# Discussions on the Processing of the Multi-Component Seismic Vector Field
## 2 Multi-Component De-Noising Methods with Respect to the Vector Seismic Field
### Multi-Component Joint De-Noising Based on the Group Sparsity Representation  
Expressing the time-domain noise-containing signals in other domains through mathematical transforms is the approach often used in de-noising seismic data and wavefield reconstruction. The mathematical transforms often used include Fourier transform, S transform, wavelet transform, and curvelet transform. These signal representation methods based on specific mathematical transformations can effectively represent the signal of a corresponding structural model according to the structural characteristics of the individual transform dictionaries. However, real seismic signals are very complex; therefore, it is difficult to achieve the sparse representation of seismic signals by only selecting a single transform method. Many overlaps in the representation coefficients of the noise and the effective signal still exist when using a single transform method. We combine the dictionary of  
Figure 5: Hodograms of two-component seismic signal segments, dashed lines are uncontaminated seismic data, and solid lines are the de-noised results using different methods. The first, second, and third columns are the results of low-pass filtering, single-component filtering, and vector order statistics-based filtering, respectively. multiple transforms to compose an over-complete dictionary or construct a new dictionary according to the signal characteristics. Through searching, we can only use a small number of elements in the over-complete dictionary for the sparse representation of complex signals [62; 63]. Because many fast and effective algorithms of sparse representation have been proposed in recent years, such as the matching pursuit algorithm [64], the orthogonal matching pursuit algorithm [65], and the base tracking algorithm [66], sparse representation has been widely applied in seismic data processing. Applications include the reconstruction of the wavefield [67], surface-wave attenuation [68; 69], and de-convolution [70]. To retain the relative amplitude relationship among the different components of multi-component seismic data after de-noising, [PERSON] et al. [32] developed and applied the time-frequency transform based on the group sparsity constrained representation to de-noise three-component micro-seismic data. They believe that because the micro-seismic events recorded by the three-component detector have the same arrival time and frequency component for the three components, the coefficient of sparse representation for the effective signal of the three components in the time-frequency domain should have the same distribution mode. In other words, the coefficients of the three components at the same time-frequency point are either all zero or nonzero. With the coefficients of three-component data at the same time-frequency point as one group, we can use the concept of group sparsity proposed by [PERSON] and [PERSON] [71], [PERSON] and [PERSON] [72], and [PERSON] [73] to process the multicomponent seismic data. The three components x, y, and z can form an N \(\times\) 3 matrix (where N is the number of time sampling points):  
\[\mathbf{S}=\left[ \begin{array}{cccc}\mathbf{x}&\mathbf{y}&\mathbf{z}\end{array} \right] \tag{1}\]  
The group sparsity constrained representation in an over-complete dictionary \(\mathbf{\Phi}\) can be expressed as the following mathematical optimization problem:  
\[\hat{\mathbf{A}}=\text{argmin}_{A}\Big{[}\|\mathbf{S}-\mathbf{\Phi}\mathbf{A} \|^{2}_{2,2}+\lambda\|\mathbf{A}\|_{2,1}\Big{]}, \tag{2}\]  
where \(\hat{\mathbf{A}}\) is the coefficient of sparse representation, and each column corresponds to a component. The symbol \(\|\bullet\|_{p,q}\) represents the mod of the matrix, which is defined as  
\[\|\mathbf{X}\|_{p,q}=\left(\sum_{n}\left(\sum_{k}\left|x_{n,k}\right|^{p} \right)^{\frac{q}{p}}\right)^{\frac{1}{q}}. \tag{3}\]  
The optimization problem of Equation (2) can be solved by adopting the fast iterative shrinkage-threshold algorithm (FISTA) proposed by [PERSON] and [PERSON] [74].