## 4 Results
### Fitness Evaluation  
To assess the decomposition performance of the proposed PGD on diverse waveform types, quantitative evaluation and qualitative analysis are implemented, the results are compared with those of CGD. Several indicators that measure fitness were calculated for quantitative assessment. The root mean squared error (RMSE) and R\({}^{2}\) are used to measure the error between the original waveform (\(y\)) and Gaussian mixture model (\(\hat{g}\)  
Figure 6: Iterative steps of PGD: (**a**) first iteration with original peaks (OPs); (**b**) second iteration with the OPs and potential peaks; (**c**) final PGD result. Figure 7: Evaluation steps of waveform decomposition. approximated through waveform decomposition. In addition, the structural similarity index map (SSIM) [43] is referenced to evaluate the structural fitness:  
\[RMSE=\sqrt{\frac{1}{w}{\sum_{i=1}^{w}}{\left({y_{i}-\hat{y}_{i}}\right)^{2}}}, \tag{9}\]  
\[SSIM=\frac{{\left({2{{u_{y}}{u_{g}}+{C_{1}}}}\right)\left({2{{\sigma_{yg}}+{C_{ 2}}}}\right)}}{{\left({{\mu_{y}^{2}+\mu_{g}^{2}+{C_{1}}}}\right)\left({{\sigma_ {y}^{2}+\sigma_{g}^{2}+{C_{2}}}}\right)}}, \tag{10}\]  
where \(w\) indicates the number of waveform samples, \({{\sigma_{yg}}}\) is the covariance, and \({{C_{1}}}={\left({{k_{1}}L}\right)^{2}}\) and \({{C_{2}}}={\left({{k_{2}}L}\right)^{2}}\) are the variables for adjusting the weak denominator. The general constant parameters are \({k_{1}}=0.01\) and \({k_{2}}=0.03\), and the dynamic range of the signal is \(L=2^{bits\ per\ signal}-1\)[43]. The smaller the RMSE value and the closer R\({}^{2}\) and SSIM are to 1, the higher the fitness. The mean and standard deviation of the fitness indicators are listed in Table 3. For a more intuitive evaluation of the RMSE magnitude, it has been expressed as normalized RMSE (nRMSE) scaled to a waveform amplitude range (65,536 DN). In both test datasets, the PGD was found to approximate the ABL waveforms better than the CGD. In test site 1, PGD exhibited better fitness than CGD in terms of error (nRMSE = 0.179), approximation (R\({}^{2}\) = 0.978), and structural similarity (SSIM = 0.907). The approximation error decreased by approximately 72% in nRMSE, and R\({}^{2}\) and SSIM improved by 0.269 and 0.372, respectively. In test site 2, the fitness of PGD was indicated by nRMSE = 0.0260, R\({}^{2}\) = 0.980, and SSIM = 0.901. In the foregoing, nRMSE decreased by approximately 66%, and R\({}^{2}\) and SSIM improved by 0.172 and 0.206, respectively, compared with those of the CGD results. To analyze how the proposed approach is applied to various ABL waveforms, experiments were performed on waveform types with different peak numbers and shapes, and the results were compared. The curves shown on the upper part of Figure 8 are the decomposition results obtained via CGD, and those on the lower part are the decomposition results obtained via the proposed PGD. Figure 8a,e show the waveform with similar intensities of water surface and bottom returns in shallow water, and each return component has its own peak. When each return is detected, both CGD and PGD successfully decompose the waveform into the same components. However, as shown in Figure 8b,f, in shallower waters, the water surface and bottom components closely overlap; hence, the bottom return peak may not be detected. Consequently, the bottom point could not be extracted by CGD, but PGD can measure even a very shallow bottom by decomposing the potential components. The bottom return signal shown in Figure 8c,g is very weakly detected compared with the water surface return, and two OPs are present. Although the bottom return was detected during the initial peak detection, the bottom component was not decomposed by CGD due to its weak intensity (Figure 8c). However, because PGD gradually estimates potential components until the initially detected peak is decomposed into separate components, the bottom return is successfully measured (Figure 8g).