# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 2 CS Theory  
CS theory was proposed by [PERSON] and [PERSON] [25; 26; 27; 28]. It is a technique that reconstructs sparse, compressible signals from under-determined random linear measurements [29]. Generally, CS theory consists of a sparse representation, a measurement matrix and a sparse promotion algorithm. The two principles of the theory are the sparsity of the signal and the uncorrelation between the measurement matrix and the sparse transformation matrix. Assuming that transform domain set \(\mathbf{\
abla}\) can sparsely represent discrete data \(\mathbf{d}{\in}R^{N}\), the matrix form is expressed as follows:  
\[\mathbf{d}=\mathbf{\
abla}^{\mathrm{T}}\mathbf{E}=\sum_{j=1}^{N}\mathbf{e}_{j}\mathbf{\psi}_{j}^{ \mathrm{T}} \tag{1}\]  
where \(\mathbf{\
abla}\) = [\(\mathbf{\
abla}_{1}\),..., \(\mathbf{\
abla}_{N}\)]\(\in\)\(R^{N}\) is the basis function vector in the transform domain, \(\mathbf{\
abla}^{\mathrm{T}}\) is the inverse transformation of \(\mathbf{\
abla}\), \(N\) is the total number of basis functions, vector \(\mathbf{e}\) = [\(\mathbf{e}_{1}\), \(\mathbf{e}_{2}\),..., \(\mathbf{e}_{N}\)], \(\mathbf{e}_{j}\) = \(\mathbf{\cdot}\)d, \(\mathbf{\
abla}_{j}\) = \(\mathbf{\cdot}\)l,..., \(N\)) is the coefficients in the sparse domain. Measurement signal \(\mathbf{d}_{\mathrm{obs}}{\in}R^{n}\) can be regarded as complete signal \(\mathbf{d}\) obtained by mapping the sampling matrix, \(\mathbf{\Phi}{\in}R^{n\times N}\):  
\[\mathbf{d}_{\mathbf{obs}}=\mathbf{\Phi}\mathbf{d} \tag{2}\]  
where \(n\) is the number of the collected data points, \(n<<N\), \(\mathbf{\Phi}\): = \(\mathbf{RM}_{\text{c}}\mathbf{M}\) is the measurement matrix and \(\mathbf{R}\) is the constraint matrix. Substitute Formula (1) into Formula (2):  
\[\mathbf{d}_{\mathbf{obs}}=\mathbf{\Phi}\mathbf{\
abla}^{\mathrm{T}}\mathbf{E}=\mathbf{\Theta}\mathbf{E} \tag{3}\]  
where \(\mathbf{\Theta}\) = [\(\mathbf{\Theta}_{1}\),..., \(\mathbf{\Theta}_{P}\)] is the model matrix, and \(\mathbf{\Theta}_{i}\): = \(\mathbf{RM}\mathbf{\
abla}_{i}^{\mathrm{T}}\mathbf{\in}R^{n\times N}\). According to the theory of CS, the premise of stably reconstructing the \(K_{i}\) maxima of the ith feature of an N dimensional signal is that \(\mathbf{\Theta}_{i}\) needs to satisfy the restricted isometry property (RIP) [30; 31]. For a \(K\) sparse vector (length is N), if \(\mathbf{\Theta}{\in}R^{M\times N}\) satisfies the following conditions:  
\[(1-\delta)\left\|\mathbf{e}\right\|_{2}^{2}\leq\left\|\mathbf{\Theta}\mathbf{e}\right\|_{ 2}^{2}\leq(1+\delta)\left\|\mathbf{e}\right\|_{2}^{2} \tag{4}\]  
where \(\delta\) is the RIP constant. Generally, if \(0<\delta<1\) is satisfied, it is considered that all submatrices of matrix \(\mathbf{\Theta}\) have at most \(K\) columns that satisfy the condition, and \(\mathbf{\Theta}\) satisfies the \(K\)-order RIP property with \(\delta\). Any matrix can be used as a measurement matrix for data reconstruction as long as it conforms to the RIP properties described in Equation (4) [32; 33], which theoretically guarantees that the \(K\) sparse signal \(\mathbf{d}\) can be completely recovered from the measurement signal \(\mathbf{d}_{\mathrm{obs}}\). In CS theory, the number of observation signals is much smaller than the length of the signal itself, and the recovery of \(K\) sparse signal \(\mathbf{d}\) by Formula (3) requires solving the underdetermined equations. If the signal \(\mathbf{d}\) is sparse, the problem of solving underdetermined equations can be transformed into a minimum zero norm problem:  
\[\min\left\|\mathbf{\
abla}^{\mathrm{T}}\mathbf{E}\right\|_{0}\text{s.t.}\,\mathbf{A}^{CS} \mathbf{E}=\mathbf{\Phi}\mathbf{\
abla}\mathbf{E}=\mathbf{D} \tag{5}\]  
where \(\mathbf{A}^{CS}\) is the model matrix, which consists of the sampling matrix and sparse transformation matrix. Combinatorial optimization is an NP problem. When \(N\) is very large, it cannot be effectively realized numerically, and the anti-noise ability is very poor.