A theoretical resolution of a historical problem in analytical chemistry

Press Releaseプレスリリース

2024.10.15

JAMSTEC

In chemical analysis, the choice between using intensity ratio or area ratio for extracting information from spectra has been the subject of extensive experimental research due to its impact on researchers' decision-making. However, no consistent conclusion has been reached. In this study, we theoretically demonstrate that the area ratio is √2 times more precise than the intensity ratio, thereby resolving this long-standing issue (Figure 1)

Traditionally, the higher precision of the area ratio over the intensity ratio in experimental results has been attributed to the fact that the area, being an integral of intensity, contains more intensity information. However, our study shows that the superior precision of the area ratio is actually due to the negative covariance between intensity and the full width at half maximum (FWHM).

By combining analytical solutions that define the limits of spectral parameter measurement with insights from spectroscopy, we have established a theoretical framework linking "analytical precision" with "instrument performance." This framework can guide the most effective equipment upgrades within a limited budget.

Yuuki Hagiwara, a researcher in the Solid Earth Geochemistry Research Group, and Tatsu Kuwatani, a senior researcher in the Data Science Research Group, both at the Volcanoes and Earth’s Interior Research Center, Institute for Marine Geodynamics, Japan Agency for Marine-Earth Science and Technology (President: Hiroyuki Yamato, hereinafter referred to as "JAMSTEC"), have provided a theoretical answer to a long-debated issue in analytical chemistry. The issue concerns which estimator, the intensity ratio or area ratio, provides a more precise prediction of concentrations from spectral data. In this study, based on the Cramér-Rao inequality^{※1}, we derived an analytical solution representing the limits of spectral parameter measurement and proved that the area ratio is √2 times more precise than the intensity ratio. Previously, experimental results showing higher precision for the area ratio were attributed to the area being an integral of intensity and thus containing more information. However, our study demonstrates that the superior precision of the area ratio is due to the negative covariance between intensity and the full width at half maximum (FWHM).

Furthermore, by combining the analytical solutions for the limits of spectral parameter measurement with insights from spectroscopy, we established a theoretical framework linking "analytical precision" with "instrument performance." This theoretical framework allows us to numerically determine which components of an instrument should be upgraded to efficiently improve analytical precision. These numerical calculations can provide compelling justification when applying for funding to update equipment and enable the most effective improvements within a limited budget. The analytical solutions for measurement limits obtained in this study not only aid in optimizing analytical systems but also serve as theoretical benchmarks for evaluating the performance of new data processing algorithms and analytical protocols. These findings are expected to significantly contribute to the advancement of analytical chemistry.

These findings were published in "Scientific Reports" on October 15(Japan Time). This research was supported by JST CREST (JPMJCR1761) and JSPS KAKENHI (JP19J21537 and JP22J00081).

Title

Precision comparison of intensity ratios and area ratios in spectral analysis

Authors

Yuuki Hagiwara^{1} and Tatsu Kuwatani^{1}

Affiliations

1. Volcanoes and Earth’s Interior Research Center, Institute for Marine Geodynamics, Japan Agency for Marine-Earth Science and Technology

Supplemental Information

※1

**Cramér-Rao Inequality**

Generally, this inequality indicates that for any unbiased estimator of an unknown parameter in a probability distribution, there is a lower bound on the variance of the estimator. In this study, we consider a more specific situation. It shows that the precision of experiments estimating parameters such as peak intensity, position, full width at half maximum (FWHM), and area in spectra using a model function is limited by this inequality.

For analytical chemists, achieving maximum precision with minimal sample usage is a critical goal. To efficiently achieve this goal, understanding the physical and statistical background governing analytical precision is essential. In many fields of science and engineering, including analytical chemistry, spectral data are collected to estimate physicochemical properties of substances using parameters such as peak intensity, area, peak position, and full width at half maximum (FWHM). Specifically, intensity ratios and area ratios are key indicators of substance concentrations. For applications such as environmental monitoring of trace element ratios or isotope ratios, where small concentration variations must be detected, these ratios need to be estimated with high precision. Therefore, the question of whether the intensity ratio or area ratio is more precise, and how to effectively improve their precision within a limited budget, has been experimentally investigated since at least the 1960s.

However, previous studies have not provided a consistent quantitative answer. This inconsistency is due to various external factors affecting experiments, such as noise, spectral distortion, instrument performance, and uncertainties in estimating population variance. It is evident that simulations and theoretical approaches are effective in eliminating these external factors, but such approaches have not been undertaken until now.

Estimating the precision of spectral parameters is critical in various fields of science and engineering, as it significantly influences decision-making in data analysis and equipment upgrades. To determine which is more precise, the intensity ratio or the area ratio, we first derived an analytical solution for the precision limits of spectral parameters using the Cramér-Rao bound. Furthermore, by combining the Cramér-Rao bound with insights from spectroscopy, we established a theoretical framework linking "analytical precision" to "instrument performance." Finally, to demonstrate how this theoretical framework can guide the optimization of instruments for high-precision analysis, we numerically calculated the improvement in precision achieved through equipment upgrades, using non-destructive local isotopic ratio measurements of CO_{2}'s carbon isotopes by Raman spectroscopy^{※2} as an example.

Based on the concept of the Cramér-Rao bound, we derived the variance-covariance matrix^{※3} of spectral parameters (Figure 2).

As a result, we found that the lower bounds of the relative standard deviations of peak intensity, position, full width at half maximum (FWHM), and area are expressed by the following equations.

(1) | |

(2) | |

(3) | |

(4) |

Here, Δ*x* represents the interval between spectral data points, *Γ* is the FWHM, and I denotes intensity. From these equations, the following conclusions were drawn:

- The relative standard deviations of intensity, FWHM, and area are proportional to {Δ
*x/Γ*}^{1/2}. This means that the more data points there are above the FWHM, the better the precision of intensity, FWHM, and area. - The precision of intensity, FWHM, and area improves with wider peaks, whereas the precision of peak position is higher for narrower peaks.

Furthermore, calculating the analytical solutions for the relative standard deviations of these differences and ratios yields the following equations.

(5) | |

(6) | |

(7) | |

(8) |

From these results, taking the ratio of the precision of intensity ratio to area ratio yields the following

(9) |

This equation shows that the area ratio is √2 times more precise than the intensity ratio (Figure 3). However, note that this relationship does not hold for all spectra; it depends on the model function and noise model. For instance, if the model function is changed from a Gaussian to a Lorentzian function, the right-hand side of the equation becomes √3 instead of √2.

As shown in Figures 2a-b, simulation results revealed a clear negative correlation between intensity and full width at half maximum (FWHM). In contrast, there was no correlation between intensity and peak position. This outcome aligns with theoretical calculations, indicating a negative covariance between intensity and FWHM, and zero covariance between intensity and peak position (Figure 2d).

To qualitatively explain why the area ratio is √2 times more precise, the key lies in the negative covariance between intensity and FWHM. Specifically, when the intensity is low, the FWHM tends to be large, and vice versa. As a result, the area, which is proportional to the product of intensity and FWHM, tends to remain stable. Therefore, the superior precision of the area ratio is due to this negative covariance, and the traditional explanation that "the area ratio is more precise because it integrates more intensity information" is not accurate.

By expressing the interval between data points (Δ*x*) and FWHM (*Γ*) in equations 1-8 in terms of instrument performance parameters, we developed a theory linking the precision of spectral parameters to instrument performance (Figure 4). This theoretical framework shows that the relative standard deviation of the area ratio can be expressed as a function of detector pixel size, diffraction grating constant, spectrometer focal length, and wavelength.

Using this theoretical framework, we numerically calculated how much the precision of non-destructive local isotopic ratio measurements of CO_{2}'s carbon isotopes using Raman spectroscopy could be improved by upgrading the equipment (Figure 5). This method estimates the isotopic ratio by using the intensity ratio and area ratio of peaks corresponding to ^{12}CO_{2} and ^{13}CO_{2} in the Raman spectrum of CO_{2}. Figure 5 shows how changes in instrument performance affect the measurement precision of the carbon isotopic ratio. The vertical axis represents the relative standard deviation of the area ratio, which corresponds to the standard deviation of the carbon isotopic ratio. The blue region indicates the range of typical instrument performance, and the vertical dotted line represents the performance of the equipment used by Hagiwara et al. (2023).

The figure shows that reducing the detector pixel size from 16 μm to around 5 μm within the blue region significantly improves the measurement precision of the isotopic ratio. Such numerical calculations allow us to quantitatively estimate which components of the equipment should be upgraded to efficiently improve the precision of parameter estimation. This provides compelling justification when applying for funding to upgrade equipment and offers guidance on using research funds more effectively.

Supplemental Information

※2

**Raman Spectroscopy**

Raman spectroscopy is a technique that utilizes the Raman scattering phenomenon, which occurs when light interacts with a material. In Raman scattering, a portion of the incident light changes energy due to molecular vibrations or rotations of the material, resulting in scattered light with different wavelengths. By detecting and analyzing the energy distribution and photon count of this scattered light, the molecular structure, chemical composition, stress, temperature, and concentration of the material can be determined.

※3

**Variance-covariance Matrix**

A variance-covariance matrix is a matrix that summarizes the variances and interrelationships of multiple variables. The diagonal elements of this matrix contain the variance of each variable, while the off-diagonal elements contain the covariance between different variables. Variance indicates the degree of spread for each variable, and covariance shows how much two variables change together. Using a covariance matrix helps in understanding the overall patterns and relationships among variables in the data.

In this study, based on the concept of the Cramér-Rao bound, we provided a quantitative answer to the long-standing debate in analytical chemistry about whether the intensity ratio or area ratio is more precise. We found that the area ratio is √2 times more precise than the intensity ratio. Furthermore, by combining the derived analytical solutions for measurement limits with insights from spectroscopy, we established a theoretical framework linking "analytical precision" with "instrument performance."

In this study, we used Raman spectroscopy as an example to establish the theoretical framework linking "analytical precision" with "instrument performance." Moving forward, we plan to collaborate with experts in other analytical techniques to extend this framework, allowing "analytical precision" to be expressed as a function of "instrument performance" in various methods. This approach will enable the optimization of equipment based on the Cramér-Rao bound framework in different fields of analytical chemistry, extending the theoretical foundation to achieve maximum precision with minimal sample across various disciplines.

Furthermore, this study aims to apply non-destructive local carbon isotope ratio measurements of CO_{2} to minute inclusions ranging from a few micrometers to several tens of micrometers within volcanic phenocrysts. In fact, using a newly acquired detector with smaller pixel size, based on the theory developed in this study, has significantly improved the measurement accuracy of the carbon isotope ratio. In the future, we aim to apply this method to natural samples to identify the source materials of CO_{2} that trigger volcanic eruptions.

For this study

Yuuki Hagiwara, Researcher, Solid Earth Geochemistry Research Group, Volcanoes and Earth’s Interior Research Center, Institute for Marine Geodynamics, JAMSTEC

Tatsu Kuwatani, Principal Researcher, Data Science Research Group, Volcanoes and Earth’s Interior Research Center, Institute for Marine Geodynamics, JAMSTEC

For press release

Press Office, Marine Science and Technology Strategy Department, JAMSTEC