Dispersion analysis of multi-modal waves based on the Reassigned Cross-S-Transform

https://doi.org/10.1016/j.soildyn.2021.106610Get rights and content

Highlights

  • A new procedure to compute dispersion curves reassigning the Cross-S-transform of two time-varying signals is presented.

  • The new procedure is not based in any assumption regarding the wave types and/or velocity of propagation.

  • A procedure capable of computing separate dispersion curves in cases of mixed modes is illustrated.

  • The application of the technique is illustrated with artificial waveforms, numerical simulations and with recorded real data.

Abstract

The quantitative description of the propagation of the energy components in a signal is a fundamental task in signal processing. The correct identification of the components is important because they carry information about the medium in which they propagate. In this article, I introduce the Reassigned Cross-S-Transform to estimate dispersion curves from time-varying signals, a technique based on function maximization principles and time-frequency cross-correlation. The RCST provides a sharpened and direct estimate of the dispersion curve for multi-modal propagating waves. Furthermore, the RCST can be used with signals that are not synchronized with the same initial time. I present examples studying the degradation effects on the slowness curves due to mode mixture and reflected waves. I illustrate the application of the RCST technique to artificially generated Ricker wavelets, and to real recordings of seismic waves generated by an active source. Finally, using 2D numerical simulations of wave propagating through an elastic layered medium, I solve a mode identification problem by combining the RCST with a previously proposed technique to extract Rayleigh waves from signals.

Introduction

Analysis of recorded signals is fundamental in the understanding of wave propagation phenomena. The Fourier analysis has been a classical technique in the field of signal processing, due to its robust mathematical foundation and ease of application [1]. The Fourier spectrum provides insight on the energy of the signal as a function of frequency, allowing the identification of its components. Spectral analysis has been the basis of quantitative analyses and modeling of wave propagation problems in many fields such as physics, applied mathematics, and chemistry. However, most signals related to physical processes are non-stationary, this is, their frequency content change with time, and simple Fourier analysis cannot address this change. That is why many efforts have been devoted in the last decades to develop Time-Frequency Representations (TFR), being the most popular the Short-Time Fourier Transform [2], the Continuous Wavelet Transform [3], the Stockwell Transform [4], and the Empirical Mode Decomposition [5]. However, according to the Heisenberg-Gabor uncertainty principle [2], most of the TFRs mentioned above are of limited time-frequency resolution, leading in some cases to misidentification of components and poor readability. One technique that has been proposed to improve the readability of TFRs consists of reassigning the TFR so that the energy is concentrated at the time-frequency coordinates where the energy itself attains its maxima. According to Meignen et al. [6], this approach was initiated in the late 1970s by Kodera [7], after which many reassigned TFRs have been proposed for both non-invertible and fully invertible distributions. A recent comprehensive review on the subject of reassigned TFRs is provided in Ref. [6].

Among the different TFRs, the Stockwell Transform (also known as the S-Transform) has distinguished itself in practical applications because (i) it preserves the absolute phase of the original time-domain signal, (ii) it can be inverted without losing information. These advantages have been exploited in recent seismological investigations, where the S-Transform has been successfully implemented for the identification and extraction of surface waves (Rayleigh and Love) from strong ground motion recordings, by characterizing the polarization characteristics of the different wave-types in the time-frequency domain [8]. Furthermore, the phase velocity of the extracted Rayleigh and Love waves was estimated in the time-frequency domain also making use of the Stockwell Transform [9]. In a previous publication [10], Stockwell himself had suggested the use of the Cross-Stockwell-Transform (CST) to estimate phase velocity among two stations, because the product of the two S-Transforms gives directly the phase difference for each point of the time-frequency space. However, this method gives correct results inasmuch as the peaks of the two signals are “located” at the same times.

In this work, I extend the ideas of Stockwell regarding the estimation of phase velocity from recorded signals to implement a time-frequency analysis that permits the estimation of the phase velocities of the waves making use of the reassignment technique. Different reassignment approaches have been already proposed for the S-Transform in order to improve its resolution, such as the generalized S-Transform [11] and the Synchrosqueezed S-Transform [12], among others. In particular Pan et al. [13] performed time-frequency cross-correlation analysis of signals using the Synchrosqueezed S-Transform, which allows the visualization of the change with time (the evolution) of the cross-correlation between two signals. Furthermore, the generalized S-Transform has been implemented by Askari et al. [14] in a slant stack procedure to compute dispersion curves of Rayleigh waves in terms of group velocities. Although the reassignment technique I propose herein indeed improves the resolution of the S-Transform, its main objective is to provide a Cross-S-Transform with energy concentrated at the time-shifts between two signals, which directly gives the relation between slowness and frequency, namely, the dispersion curve. Since spreading of energy is one of the difficulties in identifying different modes in dispersion analysis, especially when the modes are close to each other (in frequency) (see for instance, Levshin and Panza [15]), a dispersion curve obtained from reassignment could be useful to the misidentification problem. Furthermore, as I will show in the sequel, the Reassigned Cross-S-Transform (RCST) is basically a mathematical maximization technique applied to the S-Transform of the signals, and I make no assumptions regarding the physical processes represented in the signals: no assumptions regarding either the type of waves or the spatial-temporal distribution of the phase velocity. Thus, the technique can be applied to signals related to any wave-propagation problem, provided that they are correctly sampled, as any other signal processing technique requires.

In this article I consider a natural application of the reassignment technique, the dispersion analysis of Rayleigh waves, since they are waves with strong dispersive characteristics. Multimodal dispersion analysis of Rayleigh waves is a very common tool for in-situ, non-destructive testing of different types of materials. In near-surface geophysics in particular, a popular technique used in seismic surveys is the use of the dispersion curve of Rayleigh waves propagating through the surface, to derive the shear wave velocity of the different soil layers at a specific site [16]. The Rayleigh waves are recorded at an arrangement of sensors after they are excited by a blast or by a sledgehammer. However, in the obtained seismic wave field different types of waves are present (body, Love, reflected, scattered, noise), which can have a detrimental impact on the estimation of the desired dispersion curve. Different wave types and different modes of similar frequency arriving at neighboring times can interfere and degrade the accuracy of the estimated velocities. Addressing these inaccuracies is still a problem of interest in near-surface geophysics, as the correct determination and interpretation of the dispersion curve is a fundamental step in a successful surface-wave testing campaign [17].

In the following sections, I first explain the basic concepts about the S-transform and about dispersion curves based on measurements of phase velocity. I then derive the mathematical formulation to reassign the S-transform of a signal, and extend the reassignment ideas to derive the Reassigned Cross-S-Transform (RCST). Later, I illustrate the applicability of the RCST with three experiments of artificial dispersive waves: one with a single wave train, a second one with two dispersive modes propagating in the same direction, and a third one with two modes propagating in opposite direction. I implement this last synthetic experiment in order to asses the effect of reflected waves on the RCST. I also analyse a set of real recordings, acquired to be analysed with the method of Multichannel Analysis of Surface Waves (MASW) [18], and compare the obtained dispersion curve with estimations from the more classical F–K Transform technique. Finally, I illustrate the advantages of using recordings of several components (horizontal and vertical) to identify and separate different modes of surface waves, and in turn, to improve the estimation of the dispersion curve. For this, I consider a seismic wave field generated through 2D numerical simulations with the Spectral Element Method, in a layered velocity structure where several Rayleigh wave modes propagate.

Section snippets

Dispersion analysis and the S-transform

Dispersion phenomena is observed when the phase velocity of a wave is frequency-dependent. In seismological processes dispersion is of two types: material dispersion due to viscoelastic attenuation, and geometric dispersion due to heterogeneity of the medium [19]. Regardless of the type of dispersion, a frequency-dependent phase velocity can be defined as follows:vp(fn)=dtp,tp=tp(fn)for each frequency fn, where d is the propagation distance (the distance between the two nearby stations where

Reassignment of the S-transform

The idea of reassigning the S-transform consists in concentrating the energy of the TFR around the coordinates (t,f) where the energy itself reaches its maximum values. These coordinates are usually referred to as the time delay and the instantaneous frequency. The time delay is identified as the time of arrival of the peak of the envelope of an analytical signal. The instantaneous frequency is the frequency around which the spectral energy is concentrated at a specific time. These concepts are

Reassignment of the cross-S-transform

Following the ideas of the reassigned S-transform presented in the previous section, I want to reassign the Cross-S-Transform (CST) on a time-frequency distribution of the time-shift, that I denote here by Tp, in the following mannerSRC(Tp,fn)=SC(τ,fn)δ(Tpτ)dτand using a redefined Cross-S-Transform:SC(t,fn)=S1S2swhere now S1 is multiplied by the shifted S-Transform of station 2, S2s. More precisely, S2s is the result of shifting each voice of S2, so that the amplitude peaks of S1 and S2s

Example with one artificial dispersive wave train

In this example, I propagate a Ricker wavelet through different stations, imposing an artificial dispersion curve. The Ricker wavelet is given by the expression:R(t)=AR(2β1)exp(β),β=[πfc(ttc)]2where AR is the amplitude and tc is the time of its largest peak. The Ricker wavelet has a limited bandwidth with energy concentrated at its central frequency fc. This limited bandwidth is one of the reasons why the Ricker wavelet is used in many seismological applications, since registered seismograms

Example with two artificial dispersive wave trains propagating in the same direction

In this example I add another wave train to the signals, with a different artificial dispersion curve. In this way I intend to simulate the propagation of bi-modal dispersive waves. I add another Ricker to the first mode I already analysed, with a lower amplitude AR2=0.5, central frequency fc2=60Hz, and tc2=1/fc2. For this new “higher mode” the slowness law is shown in Fig. 6a with the thicker line, and the mathematical expression for the phase velocity isVp2=4exp(f17)+0.55(kms)

I adopt again

Example with two artificial dispersive wave trains propagating in opposite directions

In this experiment I consider the same two Ricker wave trains (modes) used in the previous example, but now the weaker wave train propagates in the opposite direction. With this experiment I try to simulate the presence of reflected/refracted waves in the signals, and investigate the effect on the estimated dispersion curve. The time histories of the waveforms are shown in Fig. 8a, where it can be observed that the interference between the wave trains starts at 0.06 km. In Fig. 8b I present the

Example of mode separation from numerical simulations

In this last example I investigate the benefits of mode separation in the estimation of the dispersion curves. An efficient technique based on the S-Transform to identify and separate wave trains in seismograms has been proposed in the recent years [8], which provides the time histories of the extracted waves. However, the extraction technique is based on the polarization characteristics of the waves and therefore it requires the vertical component and at least one horizontal component of the

Example with real recordings

In this example, I estimate the dispersion curve via the RCST from actual recorded data which is provided with the software Geopsy [21], an open source tool for geophysical research and applications (e.g. Ref. [22]). The 1D linear arrangement of seismic surface wave recordings is to be analysed with the active seismic experiment (MASW) tool implemented in Geopsy, based on the F–K transform. The profile of recorded waveforms is presented in Fig. 14, where for clarity I normalized each time

Conclusions

In this work, I proposed a technique to estimate dispersion, slowness curves for multimodal, non-stationary signals, based on the Reassigned Cross-S-Transform. I presented in detail the mathematical derivations and the computational steps for the construction of the RCST. The RCST I proposed is derived from simple maximization principles so that the energy is concentrated at the ridges of the time-frequency representations of the signals. Furthermore, in deriving the RCST I made no assumptions

Credit author statement

Kristel Meza-Fajardo: Single author for all relevant CRediT roles: original idea, conceptualization, methodology, formal analysis, writing original draft, preparation, visualization, investigation, writing review and editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The author would like to thank Prof. Apostolos Papageorgiou for discussions on the reassignment technique, and two anonymous reviewers for constructive comments that improved the manuscript. This research has been financed by the French National Research Agency (Agence National de la Recherche), under project MODULATE, grant number ANR-18-CE22-0017.

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