## 2 Principle of WRIM for DLS
### Inversion Principle of WRIM  
Owing to the inevitable noise of ACF and the inappropriate inversion range, it is extremely easy for the traditional single-scale optimization of Equation (7) to fall into local minimum value. Therefore, we obtained the optimal solution with much difficulty. Wavelet transform is a new multi-scales analysis method that can provide a \"time-frequency\" window with frequency change [39; 40]. It is an ideal tool for time-frequency analysis and signal processing. It gradually decomposes the signal on the multi-scales through the expansion and translation operation, and finally meets the requirements of adaptive analysis of signal. Therefore, the wavelet transform is widely used in many fields such as industry, medicine, and military. Considering the multi-scales characteristics of the wavelet transform, the wavelet inversion strategy that can accelerate the convergence, enhance the stability, and overcome the disturbance of local minima is developed [41; 42]. Using this inversion strategy, the global optimal solution can be quickly searched out. Taking the wavelet transform of IDS 4 shown in Figure 1 as an example, we introduce its specific principle. In Figure 1, scale 1 to scale 4 corresponds to the scale from fine to coarse. The original ACF A\({}_{1}\) is decomposed into coarse and fine two sub-spaces. The signal parts corresponding to the fine and coarse sub-spaces are A\({}_{2}\), D\({}_{2}\), respectively. A\({}_{2}\) corresponds to the approximation part, namely, overall trend of ACF. D\({}_{2}\) corresponds to the detail part, namely, removed noise form original ACF. Then, the approximation part A\({}_{2}\) continues to be decomposed into A\({}_{3}\) and D\({}_{3}\). Thus, the detail part is gradually removed from ACF. This process continues repeatedly until the decomposition scale reaches the coarsest scale. With the increase of decomposition scale, ACF noise of the coarser space is gradually removed, and then ACF of this space becomes relatively smoother. In Figure 1, ACF can be written as  
\[\mathrm{ACF}(\mathrm{A}_{1})=\mathrm{A}_{2}+\mathrm{D}_{2}=\mathrm{A}_{3}+ \mathrm{D}_{3}+\mathrm{D}_{2}=\mathrm{A}_{4}+\mathrm{D}_{4}+\mathrm{D}_{3}+ \mathrm{D}_{2} \tag{8}\]  
Through the multi-scale decomposition, the original ACF is composed into the approximation parts A\({}_{2}\), A\({}_{3}\), and A\({}_{4}\) in the different sub-spaces. Correspondingly, the original inversion problem is divided into many sub-problems. \[A_{i}x_{i}\ =\ b,\ \ i\ =\ 1,2,3,4 \tag{9}\]  
in which \(A_{i},x_{i}\) is similar to the matrix \(A\) and \(x\) in Equation (4). They are calculated in the diameter inversion range [\(d\)min, \(d\)max] of the different sub-spaces. Because A\({}_{4}\) has the best smoothness in the coarsest space, even within larger initial inversion range, the optimal value of this sub-problem is easy to solve. According to this global optimal value, we can get the approximate inversion range close to true value. Then, the scale becomes finer again,and the detailed part D\({}_{4}\) of ACF is gradually retrieved. The sub-problem corresponding to A\({}_{3}\) is inverted above near true inversion range. Therefore, the global optimal value of this scale can be easily found. Repeatedly, as the decomposition scales become finer, the inversion range gradually becomes closer to the true value. When the scale reaches 1, the final optimal PSD is obtained in the inversion range closest to the true values. The flow chart of this method is shown in Figure 2.  
In Figure 2, \(V_{i}\), \(i=1,2\cdots J\) denote different sub-spaces, and \(J\) is IDS. \(I_{J-1}^{J}\) represents the wavelet decomposition from \(V_{J-1}\to V_{J}\); \(I_{J}^{J-1}\) represents the wavelet reconstruction from \(V_{J}\to V_{J-1}\); \(R_{J}\) expresses the inversion operation in the \(J\)th sub-space; \([d_{\min J},d_{\max J}]\) represents the PSD range retrieved in the \(J\)th sub-space, which is used as the initial inversion range on the \(J-1\)th scale.