Automated assessment of macroseismic intensity from written sources using the fuzzy sets

Abstract

We apply a computer-aided methodology to assess macroseismic intensity from the descriptions reported by documentary material available for eight Italian earthquakes occurred around the beginning of the instrumental era. The procedure consists of three phases: (i) the identification of significant macroseismic effects on the sources and their archiving in a georeferenced database, (ii) the association between the effects and the degrees of the intensity scale by the comparison with traditional estimates made by macroseismic experts, (iii) the assessment of intensities using a multi-attribute decision-making algorithm based on the Fuzzy Sets logic. This work represents a substantial improvement of our previous efforts as we completely redesigned the three phases of the procedure in the light of the experience of the last 10 years and analyzed six further Italian earthquakes so that our database now includes more than 19,000 encoded effects. Our formalized procedure allows to tracing all of the steps of intensity assessment process so that to identify discrepancies with respect to the expert evaluations that might be possibly due to mistakes or to the incomplete account of the available information. Hence, this approach may be useful for providing a systematic and reproducible intensity assessment as well as for supporting standard man-made assessments. The database of effects we have built could also be employed for testing the internal consistency of the macroseismic scale as well as for designing an improved macroseismic scale, based on consistent statistical criteria.

Appendices

Appendix 1: English version of the MCS scale (Sieberg 1932)

See Tables 5 and 6.

Table 5 Detailed Mercalli—Sieberg scale for determining the relative earthquake intensities

Table 6 Simplified overview of the most important features of the Mercalli-Sieberg scale

Appendix 2: Formulation of MADM algorithm

The HRW algorithm (Hurwicz 1951) computes the aggregate vector D(A _i ) as a linear combination of the minimum and maximum values of exponentially weighted MFs scores using an arbitrary risk coefficient α in the range [0–1]

$$ D_{HRW} (A_{i} ) = \alpha \cdot min_{j} \left[ {\mu_{j} \left( {A_{i} } \right)^{{w_{j} }} } \right] + \left[ {1 - \alpha } \right] \cdot max_{j} \left[ {\mu_{j} \left( {A_{i} } \right)^{{w_{j} }} } \right] $$

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The risk coefficient changes the typology of the approach from “pessimistic” (maximum risk, α = 1) to “optimistic” (minimum risk, α = 0). It is easy to see that the pessimistic approach exactly corresponds to the MXMN algorithm. In our present application we assume an intermediate value α = 0.5.

The SAW algorithm (Hwang and Yoon 1981) computes the D(A _i ) scores as the sums of the products of the weights with the MF scores over all the attributes

$$ D_{SAW} (A_{i} ) = \sum\limits_{j = 1}^{m} {w_{j} \mu_{j} \left( {A_{i} } \right)} $$

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The WPM algorithm (Easton 1973) computes the aggregation function D(A _i ) as the product of exponentially weighted MFs

$$ D_{WPM} (A_{i} ) = \prod\limits_{j = 1}^{m} {\mu_{j} \left( {A_{i} } \right)}^{{w_{j} }} $$

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The TOPSIS algorithm (Hwang and Yoon 1981) computes the decision matrix D _ij as

$$ D_{ij} = w_{j} \frac{{\mu_{j} \left( {A_{i} } \right)}}{{\sqrt {\sum\nolimits_{i = 1}^{m} {\left[ {w_{j} \mu_{j} \left( {A_{i} } \right)} \right]^{2} } } }} $$

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and considers the theoretically ideal and anti-ideal alternatives that would have respectively maximum and minimum scores in the membership function of all of the effects

$$ \begin{aligned} D_{j}^{ + } = \hbox{max} \left( {D_{ij} } \right) \hfill \\ D_{j}^{ - } = \hbox{min} \left( {D_{ij} } \right) \hfill \\ \end{aligned} $$

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The Euclidean distances from both of them is then computed for every (real) alternative as

$$ \begin{aligned} S_{i}^{ + } = \sqrt {\sum\nolimits_{j = 1}^{m} {\left( {D_{ij} - D_{j}^{ + } } \right)^{2} } } \hfill \\ S_{i}^{ - } = \sqrt {\sum\nolimits_{j = 1}^{m} {\left( {D_{ij} - D_{j}^{ - } } \right)^{2} } } \hfill \\ \end{aligned} $$

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The aggregation function D(A _i ) is computed as the relative closeness to the ideal alternative

$$ D_{TOPSIS} \left( {A_{i} } \right) = \frac{{S_{i}^{ + } }}{{S_{i}^{ + } + S_{i}^{ - } }} $$

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Appendix 3: Goodness of fit estimators

In order to compare the expert and automated intensities of examined earthquakes, we used:

(1) The root mean square (r.m.s.) deviation between the N corresponding values of expert (I _e ) and automated (I _f ) intensities

$$ D_{rms} = \sqrt {\frac{1}{N}\sum {\left( {I_{e} - I_{f} } \right)^{2} } } $$

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(2) The mean absolute deviation between the two sets of estimates

$$ D_{abs} = \frac{1}{N}\sum {\left| {I_{e} - I_{f} } \right|} $$

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(3) The Pearson (or product-moment) correlation coefficient, defined as the ratio between the covariance σ _ef and the product of the standard deviations σ _e and σ _f of the two sets of intensities

$$ r_{p} = \frac{{\sigma_{ef} }}{{\sigma_{e} \sigma_{f} }} $$

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(4) The Spearman (1904) non-parametric correlation coefficient, computed by sorting in increasing order all of the N values of the two variables and defining the rank R of each observation as a progressive number from 1 for the lowermost value to N for uppermost value. In presence of ties (observations with identical values and then identical rank), the Spearman coefficient is defined as

$$ r_{s} = \frac{{\left( {N^{3} - N} \right) - 6\sum {\left( {R_{e} - R_{f} } \right)^{2} - \left( {T_{e} - T_{f} } \right)/2} }}{{\sqrt {\left( {N^{3} - N} \right)^{2} - \left( {T_{e} + T_{f} } \right)\left( {N^{3} - N} \right) + T_{e} T_{f} } }} $$

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where T _e and T _f are the total numbers of ties for the two set of intensities.

(5) The mean difference between expert and automated intensity estimates

$$ d_{mean} = \frac{1}{N}\sum {\left( {I_{e} - I_{f} } \right)} $$

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