## 2 Materials and Methods
### Wave Field Imaging
These imply that the acquired signal \(\mathbf{y}(\omega)\) could be recovered by chasing the sparse solution for \(\mathbf{\alpha}(\omega)\) based on the CS theory from the underdetermined inverse problem:  
\[\mathbf{\hat{\kappa}}(\omega)=\arg\min\left(\left\|\mathbf{A}\mathbf{\cdot}\mathbf{\alpha}( \omega)-\mathbf{y}(\omega)\right\|_{2}+\lambda\left\|\mathbf{\alpha}(\omega)\right\|_ {1}\right) \tag{10}\]  
It is noted that \(\mathbf{G}(\omega)\) is a two-dimensional function across frequency and wavenumber, thus, a filter bank is proposed to estimate the dispersion image in the various frequency bands. The benefit of the filter bank is to solve Equation (10) in each certain narrow band to focus on the sparsest representation of the wavenumber (velocity) from measurements. The dispersion curves of the wavenumber (phase velocity) are obtained from one frequency range to another. The filter bank is established by a family of bandpass filters which are used to separate the information according to their frequency components. Second-order bandpass filters with tuned frequencies are employed to assemble the bandpass filter bank to separate the information at various interested frequencies (see in Figure 2). The central frequency and the spacing are the most important parameters for the parallel bandpass filter bank. The basic bandpass transfer function of the filter is expressed as Equation (11):  
\[H_{BP}(s)=\frac{\frac{\omega_{p}}{Q_{p}}}{s^{2}+\frac{\omega_{p}}{Q_{p}}s+ \omega_{p}^{2}} \tag{11}\]  
where \(s\) is the complex variable, \(\omega\) is the angular frequency in rads\({}^{-1}\), that is, \(\omega=2\pi f\), \(\omega_{p}\) is the central angular frequency, and \(Q_{p}\) is the the quantity Q-factor of the filter; higher \(Q_{p}\) produces narrower bandwidth. The 3 dB bandwidth of the bandpass filter is defined as:  
\[\omega_{BW}=\omega_{+3 dB}-\omega_{-3 dB} \tag{12}\]  
where \(\omega_{BW}\) is the bandwidth of the bandpass filter, \(\omega_{+3 dB}\) and \(\omega_{-3 dB}\) are the half-power frequencies, which are defined as:  
\[\left|H_{BP}(j\mathbf{\cdot}\omega_{+3 dB})\right|=\frac{1}{\sqrt{2}},\ \omega_{+3 dB}=\omega_{p}\sqrt{1+\frac{1}{2Q_{p}^{2}}+\frac{1}{Q_{p}}\sqrt{1+ \left(\frac{1}{2Q_{p}}\right)^{2}}} \tag{13}\]  
\[\left|H_{BP}(j\mathbf{\cdot}\omega_{-3 dB})\right|=\frac{1}{\sqrt{2}},\ \omega_{-3 dB}=\omega_{p}\sqrt{1+\frac{1}{2Q_{p}^{2}}-\frac{1}{Q_{p}}\sqrt{1+ \left(\frac{1}{2Q_{p}}\right)^{2}}} \tag{14}\]  
In summary, the proposed FAWS is divided into five steps to carry out for the on-site experimental testings: (1) preparing step, determining the detection range and the measurement number; (2) arrangement step, generating random Bernoulli matrix and allocating the geophones based on the random matrix; (3) acquiring step, generating surface wave and recording the raw data of the ground-motion waveforms by the ununiformed geophone array; (4) processing step, filtering the raw data by a narrow-bandpass filter bank to decompose and extract the information of the surface wave from the recorded data at various interested frequencies; (5) inversion step, computing the wavenumber or velocity of surface wave at each frequency and assembling the inversed information as the final dispersive mapping of the surface wave of the subsoil to evaluate its mechanical properties. Figure 3 shows a summary flow diagram that describes the transformation process in detail.