## 2 Principles and Methods
### Hierarchical Frequency Fusion  
#### 2.2.1 Theoretical Analysis  
In microtremor array survey, different pairs of stations can be combined according to the distribution characteristics of the stations. These station pairs vary in azimuth and distance. The microtremor array survey finally obtains a dispersion curve as a uniform response below the site. Therefore, it needs to process the dispersion curves from each station pair, and the final dispersion curves can be expressed as follows:  
\[V=F(f,d,\theta) \tag{4}\]  
where \(V\) is phase velocity, \(f\) is frequency, \(d\) refers to distance between stations, and \(\theta\) represents azimuth. The oscillation source distribution at the observation site may not be uniform; the oscillation energy also varies in strength, so the response of different azimuth station pairs to the surface waves generated by different oscillation sources varies to some degree. In the SPAC method, the coherence coefficients of the center and circumference stations are azimuthally averaged to compensate for the effects of the inhomogeneity of the distribution of the oscillation sources. The azimuthal averaging treatment of SPAC can be expressed as follows:  
\[ \begin{array}{rl}\rho(r,f)&=\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{Re}\{Coh((0,0),(r, \theta),f)\}d\theta\\ &\approx\frac{1}{M}\sum\limits_{m=1}^{M}\mathrm{Re}\{Coh((0,0),(r,\theta),f)\} \end{array}  \tag{5}\]  
where \(r\) is the radius of the circular array, \(f\) means angle frequency, \(M\) is the number of stations on the circumference of the circle, and \(\mathrm{Re}\{Coh((0,0),(r,\theta),\omega)\}\) is the real part of coherence coefficient between the center station and the circumference station of the center. Since SPAC and SI are both based on the correlation theory [25], and the spatial autocorrelation coefficient is the result of the operation of center-circumference stations, there is nevertheless no strict requirement for the division of center-circumference stations when using the SI method to extract the dispersion curves. In order to lessen the nonuniformity of the source distribution, the dispersion curves of each station pair are frequently extracted first when processing the dispersion curves of small-scale arrays using SI. The mean dispersion curve is then derived using superimposing averaging. The superimposed averaging of the dispersion curves of a microtremor array based on SI can be expressed as:  
\[V(f,d)=\frac{1}{N}\sum\limits_{ \begin{array}{c}n=1,m=1,\\ n\
eq m\end{array} }^{N}V_{nm}(f,d_{mm})\qquad(d_{mm}=d) \tag{6}\]where \(N\) is the total number of stations in the array, \(n,m\) is the station number, \(d_{nm}\) is the distance between station pairs, and \(V_{nm}(f,d_{nm})\) is the dispersion curve extracted from microtremor signals from station pair. The superimposed averaging approach of processing dispersion curves is simple and quick, and it can essentially suit the needs of small-scale station array microtremor surveys. However, the accuracy and reliability of dispersion curves are somewhat diminished by overlaid averaging since the dispersion information of various depth media carried by dispersion curves differs among stations at various distances. The correlation coefficient can be used to evaluate the reliability of the dispersion information. \(J_{0}\) is conceptually similar in shape to the correlation coefficient curve obtained through SI. The accuracy and interference factor influence are weaker in the initial segment of the correlation coefficient curve that is monotonically declining. As a result, the phase velocity produced by fitting the first monotonically declining section of both the correlation coefficient curve and \(J_{0}\) is typically seen to be more accurate when extracting the dispersion curve. Figure 3 displays the correlation coefficient curves obtained using SI from the microtremor signals of station pairs with various spacing in a linear array, and Figure 3 displays \(J_{0}\).