Synthetic spectral structure of the seismic ambient vibrations generated by a distribution of superficial random sources with a finite extension

Highlights

• Seismic ambient-vibrations and respective sources are described as stochastic fields.

• Stochastic-field spatial-correlation can be used to model source's finite extension.

• Realistic synthetics for ambient-vibration displacement power spectra can be obtained.

• Satisfactorily synthetics for HVSR can be produced.

Abstract

A model of seismic ambient vibrations is presented in the form of a random wavefield generated by a uniform continuous distribution of correlated aleatory forces, located at the surface of a flat layered weakly-dissipative Earth. The frequency-dependent spatial correlation among these sources is assumed to be representative of their spatial extension and has been parametrized by considering the role played by sea-wave dynamics in generating ambient vibrations. This enables a realistic description of the average power spectral density function of ambient vibrations observed worldwide at reference soil conditions. Effectiveness of the model is tested by reproducing observed average Horizontal-to-Vertical Spectral Ratios obtained from ambient vibration measurements at a well documented site in Northern Italy.

Introduction

Passive seismic survey is nowadays a widely used technique to infer some mechanical properties of the shallow subsoil, significant in order to characterize the places by means the local seismic effects they manifest. In the last couple of decades, both single-station (e.g., Refs. [[1], [2], [3], [4]]), and multi-station (e.g., Refs. [[5], [6], [7]]), configurations have been used to extract some observables by the ambient-vibration wavefield, which make it possible, by exploiting suitable inversion procedures, to infer the shallow subsoil seismic properties (e.g., Refs. [8,9]).

The use of ambient-vibration measurements to retrieve low-strain seismic properties of the subsoil relies, however, on the development of physical models able to establish a relationship between average properties of this wavefield, measured at the Earth's surface, and the subsoil structure. These forward models, implemented in the inversion procedures, should be characterized by sound physical basis (in that they are able to capture main features of underlying physical processes) and computational effectiveness. These two aspects are in many cases conflicting, since completeness and soundness of the physical model (including, e.g., all the seismic phases expected to contribute to the ambient vibration wavefield) also implies a significant increase of computational efforts necessary to provide realistic outcomes.

In order to maximize computational effectiveness, several simplified forward models have been considered, by assuming as negligible the contribution of some seismic phases. Since array measurements of ambient vibrations focus on the characterization of surface-wave components (see, for a review, Ref. [10]), forward modelling can be performed by considering these phases only. In this case also, some limiting assumptions have to be made and, in particular, that energy partition between the propagation modes only depends on the subsoil structure and not on the generating sources [11,12]. Modelling HVSR curves appears a much more complex task, because, in principle, these curves are expected to depend on the spectral amplitudes of all seismic phases (body and surface waves) present in the ambient vibration wavefield (see, e.g., Ref. [13]). The relative contribution of these different phases is expected to depend on the distribution and characteristics of sources, propagation phenomena and local seismo-stratigraphical conditions. Experimental studies [14,15] and numerical modelling [16] indicate that surface waves (both Rayleigh and Love) dominate some parts of the spectral domain, but other phases may play an important role, in particular in the spectral range below and close to the resonance frequency of the S waves. This implies that simplified models focusing on specific phases only (e.g., Refs. [1,[17], [18], [19]]), may only capture part of the observed phenomenology.

To overcome these problems, full wavefield models have been proposed and applied in the practice of site seismic characterization. An effective model has been proposed by Sánchez-Sesma et al. [20] under the simplifying assumption that ambient vibration is a diffuse random wavefield (DFA, Diffuse Field Approach, in the following). This assumption appears appealing, since it allows to remove from the model the role of sources by focusing on the effect that the local structure plays on the ambient vibration spectral structure. Some theoretical consequences of this model have been explored [21] and effective numerical forward computations have been developed and successfully implemented in inversion protocols [22]. The DFA model has proved its effectiveness in many situations (e.g. Refs. [[23], [24], [25], [26]], included the case of offshore measurements [27].

However, disregarding the role of sources cannot allow accounting for observations showing that the shape of the HVSR curve may be, sometimes, affected by the distribution and activity level of ambient vibration sources [28,29].

This suggests that developing a more general model inclusive of ambient vibration sources could be of some interest. Such an alternative model should implement the inherent stochastic nature of the ambient vibrations (cf., e.g., Ref. [30,31]) and the physical description of seismic observations in terms of source characteristics and propagation phenomena. First attempts in this direction were proposed by Field and Jacob [32], Lachet and Bard [53] and Bonnefoy-Claudet et al. [33], based on the assumption that ambient vibrations are the effect of a distribution of independent point-like harmonic random sources located at the surface of the Earth. Lunedei and Albarello [34] provided a consistent formalization of that model, by considering a flat layered viscoelastic Earth. In that model, a formal relationship can be established between the average spectral power of the random sources and that of the resulting displacement wavefield at a site. This model allowed to simulate realistic HVSR patterns and also effectively reproduce phase correlations between ambient vibrations measured at multiple sites.

However, the assumption of independent random point sources seems in contrast with the well established oceanic/atmospheric origin of most part of ambient vibrations, at least in the frequency range below 1 Hz [14,35], which implies a finite dimension of the sources. This feature can be described by introducing a form of correlation between the considered point sources and, in particular, the typical dimension (wavelength) of the source can be simulated by imposing a suitable correlation radius. A first attempt in this direction has been provided in Lunedei and Albarello [36] by considering a correlation radius whose dimension is independent from the vibration periods of point-like sources. This model, however, seems to be unrealistic since it is plausible that source dimensions may affect the range of excited frequencies. Specifically, oceanic swells, most probably responsible for observed ambient vibrations in the low-frequency range, are characterized by a well defined relationship between wavelength λ and frequency ν in the approximate form

$$\lambda =\frac{g}{2\pi }{\nu }^{-2},$$

where g is the gravity acceleration modulus [37]. Tanimoto [38] and Webb [39] accounted for this effect at the global scale by modelling ambient vibrations in the very low frequency domain (<0.1 Hz). In order to extend this view to the frequency range of interest for engineering applications (>0.1 Hz), a frequency-dependent correlation radius is introduced in this paper to generalize our previous model [36].

The formalization of the proposed model is presented at first. The tuning of parameters controlling the frequency dependence of the correlation radius is then obtained, by considering as benchmarks Eq. (1) and the shape of average spectral powers defined by Peterson [40] for ambient vibrations measured worldwide at rock sites. Then, a case study is presented, where the effectiveness of the proposed model is evaluated in comparison with observations and outcomes of alternative formulations.