## 2 Materials and Methods
### Wave Field Imaging  
The speed of a Rayleigh wave propagating in the elastic Earth's surface is dispersive which means the velocity of the Rayleigh wave depends on its frequency. In addition, multiple modes of Rayleigh wave are involved at any frequency. Thus, it is difficult to analyze the real-world dataset with the interference of a multi-mode Rayleigh wave to obtain an accurate, high-resolution dispersion image. A geophone array containing hundreds of geophones was used to pursue the wave speed and to distinguish the involved modes of Rayleigh wave for technicians and engineers (see in Figure 1). In order to rapidly estimate the velocity of a surface wave from multiple channel data, a CS platform is proposed to improve the conventional geophone array assignment and analysis method. A randomly allocated geophone array is assigned in a line instead of the traditional uniformly spacing geophone linear array in this study. The signal measured by each geophone in the frequency domain is a sum of frequency dispersive modes of Rayleigh waves:  
\[y(r,\omega)=\sum_{j=1}^{N}\frac{1}{\sqrt{r}}G_{j}(\omega)e^{-ik_{j}(\omega)r} \tag{5}\]  
where \(G_{j}(\omega)\) is the complex-valued amplitude of each wave mode, \(k_{j}(\omega)\) represents the real-valued frequency dependent wavenumber of each mode, \(\omega\) represents the angular frequency, and \(r\) is the distance from the source to the received sensor. Based on the character of dispersion, for each mode \(j=1\), \(2\), \(3,\cdots\), \(N\), the complex-valued amplitude \(G_{j}(\omega)\) and wavenumber \(k_{j}(\omega)\) in Equation (5) are continuous functions of frequency \(\omega\) and wavenumber \(\kappa\) in the Rayleigh wave equation:  
\[G(\omega,k)=\left\{ \begin{array}{ll}G_{j}(\omega)&\text{if $k_{j}(\omega)= \kappa$ for some $j$}\\ 0&\text{otherwise}\end{array} \right. \tag{6}\]  
\[k_{j}(\omega)=\left\{ \begin{array}{ll}\kappa&\text{for some $j$ at special frequencies}\\ 0&\text{otherwise}\end{array} \right. \tag{7}\]  
where \(\kappa\) indicates the actual wavenumber in the Earth's surface, which contains the roots of total S modal Rayleigh wave of the wave equation. Considering the randomly allocated geophone array of \(m\) geophones, the signal received by the geophone array is the superposition of all modal components of the Rayleigh waves:  
\[\boldsymbol{y}(\omega)=\boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{\alpha}(\omega) \tag{8}\]  
\[\boldsymbol{y}(\omega)=\left[ \begin{array}{c}y_{1}(\omega)\\ y_{2}(\omega)\\ \vdots\\ y_{m}(\omega)\end{array} \right]=\left[ \begin{array}{cccc}\frac{1}{\sqrt{r_{1 }}}e^{-ik_{1}(\omega)r_{1}}&\frac{1}{\sqrt{r_{1}}}e^{-ik_{2}(\omega)r_{1}}& \cdots&\frac{1}{\sqrt{r_{1}}}e^{-ik_{N}(\omega)r_{1}}\\ \frac{1}{\sqrt{r_{2}}}e^{-ik_{1}(\omega)r_{2}}&\frac{1}{\sqrt{r_{2}}}e^{-ik_{2} (\omega)r_{2}}&\cdots&\frac{1}{\sqrt{r_{2}}}e^{-ik_{N}(\omega)r_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{\sqrt{r_{m}}}e^{-ik_{1}(\omega)r_{m}}&\frac{1}{\sqrt{r_{m}}}e^{-ik_{2} (\omega)r_{m}}&\cdots&\frac{1}{\sqrt{r_{m}}}e^{-ik_{N}(\omega)r_{m}}\end{array}  \right]\left[ \begin{array}{c}G_{1}(\omega)\\ G_{2}(\omega)\\ \vdots\\ G_{N}(\omega)\end{array} \right] \tag{9}\]  
Figure 1: Near-surface Rayleigh wave propagation and geophone array schematic. where \(A\) is a \(m\times N\) transfer matrix. In addition, the condition for the RIP is satisfied due to the random allocation of the geophone array. \(\mathbf{\alpha}(\omega)\) represents the solution to a discrete inverse problem and a discretized approximation of \(\mathbf{G}(\omega)\). For most frequency \(\omega\) and wavenumber \(k(\omega)\), \(\mathbf{G}(\omega,k)\) is zero based on the theoretical dispersion image and wave equation. Thus, its discretization \(\mathbf{\alpha}(\omega)\) is sparse as well.