Estimation of Error in Non-linear Least Square for Quantitative Analysis in Fourier Transform Infrared Spectrometry

The quantitative analysis of Fourier transform infrared spectrometry using non-linear least squares method has achieved a wide range of applications. At present, it is common practice to evaluate the fitting performance based on the magnitude of the residuals, which cannot quantify the inversion error of each parameter involved in the fitting. In this paper, the parameter error estimation method for inversion using the non-linear least squares method in quantitative analysis of infrared spectra is proposed based on the statistical theory of parameter estimation. The inversion errors for each fitting parameter are estimated through the Jacobian matrix of the parameters and the estimation of the variance of measurement errors, where the variance of measurement errors can be approximated using the variance of fitting residuals. Since the model adopts a series of idealized assumptions and the error estimation is an approximation at the optimal parameters, we conducted experimental validation for toxic and hazardous gases commonly found in ship's compartments to verify its applicability and stability for quantitative infrared spectroscopy. The materials used in the experiment are three gases, CBrF3, CH2Cl2, and CHCl3, which exhibit significant absorption peak overlap in the 725-795 cm(-1) spectral range. We conducted a comparative analysis of the commonly used single-beam spectra and transmittance spectra in quantitative analysis, and controlled the noise level of the spectrum by its averaging number. The acquisition of transmittance spectra relies on single-beam spectra obtained with high-purity nitrogen gas as the background. The experimental results indicate that, the primary reasons for differences in the inversion results between single-beam spectra and transmittance spectra are spectral drift, baseline fitting errors, and systematic errors. For the selfdeveloped extractive Fourier transform infrared spectrometer, an 8 averaged spectra is sufficient to meet the requirement of an inversion error of less than 3%. When using the inversion results from 64 averaged spectra in conjunction with error estimation, it is possible to achieve 100% coverage of the mean concentration. As the noise level decreases, disturbances from factors such as the instrument and the environment become the main contributors to estimation error. The differences in the convergence values of relative errors for various gas components are primarily caused by variations in the spectral accuracy of each component in the spectral database. In practical applications, the transmittance spectrum and single-beam spectrum can be reasonably selected for quantitative analysis according to the specific conditions of the monitoring scene. The estimation error of the inversion results can be obtained as reference indicators for the reliability and accuracy of the inversion results and can be used to balance the trade-off between measurement precision and time resolution. At the same time, this method has important application prospects in many aspects such as optimizing the parameter configuration of spectral analysis and guiding the design of spectral instrument systems.