The ICESat-2 satellite carries the world's first photon-counting system LiDAR system, i.e., the advanced topographic laser altimeter system (ATLAS), whose high repetition frequency and multibeam design can provide water-profile data with high spatial resolutions. However, the water-profile signals acquired by the ATLAS exhibits "after pulse" which restricts the application of the observation data and must be corrected using an inverse convolution algorithm, where an appropriate system impulse-response function is the core of the inverse convolution algorithm. In this study, the possible sources of "after pulse" are first identified, based on which the requirements of the system impulse-response function in the deconvolution are clarified through theoretical derivation, and a data-quality control algorithm is proposed on this basis, and the system impulse-response function is established using salt-marsh and desert data. Based on an analysis of ex-Gaussian fitting results, the pulse widths of the system impulse-response functions established using the desert and salt-marsh data are 0.095 and 0.142, a difference of 0.047, which can be reduced to 0.002 using quality-control methods. Subsequently, the system impulse-response functions generated from the two types of surface data are used to back-convolute the water body signals and invert the backward scattering coefficients of the water body and particulate matter. Finally, the inversion results show that in the Black Sea waters, without quality control, the inversion error of the system impulse-response function based on salt-marsh data is smaller, whereas after quality control, the inversion results using desert and salt-marsh data as the system impulse response-function indicate similar errors.