Spectral unmixing of hyperspectral lidar signals considering laser beam profile

Full-waveform hyperspectral LiDAR (FWHSL) provides range-resolved spectral measurements and enables quantitative interpretation of multiple targets within a laser footprint. Conventional spectral unmixing (SU), originally developed for passive hyperspectral imagery, estimates the fractional contribution of each endmember from a mixed spectrum and commonly relates this abundance to the area proportion of the corresponding target within pixels. However, HSL is an active sensing modality, and its measured mixed spectrum is inherently modulated by the spatial energy distribution of the emitted laser beam. As a result, in HSL data, the spectral contribution of a target depends not only on its reflectance and geometric coverage, but also on its position within the beam footprint. Directly applying conventional SU to HSL therefore ignores this intrinsic beam-profile weighting and can lead to a mismatch between energy-weighted spectral abundance and true geometric area proportion. To address this problem, this study formulates the beam-profile-induced energy-to-area mismatch and proposes a beam-profile-constrained geometric spectral unmixing method for FWHSL data. The method interprets the mixed spectrum as an energy constraint and uses a known, calibrated Gaussian beam profile together with a low-dimensional circular-arc boundary model to retrieve target area proportions. Simulations with controlled binary mixtures show that the method outperforms a linear mixing model (LMM)-only baseline, reducing mean relative error (RE) from 21.55 % to 9.64 % and composite root mean square error (CRMSE) from 25.81 % to 13.43 %. A laboratory 101-channel FWHSL system experiment further validates the method using leaf-wood board, leaf-red brick, and leaf-ceramic tile binary mixtures and image-based ground truth. Relative to the direct LMM baseline, the proposed method reduces the overall mean absolute error (MAE) from 0.2061 to 0.1321, across-group root mean square error (ARMSE) from 0.1412 to 0.0843, mean RE from 24.15 % to 15.48 %, and mean CRMSE from 0.1158 to 0.0660. These changes correspond to reductions of 35.90 %, 40.27 %, 35.91 %, and 42.97 %, respectively. These results indicate that accounting for laser beam-profile weighting can improve geometric area-proportion retrieval in FWHSL SU, particularly when the target boundary intersects beam-sensitive regions with large spatial energy gradients. The current validation is limited to binary mixtures with locally smooth boundaries and a known or calibrated beam profile. Extension to unknown or irregular beam profiles, multi-material mixtures, and more complex intra-footprint geometries remain a subject for future research.