Advanced mathematics for the definition of local seismic response: rexel and strata in a case study

Antonio Gallo, Gaetano La Bella

The choice of the “design earthquake” is a fundamental challenge for the design of strategic works, since it represents the basis for the construction of safe buildings and infrastructures in seismically active areas. This mathematical process assumes particular relevance for infrastructures with a decisive impact on social, environmental and economic levels, which must maintain their functionality even in the event of significant seismic events.

In relation to this, it appears more relevant than ever to define the “design seismic actions” consisting of the critical seismic characteristics of a given site, such as magnitude, ground acceleration, characteristic period and duration of the event, the determination of which occurs through the use of complex mathematical and physical tools, such as the Fourier transform. This paper illustrates how the use of mathematical methods, with particular reference to Fourier analysis, are important to extract significant indications from the data relating to reference seismic events in a specific geographical area. To determine the “seismic actions” it is necessary to define a “Local Seismic Response” (LSR) study consisting of the analysis of the set of changes in amplitude, duration and frequency content that a seismic motion undergoes as it passes through the layers of soil between the basic rock formation (bedrock) and the foundation soils of the work.

Seismic motion analyses can be represented in both the “time domain” and the “frequency domain”. In the “time domain”, the descriptive parameters include the maximum (or peak) value of acceleration, velocity or displacement, and the duration of the event, while in the “frequency domain” the Fourier spectrum (frequencies) is studied. The quantitative evaluation of the “Local Seismic Response” is based on the comparative analysis of the physical quantities that characterize the seismic motion at the surface level and at the bedrock. In the “time domain”, the most significant parameter is the “amplification factor”, defined as the ratio between the “peak acceleration” at the surface and that at the bedrock. In the “frequency domain”, the main parameter is the “transfer function”, expressed as H(f)=Fs(f)/Fr(f), which represents the ratio between the Fourier spectrum of the motion at the surface and that at the base. This function, being a ratio between two spectra, is distinct from an “amplitude spectrum” A(f), which provides important information on the filtering effect produced by the soil layers on the seismic motion, which, in fact, produces an increase in the amplitude of the motion for some frequencies and reduces it for others.

By analyzing the accelerograms of the seismic events recorded at the surface and at the bedrock, the corresponding “Fourier spectra” can be obtained by means of the “Fast Fourier Transform” (FFT). The ratio between these spectra, i.e. the “deconvolution”, provides the site amplification function. Furthermore, the analysis of the exponential components in the time domain allows to reduce the width of the Cauchy or Lorentz probabilistic distribution lines in the frequency domain, identifying the amplified motion components along the path of the seismic wave from the bedrock to the surface.

This study describes an integrated approach that uses data from the “Istituto Nazionale di Geofisica e Vulcanologia” (INGV) catalogues, the “Rexel” software to identify accelerograms compatible with a target spectrum, and the open-source “STRATA” software for the analysis of local seismic response in one-dimensional contexts. STRATA allows to model the linear propagation of seismic waves and the variations of the dynamic properties of the ground as a function of the deformations induced by an earthquake.

The integrated approach involves the use of Fourier analysis to characterize the spectral content of the analyzed seismic events, with the aim of determining the key parameters of the “design earthquake”. This is done through the integration of seven reference seismic events, selected for their criticality with respect to the site under consideration. The results offer a useful methodological contribution for designers and researchers of applied seismology. Recent studies have highlighted the importance of spectral analysis and advanced statistical techniques in seismic modeling. However, uncertainties and margins for improvement persist, especially for the integration methods used between empirical data and theoretical models, especially for complex phenomena such as energy defragmentation and local-scale interactions.

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