## 3 Methodology
### Optimize General Fitting Methods by Wavelet Transform
#### 3.1.1 Wavelet Transform  
As a mathematical tool initially designed for signal processing, WT provides multiresolution/multiscale analysis functions and effectively extracts global and multiscale features of images within the spatial frequency domain [45].  
The block diagram in Figure 2 explains the application of a wavelet transformation in detail. During the two-dimensional wavelet transform process, a digital image is decomposed into four images (hereafter called LL, HL, LH, and HH). Here, 'H' stands for high-pass filter, 'L' stands for low-pass filter, and 'HL' means a high-pass filter corresponding along the row direction of the input image and a low-pass filter following along the column direction. The same nomenclature applies to the LH, HH, and LL images. The spatial resolutions of these decomposed images are twice as coarse as the original one. For example, in this work, the spatial resolution of the original SM image is 0.25\({}^{\circ}\)\(\times\) 0.25\({}^{\circ}\), and the spatial resolution of its four decomposed images is 0.5\({}^{\circ}\)\(\times\) 0.5\({}^{\circ}\). Commonly applied WT techniques include the Haar, Daubechies, Coiflet, and Symmlet filters [73].  
This study uses a 2D discrete Harr WT to decompose SM image signals. Following Figure 2, the Haar decomposition process can be expressed as shown in Equations (2) and (3) [74]. Taking an original \(n\times m\) image \(A(n,m)\) as an example, first, WT applies a convolution filter to matrix \(A(n,m)\) along its row direction, and implements a downsampling process to obtain row decompositions of \(A_{L}\) and \(A_{H}\):  
\[\begin{cases}A_{L(i,j)}=\frac{A_{(i2(i-1)}+A_{(i2)j})}{2}&1\leq i\leq n,\;1\leq j \leq\frac{m}{2}\\ A_{H(i,j)}=\frac{A_{(i2(i-m-1)}-A_{L(i,j-\frac{m}{2})}}{2}&1\leq i\leq n,\; \frac{m}{2}<j\leq m\end{cases} \tag{2}\]  
where \(i\) and \(j\) are the row and column number of \(A\), respectively, and \(A_{L}\) and \(A_{H}\) represent the average and detail coefficients for image \(A\), respectively.  
Next, the Harr WT applies a convolution filter to \(A_{L}\) and \(A_{H}\) along the column direction:  
\[\begin{cases}A_{LL(i,j)}=\frac{A_{L(2i-1,i)}+A_{L(2i,j)}}{2}&1\leq i\leq\frac{ n}{2},\;1\leq j\leq\frac{m}{2}\\ A_{LH(i,j)}=\frac{A_{L(2i-n,j)}-A_{LL(i-\frac{m}{2},j)}}{2}&1\leq i\leq\frac{n}{2 },\;m<j\leq m\\ A_{HL(i,j)}=\frac{A_{HL(2i-1,j)}+A_{HL(2i,j)}}{2}&\frac{n}{2}<i\leq n,\;1\leq j \leq\frac{m}{2}\\ A_{HH(i,j)}=\frac{A_{H(2i-n,j-1)}-A_{HL(i-\frac{m}{2},j)}}{2}&\frac{n}{2}<i\leq n,\;m<j\leq m\\ \end{cases} \tag{3}\]  
where \(A_{LL}\) represents the low-frequency average characteristics, and looks most like the original image. Analogically, \(A_{LH}\) represents the detail coefficients of \(A_{L}\); \(A_{HL}\) and \(A_{HH}\) represent the average and detail coefficients of \(A_{H}\), respectively. Note that fitting models can instead be applied to the WT space--e.g., the high-frequency details or low-frequency approximations obtained in Equation (3)--to retain high-frequency or low-frequency information contained in the original image. In addition, the four decomposition images generated in Equation (3) can be inversely reconstructed to regenerate image \(A\) following the inverse process along the column and row directions.