## 2 Materials and Methods
### Image Processing
#### 2.2.2 Application of Different DAS Methods (DAS-F, DAS-f and DAS-2f)
Regarding Figure 4a,b, the maximum absorbance is a suitable measure for determining the concentration efficiently and sensitively. This procedure can be linked to the method that is called DAS-f. As illustrated in Appendix B, the pixelwise smoothed peak absorbance can be estimated by finding the maximum image along the wavenumber dimension of the DAS-f sequence. Therefore, DAS-f allows a straightforward calculation of the methane concentration image, since the smoothed peak absorbance shows a linear relationship with the concentration. The resulting concentration image and the leak situation with remarkable clarity is discussed in Section 3 in more detail. So far, concepts such as DAS-F and DAS-f based on a resource-efficient BLC, including customized data preprocessing, have been described. The promising results of the more advanced image processing of DAS-f motivate us to go a step further to DAS-2f. * DAS-2f method:  
A straightforward realization of the DAS-2f concept is to apply a second partial derivative with respect to the wavenumber dimension in terms of Equation (5). However, considering the BLL, including the wavelength or intensity modulation of the laser (e.g., Equations (1) and (2)), a baseline-free form of the second derivative of the absorbance according to the wavelength can be represented by  
\[h_{x,y,\lambda}\ast\frac{\partial^{2}}{\partial\lambda^{2}}A(x,y,\lambda) \approx\ -h_{x,y,\lambda}\ast\frac{\partial^{2}}{\partial\lambda^{2}}\ln(I(x,y, \lambda)). \tag{6}\]  
The 3D Gaussian smoothing was chosen as above for DAS-f (\(\sigma_{x}=\sigma_{y}=\)3 pixels and \(\sigma_{\lambda}=2.5\) frames). It should be noted that no BLC is required, as the derivative of the natural logarithm is self-normalizing (e.g., \(\partial/\partial x\ln(x)=1/x\)) and the laser characteristic is sufficiently linear with respect to the wavelength or intensity modulation (e.g., \(\partial^{2}/\partial\lambda^{2}\ln(I_{0}(\lambda))\approx 0\), according to Equation (2)). Note that an image normalization by a reference image as above is also not required for DAS-2f. As illustrated in Figure 4c, the methane leakage of 0.25 mL/min is clearly visible in the DAS-2f sequence in the form of a spectral red-blue-red contrast. This pattern can be attributed to the second derivative of the gas absorption line with respect to the wavelength, as also indicated by analysis of the integrated pixels \(((x,y)\ \in\ \Omega)\) in Figure 4d. As with DAS-f, a methane anomaly can be seen spatially near the leak (center of the images), including a methane plume towards the bottom and a general methane background of approximately 9 ppm*m. As previously with DAS-f, the DAS-2f signal is in excellent agreement with the corresponding simulation, considering the spectroscopic parameters provided by HITRAN and Gaussian smoothing. The latter means that the second derivative of the spectral absorbance can be analyzed by a very distinct peak-dip-peak feature. As described in Appendix B, a peak-dip-peak measure or image is calculated by a maximum image (first row), minimum image (second row) and maximum image (third row) of the DAS-2f sequence. According to Appendix B, this feature allows a direct calculation of the methane concentration image, as shown and discussed in Section 3.