Correlations between macroseismic intensity estimations and ground motion measures of seismic events

Abstract

Macroseismic intensity data are an important source to learn from historical earthquakes. Nevertheless, this data needs to be converted into more suitable intensity measures to be used in risk analyses, as well as in design practice. To this purpose, in this paper, correlations between macroseismic scales and ground motion parameters have been derived. Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV) and Housner Intensity (I_H) as instrumental measures, and European Macroseismic Scale (EMS-98) and Mercalli-Cancani-Sieberg (MCS) as macroseismic measures, have been considered. 179 ground-motion records belonging to 32 earthquake events occurred in Italy in the last 40 years have been selected, provided that for each record, macroseismic intensity in terms of either EMS-98 or MCS or both were also available. Statistical analyses have been carried out to derive both direct (i.e. macroseismic vs instrumental intensity) and inverse (instrumental vs macroseismic intensity) relationships. Results obtained from the proposed relationships have been analyzed and compared with some of the most prominent results available in the technical literature.

Appendix

See Tables 9, 10 and 11.

Table 9 Main parameters of the selected earthquakes

Table 10 Macroseismic and instrumental data of the considered events

Table 11 Correlations between instrumental parameters (PGA, PGV and I_H) and macroseismic intensity scales (MCS and EMS-98)

1.1 Total Least Squares (TLS) method

According to Zaiontz (2019), the goal of the TLS method is to minimize the sum of the squared Euclidean distances d² from the observed points ${\text{y}}_{\text{i}}$ to the corresponding ones on the regression line (which is in the form ${\text{y}} = {\text{a}} + {\text{bx}}$), as follows:

$$min\left\{ {\mathop \sum \limits_{i = 1}^{n} d_{i}^{2} } \right\}$$

that is equivalent to:

$$min\left\{ {\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {y_{i} - y'_{i} } \right)^{2} }}{{b^{2} + 1}}} \right\}$$

where ${\text{y}}_{\text{i}}$ and ${\text{y}}_{\text{i}}^{ '}$ are the observed and the corresponding estimated data (along the vertical line), respectively, and b is given by:

$$b = \frac{{w + \sqrt {w^{2} + r^{2} } }}{r}$$

where

$$w = \mathop \sum \limits_{i = 1}^{n} \left( {y_{i} - y_{m} } \right)^{2} - \mathop \sum \limits_{i = 1}^{n} \left( {x_{i} - x_{m} } \right)^{2}$$

and

$$r = 2\mathop \sum \limits_{i = 1}^{n} \left( {x_{i} - x_{m} } \right) \left( {y_{i} - y_{m} } \right)$$

x_m and y_m are the mean values of the x_i and y_i values, respectively. The intercept a can now be expressed as:

$$a = y_{m} - bx_{m}$$

An analogous procedure can be found also in Petráš and Bednárová (2010).