Calibration of Marine Pressure Sensors with a Combination of Temperature and Pressure: A Case Study of SBE 37-SM

Abstract

Accurate pressure measurement is crucial for understanding ocean dynamics in marine research. However, pressure sensors based on strain measurement principles are significantly affected by temperature variations, impacting the accuracy of depth measurements. This study investigates the SBE37-SM sensor and presents an improved calibration method based on a constant-pressure, variable-temperature scheme that effectively addresses temperature-induced deviations in pressure measurement. Experiments were conducted across a pressure range of 2000 dbar to 6000 dbar and a temperature range of 2 °C to 35 °C, establishing a comprehensive pressure–temperature calibration grid. The results show that, at a pressure of 6000 dbar, temperature-induced variations in readings for brand new SBE37-SM sensors can reach up to 9 dbar, while, for used sensors, they exceed 12 dbar, following a U-shaped trend. After applying a polynomial regression model for calibration, these variations were reduced to within ±0.5 dbar, significantly reducing the measurement uncertainty of the sensors in complex marine environments. This method underscores the necessity of further optimizing the CTD system’s temperature compensation mechanism during calibration and highlights the importance of regular calibration to minimize measurement uncertainty.

1. Introduction

In oceanographic research, Conductivity–Temperature–Depth (CTD) instruments are widely used to obtain precise oceanographic data. These data provide essential insights into the physical, chemical, and biological properties of the ocean, supporting studies of ocean circulation patterns, climate change monitoring, and marine ecosystem assessments [1,2]. By measuring temperature, conductivity, and pressure with CTD instruments and applying the 1978 International Equation of State for Seawater, seawater density can be calculated, enabling scientists to better understand and predict ocean dynamics [3]. Precise CTD data reveal subtle oceanic structures and dynamic changes, directly affecting the accuracy of ocean models and forecasts [4].

Pressure measurement is crucial in this process, as it determines the precise depth of the instrument by measuring the pressure exerted by the overlying seawater. This depth information is vital for analyzing oceanic processes and predicting ocean dynamics [5]. Regular calibration of sensors is necessary to ensure accurate pressure measurements, particularly during long-term monitoring in high-pressure deep sea environments. High-accuracy pressure sensors are essential for maintaining the integrity of scientific data and for ensuring the safety and reliability of engineering applications, such as submersible navigation, deep sea mining, and submarine communications [6]. Additionally, CTD data indirectly measure the speed of sound in seawater, which is crucial for ocean acoustics research [7].

The stability and accuracy of pressure measurements are crucial for obtaining reliable vertical profiles. Errors in pressure measurement can propagate into calculations of seawater density and depth, potentially distorting analyses of global ocean warming trends and dynamics [8]. Fixed buoy systems also rely on precise pressure measurements to correct buoy orientation and ensure data reliability [9]. Thus, the use of high-accuracy pressure sensors and advanced measurement techniques is critical for the success of marine scientific research and related engineering applications.

Accurate pressure measurement in marine environments presents significant challenges, primarily due to the complexity of oceanic conditions and the inherent limitations of existing pressure sensors. These measurements must withstand the high-pressure conditions of the deep sea and account for deviations caused by rapid temperature fluctuations. As shown in Figure 1, the water layer between 0 and 1000 m exhibits the most rapid temperature variations, primarily due to direct solar radiation. In contrast, at depths exceeding 2000 m, temperature fluctuations are significantly smaller, and the overall temperature is lower because the effects of solar radiation diminish and water mixing is less prominent [10]. During deployment, mobile survey equipment undergoes substantial temperature changes, particularly in shallow-sea environments. On the other hand, monitoring instruments deployed on the seafloor must be capable of operating at low temperatures and be resilient to the impact of minor temperature variations on measurements.

However, temperature fluctuations in marine environments can lead to errors in pressure readings, which are often overlooked in current calibration procedures. Most electrical pressure sensors typically operate based on the principle of material strain, converting applied mechanical pressure into electrical signals. Temperature variations can affect the mechanical properties of pressure-sensitive materials, causing sensor readings to be influenced by these temperature changes. To address this issue, numerous studies have sought to compensate for temperature-induced errors in pressure measurements by incorporating internal or external temperature sensors for synchronized readings [11,12,13]. Simultaneously, novel pressure sensors, such as those based on optical sensing principles, have made significant advancements in decoupled measurements within complex marine environments [14,15,16,17]. Although existing temperature compensation techniques can effectively mitigate the effects of temperature variations, achieving high-precision pressure measurements in complex marine environments remains a significant challenge [18,19].

Furthermore, pressure measurement errors may occur over time due to material aging and the accumulation of long-term mechanical stress, particularly in high-pressure deep sea environments [20]. This aging phenomenon can lead to the failure of the original temperature compensation mechanisms, thereby affecting the accuracy of both absolute and relative pressure measurements [21,22]. Therefore, regular sensor calibration is crucial for ensuring the accuracy and reliability of measurement data. To address these challenges, various correction methods have been developed, including pressure sensors with self-calibration capabilities that correct for drift, thereby reducing the impact of temperature-induced stress and mechanical aging on long-term measurements [23]. Multi-sensor fusion technology is also an effective approach, integrating data from sensors with different types and ranges to minimize measurement uncertainty and enhance long-term reliability [24].

Although temperature significantly impacts pressure sensors, current calibration standards do not fully address the temperature drift characteristics of pressure sensors used in marine environments. For example, China’s Metrological Verification Regulation JJG763-2019 [25] and the international standard BS ISO 22804:2023 [26], both of which are the latest calibration standards, typically involve comparing pressure sensors with standard pressure sources under room temperature conditions. This approach neglects the influence of temperature variations present in actual marine environments [27]. Consequently, there is a pressing need for enhanced calibration methods that can effectively account for the temperature compensation effects of marine pressure sensors, thereby ensuring accurate measurements of oceanic pressure.

This study addresses the challenges faced by pressure sensors in complex marine environments by deploying a full-depth ocean environmental simulator that provides high-precision, long-term stable conditions for ocean pressure, temperature, and salinity. A high-precision quartz resonator pressure gauge serves as the reference standard, allowing for calibration and validation of sensor measurements under the influence of multiple environmental parameters. The objective is to fully assess the effectiveness of temperature compensation in pressure sensors and to propose an improved compensation formula that meets the dynamic and static monitoring requirements of oceanic pressure measurements. Compared to existing metrological calibration methods, this approach, which more closely replicates real marine conditions, offers a more comprehensive understanding of the temperature drift characteristics of marine pressure sensors. It also enables a more thorough validation of the effectiveness of current temperature compensation methods. The new calibration formula, derived from this dynamic temperature variation calibration method, ensures higher accuracy in ocean pressure measurements under both dynamic and static temperature change conditions.

For the experimental evaluation, the SBE 37-SM CTD, a widely used high-precision oceanographic monitoring instrument manufactured by Sea-Bird Electronics (Bellevue, WA, USA), was selected. This instrument is commonly employed to measure seawater conductivity, temperature, and depth. It offers a conductivity range of 0 to 7 S/m, a temperature range of −5 °C to +45 °C, and a depth range of up to 7000 m, all housed in titanium casing. The instrument’s accuracy is specified as ±0.0003 S/m for conductivity, ±0.002 °C for temperature, and 0.1% full scale (F.S.) for depth. The resolution is 0.0001 S/m for conductivity, 0.0001 °C for temperature, and 0.002% F.S. for depth. It is worth noting that the pressure sensor of the SBE 37-SM incorporates an internal temperature compensation formula, using its own temperature module measurement results to adjust the pressure readings. More representative sensors to be tested also demonstrate the effectiveness and universality of this method. The following sections will elaborate on the limitations of existing calibration methods, provide a comprehensive introduction to the new approach, and outline the logic of improving existing sensors through practical testing.

2. The State-of-Art Calibration Method

Many countries have established standards and methods for calibrating CTD sensors. For example, Chinese standard JJG 763-2019 Calibration Regulations for Temperature–Salinity–Depth-Measuring Instruments and the Britannic standard BS ISO 22804:2023 Ocean Technology—General Requirements for Ocean Conductivity–Temperature–Depth-Measuring Instruments specify the technical requirements and testing methodologies. These calibration standards primarily focus on verifying the accuracy of temperature and conductivity (used to calculate salinity) measurements under standard atmospheric pressure as well as studying the accuracy of pressure measurements using a piston gauge. They outline technical specifications, testing procedures, and environmental adaptability tests.

According to the “JJG 763-2019 Calibration Regulations”, pressure calibration is performed by selecting a minimum of seven calibration points that are evenly distributed across the pressure range, following a sequence of both increasing and decreasing pressure. The CTD pressure sensor is connected to a standard deadweight pressure tester, with its position being carefully adjusted to ensure consistent measurements. Once the readings stabilize at each calibration point, the arithmetic mean of at least ten sets of readings is recorded as the pressure indication value. If the pressure sensor includes a temperature compensation function, its correction performance must also be verified at the specific temperature points of 0 °C and 30 °C within a temperature-controlled air chamber. However, the “BS ISO 22804:2023” standard does not specifically mention the need to verify the temperature compensation effect of the pressure gauge. This calibration method does not comprehensively study pressure accuracy at different temperatures. In practical use, the ability to accurately measure pressure depends entirely on whether the manufacturer’s temperature correction formula can reduce the pressure indication drift caused by temperature variations. It does not provide a comprehensive evaluation of the pressure measurement uncertainty within the specified temperature operating range of the device.

3. Innovative Calibration Method and Associated Instruments

To ensure accurate measurements of CTD pressure sensors in complex ocean environments, a more comprehensive pressure calibration in simulated conditions of full range of oceanic pressure and temperature is necessary. In this work, the measurement apparatus used in calibration is described below, and the calibration process is shown in Figure 2.

Reference Standard Instrument: A piston-type pressure gauge serves as the highest- accuracy reference standard. The operating principle of the piston pressure gauge relies on using standardized weights and hydraulic pressure to generate precise reference pressures. When the standard weights are placed on the piston, their weight is transmitted through the piston to the fluid. Due to the minimal friction between the piston and the cylinder, the pressure is evenly distributed, ensuring stability and minimizing measurement uncertainty. The weight-loaded piston gauge is a traditional and reliable tool for pressure calibration that is known for its high stability and accuracy.

Working Standard Instrument: A high-accuracy quartz resonator pressure gauge serves as the working standard. Quartz resonator pressure gauges exploit the resonant properties of quartz crystals to accurately convert applied pressure into measurable changes in resonant frequency. These gauges can ensure a high degree of measurement accuracy, with a consistency of up to 0.01% of full scale (F.S.). The low thermal coefficient of quartz ensures that high measurement accuracy is maintained across varying temperature conditions. The measured pressure data are transmitted to the control system for real-time monitoring and adjustment of the system pressure, ensuring the stable operation of the pressure control equipment within the specified pressure range. This high-accuracy pressure sensor was calibrated at the National Institute of Metrology (NIM) prior to the experiment.

CTD Sensor Calibration: Once the quartz resonator pressure gauge has been calibrated, the CTD sensor to be tested is placed in a controlled temperature and pressure environment for final calibration. By conducting prolonged, stable measurements and comparisons under varying temperature and pressure conditions, the performance data of the CTD sensor in complex environments can be obtained.

In calibrating and validating pressure sensors, selecting a reference standard with high stability and reliability is essential. This study uses the quartz resonant pressure gauge (model 745-20K, Paroscientific, Inc., Redmond, WA, USA) as the working reference device due to its exceptional stability across a wide temperature range [28]. Quartz resonant pressure sensors are widely recognized for their performance in marine environments, offering high accuracy and long-term stability. They have been successfully applied in fields such as tsunami detection, wave and tide measurements, volcanic monitoring, subsea drilling, and deep sea measurements [29].

To ensure that the reference pressure gauge accurately represents true pressures under varying temperature conditions, the stability of the quartz resonant pressure gauge was systematically verified within a temperature range of 2 °C to 40 °C. A piston-type pressure standard device with an uncertainty of 0.005 dbar was connected to the quartz resonant pressure gauge to apply known pressures ranging from 2000 dbar to 12,000 dbar. The pressure gauge was placed for extended periods in a thermostatically controlled environment to allow both the hydraulic oil and the resonator core to fully acclimatize to the ambient temperature. Verification was achieved through repeated measurements and the recording of pressure readings at various temperature points. The experimental setup is depicted in Figure 3.

In the in-depth statistical analysis of the temperature stability of the quartz resonant pressure gauge, raw data were directly obtained from the internal temperature sensor of the gauge. Known pressure values, generated by a standard weight disk, were recorded only when the sensor indicated stable environmental temperatures. To ensure the reliability and accuracy of the data, at least 20 data sets were collected at each temperature point, and their means were calculated. These means were then compared with the corresponding known standard pressures. Using the measurement error at 2 °C as a reference, the mean (μ) and standard deviation (σ) of the pressure readings for each pressure level within the specified temperature range were computed, as shown in Figure 4. The results showed that the pressure readings of the quartz resonant pressure gauge exhibited remarkably low fluctuations across the entire tested temperature range. As the temperature increased, the reading errors of the working reference pressure gauge followed a linear growth pattern. However, at two pressure points (10,000 dbar and 12,000 dbar) at 40 °C, the reading errors increased abnormally by 0.2 dbar, deviating from the linear trend. This anomaly was attributed to insufficient temperature control in the high-temperature chamber at elevated temperatures, which introduced additional temperature measurement errors. At a pressure level of 2000 dbar, as the temperature increased from 2 °C to 40 °C, the standard deviation of the pressure readings remained as low as 0.062 dbar. For the 12,000 dbar pressure condition, temperature increases from 2 °C to 35 °C (the selected experimental range) resulted in a measurement error of only 0.15 dbar. The pressure reading errors due to temperature variation were less than 0.0015% F.S, demonstrating the high precision and stability of the working reference pressure gauge. This can be attributed to the temperature stability of the quartz resonant pressure measurement method, the temperature compensation mechanism within the core, and the improved temperature compensation formulas. While the impact of temperature on the working reference is minimal, the linearly increasing error trend will be accounted for in subsequent calibration processes to further ensure the accuracy of the data.

These statistical results further confirm that the quartz resonant pressure gauge maintains high accuracy even without strict control over operating temperature. Its inherent stability and accuracy make this sensor highly suitable for calibrating CTD sensors and other marine measurement devices. As a result, the quartz resonant pressure gauge is a reliable working reference in calibration processes involving variable temperatures under constant pressure conditions.

In current marine pressure sensor calibration processes, the indication error of sensors is determined by comparing their readings to a piston-type pressure standard [25,26]. Incorporating variable temperatures in high-accuracy calibrations enables a thorough evaluation of temperature effects on sensor performance, thus reducing the need for corrections related to temperature-induced uncertainties. Accurate documentation and analyses of these temperature-related uncertainties are crucial during calibration to ensure the accuracy and stability of CTD measurements under diverse temperature conditions.

When the environmental pressure is set to $P_i$, the temperature-induced indication error at different pressures is defined as:

$$\Delta P_{iT}=P_{iT}-P_{is}$$

(1)

Here, $\mathsf{\Delta }{P}_{\mathrm{iT}}$ represents the pressure measurement error at the calibration point $P_i$ when the environmental temperature is $T$; $P_{\mathrm{iT}}$ denotes the pressure reading of the target sensor at the calibration point $P_i$ under environmental temperature $T$; and $P_{is}$ refers to the corresponding reading from a higher-precision reference pressure gauge taken simultaneously with the target sensor.

As maintaining a constant pressure is challenging due to potential minor fluctuations in the system, and because asynchronous sampling between the target sensor and the reference pressure gauge may introduce additional errors, a standard deviation analysis of multiple measurements of $\Delta P_{iT}$ is necessary. By calculating the standard deviation of $\mathsf{\Delta }{P}_{\mathrm{iT}}$, we can assess the synchronization between the reference pressure gauge and the target sensor, as well as the reliability of the measurement data. This approach ensures the accuracy of the pressure sensor across different temperature conditions. The formula is as follows:

$$\sigma \Delta P_T=\sqrt{{\displaystyle \frac{{\Sigma }_{i=1}^n{\left(\Delta P_{Ti}-\overline{\Delta P_T}\right)}^2}{n-1}}}$$

(2)

Here, $\sigma \Delta P_T$ represents the standard deviation of the pressure measurement error at a given calibration temperature $T$, with units in dbar. $\Delta P_{Ti}$ denotes the pressure measurement error from the i-th measurement at the calibration temperature $T$, also in dbar. $\overline{\Delta P_T}$ is the arithmetic mean of the pressure measurement error at the calibration temperature $T$, measured in dbar, and $n$ refers to the number of measurements taken.

The pressure measurement errors$\Delta P$ under different temperature and pressure conditions organized into matrix form:

$$\Delta P_{T,P}=\left[\begin{array}{ccc}\Delta P_{T1,P1}& \cdots & \Delta P_{T1,Pn}\\ ⋮& \ddots & ⋮\\ \Delta P_{Tm,P1}& \cdots & \Delta P_{Tm,Pn}\end{array}\right]$$

(3)

where $T_1$,……, $T_m$ are different temperature points and $P_1$,……, $P_n$ are different pressure calibration points.

Using the error matrix analysis, a comprehensive multi-term polynomial regression model was developed to create a unified calibration equation that accounts for measurement errors across all pressure and temperature conditions. This model describes the specific relationship between deviations in pressure readings and ambient temperature. Once the repeatability of this relationship is confirmed, it can be used to apply temperature compensation to the sensor or to correct historical observational data. This method adjusts raw pressure readings by eliminating temperature effects, thereby yielding values that more accurately reflect the actual pressure. The calibration equation is as follows:

$$\begin{gathered} \mathsf{\Delta }{P}_{T,P}=a+b·T_{read}+c·P_{read}+d·{T_{read}}^2+e·T_{read}·{P_{read}}^2+f·{P_{read}}^2+g·{T_{read}}^3 \\ +h·{T_{read}}^2·P_{read}+i·T_{read}·{P_{read}}^2+j·{P_{read}}^3 \end{gathered}$$

(4)

where $\Delta P_{T,P}$ represents the pressure measurement error at temperature $T$ and pressure $P$; $a$, $b$,……, $j$ are the regression coefficients; $T_{read}$ is the temperature reading of the sensor; $P_{read}$ is the pressure reading of the sensor; and m and n represent the polynomial orders.

The polynomial regression model described above allows us to accurately represent the measurement error of the pressure sensor under different temperature and pressure conditions. When applying this model for calibration, temperature compensation can be performed using the following Formula (5):

$$P_{corrected}=P_{read}-\mathsf{\Delta }{P}_{T,P}$$

(5)

where $P_{corrected}$ represents the corrected pressure value, $P_{read}$ is the pressure reading measured by the sensor, and $\Delta P_{T,P}$ is the measurement error obtained from the regression equation.

To gain a deeper understanding of the characteristics of temperature-induced pressure deviations during measurements, we conducted a systematic analysis of the pressure sensor’s performance under various temperature conditions. This analysis involved a comprehensive experimental design and optimized calibration methods. The experiments enabled the identification and compensation of temperature-induced errors in pressure measurement, thereby ensuring the stability and accuracy of pressure data in complex marine environments. The following sections will discuss the experimental setup, testing procedures, and result analysis, validating the effectiveness of the proposed calibration method.

4. Experiment Setting

To effectively illustrate the temperature-compensated pressure measurement, seven SBE 37-SM CTD instruments (manufactured by Sea-Bird Scientific, Bellevue, WA, USA) were selected as target samples. Two of the sensors (serial numbers 25526 and 22749) had been deployed on the seafloor for over six months and the remaining five (serial numbers 24365, 24744, 24820, 24833, and 24874) were brand new. This sample size is adequate and representative, ensuring the reliability and reproducibility of the experimental results.

The SBE37-SM sensor utilizes a high-precision strain-gauge pressure mechanism to detect resistance changes caused by pressure, leveraging the piezoresistive effect. Its operating principle involves strain gauges attached to a sensing diaphragm. As pressure deforms the diaphragm, the resistance of the sensing material changes, and this change is converted into an electrical signal via a Wheatstone bridge circuit.

Given the significant impact of temperature on pressure readings, the SBE37-SM CTD sensor includes an internal temperature compensation formula. An internal temperature sensor collects temperature data, and analog-to-digital (AD) readings are adjusted to pressure values using a formula that incorporates temperature coefficients. This approach effectively reduces temperature-induced measurement errors.

Achieving precise pressure data requires both accurate sensor calibration and effective temperature compensation, as temperature variations can alter the physical properties of the sensor material, leading to signal deviations.

Conducting variable-temperature calibrations under different pressure conditions enables a precise assessment of temperature effects on sensor performance, providing several key advantages:

Controlled Single Variable: By maintaining constant pressure while varying temperature, the specific influence of temperature on sensor readings can be isolated and studied without the interference of pressure variations, ensuring an accurate evaluation of temperature effects.

Optimized Temperature Compensation: Testing the sensor under different temperature conditions enables verification and optimization of the built-in temperature compensation mechanism, ensuring high accuracy during actual operational use.

Simulation of Real-Ocean Conditions: The experimental setup replicates the temperature variations that the sensor would encounter in real oceanic environments, thereby increasing the practical applicability of the experimental results.

Error Analysis and Compensation: Systematic recording and analysis of error data under varying temperature conditions allow for the development of error compensation models, which can be used to correct historical data sets and improve the overall quality of observations.

In this study, we used an experimental system that maintained constant pressure while varying the temperature of a thermostatic water bath. This setup enabled investigation of temperature-induced pressure measurement errors, especially focusing on the SBE 37-SM sensors. To accurately measure water temperature without introducing pressure-related bias to the temperature sensor, the SBE 37-SM was only partially submerged, with pressure being applied exclusively to the pressure core via a pressure-transmitting hose. This full-depth ocean simulation chamber generates pressure conditions ranging from 0 to 12,700 dbar, simulating depths of up to 12,700 m. The system ensures precise control of environmental temperature and pressure, as described in the following experimental procedures:

Temperature range: −2 °C to 35 °C; temperature fluctuation: ±0.0006 °C

Pressure range: 0 to 12,700 dbar; pressure stability: ±0.005% F.S

As shown in Figure 5, the internal medium of the pressure stabilization system is filled with seawater, and a quartz resonator pressure gauge, known for its high measurement precision and minimal variability across measurements (Paroscientific, Redmond, WA, USA, 745-20) is used as the working reference. The readings from both the standard pressure gauge and the SBE 37-SM are compared through a measurement and control feedback system. Pressure is maintained via a pressure buffer chamber to ensure the sensor operates under constant pressure conditions. The pressure levels set for the experiment were 2000 dbar, 4000 dbar, and 6000 dbar, with a permissible pressure fluctuation of ±0.005% F.S. The pressure probe of the SBE 37-SM was connected to the pressure buffer chamber via a pressure hose. To avoid measurement errors due to liquid height differences, both the standard pressure gauge and the SBE 37-SM were positioned at the same height during the experiment.

As shown in Figure 5, the SBE37-SM was completely submerged in the thermostatic water bath to simulate operational conditions at different temperatures. By controlling the water bath’s temperature, various environmental conditions were replicated. Once the pressure stabilized, the temperature of the bath was incrementally adjusted to the following set points: 2 °C, 5 °C, 10 °C, 15 °C, 20 °C, 25 °C, 30 °C, and 35 °C. At each temperature point, the temperature was maintained with a fluctuation of less than 0.001 °C. After achieving stable temperature conditions, data were continuously sampled for 30 min, recording the readings of both the pressure sensor and the standard pressure gauge. The measurement error was calculated by comparing the readings from the pressure sensor to those from the standard pressure gauge, allowing us to assess the impact of temperature variations on the accuracy of the pressure sensor.

In this study, a high-stability standard quartz pressure gauge was used as the reference instrument to ensure the precision and reliability of the experimental results. Before the formal experiment, an SBE 37-SM was connected to the system for a preliminary test that was aimed at verifying whether the reference pressure gauge could exhibit the required level of stability throughout the experiment. This stability is critical for synchronized measurements with the Seabird 37-SM pressure gauge, ensuring the credibility of the experimental data.

Figure 6 illustrates that pressure fluctuations within the pressure chamber are minimal. These fluctuations are caused by the feedback mechanism, where the quartz resonator pressure gauge measures the internal pressure of the chamber and transmits the data to the control system. The control system, in turn, adjusts by driving the servo cylinder, resulting in minor pressure fluctuations within a range of approximately 1 dbar. Given the SBE 37’s resolution of 0.14 dbar, it is capable of accurately capturing these fluctuations, which align perfectly with the readings from the quartz resonator pressure gauge. These fluctuations do not affect the calculation of measurement error. Furthermore, during the process of increasing temperature, the pressure control system responds swiftly, ensuring that the internal pressure remains stable within a fluctuation range of 1 dbar.

Comparative measurements between the SBE 37-SM pressure gauge and a standard quartz pressure gauge demonstrate that the SBE 37-SM exhibits noticeable measurement deviations under different temperature conditions. These deviations indicate that environmental temperature changes, even under constant pressure conditions, can significantly alter the pressure readings of the SBE 37-SM. To improve the accuracy of pressure measurements, further optimizations in temperature compensation and sensor design are necessary. Additional comparative results and conclusions regarding the SBE 37-SM will be discussed in the following sections.

5. Result and Discussion

The performance of the standard pressure gauge and multiple SBE 37-SM pressure gauges under controlled temperature conditions is shown in Figure 7. A detailed statistical analysis was conducted to assess the response of these instruments to temperature fluctuations, evaluating their reliability and accuracy in complex marine environments.

As indicated by the readings from the standard pressure gauge, the pressure chamber exhibited exceptional stability throughout the experiment, with fluctuations being limited to a narrow range between 3999.9 dbar and 4000.2 dbar. This minimal variation demonstrated the high stability of the experimental setup and the reliability of the measurements, providing a robust reference for the comparison and analysis of the SBE 37-SM pressure gauges.

For the SBE 37-SM pressure gauges, their pressure readings were influenced by the changes in water temperature. The data revealed a distinct non-linear trend in the pressure measurement errors across a broad temperature range (2 °C to 35 °C) under a constant pressure of 4000 dbar. As the temperature increased, the pressure measurement errors initially decreased and then increased, showing a stepwise change. This indicates that the sensors are highly sensitive to temperature, likely due to thermal expansion of piezoresistive materials or other thermally related mechanisms causing slight deformations in the sensor diaphragm, ultimately affecting resistance values and pressure readings.

From the perspective of individual sensor performance, pressure variations of 8 dbar were observed during temperature changes, with the maximum error occurring near 15 °C. Despite the manufacturer’s temperature compensation algorithm, the temperature effects on pressure measurements were not fully corrected. Specifically, at a constant pressure of 4000 dbar, when the temperature ranged from 2 °C to 35 °C, the pressure readings showed a negative drift that exceeded −0.1% F.S. specified in the Sea-Bird datasheet.

The three-dimensional plots presented in Figure 8a–g illustrate the variations in pressure measurement errors under different temperature and pressure conditions, with the vertical axis displaying the pressure measurement errors in decibars (dbar). These 3D error plots exhibit a “saddle shape” across varying pressure and temperature conditions. Under constant pressure conditions, the pressure readings follow a U-shaped curve: they initially decrease and then increase as the temperature rises. This overall trend reflects the thermal sensitivity of the pressure sensors used in CTD instruments. A stable range is observed in the mid-temperature region (15 °C to 25 °C), where the pressure measurement errors remain relatively constant. In contrast, temperature variations have a more significant impact on pressure measurement errors in the low-temperature (2 °C to 15 °C) and high-temperature (25 °C to 35 °C) regions.

At high-pressure levels, such as 6000 dbar, the magnitude of reading errors increases. Even minor thermal stress variations induced by temperature changes can significantly affect the reading error under these conditions. Conversely, at lower pressure levels like 2000 dbar, temperature-induced error variations are smaller, but the discrepancy between the sensor readings and the reference values is larger.

Figure 8f,g present the measurement results for two used SBE37-SM sensors, identified by the serial numbers 25526 and 22749, while Figure 8h illustrates the differences in error between new and used sensors. Under a constant pressure of 6000 dbar, temperature variations cause pressure readings to exceed 9 dbar for new sensors, while, for used sensors, the variations exceed 12 dbar. As the temperature increases, measurement errors become more irregular across varying pressure levels, with the most pronounced fluctuations being observed in the mid-pressure range (3000 dbar to 4000 dbar). At higher temperatures, the error patterns of the used sensors resemble those of the new sensors, albeit with slightly larger magnitudes, confirming that prolonged usage contributes to increased inaccuracies.

Although the same formulas and calibration methods were applied to all SBE37-SM sensors, both had been in operation in deep sea, high-pressure environments for six months, likely contributing to performance degradation. Prolonged exposure to seawater can cause sensor material aging, impacting their physical properties and measurement accuracy [30]. Additionally, extended operation under high-pressure conditions may lead to material fatigue in the sensor diaphragm and other critical components, altering the sensor’s modulus of elasticity and affecting its pressure response [31]. Permanent deformation of the sensor’s mechanical structure under high pressure can also increase reading errors, as even minor deformations can accumulate over time. Meanwhile, repeated temperature variations induce thermal stress cycles, accelerating material fatigue and aging, which further impacts measurement stability and accuracy [32].

The performance of the new sensors under high pressure and varying temperature conditions surpasses that of the used sensors. These findings underscore the importance of regular calibration and maintenance to ensure the long-term stability and accuracy of sensor measurements.

Figure 9 illustrates the pressure measurement errors as a function of temperature for the CTD instrument with the serial number 24744. The results indicate that in low-temperature seawater environments, the relationship between pressure measurement errors and temperature is as follows: at 6000 dbar, the slope of the pressure reading error with temperature is −0.67 dbar/°C; at 4000 dbar, the slope is −0.5 dbar/°C; and at 2000 dbar, the slope is −0.33 dbar/°C. The largest absolute slope is observed at 6000 dbar, which is consistent with expectations. This indicates that, at 2 °C, the higher ambient pressure correlated with a greater rate of change in pressure measurement errors as the temperature varied. As the SBE 37 is used as a moored CTD, where seawater temperature fluctuations are significant, these temperature variations may lead to inaccuracies in pressure measurements, thereby affecting depth calculations. These findings are crucial for understanding how temperature variations impact deep sea measurement accuracy, highlighting the need for more comprehensive calibration processes that account for the coupling of environmental variables and improved error compensation methods in the design and application of these instruments.

To demonstrate that the reading errors caused by temperature variations in SBE 37-SM sensors can be calibrated, we subjected five SBE 37-SM pressure sensors to two temperature-cycling experiments at a constant pressure of 4000 dbar. This was conducted to verify the repeatability of their pressure measurement errors with respect to temperature changes. The average readings and standard deviations for each sensor from the two tests are presented in

Table 1. Mean values and standard deviations from two tests at 4000 dbar.

Temperature (°C)		2 °C	5 °C	10 °C	15 °C	20 °C	25 °C	30 °C	35 °C
SBE37-24365	$\overline{P}$ (dbar)	4000.9	3998.2	3994.6	3992.6	3991.8	3992.2	3993.5	3995.6
	σP (dbar)	0.005	0.0513	0.0017	0.0102	0.0418	0.0715	0.005	0.01
SBE37-24744	$\overline{P}$ (dbar)	3997.8	3995.6	3992.8	3991.6	3991.6	3992.5	3994.8	3997.7
	σP (dbar)	0.0541	0.0562	0.0663	0.0179	0.0456	0.0213	0.0201	0.0190
SBE37-24820	$\overline{P}$ (dbar)	3999.2	3996.9	3993.8	3992.2	3991.8	3992.4	3994.1	3996.6
	σP (dbar)	0.0437	0.0210	0.0377	0.0358	0.0615	0.0713	0.0427	0.0890
SBE37-24833	$\overline{P}$ (dbar)	3997.8	3995.5	3992.7	3991.5	3991.6	3992.5	3994.8	3997.7
	σP (dbar)	0.0113	0.0362	0.0207	0.0048	0.0030	0.0433	0.0512	0.0482
SBE37-24874	$\overline{P}$ (dbar)	3997.7	3995.5	3992.6	3991.3	3991.2	3991.9	3994.0	3996.7
	σP (dbar)	0.0174	0.0435	0.0629	0.0109	0.0445	0.0162	0.0085	0.0398

.

During the experiments, the pressure was consistently maintained at 4000 dbar, and a thermostatic water bath was used to perform one temperature cycle ranging from 2 °C to 35 °C. The water bath was then cooled back to 2 °C, and a second identical temperature cycle was conducted.

As shown in Table 1, the results for each sensor at different temperature points across the two experiments are very similar, with small standard deviations indicating minimal fluctuations and good repeatability in the measurements. The experimental results demonstrate that, although the SBE 37-SM series pressure sensors exhibit noticeable deviations in pressure readings under different temperature conditions, these deviations show a highly consistent pattern in both tests. This consistency indicates that, despite the temperature-induced deviation issue, the sensors’ temperature-dependent errors at 4000 dbar are stable and can be corrected using more refined compensation methods.

Building on the research discussed earlier, we employed a constant-pressure, variable-temperature scheme to conduct a temperature–pressure grid calibration of the CTD instrument. The results revealed a temperature-induced pressure deviation phenomenon in the SBE 37-SM sensors and confirmed its repeatability and calibratability. After collecting and preprocessing the data, we comprehensively analyzed the matrix of temperature-related pressure errors using the following steps. A polynomial regression model was fitted using the least squares method and the regression coefficients were determined (a = −5.791; b = −0.431; c = −1.106 × 10⁻⁹; d = 0.023; e = −2.504 × 10⁻⁴; f = 1.086 × 10⁻⁶; g = −3.325 × 10⁻⁴; h = 4.345 × 10⁻⁶; i = 1.256 × 10⁻⁸; j = 1.313 × 10⁻¹⁰). These coefficients provide a detailed understanding of how temperature and measured pressure specifically affect sensor errors.

By applying the experimental data to the previously defined polynomial regression model, we obtained corrected pressure readings, as illustrated in Figure 10. The calibration process did not affect the resolution of the SBE 37-SM pressure sensors, which remained capable of detecting subtle pressure variations within the system with high accuracy. The errors in pressure readings caused by temperature variations were significantly reduced, resulting in a notable improvement in measurement accuracy compared to the uncalibrated data.

After calibration, the influence of temperature variations on the SBE 37-SM sensors was effectively eliminated, with temperature-induced pressure measurement errors being confined within ±0.5 dbar. These findings underscore the critical importance of accounting for the effects of temperature on pressure readings to achieve higher measurement accuracy in real marine environments. The results further highlight the potential impact of temperature variations on pressure measurements and emphasize the need for developing robust quantifiable metrics for temperature–pressure cross-sensitivity, as well as effective temperature compensation strategies, in future CTD instrument calibration processes. Implementing these measures can enhance data accuracy, ensure reliable measurements in complex marine environments, and ultimately improve the quality of foundational data used in oceanographic research.

6. Conclusions

This study proposes an improved calibration method for Conductivity–Temperature–Depth (CTD) pressure sensors, addressing the limitations of existing calibration standards. The method incorporates a traceable metrology system capable of maintaining long-term pressure and temperature stability and utilizes a combined pressure calibration process with temperature compensation for more accurate deep sea pressure measurements.

For the SBE 37-SM pressure sensors, the study reveals significant deviations in pressure readings under temperature variations, indicating that the sensors’ internal physical or chemical properties are highly sensitive to temperature changes. In particular, thermal expansion of materials or other temperature-related mechanisms may cause slight deformations in the sensor diaphragm, ultimately affecting resistance values and pressure readings. Although the sensors include embedded temperature compensation formulas, they were not able to completely eliminate the influence of temperature on pressure measurements. The temperature-related pressure reading errors shifted unidirectionally, exceeding 0.1% full scale.

In the used sensors, material aging and stress fatigue from prolonged exposure to high-pressure environments affected their physical properties and measurement accuracy, resulting in pressure reading errors exceeding 12 dbar under variable temperature conditions. In contrast, uncalibrated new sensors exhibited pressure deviations exceeding 9 dbar due to temperature variations. However, after applying the calibration proposed in this study, the fluctuation range was reduced to within ±0.5 dbar. This demonstrates that the polynomial regression model proposed in this paper was highly effective in fitting and calibrating pressure measurement errors, significantly reducing the discrepancies between the calibrated readings and the reference pressure values.

Furthermore, this study highlights the critical need to address the temperature drift characteristics of pressure sensors when collecting pressure data in marine environments using CTD instruments. Extended exposure to high-pressure conditions can result in significant reading deviations, underscoring the necessity of regular calibration to maintain measurement accuracy. The proposed constant-pressure, variable-temperature calibration method offers a highly realistic simulation of actual marine conditions, effectively minimizing the influence of temperature and its variations on pressure measurements. Compared to traditional approaches, the advantages of this method in enhancing pressure sensor accuracy are summarized in

Table 2. Comparison of JJG-763 calibration method and the proposed constant-pressure, variable-temperature calibration approach.

Comparison Items	JJG-763 Pressure Calibration	Proposed Calibration Method
Principle	Static, single-point calibration.	Dynamic, multi-factor nonlinear calibration.
Error Correction Method	Relies on error tables under fixed conditions.	Uses self-adaptive algorithms for dynamic adjustment.
Temperature compensation capability	Dual-point calibration at 0 °C and 30 °C.	Covers the full temperature range of the sensor.
Pressure Accuracy	Accurate at ambient temperature during verification.	Accurate across the entire temperature range.
Temperature-Induced Error Range (for SBE 37-SM)	9 dbar (2 °C to 35 °C).	±0.5 dbar (2 °C to 35 °C).
Operational Complexity	Manual, operator dependent.	Fully automated, reducing human error.
Adaptability to Environmental Variations	Cannot handle temperature fluctuations.	Adapts to varying temperatures, ensuring accuracy.
Response to Sensor Aging and Fatigue	Reflects errors caused by aging at calibration points.	Tracks 3D error changes due to aging and fatigue.
Expandability	Limited to static pressure calibration.	Scalable to other factors like humidity and salinity.

. In summary, this calibration method, characterized by its high adaptability and environmental fidelity, not only significantly improves the accuracy of CTD pressure measurements but also facilitates the correction of historical datasets. These improvements enhance the reliability and consistency of oceanographic data, thereby providing a robust foundation for advancing high-quality marine scientific research.