor. This effect was ignored until several assessments were carried out. From in situ comparisons of SBE 3 and SBE 35, Uchida et al. concluded, in 2007, that three probes were not very sensitive to pressure and eight had errors of 1 to 2 mK at a pressure of 600 bar [21]. In another study carried out in 2015, Uchida et al. found one SBE 3 with an error of −0.06 mK at 600 bar and another one (already tested in 2007) with an error of 5.07 mK instead of 1.94 mK in situ [22]. In 2017, experiments carried out in a pressure chamber on one SBE 3 showed a sensitivity of −77 μK MPa⁻¹ from 0.1 to 60 MPa [23]. That is equivalent to an error of −4.6 mK at 600 bar. Therefore, it appears that the pressure sensitivity of SBE 3 is variable from one probe to the other and varies from neglectable to 5 mK at 600 bar. The SBE 3 sensors used during our measurements had never been tested under pressure. The maximal uncertainty caused by the pressure effect is then u_p = 8 μK bar⁻¹. According to the Guide to the Expression of Uncertainty in Measurement [24], the standard uncertainty of this effect can be evaluated to be 4.8 μK bar⁻¹ or 2.9 mK at 600 bar.

The acceleration of flow as it meets the surface of the sensor provokes an increase in temperature related to the fluid viscosity. This effect was studied and quantified by Larson and Pedersen in 1996 [25]. According to their communication, the viscous heating uncertainty can be assessed with the relation dt = 1.263 × 10⁻⁴ Pr^0.5U², where Pr is the Prandlt number and U is the speed of the flow. If U = 1 m s⁻¹, dt ≈ 1 mK. If the flow changes or is not the same between the calibration and the time the sensor is used, this uncertainty should be considered. As it cannot be corrected, it corresponds to a standard uncertainty, u_v, of 1 mK.

The drift over time of sensors between two calibrations must also be considered in the uncertainty budget. Figure 2 presents the drift of the SBE 3 sensors used in this study. Explanations of how these drifts are obtained can be found in reference [26]. It is generally very low: between 6 μK/year (sn 4409(2)) and 0.29 mK/year at 15 °C for the older SBE 3s (sn 4398 and sn 4409(1)). However, depending on use, the drift between two calibrations may be greater; for example, sn 6528 (pale blue) had been used in hard conditions, on a towed fish SeaSoar. For more information about the using of this towed fish, see reference [27]. This one was not used for the double sensor measurements. The drift uncertainty, u_d, can be considered to be 0.29 mK/year, with a maximum time between two calibrations of 12 months.

During temperature measurements at sea, and particularly as the sensor passes through temperature gradients, another parameter can introduce measurement uncertainties: the depth positioning. This uncertainty depends on the pressure sensor measurement trueness. True depth positioning is mandatory to follow the evolution of the thermal content of oceans. For the World Ocean Circulation Experiments (WOCE), the specification is 1 dbar at 6000 dbar. At low depth, constraints are more important because of temperature gradients and thus are not well defined. Both the resolution and accuracy of pressure sensors decrease as their measurement range increases. SBE 9⁺s are equipped with Piezoelectric sensors (Digiquartz from ParoScientific). These sensors have a very good repeatability of 0.005% of the full scale, corresponding to a standard uncertainty of u_r = 0.34 dbar for a 6800 dbar range. Their thermal sensitivity is 0.0008% of the full scale °C⁻¹, with a thermal variation range of ±20 °C that leads to an uncertainty of u_Ts = 0.98 dbar. Their initial accuracy is ±0.015% or ≈1.02 dbar for a 6800 dbar range, and their typical stability is ±0.02% per year. This can be considered as a maximal drift uncertainty, u_d. For one annual calibration, it corresponds to ≈1.4 dbar/year (6800 dbar).

However, with an automated pressure balance, such as the PG7502 from Fluke Calibration, it is possible to obtain a calibration standard uncertainty of u_c = 0.2 dbar. The pressure sensor’s drift of the SBE 9⁺ sn 766 used on our carousel is 0.25 dbar year⁻¹.

In the case of the error curves, which can be obtained with the double sensor measurements, this uncertainty is cancelled out to the nearest error due to sensor hysteresis. The typical hysteresis of a Digiquartz is 0.005% of the full scale, or 0.34 dbar, but measurements with a pressure balance show that this number is rarely met. For the SBE 9⁺ sn 766 used on our carrousel, the hysteresis standard uncertainty was evaluated to be u_h = 0.02 dbar (Students distribution with five degrees of freedom). Table 1 gives the standard uncertainty budget of the SBE 9⁺ sn 766 pressure sensor in the case of a single downward profile and in the case of the double sensor configuration. As the influence quantities are independent of each other, the combined standard uncertainty on the pressure measurements, u_c(p), is obtained by a simple quadratic sum:

u_{c} (p) = \sqrt{{(\frac{u_{r}}{\sqrt{6}})}^{2} + {(\frac{u_{T s}}{\sqrt{6}})}^{2} + {(\frac{u_{d}}{\sqrt{3}})}^{2} + + u_{c}^{2}}

(1)

The last uncertainty to evaluate is in relation to the response time of the temperature sensor. This uncertainty, called u_τ, is studied in detail in Section 4. Considering the details given in this section, the combined standard uncertainty on temperature measurements can be assessed by Equation (2):

u_{c} (T) = \sqrt{{u_{l}^{2} + (\frac{u_{s h}}{\sqrt{3}})}^{2} + u_{c}^{2} + {u_{v}^{2} + (\frac{u_{d}}{\sqrt{3}})}^{2} + {(\frac{u_{p} \times p}{\sqrt{3}})}^{2} + {{(u_{c} (p) \times g r a d (T))}^{2} + u}_{τ}^{2}}

(2)

To conclude this section, the uncertainty budget of SBE 9⁺ temperature measurements is largely dominated by the uncertainty of the depth positioning in the surface oceanic layers. Table 2 shows the uncertainty budget calculated from Equation (2), at 0 dbar and at 600 dbar, where the positioning effect is null and where the temperature gradients are very low. The result shows that, at the surface, it is possible to obtain a standard uncertainty of 1.43 mK and, at a depth corresponding to 600 bar, we obtained 3.22 mbar for an SBE 3 where the pressure effect on its sensor was not known. It appears that this uncertainty is by far the largest and that it is important to determine and to correct the effect of pressure on each SBE 3 sensors in order to improve the uncertainty budget at great depths.

4. The Equation of the Measured Temperature

In order to quantify the errors related to the response time of the temperature sensor to implement the relation (2) and to better understand the existing deviations between observed and analysed fields, we set up a model of the measured temperature, based on the general response of temperature sensors to temperature changes.

The carousel water sampler is lowered to depth using a winch that unwinds constantly. Before being lowered, it remains under the surface for 10 min in order to gain an equilibrium between the instrument temperature and the environment temperature. The same type of plateau is also performed before the ascent. In the ocean, radiation phenomena can be neglected, and the heat flow exchange between the sensor and the fluid is essentially made by convection. According to [28], the response of the sensor to a variation in temperature is governed by the following equation:

T_{1} - T = τ \frac{d T}{d t}

(3)

where T₁ is the temperature of the medium, T is the measured temperature, τ is the response time of the sensor, and t is the time. If T₁ remains constant, the solution of Equation (3) is [29]:

T - T_{1} = (T_{0} - T_{1}) e^{\frac{- t}{τ}}

(4)

where T₀ is the initial temperature of the sensor.

First, let us consider a quiet sea. In this case, the lowering is carried out at a constant speed, v, expressed in °C s⁻¹, and the evolution of the temperature seen by the sensor can be considered as being linear or piecewise linear. Figure 3b and Figure 4b give two examples of profiles segmented into segments over which the temperature is considered linear. In a segment, the sensor must measure a temperature, T₁, that changes according to the following equation:

T_{1} = T_{10} + v t

(5)

where T₁₀ is the initial the temperature of the medium. After replacing T₁ by its value in Equation (3), the solution of this differential equation is basically:

T = (T_{0} - T_{10} + τ v) e^{\frac{- t}{τ}} + v t + T_{10} - τ v

(6)

The first terms of Equation (6) describe the temperature rise of the sensor. The last term, τ v, describes the difference that will remain between the temperature of the medium during the sensor’s ascent or descent and the measured temperature, T. This will allow one to quantify the uncertainty introduced by the response time of the temperature sensor. According to the Sea-Bird Scientific datasheet, for SBE 3 probes, τ = 65 ± 10 ms. v can be obtained by using the pressure records.

Second, let us now consider the case of a sea state producing regular rolling movements of the boat. The lowering is always carried out at a constant speed, v, but the carousel is subjected to a pulsation of ω = 2πf, where f is the frequency of oscillations. Figure 4d gives an example of a pressure profile where oscillations are visible. Equation (5) becomes:

T_{1} = T_{10} + v t + ∆ T_{1} (t) s i n (ω t)

(7)

where ΔT₁(t) describes the thermal amplitude of oscillations. This will vary in time with the thermal gradients and the amplitude of the boat oscillations according to Relationship (8):

T_{1} (t) = g r a d (T) \times ∆ p (t, ω)

(8)

The sensor will measure the temperature, T, given by Equation (9):

T = T_{1 M} + ∆ T (t) s i n (ω t + φ)

(9)

where φ is the phase shift introduced by the sensor response, ΔT is the amplitude of the temperature variation measured by the sensor, and T_1M represents the terms

T_{10} + v t

. The solution is obtained by replacing the terms of Equation (3) by Equations (7) and (9), and the solution is:

T = (T_{0} - T_{10} + τ v + \frac{τ ω ∆ T_{1}}{1 + {(ω τ)}^{2}}) e^{\frac{- t}{τ}} + v t + T_{10} - τ v + \frac{∆ T_{1} (t)}{\sqrt{1 + {(ω τ)}^{2}}} s i n (ω t + φ)

(10)

In Relationship (10), the first terms again describe the temperature rise of the sensor. The last two terms,

- τ v + \frac{∆ T_{1} (t)}{\sqrt{1 + {(ω τ)}^{2}}} s i n (ω t + φ)

, describe the difference that will remain between the temperature of the medium and the measured temperature, T. The value of φ can be determined with Equation (11):

φ = a s i n (\frac{- τ ω}{\sqrt{1 + {(ω τ)}^{2}}})

(11)

Table 3 gives values of the response time uncertainties obtained from a CTD profile made in the Mediterranean Sea in April 2024 (average longitude: 21.88, average latitude: 35.37), when the sea was quiet. The profile was made at a depth of 2026 m. From 0 to 2000 dbar, the values of the response time uncertainty were obtained by using the product τ v. The table is extrapolated to 600 bar to obtain values of the total standard uncertainty that take into account the positioning and the pressure effect uncertainties. The column ‘Constant uncertainties’ presents the sum of uncertainties: linearity, self-heating, calibration, viscous heating, and drift over time, as given in Table 2. It is clear that the response time uncertainty is not dominating the budget, even in the first layers.

5. Implementation of the Equations of the Measured Temperature

Relationship (10) was programmed using Python 3.12.5 software (Packaged by Conda-forge). As the carousel remains at a constant depth before the descents or the ascents, the initial sensor temperature is the same as the medium. The term

(T_{0} - T_{10})

is null.

In Relationship (8),

g r a d (T)

is the temperature difference between two depths given by the pressure sensor. The problem is to automatically detect slope changes in the temperature record. This detection was made from the method described in [30]. This publication used a Python package called Ruptures (v1.1.9) (http://ctruong.perso.math.cnrs.fr/ruptures, accessed on 8 January 2025), based on the combination of three factors: a cost function, a research method, and a constraint. This package can be loaded at the following address: https://github.com/deepcharles/ruptures (v1.1.9, accessed on 8 January 2025). The cost function measures the number of breakpoints in the signal. The constraint corresponds to the number of breakpoints to detect. After trials and evaluations of different solutions, we used the cost function “Continuous Linear (CLinear)”, which calculates the fitting error between the signal and a straight line. The research function we used is “Dynamic programming (Dynp)”, which finds the minimum sum of costs by calculating the cost of all the sub-sequences of the signal under study. There can be a maximum of 29 breakpoints per descent and 29 per ascent, corresponding to 60 gradients per profile. To avoid false detections due to the oscillating movements of the boat, the time between two detections is filtrated to 24 × 1/f, where 24 is the sampling frequency of the CTD profiler. A threshold is also applied to eliminate quick temperature variations during the detection. The study of profiles acquired during a campaign, called GDG22 (see Section 7), made in the Bay of Biscay with a quiet sea, allowed the detection of thermal gradients close to 1 °C dbar⁻¹. Another, carried out near the Faroe Islands (GN23) in very rough seas, found gradients of less than 0.3 °C dbar⁻¹. Therefore, thresholds have been fixed to 1 °C dbar⁻¹ for sea states ≤ 2 and 0.3 °C dbar⁻¹ for sea states > 2.

Relationships (6) and (10) require the calculation of v, the speed at which the cage rises or falls. v is obtained by calculating the ratio of the difference between two consecutive temperatures and the difference in time, from the original signal sampled at 24 Hz.

Relationships (10) and (11) require the determination of ω. ω is extracted from the pressure signal at the time when the carousel is in standby at maximal depth for a few minutes. The boat oscillations are visible (see Figure 4d), and after using a low pass filter, Fourier Transform (FT) can be applied. The positive frequencies of this FT are kept in order to calculate the power spectral density, which allows one to determine the principal frequency, f, of the signal.

Knowing the amplitude of oscillations is also required to calculate

∆ p (t, ω)

in Relationship (6). In standby phases, this amplitude essentially depends on the boat movements and on the lift of the carousel in the water. During the descent and ascent phases, this is superimposed on the speed at which the cable unwinds. A sliding window, FT, is therefore required to measure this amplitude along the descent and ascent profiles, giving a spectrogram from which we can extract the amplitude values at the frequency determined previously.

6. Results of the Implementation

Relationships (6) and (10) have been tested on a significant amount of profiles in order to fill a table of results. To choose between these two relationships, the programme first detects the presence or absence of oscillations in the pressure signal when the profiler is at the bottom. As examples, Figure 3 shows the results of the application of Relationship (6) on a profile of 1500 m obtained in the Mediterranean Sea with a quiet sea state of 2, and Figure 4 shows the application of Relationship (10) on another one of 1000 m depth obtained in the North Sea, selected because the ‘sea state’ was of level 4. Figure 3a shows the measured temperature during the downcast and the upcast and the error signal obtained from the two temperature sensors. This error signal,

δ T

, takes into account the difference in height of the two temperature sensors according to the following relationship:

δ T = T_{b o t t o m} (p) - T_{t o p} (p - 1.4)

(12)

where T_bottom is the temperature measured by the sensor at the bottom of the carousel, T_top is the temperature measured by the sensor located at the top, and 1.4 is the distance between the two sensors. The combined expanded uncertainty of

δ T

per water layer is superimposed on some values of the error signal located at 7, 24, 50, 75, 100, 225, 380, 500, 750, 1000, 1250, and 1500 dbar. It is obtained with Equation (2), multiplied by

2 \sqrt{2}

to take into account the two independent temperature values measured during the downcast and the upcast. The downcast lasted 32 min, while the upcast lasted 45 min. At 0 m depth, the maximum delay is then 77 min and the maximal deviation is 0.198 °C. Down to a depth of about 700 m, the amplitude of the deviation signal is much greater than the instrumental uncertainty, demonstrating the natural variability of the medium between the moment of descent and the moment of ascent. The variability of shallow waters is in relation to tides, currents, internal waves, sun, cloud activity, etc. This variability is known, but it was hardly quantified. Figure 3b shows the temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (6). The two curves overlap perfectly. The 18 stars superimposed on the signal delimit the segments found by automatic slope change detection. Figure 3c shows the same results for the downcast temperature profile as a function of pressure, with added signal of error between the model and the measured temperature. This error signal is significant in the first 10 m, but its maximal amplitude is between −0.012 °C and 0.05 °C so that the deviation signal of the figure (a) is between −0.2 °C and +0.10 °C. If this profile was used as a reference in data assimilation, its uncertainty or representativity would be 0.3 °C at this depth, so the model error is 0.065 °C at the same depth. The origin of these errors can be partially attributed to the assumption that the term

(T_{0} - T_{10})

is null. This assumption is true at the beginning of the descent and at the beginning of the ascent, but not at the start of major temperature gradients. Table 4 gives the ratio values between these two signals per water layer. Finally, Figure 3d shows the amplitudes of the different terms of Equation (6). The violet colour represents the first part of Equation (6), i.e., the transient part of the sensor temperature rise. The red colour represents the sensor’s response time error, τv. The orange colour represents the temporal thermal variations obtained with the vt term.

Figure 4a also shows the measured temperature values during the downcast and the upcast and the deviation signal calculated from the two temperature sensors, but on a profile obtained with a sea state of 4. The downcast lasted 19 min and the upcast 16.5 min. At 0 m depth, the maximum delay is then 35.5 min, and the maximal deviation is 0.01 °C in the 0–10 m layer. It should be noted that, for this profile, the maximum deviation is in the 100–500 m range and is therefore independent of the time between the measurement on descent and the measurement on ascent of the cage. The uncertainty bars are again smaller than the deviation signal in the first 400 metres. They are represented by the pressures 7, 24, 50, 75, 100, 225, 380, 500, 666, 804, and 998 dbar. Figure 4b shows the temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (10). Once again, the two curves overlap perfectly. Fourty-two changes of slope were required to describe the gradients. Figure 4c shows the downcast temperature profile as a function of pressure, with the numerical model superimposed, and in green the point by point errors between the model and the measured signal. It appears that the instantaneous errors are about ten times smaller than the spikes of the deviation signal of the Figure 4a, and probably in relation to the water mixing generated by the upward and downward movements of the carousel (see discussion in Section 8). Table 4 gives the ratio values between these two signals per water layer. Figure 4d shows the pressure signal with the time during which the cage is at rest before ascending. The oscillations associated with the movement of the boat are clearly visible. It is this part of the signal that is used to measure the pulsation, ω, of oscillations.

The same parameters are given for the deviation signal (downcast–upcast), and the ratio between the deviation signal and the model is calculated. The maximal errors of the model are between 2 and 21.4 times smaller than the deviation signal, and the standard deviations of the model are between 3.8 and 42.6 times smaller than the deviation signal.

7. Location and Description of Measurement Sites Used

The data on which this study rely were acquired in different environments regarding latitude, period of the year, and oceanic processes. An overview of the dataset gathered is given in Table 5 below.

The general comments about the data acquisition summed up in Table 5 are as follows:

-: All the profiles were acquired with the dual CT described previously.
-: All the data acquired during GdG22 are weakly influenced by large internal tides.
-: The only rather rough sea state conditions were met during the GN23 experiment and used as a reference to assess the influence of the rolling movement on the measured temperature model.
-: The most marked seasonal thermocline (~35 m/ΔT 6 °C) was met during the GdG22 and is likely to demonstrate vertical oscillations of the order of 5 to 30 m amplitude (due to internal tide and internal solitary waves).
-: GIB20 and PR24 where performed in the vicinity of a large mesoscale gyre (Alboran gyre) or eddies (Pelops anticyclone).

8. Discussion

Ship-based CTD profiles are currently used as references to qualify temperature and salinity data from other instruments and in data assimilation with optimal estimation, but the uncertainties associated with these profiles are generally considered to be equivalent to the calibration uncertainties of the temperature or conductivity sensors of these instruments. The effects of the pressure measurement uncertainties are rarely considered, even though their contribution to the uncertainty budget of temperature measurements may be high, particularly in areas where the vertical temperature gradients are significant.

We could wonder if this problem is the same with Argo profiling floats. As most drifting floats are fitted with Sea-Bird Scientific SBE 41 CTDs, it is possible to assess the uncertainties of their temperature measurements from the documents available on the Sea-Bird Scientific website. As the temperature sensor is identical to that of the CTD SBE 9⁺ profilers, its constant measurement uncertainties are identical to those given in Table 2, except for the drift over time, u_d. In the Sea-Bird Scientific documentation, this is calculated at 0.2 mK/year instead of the measured 0.29 mK/year. If we take this into account over 4 years, the final uncertainty is 0.16 mK instead of 0.17 mK, but it does not change the final result.

Its response in temperature is given by Equation (4). As the instrument ascends at a very low speed (≈0.1 m/s), the uncertainty linked to its response time during the crossing of temperature gradients will be smaller. For example, with a temperature gradient of 0.15 °C dbar⁻¹, its theoretical uncertainty is 0.001 °C so that, for an SBE 9 at 1 m s⁻¹, the theoretical uncertainty is 0.010 °C.

Most SBE 41s are equipped with the DRUCK PDCR 1830 pressure sensor [31]. The characteristics of this sensor can be retrieved from the Sea-Bird Scientific and DRUCK documents, and the pressure measurement standard combined uncertainty, u_c(p_float), can be calculated thanks to Equation (13):

u_{c} (p_{f l o a t}) = \sqrt{{(\frac{u_{r}}{\sqrt{3}})}^{2} + {(\frac{u_{T s}}{\sqrt{6}})}^{2} + {(\frac{u_{d}}{\sqrt{3}})}^{2} + {(\frac{u_{h}}{\sqrt{3}})}^{2} + {(\frac{u_{c}}{\sqrt{6}})}^{2}}

(13)

Table 6 gives the results of this equation and the values of each term. The repeatability is taken from [31], Figure 3 (u_r = 0.25 dbar), and divided by the square root of 3 because 0.25 dbar is a maximum value. The lifetime drift (u_d = 1.5 dbar) is also taken from Figure 1 of this document and divided by the square root of 3. The value of the hysteresis has not been found, but we can consider that it is included in the lifetime drift as the values of Figure 1 come from measurements made when the floats were close to the surface after an ascent from 2000 m. The value of the thermal sensitivity can be found from DRUCK GE Sensing documents [32]. The temperature effect is 0.3% of the full scale in the temperature error band, and that gives u_Ts = 0.60 dbar. The uncertainty in relation to the calibration can be found in the datasheet SBE 41/41CP Argo CTD, under the term of initial accuracy (u_c = ± 2 dbar). The thermal sensitivity and the initial accuracy have been divided by the square root of six, according to paragraph F.2.3.3 of reference [24]. The pressure combined a standard uncertainty of u_c(p_float) = 1.22 dbar, so that, for an SBE 9⁺, it is 0.49 dbar. Argo floats perform more accurate temperature measurements because of their low ascent speed, which makes the error term, τ v, very small, but they are clearly disadvantaged by the uncertainty budget of their pressure sensor when they meet temperature gradients. This is shown in Table 7, which uses the temperature gradient values from Table 3 up to 2000 m. In the 10–50 dbar and 50–100 dbar layers, the temperature combined standard uncertainty is multiplied by more than 2 compared to what is obtained with the SBE 911+.

The other problem raised by this study concerns the large discrepancies observed in the first layers of water between the temperature measurements taken on descent and on ascent. These differences can be seen in Figure 3a and Figure 4a. They are generally much greater than the combined standard uncertainties calculated for the corresponding water layers. Figure 5 is a zoom on one part of profile GN23_006 of Figure 4b, between 140 and 230 s. We can see that rapid temperature variations of up to 0.26 °C are taken into account by the numerical model, but they are not considered in the gradient’s detection. It is likely that, with these oscillations, the carousel is dragged downwards then upwards (between 140 and 150 s) then upwards only (between 160 and 170 s), generating a mixing of the medium and large temperature measurement errors, compared with the gradients that should be measured. The sea state must therefore be accounted for when processing temperature profiles, as it can cause significant measurement errors, and in this case it is difficult to distinguish the part of the measurement uncertainty linked to the natural variability of the environment and the part linked to the mixing of the medium by the carousel.

However, Figure 3a shows that when the sea is calm (sea state 2), in the first layers of water the differences in temperature between descent and ascent can also be significant compared with the calculated uncertainty. These differences can only be interpreted as the natural variability of the environment in the time between descent and ascent to and from a given depth. The uncertainty in the measurement of this variability can be estimated as the combined instrumental uncertainty calculated for each water section.

In either case, this error signal can be interpreted as a measure of the representativeness of the CTD profile to within the combined instrumental uncertainties. However, these uncertainties may be greatly underestimated in gradient areas, in cases where the vessel is oscillating due to the sea state.

The results of four campaigns have been compiled in an Excel table that contains the following:

-: The characteristics of the profile studied: coordinates, maximal depth, downcast duration, upcast duration;
-: The water layers, such as those given in the first column of Table 3;
-: The pressure measurement standard uncertainty corresponding to the uncertainty budget of Table 1;
-: The temperature maximal gradient per water layer, calculated with the Python programme;
-: The positioning uncertainty in depth, obtained by multiplying the two previous columns. As the maximum gradient is considered, the result is divided by the square root of 3;
-: The pressure effect uncertainty on the temperature sensor per water layer;
-: The temperature constant standard uncertainty of the temperature sensor, as described in Table 3;
-: The maximal τ v values per water layer, found during the downcast and the upcast and divided by the square root of 3;
-: The uncertainty linked to the sensor response time and the boat movements. It corresponds to the difference between Equation (7) and the last term of Equation (10);
-: The expanded combined instrumental uncertainty calculated with the previous column, per water layers;
-: The maximum downward/upward error and the standard deviation of this error per water layer calculated with Relationship (12);
-: The amplitude of the horizontal current component per water layer when available;
-: The presence or absence of internal waves.

Table 8 gives an example of how to use the data of the Excel table calculated from a campaign comprising six profiles carried out between 63.855 and 64.397 in latitude and 8.120 and 8.135 in longitude. The average durations of downcasts were of 13.5 ± 6 min and for upcasts 13.0 ± 8 min. This indicates the mean maximum deviation, the mean standard deviation, and the expanded combined instrumental uncertainty (to 95%), making it possible to quantify the representativeness of the data in this zone. It also shows that, for this area, the RE defined in the introduction presents the same non-uniformity horizontally and vertically for depths between 10 and 500 m.

However, the ultimate aim of the Excel table is to create a database that can be used to quantify the representativeness of CTD profiles carried out in areas close to the recorded profiles. Combined with a program based on artificial intelligence, it should make it possible to associate values of instrumental uncertainty and natural variability by water level with profiles obtained with a single pair of sensors.

9. Conclusions

The dual-sensor CTD configuration is an innovation that allows the assessment of the representativeness of temperature profiles obtained during campaigns at sea. The work carried out on the temperature profiles obtained using this method and the methodology proposed to assess uncertainties of temperature profiles have shown that the representativeness of the profiles includes an instrumental uncertainty, but also a contribution of natural variability that is often much greater than the instrumental uncertainty in the first layers of water. Ship-based CTD profiles are generally considered as perfect or without uncertainty in data assimilation and in the qualification per comparison of the other instruments (XBT, Argo profiles, etc.). Our findings imply that this hypothesis will have to be reconsidered.

It has been possible to quantify the difference between the two by assessing the uncertainty of the measured temperature and establishing a model of the sensor’s response to temperature variations. This study revealed that the factor that has the greatest influence on measurement uncertainty is the positioning of the temperature value as a function of pressure in areas where pressure gradients are significant. It is also revealed that, while Argo floats perform more accurate temperature measurements because of their low ascent speed, they are clearly disadvantaged by the uncertainty budget of their pressure sensor when they meet temperature gradients.

It is also shown that another factor could be investigated to improve measurements at greater depths (>1100 m): the pressure sensitivity of SBE 3 sensors. A pressure characterisation of each sensor is recommended in order to reduce this uncertainty to one and a half mK, as shown in the budget set out in Table 2.

The equation was used to calculate the uncertainty associated with the response time of the temperature sensor and to show that this uncertainty does not dominate the final result. It also shows that the sea state must be taken into account when processing temperature profiles, as it can lead to significant measurement errors, and that, in this case, it is difficult to distinguish between the part of the measurement uncertainty linked to the natural variability of the environment and the part linked to the mixing of the medium by the carousel.

Author Contributions

Conceptualization, methodology, F.D.; writing—original draft preparation, writing—review and editing, M.L.M. and F.D.; software, validation, B.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by Shom.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

To make the data used in this publication available, send an e-mail to [email protected] or [email protected].

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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Figure 1. Carousel water sampler used for the measurements at sea. The SBE 9⁺ CTD profiler is located at the bottom with the first pair of CT sensors. The second pair (not visible on this photo) is located at the top. The pump and the pipe connecting it to the conductivity sensor are visible in the foreground.

Figure 2. Drift over time of SBE 3 sensors used during the campaigns at sea. During acquisitions, data are corrected for these drifts.

Figure 3. (a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The combined expanded uncertainty per water layer is superimposed on the error signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (6). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and, in green, the error signal between the model and the measurement. (d) Amplitudes of the different terms of Equation (6). The green colour represents the transient part of the sensor temperature rise of the equation. The orange colour represents the sensor’s response time error, τ v. The blue colour represents the temporal thermal variations obtained with the v t term.

Figure 4. (a) Measured temperature during the downcast and the upcast and deviation signal obtained from the two temperature sensors. The total expanded uncertainty per water layer is superimposed on the deviation signal (black bars). (b) Temperature profile as a function of time and the result of the numerical modelling of this signal obtained with Equation (10). The stars show the segments detected by the automated detection of slope changes. (c) Downcast temperature profile as a function of pressure with the overlapped numerical model and the instantaneous errors between the model and the measured signal. (d) Zoom of the pressure signal between the descent phase and the ascent phase to illustrate the oscillations due to the movement of the boat.

Figure 5. On the left, enlargement of one part of the profile no. 0006 descent (orange line) and visualisation of the numerical model obtained with Equation (8) (green line). Large and rapid variations in temperature appear that do not follow the detected temperature gradients (in red). On the right, visualisation of the corresponding recorded pressure variations.

Table 1. Uncertainty budget of the SBE 9⁺ sn 766 pressure sensor calculated from Equation (1), in the case of a double sensor configuration.

	Distribution	Double Profile
Repeatability	Triangular	0.14
Thermal sensitivity	Triangular	0.40
Drift (sn 766)	Rectangular	0.14
Hysteresis (sn 766)	t-distribution	0.02
Calibration	Gaussian	0.20
Pressure sensor standard combined uncertainty:		0.49

Table 2. Temperature uncertainty budget obtained from Equation (2), at the surface and at 600 bar for an SBE 3 probe where the pressure effect on the sensor has not been measured and compensated.

	pdf	At 0 bar (mK)	At 600 bar (mK)
Linearity	Gaussian	0.1	0.1
Self-heating	Rectangular	0.06	0.06
Calibration	Gaussian	1.00	1.00
Viscous heating	Gaussian	1.00	1.00
Drift over time	Rectangular	0.17	0.17
Pressure effect	Rectangular	0.00	2.90
Pressure positioning	Triangular	0.00	0.00
Response time	Gaussian	0.00	0.00
Temperature combined standard uncertainty:		1.43	3.22

Table 3. Calculation of the combined standard uncertainty with values taken from a profile (PR24_026 0–2026 dbar) obtained in the Mediterranean Sea in April 2024. The calculations are extrapolated to 6000 dbar, with values close to real temperature gradients, to see the influence of the pressure effect uncertainties and of the pressure positioning uncertainties on the total uncertainty at 600 bar.

Water Layers (dbar)	Pressure Sensor Uncertainty (dbar)	Temperature Gradient (°C dbar⁻¹)	Positioning Uncertainty (°C)	Response Time Uncertainty (°C)	Pressure Effect Uncertainty (°C)	Constant Uncertainties (°C)	Comnined Standard Uncertainty (°C)
0–10	0.49	0.096	0.0469	0.003	0.0000	0.0014	0.0470
10–50	0.49	0.067	0.0330	0.005	0.0000	0.0014	0.0334
50–100	0.49	0.008	0.0038	0.001	0.0000	0.0014	0.0042
100–500	0.49	0.005	0.0025	0.003	0.0002	0.0014	0.0041
500–1000	0.49	0.004	0.0017	0.002	0.0005	0.0014	0.0030
1000–2000	0.49	0.009	0.0044	0.001	0.0010	0.0014	0.0048
2000–3000	0.49	0.000	0.0000	0.000	0.0014	0.0014	0.0020
3000–4000	0.49	0.000	0.0000	0.000	0.0019	0.0014	0.0024
4000–5000	0.49	0.000	0.0000	0.000	0.0024	0.0014	0.0028
5000–6000	0.49	0.000	0.0000	0.000	0.0029	0.0014	0.0032

Table 4. Maximum and standard deviation of the model vs. the measured signal per water layer.

		Downcast		Upcast				Downcast		Upcast
Profile	Layer	Model Max Deviation (°C)	Model Standard Deviation	Model Max Deviation (°C)	Model Standard Deviation	Max Signal Deviations (°C)	Standard Deviation Signal Deviations	Max Signal Dev./Model Error Max	s.d. Signal Deviations/Model s.d. (°C)	Max Signal Dev./Model Error Max	s.d. Signal Deviations/Model s.d. (°C)
GN23_006	0–10	0.002	0.000	0.002	0.000	0.010	0.002	4.8	4.8	4.9	3.8
	10–50	−0.005	0.001	0.058	0.003	0.115	0.002	−21.4	40.2	2.0	7.1
	50–100	−0.055	0.006	0.027	0.003	0.434	0.086	−7.8	13.8	15.9	25.6
	100–500	0.071	0.004	−0.044	0.003	−0.566	0.118	−8.0	30.2	13.0	42.6
	500–1000	0.005	0.000	−0.003	0.000	−0.041	0.006	−8.9	15.3	15.0	12.0
PR24_033	0–10	0.051	0.004	−0.042	0.003	−0.198	0.037	−3.9	8.4	4.7	12.6
	10–50	0.037	0.004	−0.024	0.002	−0.173	0.082	−4.7	22.0	7.2	34.6
	50–100	0.016	0.002	−0.009	0.002	−0.150	0.026	−9.4	16.4	16.3	16.6
	100–500	0.015	0.001	0.011	0.001	−0.084	0.015	−5.7	16.0	−7.6	20.7
	500–1000	0.006	0.000	−0.009	0.001	−0.022	0.006	−3.5	14.7	2.4	11.4
	1000–2000	0.002	0.000	−0.002	0.000	0.011	0.002	6.5	10.1	−6.7	6.5

Table 5. Summary of campaigns used in this study.

Name of the Experiment/Acronym	Period/Year	Area	Zone	Number of Dual Profiles Performed/ Typical Depth
Protevs Gibraltar/GIB20	Early fall October 2020	Gulf of Cadiz, Gibraltar Strait, Alboran Gyre	8° W–3° W 35° N–37° N	30/1000 m
Protevs Golfe de Gascogne/GdG22	Late summer September 2022	Continental shelf to shelf break of Bay of Biscay	6° W–2° W 46° N–48° N	534/200 m
Grand Nord/GN23	Late summer September 2023	Faroe Islands, Norway	18° W–4° E 59° N–66° N	6/1500 m
Proteion/PR24	Early Spring March/April 2024	Ionian Sea (Mediterranean Sea)	15° E–24° E 35° N–39° N	35/2000 m

Table 6. Standard combined uncertainty of the DRUCK PDCR 1830 pressure sensor installed on the CTD SBE 41 of Argo floats, calculated with Equation (13).

	Distribution	Standard Uncertainty (dbar)
Repeatability	Rectangular	0.14
Thermal sensitivity	Triangular	0.24
DRUCK lifetime drift	Rectangular	0.87
Hysteresis	Rectangular	0.00
Initial accuracy	Triangular	0.82
Pressure standard combined uncertainty:		1.22

Table 7. Uncertainty budget of an Argo float calculated from the temperature gradients of Table 3 (profile PR24_026, 0–2026 dbar obtained in the Mediterranean Sea on April 2024 with a CTD SBE 911+).

Water Layers (dbar)	Pressure Sensor Uncertainty (dbar)	Temperature Gradient (°C dbar⁻¹)	Positionning Uncertainty (°C)	Response Time Uncertainty (°C)	Pressure Effect Uncertainty (°C)	Constant Uncertainties (°C)	Comnined Standard Uncertainty (°C)
0–10	1.22	0.096	0.1171	0.0006	0.0000	0.0016	0.1171
10–50	1.22	0.067	0.0824	0.0004	0.0000	0.0016	0.0824
50–100	1.22	0.008	0.0094	0.0001	0.0000	0.0016	0.0096
100–500	1.22	0.005	0.0061	0.0000	0.0002	0.0016	0.0063
500–1000	1.22	0.004	0.0043	0.0000	0.0005	0.0016	0.0046
1000–2000	1.22	0.009	0.0109	0.0001	0.0010	0.0016	0.0111

Table 8. Example of how to quantify the representativeness of data per water layers, for a campaign comprising six profiles located between 63.855 and 64.397 in latitude and 8.120 and 8.135 in longitude and of average duration 26.5 ± 7 min.

Water Layers (m)	Average of the Maximal Deviation	Average of the Standard Deviations	Expanded Combined Instrumental Uncertainty
0–10	0.010	0.003	0.002
10–50	−0.277	0.131	0.063
50–100	−0.005	0.137	0.040
100–500	−0.325	0.109	0.043
500–1000	−0.016	0.003	0.006

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Le Menn, M.; Dumas, F.; Calvez, B. Assessment of the Representativeness and Uncertainties of CTD Temperature Profiles. J. Mar. Sci. Eng. 2025, 13, 213. https://doi.org/10.3390/jmse13020213

AMA Style

Le Menn M, Dumas F, Calvez B. Assessment of the Representativeness and Uncertainties of CTD Temperature Profiles. Journal of Marine Science and Engineering. 2025; 13(2):213. https://doi.org/10.3390/jmse13020213

Chicago/Turabian Style

Le Menn, Marc, Franck Dumas, and Baptiste Calvez. 2025. "Assessment of the Representativeness and Uncertainties of CTD Temperature Profiles" Journal of Marine Science and Engineering 13, no. 2: 213. https://doi.org/10.3390/jmse13020213

APA Style

Le Menn, M., Dumas, F., & Calvez, B. (2025). Assessment of the Representativeness and Uncertainties of CTD Temperature Profiles. Journal of Marine Science and Engineering, 13(2), 213. https://doi.org/10.3390/jmse13020213

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