# Surface-Wave Extraction Based on Morphological Diversity of Seismic Events
## 3 New Method: Sparse Representations of Wavefields Based on MCA  
MCA is a method for signal separation based on sparse representations [23; 24]. It is assumed that the original signal is a linear mixture of several different parts and for each of them, there exists a dictionary that enables its construction using a sparse representation. Additionally, the dictionary can only sparsely represent the corresponding part rather than the others. For seismic data consisting of surface waves and reflections, there is  
\[\mathbf{y}=\mathbf{y}_{\text{g}}+\mathbf{y}_{\text{r}} \tag{4}\]  
where \(\mathbf{y}\) is the seismic data set, \(\mathbf{y}_{\text{g}}\) is the surface-wave part, and \(\mathbf{y}_{\text{r}}\) is the reflection part. We choose \(\mathbf{D}_{\text{g}}\) and \(\mathbf{D}_{\text{r}}\) as the sparse representation dictionaries of surface waves and reflections, respectively, i.e., \(\mathbf{y}_{\text{g}}=\mathbf{D}_{\text{g}}\mathbf{z}_{\text{g}}\), \(\mathbf{y}_{\text{r}}=\mathbf{D}_{\text{r}}\mathbf{z}_{\text{r}}\). The separation of surface waves and reflections can be formulated as [25]:  
\[\underset{\{\mathbf{z}_{\text{g}},\mathbf{z}_{\text{r}}\}}{\text{argmin}} \frac{1}{2}\|\mathbf{y}-\mathbf{D}_{\text{g}}\mathbf{z}_{\text{g}}-\mathbf{D} _{\text{r}}\mathbf{z}_{\text{r}}\|_{2}^{2}+\mu(\left\|\mathbf{z}_{\text{g}} \right\|_{1}+\left\|\mathbf{z}_{\text{r}}\right\|_{1}) \tag{5}\]  
where \(\mathbf{z}_{\text{g}}\) and \(\mathbf{z}_{\text{r}}\) are the representation vectors for the surface waves and reflections, respectively, and \(\mu\) is the regularization parameter.  
Surface waves are generally modeled as broom-like events characterized by low frequency, low velocity, and dispersion. In addition, their dispersive energy is around theoretical dispersion curves in the f-v domain [19; 20]. Therefore, frequency-domain high-resolution LRT can, in theory, sparsely represent surface waves. Reflections are approximated by hyperbolas, and time-domain high-resolution HRT can sparsely represent reflections with the assumptions that velocities change little horizontally and reflection interfaces are almost horizontal [26]. We choose frequency-domain high-resolution LRT and time-domain high-resolution HRT to represent surface waves and reflections, respectively. To match with the matrix definition of sparse representation dictionaries in our sparse representation problem, the inverse LRT and inverse HRT correspond to the matrix signs \(\mathbf{D}_{\mathrm{g}}\) and \(\mathbf{D}_{\mathrm{r}}\), respectively, while the forward LRT and forward HRT are respectively the matrices \(\mathbf{D}_{\mathrm{g}}^{+}\) and \(\mathbf{D}_{\mathrm{r}}^{+}\). The matrices \(\mathbf{D}_{\mathrm{g}}^{+}\) and \(\mathbf{D}_{\mathrm{r}}^{+}\) are the pseudo inverse of the representation dictionaries, i.e., \(\mathbf{z}_{\mathrm{g}}=\mathbf{D}_{\mathrm{g}}^{+}\mathbf{y}_{\mathrm{g}}\), \(\mathbf{z}_{\mathrm{r}}=\mathbf{D}_{\mathrm{r}}^{+}\mathbf{y}_{\mathrm{r}}\). By putting \(\mathbf{z}_{\mathrm{g}}=\mathbf{D}_{\mathrm{g}}^{+}\mathbf{y}_{\mathrm{g}}\) and \(\mathbf{z}_{\mathrm{r}}=\mathbf{D}_{\mathrm{r}}^{+}\mathbf{y}_{\mathrm{r}}\) into (5), we obtain  
\[\underset{\{\mathbf{y}_{\mathrm{g}},\mathbf{y}_{\mathrm{r}}\}}{\mathrm{argmin} }\frac{1}{2}\|\mathbf{y}-\mathbf{y}_{\mathrm{g}}-\mathbf{y}_{\mathrm{r}}\|_{2 }^{2}+\mu(\|\mathbf{D}_{\mathrm{g}}^{+}\mathbf{y}_{\mathrm{g}}\|_{1}+\| \mathbf{D}_{\mathrm{r}}^{+}\mathbf{y}_{\mathrm{r}}\|_{1}) \tag{6}\]  
which is solved by the generalized block coordinate relaxation algorithm [23].  
Surface waves are extracted by solving Equation (6). Different dictionaries respectively represent surface waves and reflections so that the influence of reflections on surface-wave dispersive energy is reduced.