# A High-Resolution Spectral Analysis Method Based on Fast Iterative Least Squares Constraints
## 2 Theory  
### Principles of Least Squares Constrained Spectral Analysis  
Seismic data can generally be represented using a convolution model as a superposition of wavelets with different amplitudes, dominant frequencies, and time shifts. Therefore, the seismic trace **d** can also be expressed as follows:  
\[\textbf{Fm}=\textbf{d} \tag{1}\]  
where **m** represents the frequency spectrum of the seismic signal, which is equivalent to the Fourier transform coefficients. The matrix **F** is a kernel matrix composed of sine signals within the truncated frequency and time range. Its row vectors represent the range of frequency values, while its column vectors indicate the relative time of the window function. This can be expressed as follows:  
\[\textbf{F}(t,f)=\cos(2\pi ft)+i\sin(2\pi ft) \tag{2}\]  
From Equation (2), it can be seen that Equation (1) can be considered the matrix representation of the discrete Fourier transform. Ref. [27] proposed the CLSSA method based on Equations (1) and (2) [27]. This method can be formulated as a least squares problem, expressed as follows:  
\[\min\Bigl{(}\|\textbf{F}_{w}\textbf{m}_{w}-\textbf{W}_{d}\textbf{d}\|^{2}+ \lambda\|\textbf{m}_{w}\|^{2}\Bigr{)} \tag{3}\]  
The constrained kernel matrix \(\textbf{F}_{w}=\textbf{W}_{d}\textbf{F}\textbf{W}_{\text{m}}\) represents the complex sinusoidal signals of the model \(\textbf{W}_{\text{m}}\) and the data weights \(\textbf{W}_{\text{d}}\) and the symbol * stands for the matrix transpose operator. The vector \(\textbf{m}_{w}\)= \(\textbf{W}_{\text{m}}\)\({}^{-1}\textbf{m}\) is the modified model parameter vector (which requires the introduction of model weights to calculate the frequency coefficients), where \(\textbf{W}_{\text{d}}\) denotes the data weights. In Equation (3), \(\lambda\|\textbf{m}_{w}\|^{2}\) is the regularization term, which is introduced to address the problem of solving ill-posed matrices. \(\lambda\) is the regularization parameter, which can be adjusted to control the sparsity and stability of the solution. As \(\lambda\) increases, the model's variance decreases, but the bias of the estimates from the true values increases, leading to a greater model bias. Therefore, it is essential to choose a reasonable \(\lambda\) to balance the model's variance and bias. When selecting \(\lambda\), one should ensure that the method is robust to noise in the data. Typically, it is set to 0.001 times the maximum value on the diagonal of the weighted Gram matrix. Generally, models are constructed with small parameters, allowing them to easily adapt to different datasets. This way, even if the data have slight deviations, they will not significantly impact the results, and it can help mitigate overfitting to a certain extent.  
By solving Equation (3), the frequency spectrum of the seismic signal can be obtained as follows:  
\[\textbf{m}_{w}=\textbf{F}_{w}^{*}(\textbf{F}_{w}\textbf{F}_{w}^{*}+\lambda I )^{-1}\textbf{W}_{d}\textbf{d} \tag{4}\]  
The matrix inversion in this equation is computed using the Gaussian elimination method. Iteration is achieved by updating the model weights of the frequency spectrum at the center of the window. Increasing the number of iterations will make the solution sparser and yield a more compact frequency spectrum.