## Abstract

The effect of thermal mass on the salinity estimate from conductivity–temperature–depth (CTD) tags sensor mounted on marine mammals is documented, and a correction scheme is proposed to mitigate its impact. The algorithm developed here allows for a direct correction of the salinity data, rather than a correction of the sample’s conductivity and temperature. The amplitude of the thermal mass–induced error on salinity and its correction are evaluated via comparison between data from CTD tags and from Sea-Bird Scientific CTD used as a reference. Thermal mass error on salinity appears to be generally *O*(10^{−2}) g kg^{−1}, it may reach *O*(10^{−1}) g kg^{−1}, and it tends to increase together with the magnitude of the cumulated temperature gradient (*T*_{HP}) within the water column. The correction we propose yields an error decrease of up to ~60% if correction coefficients specific to a certain tag or environment are calculated, and up to 50% if a default value for the coefficients is provided. The correction with the default coefficients was also evaluated using over 22 000 in situ dive data from five tags deployed in the Southern Ocean and is found to yield significant and systematic improvements on the salinity data, including for profiles whose *T*_{HP} was weak and the error small. The correction proposed here yields substantial improvements in the density estimates, although a thermal mass–induced error in temperature measurements exists for very large *T*_{HP} and has yet to be corrected.

## 1. Introduction

The Conductivity–Temperature–Depth Satellite Relay Data Logger (CTD-SRDL) tags (referred as “tag” in the following), developed at the Sea Mammal Research Unit (SMRU; University of St. Andrews, United Kingdom), are routinely deployed on various species of seals, such as southern elephant seals, *Mirounga leonina*; Steller sea lions, *Eumetopias jubatus*; or ribbon seals, *Histriophoca fasciat*. They represent a tremendous source of hydrographic data in largely undersampled areas, such as the Southern Ocean or the northern subpolar regions (Roquet et al. 2014; Treasure et al. 2017; see http://meop.net for more information). The temperature and conductivity sensors fitted on tags, manufactured by Valeport Ltd. (Totnes, United Kingdom), yield high precision ( for temperature and ms cm^{−1} for conductivity; see Boehme et al. 2009) and reasonable accuracies (±0.02°C for temperature and psu for salinity) after delayed-mode calibration (Roquet et al. 2011). However, a recent study by Nakanowatari et al. (2017) demonstrated that the tags are also affected by a thermal mass error—a phenomenon resulting from the transfer of heat from the sensor’s walls to the sample being measured—on both their temperature and conductivity cells. Salinity being estimated via measurements of conductivity and temperature, the error in these measurements reflects on the salinity estimates, which display large discrepancies across sharp thermoclines.

The thermal mass phenomenon and its effect on salinity data have been well documented for the Sea-Bird Scientific SBE4 conductivity sensor (Lueck 1990), and they manifest in areas of large temperature gradients, such as the seasonal thermocline, where large salinity spikes of *O*(10^{−2})–*O*(10^{−1}) psu appear, followed by a slow decaying hysteresis. A correction model has been developed by Lueck and Picklo (1990), and adjustments to the correction coefficients have subsequently been implemented by Morison et al. (1994), Mensah et al. (2009), Garau et al. (2011), and Liu et al. (2015). Nakanowatari et al. (2017) successfully applied the correction method on a set of eight tags deployed on seals in the Okhotsk Sea between 2011 and 2014, proposing a set of correction coefficients validated by comparing corrected salinity results with spatially and temporally averaged historical data. However, the effectiveness of this correction methodology in various oceanic conditions and geographical locations merits further assessment.

In this paper we first document the effects of thermal mass error on tag data by comparing results of temperature, conductivity, and salinity profiles obtained simultaneously by tags and by SBE CTDs attached together on the same frame. We then develop and implement a thermal mass correction model loosely based on Lueck and Picklo (1990)—but applied directly to the salinity data—and we estimate its effectiveness on our comparison dataset. The data tested for this study having been sampled under various hydrographic and thermocline conditions, we can therefore correct each tag’s data with two different sets of correction coefficients: 1) a set of coefficients optimized for each specific tag sensor and 2) a unique set of coefficients (called generic coefficients) valid for any tag sensors and in any oceanic condition. The current study builds on Nakanowatari et al. (2017), proposing a comprehensive assessment of the effects of thermal mass error on CTD-SRDL tag measurements, and introduces a generic method to optimally reduce thermal mass–induced errors.

The thermal mass error affecting the tags and the salinity correction method are introduced in section 2. Section 3 presents the implementation of the correction scheme and a comparison of corrected tags’ data versus reference CTD data, as well as a discussion on the effect of the correction obtained with the optimized and generic sets of correction coefficients. The generic coefficients are further tested on tens of thousands of in situ profiles in the Southern Ocean using upcast and downcast data as a mean of comparison, and these results are presented in section 4. A summary and conclusions are proposed in section 5.

## 2. Thermal mass–induced errors and its correction for CTD and tag sensors

### a. Theory

The thermal mass is a well-known phenomenon that affects primarily the conductivity cells of various CTD sensors, especially when the cell is unpumped as is the case for the tags. Inductive conductivity cells are made of a cylinder through which the water flows as the CTD conducts its profile. Depending on the constructor, the cell is made of glass or ceramic and is typically surrounded by a layer of epoxy for protection. During profiling, the heat capacity of the sensor’s walls and protective layer causes heat to be stored within the sensor. This heat or “thermal mass” is exchanged through the sensor’s walls, thus contaminating the temperature—and hence the conductivity—of the water sample. While the temperature is accurately measured by the separate CTD temperature sensor, the sample’s conductivity is modified because of the thermal mass, which yields a significant discrepancy in the salinity estimation. This error has been observed on the Sea-Bird Scientific SBE4 conductivity cell, which is part of the SBE9 CTD system, and depends on the temperature gradient (function of depth or time). It is particularly evident in situations of sharp thermocline (Lueck and Picklo 1990; Morison et al. 1994; Mensah et al. 2009; Liu et al. 2015). This issue has been addressed in several works, with a thermal correction model developed by Lueck (1990) specifically for the SBE sensor. In that study the thermal mass error is modeled as an error amplitude , decaying within a relaxation time 1/ (Lueck 1990). The conductivity is then corrected via

where is the correction of conductivity added to the conductivity of the *n*th sample; is the high-pass filtered sample’s temperature [see Eq. (A4)], using a first-order discrete-time filter with a time constant ; *n* is the sample index; is the coefficient of sensitivity of conductivity to temperature; and is the sampling time interval. This model has been successfully implemented with various sets of and coefficients for the SBE4. In the limit case (i.e., when the response time is much larger than the sampling interval), the correction simply becomes with a time constant . Note that the formulation of the correction given here differs from the one given in Lueck and Picklo (1990); however, both are formally equivalent as shown in the appendix. The formulation given here is preferred because it is more readily interpretable in terms of a standard discrete-time high-pass filter.

The setting and technology of the tag sensor differ from those of the SBE4 cell, in that the wall of the conductivity sensor is made of ceramic for the tag instead of glass for the CTD cell, and the latter is an electrode cell, whereas the tag cell is inductive. Despite these differences in design, the tags are likely to show similar signs of thermal mass–induced anomalies as a result of the water sample passing through a few centimeters of long pipe, itself covered by epoxy resin. The thickness of the epoxy layer is sensibly larger than on the SBE4 cell and, should the tag sensor indeed be affected by a thermal mass error, longer relaxation time than for the SBE cell are expected. Importantly, with the platinum resistance temperature sensor being located in the immediate vicinity upstream of the conductivity cell and surrounded by epoxy, a thermal mass error may also affect the temperature measurements, contrary to the SBE CTD.

Following Morison et al. (1994), temperature could be corrected with a similar scheme as conductivity according to

where the only formal difference with (1) is that no sensitivity coefficient is required in the case of a temperature correction.

### b. Illustration of the thermal mass error on tag data

To assess the possibility of a thermal mass error affecting both the tags’ temperature and conductivity sensors, we tested the response of four tags to high-temperature gradients in in situ situations. As part of the Bouée pour l’acquisition de Séries Optiques à Long Terme (BOUSSOLE) program (Antoine et al. 2006, 2008) in the Ligurian Sea, the four sensors were attached together with an SBE9 CTD system, which is used as a reference, and seven casts were conducted. Each tag’s temperature, conductivity, and salinity profile is corrected for bias and pressure-induced slope following Roquet et al. (2011). The test was conducted at the BOUSSOLE mooring site (43°20′N, 7°54′E) in the northwestern Mediterranean Sea, on board the Sailing School Vessel (SSV) *Tethys II*. The experiment was carried out on 11 and 12 June 2008, during which a seasonal thermocline of gradient ~0.2°C m^{−1} occurred between ~10- and ~50-m depth, and with local maximum gradient of ~0.6°C m^{−1}. Our test is therefore suited for detecting and characterizing errors in a nearly idealized, steplike environment, as it was done in Lueck and Picklo (1990), Morison et al. (1994), and Mensah et al. (2009). The results of this experiment are illustrated in Fig. 1, where profiles of temperature, conductivity, and salinity (Figs. 1a–c) are plotted for both CTD and tags, whereas the difference between the sensors are plotted in Figs. 1d–f. The presence of thermal mass–induced error is highlighted by the 30-m low-pass filtered curves (green lines) in Figs. 1d–f. Strong anomalies exist for both the temperature and conductivity, with a low-frequency error *O*(10^{−1})°C and (10^{−2}) ms cm^{−1} for temperature (Fig. 1d) and conductivity (Fig. 1e), respectively. These errors reflect on the salinity estimation, yielding a maximum error *O*(10^{−1}) psu (green line in Fig. 1f). While the scale of the temperature error will be shown to be exceptional as a result of the extreme magnitude of the temperature gradient, the order of magnitude for the conductivity error is usual for temperature gradients greater than 0.1°C m^{−1} (section 3). Also, the rather extreme temperature gradients observed in this experiment are not unusual in some of the regions sampled by the marine mammals carrying the tags, such as the Okhotsk Sea (Nakanowatari et al. 2017).

Besides the typically large-scaled and long-term thermal mass error, discrepancies of smaller scale and shorter term are evidenced from the profiles of conductivity difference (Fig. 1e) and temperature difference (Fig. 1d). These errors do not show clearly on the profiles of temperature and conductivity but manifest on the salinity profile (Fig. 1c) as spikes of *O*(10^{−2}) psu. Such high-frequency error may be caused by the irregular flow within the tag sensors; as contrary to the SBE4 cell, the tag is not fitted with a pump to stabilize the inflow. In contrast with the terminology used in the rest of this paper, salinity results in Fig. 1 are expressed in practical salinity units [practical salinity scale 1978 (PSS-78) to illustrate the direct link between conductivity measurement and salinity estimate. However, in the following chapters, all salinity results in the tables, text, and figures will be expressed as Absolute Salinity (g kg^{−1}) in order to follow the new standard recommendations (McDougall et al. 2012). While the values in an Absolute Salinity profile are generally shifted by ~0.16 compared to those of a practical salinity profile, our correction scheme yields nearly identical results whether conducted on practical or Absolute Salinity profiles.

### c. An independent correction scheme for salinity

As a preliminary test, both the conductivity and temperature profiles of each of the four tags were corrected using (1) and (2), respectively, with the arbitrary values , , and s^{−1}. This test produced a significant reduction of the error (not shown) in the temperature, conductivity, and salinity data of profiles such as the one displayed in Fig. 1. However, correcting the temperature and conductivity separately may lead to some ambiguity in the correction of the salinity estimate. Residual discrepancies may remain as a result of various causes—for example, a misestimate of the coefficient values for the thermal mass correction or a slight misalignment of the CTD and tag pressures—making it possible for the temperature and conductivity residual errors to compensate each other and to yield a correct salinity estimate. Thus, the search for optimal correction coefficients for temperature and conductivity would be hampered by such considerations. Also, the error in salinity of *O*(10^{−2})–*O*(10^{−1}) g kg^{−1} is generally between one order below and the same order of magnitude as the salinity variations within any given cast. In contrast, the temperature error is most often of *O*(10^{−2})°C, which is two to three orders of magnitude below the typical variations observed within one profile, making the error signal less evident to detect and to correct following the methods described in the next section.

For the sake of practicality, finding a way to correct the salinity estimate directly becomes necessary, and a correction scheme based on (1) could be implemented following the small-amplitude assumption that the salinity correction is a linear combination of the effect of conductivity and temperature corrections:

As the causes of the temperature and conductivity thermal mass are similar, we may assume that the time constant that is included in the correction schemes (1) and (2) are identical, and we can establish the following salinity correction:

where is the coefficient of sensitivity of salinity to temperature, at fixed conductivity and pressure. The validity of (4) is ensured if within the range of salinity measured, the deviation of is small. This is demonstrated in Fig. 2a and 2b, which display the values of (coefficient of sensitivity of conductivity to temperature) and at various values of temperature and conductivity (salinity), respectively. Applying temperature and conductivity corrections following (1) and (2) is equivalent to directly correcting salinity using (4) with an error magnitude . One major drawback of correcting directly salinity instead of temperature and conductivity separately, however, is that it might lead to uncorrected biases in both the temperature and density. The magnitude of residual density errors will be discussed later in the study.

## 3. Results

### a. Optimized correction coefficients

#### 1) Determination of coefficients

We collected data from six different calibration cruises, called “experiments,” during which a set of tags is attached to a CTD frame to conduct profiles simultaneously with an SBE9 CTD system, which is used as a reference. During each of the experiments, the tags were positioned upward (i.e., the temperature and conductivity sensors facing up) about 50–100 cm above the CTD, which was installed horizontally at the bottom of the CTD rosette. The profiles are compared during the upcast only, in order to match the in situ profiling conditions during which the tags are deployed over the head of marine mammals. One drawback in using upcast data, however, is that the CTD frame, being a source of turbulence, may generate minor discrepancies when comparing data acquired by tags and CTD. Both CTD and tags data are processed following their respective standard postprocessing procedures; the tags’ temperature and conductivity sensors’ pressure-dependent drifts were previously corrected via the method of Roquet et al. (2011). The root-mean-square (RMS) difference between the salinity obtained from the CTD and that obtained from the tags is called an error, and this is calculated before and after implementation of the correction scheme. The salinity correction delineated by (4) is tested through a least squares regression scheme, in which we look for the pair , which minimizes the RMS error in salinity , where

The test was carried out separately for each of the *n* tags in order to obtain a set of optimum correction coefficient and ; *z =* 1,2, *…* , *N* is the maximum depth of the measurement (dbar) and *i* = 1, 2, …, nc is the number of casts tested on a given tag. Prior to carrying out the least squares regression, all temperature and salinity profiles went through a low-pass filter with a cutoff value set at 10 m, thus avoiding the RMS difference to include high-frequency errors unrelated to thermal mass. During these six different experiments, a total of 113 profiles belonging to 42 tags were tested. Experiments Boussole08 and Boussole09 were conducted in the western Mediterranean Sea at the BOUSSOLE mooring location in June 2008 and November 2009, respectively. Both of these experiments present strong thermoclines, where the time temperature gradient largely exceeds 0.1°C s^{−1} and where tag data are expected to exhibit strong signs of thermal mass errors. Experiment Carols08 was conducted in the Bay of Biscay in November 2008 and presents only a moderate thermocline as a result of the strong winds occurring in the eastern Atlantic in late autumn. Experiments Albion08 and Albion09 were conducted during the austral summer in the Dumont d’Urville Sea, off the coast of Adélie Land (Terre Adélie), Antarctica, and experiment iStar14 (https://www.istar.ac.uk/) occurred in the Amundsen Sea, Antarctica, in February 2014. The environment for the latter three experiments is characterized by water temperature between −1.8° and +2°C, and generally weaker temperature gradients than those encountered during the Mediterranean Sea or the Bay of Biscay experiments. Of these three experiments, iStar14 presents the largest gradients, with maximum values around 0.1°C s^{−1}. The values of the maximum temperature gradients (smoothed over a 10-m window) encountered during each of these experiments are indicated in Table 1. We also introduce in this table the value of the maximum high-passed temperature, *T*_{HP} [(1), (4)]**,** which can be interpreted as the cumulated effect of the thermal mass on the conductivity sensor. In fact, the thermal mass effect may be more important for a profile where a weak temperature gradient is met by the sensors for a long time/distance than during a cast where a greater gradient occurs over a shorter time or distance. Therefore, *T*_{HP} enables the combined effect of the temperature gradient and its duration to be appropriately expressed. The variable *T*_{HP} depends on the value of [section 2a and Eq. (A4)] and throughout this paper we calculate *T*_{HP} with s^{−1}, this being the generic value found for coefficient (section 3b).

#### 2) Effects of the correction on salinity data

The value of optimum correction for each of the different experiments is indicated in Table 1, where the correction is defined as , being the RMS difference between the CTD salinity and the uncorrected tag salinity, and the suffix MaxT_{HP} indicates that *F* was calculated within ±50 m of the maximum *T*_{HP}. Thus, the effect of the correction is evaluated where the maximum thermal mass error is expected to be. A positive (negative) value of *D* indicates an improvement (degradation) of the data quality. Figure 3 presents a typical CTD and uncorrected and corrected tag profiles for each of the six experiments. The range of salinity error is large, varying between *O*(10^{−3}) and *O*(10^{−2}) g kg^{−1} depending on the experiment. In particular for the Boussole08 experiment, where the highest *T*_{HP} were encountered, the RMS error reaches values as high as 0.053 g kg^{−1} and the maximum error within a profile exceeds 0.1 g kg^{−1} (Fig. 3f). Conversely, the two Albion experiments, with their weaker temperature gradients and T_{HP}, exhibit the smallest RMS error (0.009 and 0.012 g kg^{−1}, respectively). Our results suggest that large discrepancies between the tag and CTD can be expected for *T*_{HP} values exceeding ~1°C (Table 1). These discrepancies are largely reduced for the three experiments presenting the highest *T*_{HP} values by the correction scheme of Eq. (4) used with optimally tuned coefficients, with a correction amounting to at least 25% and up to 62% of the original error. This implies that a large part of the error for high *T*_{HP} experiments is due to thermal mass and that it is effectively removed by the correction scheme adopted here. Conversely, the other three experiments see only a small error decrease, suggesting that most of their RMS difference is due to issues unrelated to thermal mass. In particular for the Albion08 and Albion09 experiments, the very low RMS difference and optimal correction value of *O*(10^{−3}) g kg^{−1} indicate that an RMS difference of ~0.01 g kg^{−1} represents a nearly irreducible error, considering the accuracy of the tag and CTD sensors. Note that following this optimal coefficient test, 9 out of the 113 profiles tested are found to show insignificant correction in spite of a large initial error. These large error profiles are assumed to be affected by nonthermal mass–related error and are excluded from analysis in subsequent sections.

#### 3) Optimum coefficient values

The optimum coefficient values vary depending on the experiment, with the experiment averaged initial error ranging between 0.9% and 8.4% and the relaxation time () ranging between 11 and 170 s. The reason for the particularly large range of values for the coefficients is unclear although all but one (Boussole09) experiment suggests that and compensate for each other; that is, when the initial error is high (low), the relaxation time is short (long). This compensation between the two coefficients is confirmed later [section 3b(3); Fig. 6] and likely causes a large range of coefficients to yield nearly equal corrections, hence the various values found in Table 1. In addition, the properties of the environment in which profiles are conducted could also be partly responsible for the coefficient values. To explore this possibility, we test the ability to predict the values of and from environmental predictors. First, coefficients optimized for each individual profiles following (5) are estimated, yielding 104 pairs of and . Then predicted values of are fitted linearly with 10 environmental predictors obtained from each profile according to

Here, the coefficients *b*_{0}–*b*_{10} are solved by minimizing the sum of residuals squared . Terms X_{1}–X_{10} correspond to the 10 following environmental predictors: maximum *T*_{HP}, depth, and temperature of the maximum *T*_{HP}; and maximum, mean, and standard deviation of the temperature gradient, minimum, maximum, and mean temperature, and mixed layer thickness. A similar operation is conducted for using the same environmental predictors. Results show that the predicted values and are linearly fitted with and with a coefficient of determination *R*^{2} = 0.57 and *R*^{2} = 0.53, respectively. However, if is added as X_{11} to determine , then the coefficient of determination becomes 0.91. Similarly, if becomes X_{11} in the determination of , the coefficient of determination for the latter becomes 0.92. This highlights the relative importance of the – compensation with regard to environmental properties in the profile. To summarize, while and compensate each other and an effective correction can be obtained from a large range of coefficient values [section 3b(3)], environmental properties may ultimately decide the best coefficient values within this large range of effective correction coefficients. Following these assumptions on the variability of the coefficients, it seems reasonable that a set of generic coefficients could yield sufficient correction for any kind of profiling environment.

### b. Generic correction coefficients

#### 1) Determination of coefficients

To determine a set of generic coefficients, we adapted the method delineated by (5), setting nc = 60. The nc includes 10 randomly chosen profiles from each of the six cruises, in order to avoid a bias generated by the different number of profiles tested during each experiment. This test is repeated 200 times, and we average these 200 pairs of and coefficients to obtain our generic coefficients. The coefficients obtained via this method are and s^{−1}, or an initial error of 4.1% and a relaxation time of ~17 s, respectively, and yield an average correction of 0.011 g kg^{−1} out of an original averaged error of 0.035 g kg^{−1}.

#### 2) Effects of the generic correction on salinity data

The set of generic coefficients performs particularly well for the Boussole08 dataset, which exhibits the strongest temperature gradient. In this case, around 50% of the error is resorbed through the use of generic coefficients, a figure that compares well with the ~60% error decrease obtained with the optimum coefficients. Aside from this experiment, the improvement brought by the generic coefficients is more modest but still significant when the initial discrepancy is high. The salinity data from Boussole09 are corrected by about 20% (Table 1) and while the average value of correction for the profiles of the Carols08 experiment is null, a large number of these profiles are well corrected by the generic coefficients (Fig. 3e). The high standard deviation for the correction of the Carols08 experiment demonstrates, however, that the changes brought to the profiles are unequal in quality depending on the tag it applies for. On the lower end of the salinity error range, the generic set of coefficient yields either insignificant improvement or, in the case of Albion08, a moderate degradation of the data. In this case, illustrated in Fig. 3a, the maximum discrepancy of ~0.03 g kg^{−1} is reached around the halocline at 45-m depth and indicates an overshoot of the correction. This overshoot is resorbed following the halocline as the tag and CTD profiles converge from ~40-m depth to the surface. Some profiles of the Carols08 experiment follow a similar pattern of degradation.

To further asses the performance of the generic coefficients on this dataset, the values of uncorrected and corrected salinity error—as defined in Table 1—for each individual profile are sorted according to their maximum *T*_{HP} and averaged per *T*_{HP} bins of 0.5°C. The results of this experiment, displayed in Fig. 4, demonstrate that the correction performance for *T*_{HP} values less than 2.0°C is null on average, and with a particularly low standard error. The RMS error increases sharply for the uncorrected data beyond this *T*_{HP} value, and it systematically exceeds 0.05 g kg^{−1}. The RMS error for the salinity corrected with the generic coefficient is strongly reduced, however, and for each *T*_{HP} bin, it generally becomes half the value of the original error. The results in Fig. 4 demonstrate that the correction applied with a generic coefficient does not degrade the data when a correction may not be needed (very low *T*_{HP}) and significantly improves the data quality otherwise. Besides slightly degraded profiles, such as in Fig. 1a, a phenomenon independent of the correction performance may lead to an apparent degradation of the data in statistics of the two Albion experiments. Since high-frequency errors have been eliminated by the use of a 10-m low-pass filter prior to all our statistical tests, a likely cause could be a slight misalignment of the CTD and tag pressure sensors, or slight changes of positioning of the tags in between some of the casts. Such misalignment may lead to the temperature and salinity profiles being slightly offset, which could artificially cancel the effect of a small thermal mass error or conversely artificially inflate the error of properly corrected profiles. The case study in section 4 will enable the performance of the generic coefficients in a situation of low *T*_{HP} values to be more accurately evaluated.

#### 3) Impact of the generic correction on the density error

To assess the potential contribution of uncorrected thermal lag errors of the temperature sensor on the density results, we estimate the values of RMS error for the raw and corrected density profiles (Table 1). For all but one experiment the density error (kg m^{−3}) is close to the salinity discrepancy (g kg^{−1}); therefore, it seems reasonable to assume that the salinity contribution to the density error largely dominates that of temperature. To confirm this assumption, the density error for each individual profile is converted to an equivalent salinity error after calculation of the coefficient of sensitivity of density to salinity, at fixed temperature and for a change of salinity of ±0.5 g kg^{−1}, . This coefficient value is close to 0.77 for all the tags, which means that if salinity varies by 1 g kg^{−1}, then density will see a change of ~0.77 kg m^{−3}. The equivalent salinity error is then obtained by dividing each profile density error by its coefficient, and it is subsequently plotted against the actual salinity error [which equals ] in Fig. 5a. In this figure is also plotted a line of equation that represents the value that should take if it is entirely caused by . Points located above (below) that line present a negative (positive) temperature error proportional to the vertical distance between the equivalent salinity error and the line. The results from this figure show that most of the points with a salinity error lesser than 0.03 g kg^{−1} fall very close to the *y* line, thus confirming that the temperature error has little significance in those cases. For errors larger than 0.03 g kg^{−1}, those profiles with *T*_{HP} values greater than ~2°C (largest dots in Fig. 5a) present a significant temperature error, as shown by the large vertical distance between the line and the dots in Fig. 5a. For each profile, the percentage of density error caused by temperature can be calculated from this vertical distance via

For each profile, the RMS density error and are used to obtain a value (kg m^{−3}) of density error caused by temperature *E*_{T}. Both *E*_{T} and are plotted against *T*_{HP} in Fig. 5c, and these results demonstrate that significant density error (as a result of temperature) can be found only for profiles whose maximum *T*_{HP} values exceed 2°C, which corresponds to temperature gradients around 0.20°–0.28°C s^{−1}. In these cases *E*_{T} values greater than 0.01 kg m^{−3} and amounting to ~20%–30% of the total density error are expected.

Results for the data corrected with the generic coefficients show a general decrease of the salinity and equivalent salinity errors, demonstrating that the correction scheme adopted here with a generic set of coefficient improves both salinity and density estimations. In those cases where *E*_{T} is large on the uncorrected profiles, the equivalent salinity error also decreases after correction as a result of the role of the salinity correction scheme, but the temperature-related errors remain essentially unchanged as can be seen from the large vertical distance between each dot and the *y* line (Fig. 5b).

#### 4) Generic correction coefficient values

Figure 6 displays the amount of correction for any pair of coefficients located within a large interval of and . The values of the generic set of coefficients (, s^{−1}) and each optimum coefficient for individual profiles are also displayed in the figure. As shown in this graphic, a large beam exists—whose limits are defined by the 0.010 g kg^{−1} isoline—within which pairs of coefficients yield a correction close to , . We can therefore assume that the generic coefficients will provide a correction close to the optimum correction for those profiles whose , are within the high-correction beam. This is the case for nearly half of the Boussole08 profiles and a third of the Carols09 profiles. This should also be the case for the Nakanowatari et al. (2017) experiment, where the pair of optimum coefficients (, s^{−1}) was determined via a comparison of tag data and historical data from the *World Ocean Atlas 2013* (*WOA13*; Locarnini et al. 2013; Zweng et al. 2013) in the Okhotsk Sea, and yielded a decrease of salinity error of 0.07 psu over the uppermost 20 m of the water column. In contrast, those profiles whose , are located far from this beam will see the generic coefficients perform notably less well compared to the optimum correction. Note that contrary to the per-experiment statistics, the individual profile statistics show a trend where profiles with *T*_{HP} < 2.0°C (empty symbols in Fig. 6) are generally corrected by small values of and , while profiles with large *T*_{HP} values and error tend to be corrected by large values of and proportionally large values of . Also, 26 profiles are located out of the figure limits, with nearly two dozen coefficient pairs of low *T*_{HP} profiles having a slightly negative value of or . This fact illustrates the difficulty in determining correction coefficients for very low *T*_{HP} conditions where little or no thermal mass error exists and where other errors might hamper the determination of coefficients. This issue is essentially solved when optimum coefficients are calculated per tag instead of individual profiles. While the variety of optimum coefficients should encourage users to determine their own optimum coefficients in the areas of deployment whenever possible, this is often not possible or ideal, as the seals may travel long distances throughout the deployment period and seasonal changes of water properties will also change the profiling environment. Our results have shown that the use of a set of generic coefficient provides satisfactory results for the conditions of the six aforementioned experiments, and the case study in the next section will further demonstrate the ability of this set of coefficients to provide an effective correction in operational conditions even when the *T*_{HP} is low.

## 4. Application of the generic correction to CTD-SRDL biologged data

### a. Dataset

The thermal lag correction scheme described in (4) and with the generic coefficients (, s^{−1}) is applied to five full-resolution CTD-SRDL tags deployed on southern elephant seals (SES) around the Kerguelen Islands in the Indian sector of the Southern Ocean during the austral summer. A unique dataset made of full-resolution CTD-SDRL hydrographic profiles acquired by five female SES from October 2014 to January 2015 has been used to test the robustness and efficiency of the thermal lag procedure and generic coefficients. Full-resolution tags record temperature, salinity, and pressure at a frequency of 0.5 Hz for every dive of the seal’s journey. Besides the high resolution of this dataset, another advantage resides in the acquisition of both ascending and descending phases for every dive, which allows a comparison of the data and evaluation of the correction to be done. The environment in which the SES profiles were taken is marked by low temperatures and mild temperature gradients, successively negative between 0 and 150 m, then positive between 150 and 350 m (Fig. 7a). Accordingly, these weak temperature gradients are associated with weak *T*_{HP} values with the 20th–80th percentile envelope ranging between −0.17° and 0.22°C (Fig. 7c); this places this dataset below most of the *T*_{HP} values tested in the six experiments described earlier. Steady salinity changes occur throughout the water column.

### b. Implementation of the correction

The thermal lag procedure was applied on the ascending and descending phases of every dive. A total of 22 308 dives ranging from depths of 150 m up to 1000 m have been used. The ascending phase of any *n* dive is then compared to the descending phase of dive *n* + 1 for dives occurring within a 5-min interval in order to compare similar water masses. For each tag, the RMS error of ascent versus descent is first calculated at each depth *z* according to

where np is the total number of dives, and *S*_{a} and *S*_{d} are the salinity during ascent and descent, respectively. The term is calculated for both the uncorrected () and corrected () versions, and it shows values between ~0.02 and 0.03 g kg^{−1} for all but one tag (Table 2). These relatively large values are for the most part not caused by thermal lag and are likely related to slight changes in the geographic location of the ascent compared to the descent. The occurrence of a relatively large temperature RMS for all tags (Table 2) in spite of the weak temperature gradients confirms that environmental changes account for a large part of the RMS error. Therefore, the performance of the correction is evaluated through the difference of and , with positive values indicating a decrease of the RMS error. All five tags see a significant improvement after the application of the thermal lag correction, where the difference between the ascending and descending phases for both the salinity and density field is minimized, as can be seen from the profile averaged for all tags (Fig. 7d). The impact of the thermal lag correction is more pronounced in the areas where stronger gradients of temperature are located, that is, between the surface and 300-m depth. The results obtained for this depth range are presented in Table 2. RMSs of salinity and potential density are the greatest for tag 35 because of some technical issues flagged with the salinity sensor of this particular tag and independent of the thermal lag procedure. This issue does not affect the values and profile of . Besides the averaged effect of the correction procedure over the water column presented in Fig. 7d, a typical pre- and postcorrection profile is shown in Fig. 8. The profile that occurs in a region presenting sharper temperature variations sees its salinity difference between ascent and descent reduce by nearly 0.04 g kg^{−1}, which is a rather considerable improvement that is also reflected in the density profile (Fig. 8c) and in the temperature–salinity (TS) plan (Fig. 8d).

## 5. Summary and conclusions

The SRDL-CTD tag sensors are subject to the thermal mass phenomenon that affects other conductivity cells, such as the Sea-Bird Scientific SBE4. This paper has documented the effect of thermal inertia on the tags’ conductivity cell and provided evidence of thermal mass–induced errors increasing with the magnitude of the temperature gradient, and more specifically with the magnitude of the cumulated effect (*T*_{HP}) of the temperature gradient within a profile. The thermal mass applied on the conductivity cell reflects as a significant error in salinity estimates. Salinity error—defined here as the root-mean-square difference between a standard CTD upcast and a concurrent tag profile—amounting to ~0.02 g kg^{−1} for *T*_{HP} < 2°C, and >0.05 g kg^{−1} for larger *T*_{HP} occurs. A correction scheme was therefore developed to improve the salinity estimates. The main part of the correction methodology is a further development of the conductivity correction scheme of Lueck (1990), where correction coefficients represent the initial measurement error and the inverse relaxation time . However, the tag’s temperature sensor is also affected by thermal mass, which implies that to obtain an accurate estimate of salinity, both the conductivity and temperature data are to be perfectly corrected simultaneously. For practical reasons described further in this section, this is difficult to achieve and thus we developed a correction algorithm to be applied directly to salinity. The correction algorithm is successfully implemented and tested on 42 different tags profiling in various hydrographic conditions and experiencing different ranges of temperature gradients, *T*_{HP}, and error. A comparison between tag and CTD profiles conducted simultaneously allowed to calibrate the coefficients and for each of the tags. These optimum coefficients systematically lead to a significant improvement for all tags whose *T*_{HP} is 2°C or above, with an error decrease of up to 60%. However, the range among which these optimal coefficients varies is large, suggesting a compensation effect between the magnitude of the initial error and the relaxation time , with possible further influence from the environment in which a given profile is conducted. A generic set of correction coefficients was therefore determined to account for the coefficients’ variability. This set of coefficients (, s^{−1}) yields an error decrease of nearly 50% for those profiles with a *T*_{HP} greater than 2°C (the latter that on average corresponds to temperature gradients of 0.20°–0.28°C s^{−1}). Results for lower *T*_{HP} or gradients were less clear because of intrinsic difficulties in evaluating small errors in CTD versus tag experiments. To further assess the performance of and , more than 22 000 profiles acquired from five different tags deployed in the Southern Ocean were evaluated. The difference in salinity between each downcast and upcast was used to assess the effectiveness of the generic coefficients, and was found to reduce the error on average by 0.006 g kg^{−1} for profiles whose maximum *T*_{HP} was extremely low with averaged values of ~0.1°C. Those profiles presenting larger *T*_{HP} or temperature gradients saw very significant improvements with examples of upcast–downcast salinity difference reducing from 0.06 to 0.02 g kg^{−1}. Both this test and the CTD-versus-tags experiments demonstrate that the salinity correction leads to a substantial decrease of the density error.

Besides the effects of thermal mass on conductivity measurements/salinity estimates, temperature measurements also appear to be affected by thermal mass–induced errors. Temperature discrepancies are insignificant for *T*_{HP} < 2°C but become large for *T*_{HP} > 2°C, amounting to ~25% of the error in density. While temperature gradients in excess of 0.20°–0.28°C s^{−1} (which is roughly equivalent to a *T*_{HP} of 2°C) are less frequently met in the ocean, they do still occur in some of the areas typically sampled by tag-equipped mammals (Nakanowatari et al. 2017) and call for an appropriate correction. However, the rather high limit above which the temperature error becomes significant (limit met by only 24 profiles) combined with a larger sensitivity of the least squares regression scheme used to determined correction coefficients make our dataset inadequate to define correction coefficients for temperature. Future development for the improvement of the temperature data requires a larger number of profiles acquired in high *T*_{HP} conditions as well as perfectly aligned pressure measurements for the tag and CTD used in the experiments.

Another possible improvement of the correction scheme could consist of adapting the correction coefficients according to the ascent velocity of the tag, as was done in Liu et al. (2015) for glider data. Different profiling speeds are expected to be met depending on the species or body condition of the mammals on which the tags are deployed, and these are likely to affect the value of the coefficients. However, as the results of Fig. 6 suggest, the range within which the correction yields similar results is large, allowing for performances of the generic coefficients to be satisfactory even when the profiling speed differs significantly from ~1 m s^{−1}.

It is noteworthy that while the scheme described in this study applies directly to salinity data, the generic coefficients found here can be used to correct the conductivity following (1) and using and , following the assumptions developed in section 2c. A test has been conducted on the data from the six abovementioned experiments and yielded insignificant differences between the conductivity and salinity versions of the correction. It was, however, crucial to use the salinity to determine our coefficients in this study. The nature and relative magnitude of the salinity error indeed made this determination more robust than using conductivity, whose signal and error are correlated with those of temperature. Last, while this paper documents the effects of our correction scheme on full-resolution tag data (1- or 0.5-Hz sampling rate), most of CTD-SRDL profiles available to date are heavily compressed because of satellite transmission constraints with a typical number of 20 data points per temperature/salinity profile (Boehme et al. 2009). The slow response nature of the thermal mass effect, and the low-pass filtering conducted on the data for the tags versus CTD test make it highly likely that the correction will be useful on low-resolution salinity data. Nakanowatari et al.’s (2017) work is an example of the successful implementation of this thermal mass correction algorithm on postprocessed low-resolution data in the Okhotsk Sea. Users are therefore encouraged to apply the thermal mass algorithm on their low-resolution salinity data, which should yield a significant reduction of the thermal mass–induced errors.

## Acknowledgments

The present study is a contribution to the Observing System–Mammals as Samplers of the Ocean Environment (SO-MEMO) with funding and logistic support from CNES-TOSCA, IPEV, and CORIOLIS in situ ocean observation program. We are very grateful to D. Antoine for giving us access to the R/V *Tethys II* data, G. Reverdin for access to the R/V *Cotes de la Manche* data, and M.-N. Houssais for access to the R/V *L’astrolabe* data. The iStar14 data, obtained on the RSS *James Clark Ross* during the iSTAR JR294/295 cruise, were kindly made available to us by Karen Heywood and Helen Mallett. We also thank two anonymous reviewers for their constructive criticism. We extend our warm thanks to the crews and captains of the different research vessels.

### APPENDIX

#### Equivalence between the Lueck and Picklo (1990) Recursive Filter and a Standard First-Order High-Pass Filter

Here it is shown that the recursive filter scheme devised by Lueck and Picklo (1990) to correct the thermal mass effect on a measured variable *X* (conductivity in their case) is formally equivalent to a standard first-order high-pass filter applied on the temperature discrete signal, once suitably rescaled.

The recursive filter of Lueck and Picklo (1990) is given by

where is the correction of conductivity added to the conductivity of the *n*th sample, *T* is the sample’s temperature, *n* is the sample index, is the sensitivity of *X* to temperature, and *a* and *b* are coefficients depending on and according to

and

where is the Nyquist frequency function of the sampling interval .

Define the high-pass filtered temperature signal *T*_{HF} as

where is the time constant of the filter. Assuming that the correction is proportional to the high-pass filtered temperature signal, , and neglecting variations of the factor between two consecutive samples (an excellent assumption in practice; see Fig. 2), a relation can be found between (A1) and (A4) providing the two identities and . Using (A2) and (A3) and after some rearrangement, it comes that the recursive filter of Lueck and Picklo (1990) is strictly equivalent to rescaling the high-pass filtered temperature signal using a time constant and a factor .

Note that the filter is defined only if and that in the limit case , the time constant of the filter tends toward and the correction simply tends toward .

## REFERENCES

*Coastal Ocean Observing Systems*, Y. Liu, H. Kerkering, and R. Weisberg, Eds., Academic Press, 305–325, https://doi.org/10.1016/B978-0-12-802022-7.00017-1.

## Footnotes

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