With the increasing environmental challenges posed by climate change, the need to adopt accurate and effective strategies to monitor greenhouse gas emissions, particularly carbon dioxide (CO₂) and carbon monoxide (CO), has become more critical. These gases are among the primary contributors to the exacerbation of global warming. Accurate and reliable monitoring of these emissions is a vital tool to support environmental and regulatory policies. Industrial and environmental entities rely on measurement data to assess environmental impact, improve industrial processes, and comply with international environmental standards. However, the accuracy and reliability of carbon emissions measurement devices are not constant, as they are influenced by several technical factors, necessitating a deep and comprehensive study of the uncertainty sources affecting the overall performance of these devices. Carbon emissions measurement devices are pivotal tools for monitoring greenhouse gas concentrations in the atmosphere, whether in industrial or environmental settings. These devices rely on calibration using Certified Reference Materials (CRMs) [4] - [7] to ensure measurement accuracy. However, the actual performance of these devices is affected by multiple factors, such as the accuracy of the reference materials used, the sensitivity of the devices to slight changes in gas concentrations, repeatability and precision, as well as the linear relationship between reference values and measured values. Therefore, improving the accuracy of these devices requires a systematic study of the sources of uncertainty and an analysis of their impact on the reliability of the results [8] . In this context, Certified Reference Materials (CRMs) [4] - [7] play a fundamental role in improving measurement accuracy, as they are prepared according to the highest international standards, such as ISO 6143 [2] , ISO 6142 [1] , and ISO 17034 [3] . These standards ensure the preparation of reference materials with a high degree of accuracy and stability, making them an essential tool for device calibration [9] . Furthermore, analyzing uncertainty sources using the international standard ISO GUM [10] (Guide to the Expression of Uncertainty in Measurement) is a critical step to understanding the factors affecting measurements and working to minimize them. Compliance with ISO 17025 [11] requirements also ensures that laboratory measurements are conducted according to the highest standards of quality and reliability. This study aims to provide a comprehensive and accurate evaluation of the performance of carbon emissions monitoring devices by systematically analyzing the sources of uncertainty, with a focus on improving the accuracy and reliability of measurements. To achieve this goal, experiments were designed to cover a wide range of gas concentrations corresponding to the operational range of the devices under study. Additionally, five Certified Reference Materials were used, consisting of carbon monoxide gas mixtures in nitrogen, produced in a Gas Metrology Laboratory with diverse concentration ranges to ensure accurate and comprehensive calibration [7] [8] [12] , as shown in Table 1 .

This study was based on the OIML R 99- [13] 1 & 2 standard to ensure that the results comply with international requirements, enabling the evaluation of the performance of the monitoring devices used. This enhances the reliability of the resulting data and ensures its alignment with global standards.

Table 1. CO Concentration ranges.

Studying the sources of uncertainty in the calibration process of carbon emission monitoring devices is a fundamental step to ensure the accuracy and reliability of measurements [19] [20] . These devices are primarily used to monitor and determine carbon emission levels, which directly influence environmental and regulatory decisions. Many countries and institutions rely on these measurements to develop effective strategies for reducing carbon emissions in alignment with international agreements, such as the Paris Climate Agreement. Therefore, accurate measurements and proper calibration significantly contribute to reducing harmful emissions and achieving compliance with international environmental standards. The ISO GUM [10] (Guide to the Expression of Uncertainty in Measurement) standard provides a systematic framework for analyzing and estimating all sources of uncertainty associated with the measurement process, thereby enhancing confidence in the results. By applying this standard, it is possible to identify factors affecting measurement accuracy, analyze their quantitative impact, and provide a comprehensive estimation of total uncertainty. This approach is considered an essential scientific tool to ensure the quality of measurements and support decision-making. Figure 2 illustrates the sources addressed in this research.

1) Slope:

The slope in the calibration equation represents the relationship between the instrument’s response (signal) and the concentration of the target gas. It reflects how sensitive the instrument is to changes in gas concentration. Maintaining a consistent and accurate slope is essential for reliable gas analysis, as any variation in the slope may lead to measurement errors—especially when dealing with varying concentration levels. The linear calibration is typically modeled using the equation: y = ax + b ( Figure 3 & Table 2 ).

Where:

To evaluate the accuracy of the slope, the residual standard deviation (s) is calculated using the following equation:

$S=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(Y_i-b-a{X}_i\right)}^2}{N-2}}$

Explanation of the Parameters:

Once these parameters are established, the calibration equation is applied as illustrated in Table 3 .

a) Collect Calibration Data: Prepare a table of values with known concentrations x_i (from certified reference materials) and corresponding instrument responses y_i.

b) Calculate the Calibration Line (a and b): Use statistical software or Excel regression functions (e.g., LINEST, TREND) to determine:

- a: Slope

- b: Intercept

c) Calculate the Predicted Response for Each Point: Use the formula: ${\hat{Y}}_i=a{X}_i+b$

d) Determine the Residual for Each Point: Subtract the predicted value from the actual response: Residual = $Y_i-{\hat{Y}}_i$

e) Square Each Residual and Sum All Values: ${\displaystyle \sum {\left(Y_i-\left(a{X}_i+b\right)\right)}^2}$

f) Divide by (N − 2).

2) Intercept:

The intercept on the calibration line represents the value shown by the instrument when the gas concentration is zero. Any deviation from this intercept indicates a systematic error in the calibration, which can result in inaccurate measurements.

Equation for Standard Uncertainty of the Intercept ( Table 4 )

$u\left(b\right)=\sqrt{\frac{S^2{{\displaystyle \sum }}_{i=1}^n{X}_i^2}{{{\displaystyle \sum }}_{i=1}^n{\left(X_i-\bar{X}\right)}^2}}$

Explanation of Terms

a) Collect Calibration Data: Measure the instrument response for at least 3 - 5 known gas concentrations.

b) Fit a Linear Calibration Line: Use linear regression to fit the data, determine the slope, and intercept.

c) Calculate the slope -Residual Variance s²:

d) $s^2={\displaystyle \sum {\left(y_i-{\hat{y}}_i\right)}^2}/\left(n-2\right)$, where $y_i$ is the observed value and ${\hat{y}}_i$ is the predicted value from the line.

e) Calculate ${\displaystyle \sum x_i^2}$ and ${\displaystyle \sum {\left(x_i-\bar{x}\right)}^2}$ based on your input concentrations.

f) Substitute all values into the formula to calculate u(b), the standard uncertainty of the intercept.

3) CRM (Certified Reference Materials):

Certified Reference Materials are used as a standard for device calibration. The uncertainty associated with the quality and accuracy of the CRMs directly affects calibration accuracy, as any error in the CRM will be reflected in the final measurements.

Most certificates give the expanded uncertainty (U-CRM) with a coverage factor k (typically k = 2 for 95% confidence). Convert it to a standard uncertainty (u_CRM) with:

$u_{CRM}=U\text{-}CRM/k$

(For k = 2, this is simply U-CRM/2) ( Table 5 )

4) Repeatability

Repeatability refers to the instrument’s ability to produce consistent results when the same measurement is repeated under identical conditions. It is a key indicator of measurement reliability and precision. Any variation observed during repeated measurements directly contributes to the overall measurement uncertainty.

Standard Uncertainty from Repeatability

Standard Uncertainty = $SD/\sqrt{n}$ ( Table 6 )

Explanation of Terms

Guide to Applying the Equation

a) Calculate the standard deviation (SD) of the results.

b) Count the total number of measurements (n).

c) Compute the square root of n.

d) Divide the standard deviation by the square root of n to get the standard uncertainty.

5) Resolution:

Resolution is considered a source of Type B uncertainty because it does not depend on repeated measurements, but rather on the specifications of the instrument. The value is divided by $\sqrt{3}$ ( Table 7 ).

6) Accuracy:

Accuracy refers to how close the measurements are to the true value. Uncertainty in accuracy includes errors resulting from the device itself or the measurement method, serving as an indicator of the quality of results ( Table 8 ).

We determined sensitivity coefficients as follows ( Table 9 ):

We determined relative uncertainty as follows ( Table 10 ):

The Device Response and CRM with Uncertainty are displayed in Figure 4 & Table 11 .

In conclusion, this study highlights the vital role of accurate gas monitoring in supporting environmental protection and climate action. It demonstrates how advanced emission-measuring devices—when calibrated using Certified Reference Materials (CRMs) [4] - [7] prepared in line with ISO 6142 [1] , ISO 6143 [2] , and ISO 17034 [3] —can deliver reliable and traceable results. By examining key sources of uncertainty, including calibration linearity, CRM certificate accuracy, repeatability, and sensitivity, the study offers valuable insight into factors that influence measurement performance. All evaluations followed ISO GUM [10] guidelines and complied with ISO 17025 [11] standards, ensuring high data quality and international credibility. Importantly, this study applies to all types of gas-monitoring devices, regardless of application. However, the CRMs used in calibration are tailored based on each device’s measurement range—from the lowest detectable concentration to the highest operational limit. This ensures that the reference materials are fit for purpose and reflect real-world measurement conditions. The results, aligned with OIML R 99- [13] 1 & 2 standards, confirm that the use of high-quality CRMs [4] - [7] and systematic uncertainty analysis significantly improves measurement accuracy and supports compliance with regulatory requirements. Overall, this study presents a comprehensive framework for enhancing gas emission monitoring, contributing to more effective environmental decision-making and policy implementation.