Insights on surface wave dispersion and HVSR: Joint analysis via Pareto optimality

Abstract

Surface Wave (SW) dispersion and Horizontal-to-Vertical Spectral Ratio (HVSR) are known as tools able to provide possibly complementary information useful to depict the vertical shear-wave velocity profile. Their joint analysis might then be able to overcome the limits which inevitably affect such methodologies when they are singularly considered.

When a problem involves the optimization (i.e. the inversion) of two or more objectives, the standard practice is represented by a normalized summation able to account for the typically different nature and magnitude of the considered phenomena (thus objective functions). This way, a single cost function is obtained and the optimization problem is performed through standard solvers.

This approach is often problematic not only because of the mathematically and physically inelegant summation of quantities with different magnitudes and units of measurements. The critical point is indeed represented by the inaccurate performances necessarily obtained while dealing with problems characterized by several local minima and the impossibility of a rigorous assessment of the goodness and meaning of the final result.

In the present paper joint analysis of both synthetic and field SW dispersion curves and HVSR datasets is performed via the Pareto front analysis. Results show the relevance of Pareto's criterion not only as ranking system to proceed in heuristic optimization (Evolutionary Algorithms) but also as a tool able to provide some insights about the characteristics of the analyzed signals and the overall congruency of data interpretation and inversion.

Possible asymmetry of the final Pareto front models is discussed in the light of relative non-uniqueness of the two considered objective functions.

Introduction

Site characterization requires the determination of geomechanical properties often accomplished via non-invasive geophysical investigations. The employ of a single methodology necessarily brings some degree of uncertainty due both to possible ambiguity in data interpretation and non-uniqueness of the solution (e.g. Scales et al., 2001, Ivanov et al., 2005a, Ivanov et al., 2005b, Dal Moro, 2008, Dal Moro, 2010a, Palmer, 2010). Consequently, acquisition and joint analysis of further datasets is always highly recommended.

Surface Wave (SW) analysis is nowadays a popular method not only, as in past, for crustal studies (e.g. Evison et al., 1959) but for near-surface investigations as well (Glangeaud et al., 1999, Park et al., 1999, Dal Moro et al., 2007). The acronym MASW (Multichannel Analysis of Surface Waves), although could in principle indicate any kind of geophone-array-based study, is normally used for the active case while ReMi (Refraction Microtremors) commonly refers to linear-array passive experiments (e.g. Louie, 2001).

Though SW analysis potentially represents a fast and effective way to reconstruct the vertical V_S profile, problems connected with non-uniqueness of the solution (Luke et al., 2003), misinterpretation of data due both to mode misidentification (Zhang and Chan, 2003, Dal Moro, 2010a, Dal Moro, 2010b) and influence of guided waves (O'Neill et al., 2004, O'Neill and Matsuoka, 2005) pose some often poorly considered problems in SW analysis.

In order to put in evidence possible problems in SW dispersion curve interpretation, some synthetic seismograms were computed (Carcione, 1992). Data are reported in Fig. 1, Fig. 2 and show how complex the energy distribution among different modes can be (see also Dal Moro, 2010a, O'Neill et al., 2004, O'Neill and Matsuoka, 2005). For the first case (Fig. 1) fundamental and first higher modes merge together thus simulating a signal that would be easily misinterpreted as a single mode. Example reported in Fig. 2 shows the presence of the fundamental mode only up to about 33 Hz. The signal between 35 and 52 Hz actually pertain to the second higher mode while the energy for frequencies higher than about 52 Hz is basically related to the first higher mode.

Problems are particularly severe when, due to the site characteristics and the acquisition setting, different modes coalesce into a unique signal that inevitably results hard to interpret (see for instance Fig. 1). The simplistic assumptions that fundamental mode is the most energetic and/or that higher modes appear typically at higher frequencies often result in severe mistakes in velocity spectra interpretation, thus leading to erroneous V_S profiles (see Fig. 3 for a field MASW dataset).

Furthermore, a typical source of misleading signals may be also represented by guided waves that may sometimes generate very high amplitude signals with dispersive character (e.g. Robertsson et al., 1995, Roth and Holliger, 1999).

Reported data clearly show how complex and seemingly puzzling energy distribution can actually be. It results consequently apparent that the common practice of picking maxima in the f-k or v-f domain and consider them as related to simple mode distributions (very often it is assumed that most of the energy pertains to the fundamental mode) can lead to severe mistakes in the retrieved V_S profile. On the other side, when properly interpreted, higher modes provide valuable information about the actual model, highly reducing the non-uniqueness of the solution (Dal Moro, 2010b). It should be then properly acknowledged that velocity spectra interpretation is a task that must be tackled very cautiously also considering that any kind of automatic procedure now and then proposed for SW analysis or inversion is necessarily based on assumptions that, even though valid under some circumstances, cannot be universally generalized.

On the other side, Horizontal-to-Vertical Spectral Ratio (HVSR) has been traditionally used for determining site resonance frequency (Nakamura, 1989, Nakamura, 1996, Nakamura, 2000) but more recently some authors attempted to use it as contributory tool for retrieving information about vertical V_S profile (e.g. Fäh et al., 2003). The main problem with HVSR is the unclear and highly debated physical model to adopt for its modelling. Relative contribution of Rayleigh Love and body waves is an “unsolvable” problem as the relationships vary with the frequencies and is clearly site-dependant (Fäh et al., 2001, Bonnefoy-Claudet et al., 2008). Furthermore the contribution of the number of modes while considering the surface wave contribution and the influence of quality factors are further problems (Lunedei and Albarello, 2009, Albarello and Lunedei, 2010) as also the severe non-uniqueness of the solution (Fig. 4). This latter problem is basically due to the fact that HVSR is sensitive to V_S contrasts and not to absolute V_S values. Further sources of problems are related to the possible presence of anthropic components and the stability and statistical robustness of average HVSR.

All of these problems eventually prevent HVSR from being a fully stand-alone solution for V_S profiling.

Given such a scenario, an efficient joint inversion tool capable of overcoming problems and limits of both methodologies and provides a reliable subsurface model is clearly highly desirable.

When a problem involves the minimization of two or more objectives, the standard approach is represented by a normalized summation that provides a single-objective able to account for the typically different nature and magnitude of the considered phenomena. A single cost function is consequently obtained and the minimization problem is then performed through standard solvers.

Such approach is often problematic not only because of the mathematically and physically inelegant summation of quantities with different magnitudes and units of measurements (in our case m/s for the dispersion curve and the unitless Horizontal-to-Vertical Spectral Ratio). The critical point is indeed represented by the bad performances that such approach implies when dealing with problems characterized by high non-uniqueness and the impossibility to eventually assess the goodness of the retrieved model(s) in a rigorous way. In fact, using a single cost function results in the impossibility of evaluating the actual validity of the retrieved model. Some minimum will be necessarily reached but there is no way to evaluate the overall congruency of the designed and parametrized inversion (and retrieved models). This is a particularly heavy problem especially when the objectives are characterized by several local minima. In this respect SW dispersion curves and HVSR are two extremely tricky functionals as both suffer from intrinsic non-uniqueness and data interpretation problems. Their joint inversion through standard single-objective genetic procedures has already been faced by some author. Here we present a joint inversion carried out through a Multi-Objective Evolutionary Algorithm (MOEA) exploiting the Pareto dominance criterion (Van Van Veldhuizen and Lamont, 1998a, Van Veldhuizen and Lamont, 1998b, Van Veldhuizen and Lamont, 2000, Dal Moro and Pipan, 2007, Dal Moro, 2008).

A number of synthetic cases were considered to evaluate the behavior of the cost functions under different assumptions. Eventually a field dataset is analyzed and discussed.