## 2 Materials and Methods
### Image Processing
#### 2.2.2 Application of Different DAS Methods (DAS-F, DAS-f and DAS-2f)  
The starting point for further image processing is the contrast-corrected intensity sequence according to Figure 2a and Equation (1). Regarding the data quantity and data quality, more complex non-linear fit procedures for a pixel-by-pixel methane concentration determination were not considered in terms of computational complexity and robustness. In the following, three methods called DAS-F, DAS-f and DAS-2f are introduced in order to obtain a quantitative methane concentration image in a calibration-free manner. * DAS-F method: The DAS-F method is intended to analyze an area (F) below the absorbance (cf. Figure 2c) as a feature or measure of the gas absorption. First, an absorption sequence is calculated. Provided that the different reflections are corrected by normalization using a reference image, as explained in Appendix A and illustrated in Figure 2a, a pixelwise baseline correction (BLC) of the laser signal (e.g., 200 \(\times\) 200 baseline) can be avoided. The BLC can be globally described (independent of the pixels) by a first-order polynomial. Using Equation (1), the absorption sequence \(A(x,y,\lambda)\) can be calculated by the negative natural logarithm of the transmission:  
\[A(x,y,\lambda)=-\ln\left(\frac{I(x,y,\lambda)}{I_{BLC}(\lambda)}\right) \tag{3}\]  
The resulting absorption sequence is shown in Figure 3a. It shows that the approach is suitable to correct the laser baseline. As spectrally and spatially expected, the absorption sequence shows the highest absorption value in the middle of the sequence, as well as in the center of the individual images. Assuming the standard pressure and temperature (P and T) and knowing the spectroscopic HITRAN parameter (e.g., absorption line strength, S) for methane at the target wavelength, the gas concentration image \(c(x,y)\) in ppm*m can be calculated directly from the integral of the absorbance sequence (called the DAS-F method) in a pixelwise manner as follows:  
\[c(x,y)=-\frac{k_{B}T}{S\,P}\,\int d\lambda\ln\left(\frac{I(x,y,\lambda)}{I_{ BLC}(\lambda)}\right), \tag{4}\]  
where \(k_{B}\) denotes the Boltzmann constant. The resulting concentration image is illustrated in Figure 3b. Obviously, the performance of this rather simple DAS-F method is not sufficient to fulfill the demand of a methane image in ppm*m for the detection and identification of small methane leaks. More precisely, the methane concentration in the ambient air in the range of 10 ppm*m and the anomaly of the methane concentration in the vicinity of the leak cannot be determined with a sufficient signal-to-noise ratio. The graphical representation of the probability distribution of the concentration values of the methane image (\((x,y)\)\(\in\)\(\Omega\)is shown in the histogram in Figure 3c. The histogram includes all calculated concentration values obtained using the DAS-F method, including the negative values caused by data quality issues in combination with the method (refer to the later Discussion section for details). The color bar of the methane image was deliberately chosen between 0 and 50 ppm*m to consider the expected low concentration anomaly (increase) compared to ambient methane due to the leak scenario. In consequence, unphysical negative concentrations are limited to 0 ppm*m (dark blue or black), as well as the region that is not illuminated by the laser. Methane concentrations greater than 50 ppm are also color-limited (dark red). However, the strong noise of the methane concentration image can be represented quantitatively via the histogram and expressed with the standard deviation and the mean value of the methane image by \(\mu\pm~{}\sigma=(8~{}\pm~{}31)\) ppm*m. Since the standard deviation is at least one order of magnitude too high, neither ambient methane nor a methane anomaly caused by the leak can be detected.