Statistical correlation of earthquake and ambient noise spectral ratios

Abstract

The Horizontal-to-Vertical Spectral Ratio from earthquake (HVSR) and from ambient noise (HVN) recordings realistically indicate the fundamental frequency of soil response but, for the majority of the worldwide examined sites, they do not provide reliable amplification curves as calculated by the earthquake standard Spectral Ratio (SSR). Given the fact that HVSR and especially HVN can be easily obtained, it is challenging to search for a meaningful correlation with SSR amplification functions for the entire frequency band and to use the results for the SSR estimate at a further site where only noise measurements are available. To this aim we used recordings from 75 sites worldwide and we applied a multivariate statistical approach (canonical correlation analysis) to investigate and quantify any correlation among spectral ratios. The canonical correlation between SSR and HVN is then used to estimate the expected SSR at each site by a weighted average of the SSR values measured at the other sites; the weights are properly set to account more for sites with similar behaviour in terms of the canonical correlation results between HVN and SSR. This procedure, repeated for all sites in turn, constitutes the basis of a cross validation. The comparison between the inferred and the original SSR highlights the improvements of site response estimation with respect to the use of ambient noise techniques. The goodness and limitations of the reconstruction procedure are explained by specific geological settings.

Appendix

The SSR estimation is based on the following reasonable assumption: the relation of SSR and HVN spectral ratios of a testing site is similar to the one derived from the other sites used for the canonical correlation analysis. From a statistical point of view, this means that the original dataset is representative and includes several sites similar to the one under scrutiny.

As explained in Sect. 4, we estimate each bin value expected for the SSR at the target site $a$ as a weighted average of the seven bin SSR values measured at the other sites; because the weights depend on the canonical couples, the final estimates is an average of what is obtained by each reliable couple (Eq. 4 in the main paper):

$$\begin{aligned} \overline{SSR}\left( k\right) _a =\displaystyle \mathop \sum \limits _{i=1}^c \displaystyle \mathop \sum \limits _{m=1}^n \frac{u_{kmi} }{U(k)}\times \textit{SSR}(k)_m ,\quad m\ne a \end{aligned}$$

(8)

where the index $i$ identifies the used canonical couple, $a$ represents the target site and $m$ each of the other $n$ sites, $k$ refers to frequency bins, $u_{kmi}$ are the weights and U($k$) is the sum of the weights over all the significant canonical couples $c$ and all the sites $n$:

$$\begin{aligned} U(k)=\sum _{i=1}^c {\sum _{m=1}^n {u_{kmi} ,\;m\ne a} } \end{aligned}$$

(9)

The weights are crucial to define the SSR expected at the target site $a$ and they are built following two criteria:

• (1) they are set in order to account more on sites characterized by similar behaviour on the canonical plane; a satisfactory measure of the degree of similitude between two sites is provided by the distance $d$ in the Xcan–Ycan canonical plane (Fig. 7 in the main paper) for each canonical couple $i$: the shorter the distance, the more similar the behaviour;

• (2) as the contribution to the correlation is mostly due to some specific bins, the weights should account also for the contribution of each bin to the canonical variable.

Both above mentioned characteristics are taken into account in the Moran’s index (Moran 1950; Smith et al. 2007), which is the ratio between the covariance and the variance of SSR($k$) of all couple of sites having a distance $d_{mj}$ within the range $d_{p}-\varDelta \hbox {d}< d_{mj}<d_{p}+\varDelta \hbox {d}$, for a given canonical couple $i$:

$$\begin{aligned} \textit{MI}_i \left( {k,d_p } \right)&= \frac{n}{\mathop \sum \nolimits _{m=1}^{n-1} \mathop \sum \nolimits _{j=m+1}^n w_{mj} \left( {d_p ,i} \right) }\nonumber \\&\times \frac{\mathop \sum \nolimits _{m=1}^{n-1} \mathop \sum \nolimits _{j=m+1}^n w_{mj} (d_p ,i) \!\times \! [\textit{SSR}(k)_m \!-\! \overline{SSR}(k)]\!\times \! [\textit{SSR}(k)_j \!-\! \overline{SSR}(k)]}{\mathop \sum \nolimits _{m=1}^n [\textit{SSR}(k)_m \!-\! \overline{SSR} (k)]^{2}}\nonumber \\ \end{aligned}$$

(10)

where $k$ is the bin, SSR$(k)_{m}$ or SSR$(k)_{j}$ is the SSR value at a site $m$ or $j, \overline{SSR}(k)$ is the average for all $n$ sites. The weight $w_{mj}(d_{p},i)$ selects the couple of sites having a distance within the range [$d_{p} -\varDelta \hbox {d}, d_{p}+\varDelta \hbox {d}$[ in the canonical plane (Fig. 7 in the main paper): $w_{mj}(d_{p},i)$ $=$ 1 if the distance between the two sites $m$ and $j$ is inside the range, and $w_{mj}(d_{p},i) =0$ otherwise, where $\varDelta d$ is a fraction of the maximum distance range (Smith et al. 2007). The Moran index of the first canonical couple of the canonical correlation SSR–HVN described in Sect. 4 is shown in Fig. 11: the central bins 3–5 (0.6–3.3 Hz) are characterized by a high correlation at close sites in the canonical plane which quickly decreases for larger distances. This behaviour means that the spectral ratios in this frequency range are very similar when the sites have similar Xcan–Ycan values, and diverge for sites located at larger distance positions on the canonical plane.

Fig. 11

Moran Index $\hbox {MI}_\mathrm{i}$(k, d) for the 7 SSR bins (dots) for the 1st canonical couple of the canonical correlation SSR–HVN described in Sect. 4, as a function of the distance in the canonical correlation plane (Fig. 7, main paper). The continuous lines are the 2nd order polinomial fit. The Moran Index can assume values from +1 (perfect correlation) to $-$1 (perfect inverse correlation)

The Moran Index is then a suitable quantity to assess the importance of a site within the estimate of the spectral ratio at the target site, that is the weights $u_{kmi}$ of the Eq. 8 (Eq. 4 in the main paper):

$$\begin{aligned} \left\{ \begin{array}{ll} u_{kmi} =\textit{MI}_i (k,d)_{fit} &{}\quad \textit{if}\,\textit{MI}_i (k,d)_{fit} >0 \\ u_{kmi} =0&{} \quad \textit{if}\, \textit{MI}_i (k,d)_{fit} \le 0 \\ \end{array} \right. \end{aligned}$$

(11)

where $MI_{i }(k,d)_{fit}$ is a second degree polynomial fit of ($MI (k,d_p))$ for each bin and canonical couple (continuous lines in Fig. 11). Note that the position of each site in the canonical space (hence the distance range) varies depending of the canonical couple, accounting for the different correlation behaviour, and consequently the bin weights associated to each site change from one canonical couple to the other (Eqs. 10, 11).

The measure of the variability associated to each reconstructed bin $\overline{SSR} (k)_a$ (Eq. 8 or Eq. 4 in the main paper) is evaluated from the differences between the $\hbox {SSR(k)}_{m}$ recorded at the other sites and the estimated value at the target site:

$$\begin{aligned} \sigma (k)_a =\sqrt{\mathop \sum \nolimits _{i=1}^c \sigma ^{2}(k)_{ai} }=\sqrt{\mathop \sum \nolimits _{i=1}^c \mathop \sum \nolimits _{m=1}^n \frac{u_{kmi} }{U\left( k \right) }[\textit{SSR}\left( k \right) _m -\overline{SSR}_i \left( k \right) _a ]^{2}},\quad m\ne a\nonumber \\ \end{aligned}$$

(12)