# A High-Resolution Spectral Analysis Method Based on Fast Iterative Least Squares Constraints
## 2 Theory
### Fast Iterative Shrinkage-Thresholding Algorithm  
Among the numerous gradient-based algorithms, the Iterative Shrinkage-Thresholding Algorithm (ISTA) is a highly regarded method. The ISTA updates the least squares solution in each iteration through a shrinkage/soft-thresholding operation, and its specific iterative format is expressed as follows:  
\[\mathbf{m}_{k+1}=soft_{\lambda t}\Big{(}\mathbf{m}_{k}-2t\mathbf{F}_{w}{}^{T}( \mathbf{F}_{w}\mathbf{m}_{k}-\mathbf{W}_{d}\mathbf{d})\Big{)} \tag{6}\]  
The threshold operation function is  
\[soft_{T}(\mathbf{m}_{i})=\text{sgn}(\mathbf{m}_{i})(|\mathbf{m}_{i}|-T). \tag{7}\]  
In Equation (6), \(t\) represents the iteration step. As can be seen from Equation (6), during each iteration process, the ISTA only considers the information at the current point to update the iteration starting point, which results in a relatively slow convergence speed for the algorithm. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) builds upon the ISTA by considering both the current point and the previous point to update the iteration starting point, thereby achieving a faster convergence speed compared to the ISTA. The flow of the FISTA is as follows:  
1. Input time-space domain seismic data \(\mathbf{d}\) and the maximum number of iterations \(K\);
2. Initialize the model \(\mathbf{m}_{0}\), current iteration count \(k=0\), and step \(t>0\), \(p_{0}=1\), \(\mathbf{y}=\mathbf{m}_{0}\);
3. If \(k\leq K\), perform iterative updates;
4. Output the frequency spectrum of the seismic data \(\mathbf{mk}\). Compared to the ISTA, the FISTA uses \(\mathbf{y}_{k}\) instead of \(\mathbf{m}_{k}\). In the iterative process, the iteration point \(\mathbf{mk}\) of the FISTA not only depends on the previous iteration point \(\mathbf{m}_{k-1}\) but also utilizes a linear combination of the two previous points \(\{\mathbf{m}_{k-1},\mathbf{m}_{k-2}\}\), thereby improving convergence speed. The inertia parameter \(\text{a}_{k}\) controls the growth rate of \(\mathbf{m}_{k}-\mathbf{m}_{k-1}\). The ISTA is straightforward to compute but converges slowly. Although the FISTA converges faster than the ISTA, it can exhibit oscillations when \(a_{k}\to 1\), which slows down convergence. Handling massive seismic datasets imposes higher efficiency requirements on algorithms. The Greedy Fast Shrinkage Algorithm (GFISTA) [30] maintains computational simplicity while achieving faster global convergence rates than the ISTA and FISTA, with improved convergence performance [30]. This paper applies the GFISTA to spectral analysisalgorithms, proposing a least squares constrained spectral analysis method based on fast iterative shrinkage. The GFISTA proceeds as follows:  
1. Input time-space domain seismic data \(\mathbf{d}\) and the maximum number of iterations \(K\);
2. Initialize the model \(\mathbf{m}_{0}\), current iteration count \(k=0\), and iteration step \(t>0\), \(\mathbf{y}_{1}=\mathbf{m}_{0}\), the shrinkage factor of the step \(\xi<1\), and the control factor for algorithm convergence \(S>1\);
3. If \(k\leq K\), perform iterative updates: \[ \begin{array}{c}\mathbf{y}_{k}=\left(\mathbf{m}_{k}-\mathbf{m}_{k-1}\right) \\ \mathbf{m}_{k+1}=soft_{\lambda t}\Big{\{}\mathbf{y}_{k}+2t\mathbf{F}^{-1}\Big{\{} \big{(}\mathbf{F}^{H}\mathbf{F}\big{)}^{-1}\times\mathbf{F}^{H}(\mathbf{F}_{w} (\mathbf{y}_{k})-\mathbf{W}_{d}\mathbf{d})\big{\}}\Big{\}}\\ k+1\to k\end{array} \] (8) Restart condition: if \((\mathbf{y}_{k}-\mathbf{m}_{k+1})^{T}(\mathbf{m}_{k+1}-\mathbf{m}_{k})\geq 0\), \(\mathbf{y}_{k}=\mathbf{m}_{k}\);  
Convergence condition: if \(\|\mathbf{m}_{k+1}-\mathbf{m}_{k}\|\geq S\|\mathbf{m}_{1}-\mathbf{m}_{0}\|\), \(t=\max\{\xi t,t\}\ k+1\to k\). 4. Output the frequency spectrum of the seismic data \(\mathbf{m}\mathbf{k}\). In this algorithm, \(t\in\big{[}1/\lambda_{\max}\big{(}\mathbf{F}^{H}\mathbf{F}\big{)},2/\lambda_{ \max}\big{(}\mathbf{F}^{H}\mathbf{F}\big{)}\big{]}\).