Improvement of HVSR technique by wavelet analysis

doi:10.1016/j.soildyn.2007.06.006

Soil Dynamics and Earthquake Engineering

Volume 28, Issue 4, April 2008, Pages 321-327

https://doi.org/10.1016/j.soildyn.2007.06.006 Get rights and content

Abstract

In this work we investigate the application of the wavelet analysis to improve the well-known horizontal to vertical spectral ratio (HVSR) technique for site effect estimation. Wavelet analysis has been widely used in the last decade for its capacity to localise the signal in both the time and frequency domains. The wavelet packet transform and the evaluation of a cost function for every node permit one to remove from the signal the least important components and lead to significant improvements in the identification of the fundamental frequency. In this paper a new algorithm is presented based on wavelet analysis, that allows, in some cases, to improve the identification of the peak in the QTS ratio. A number of examples of application are also presented.

Introduction

The horizontal to vertical spectral ratio (HVSR) technique, proposed by Nakamura [1], [2], is one of the most logistically effective ways of characterising local seismic amplifications. The application of the wavelet analysis can, in some cases, improve the results of the simple Nakamura ratio. The wavelet packet transform (WPT) decomposes the signal in several nodes each of which is the result of a sequence of applications of high-pass or low-pass filters. The wavelet analysis allows one to remove from the time series the noise components by setting to zero the smallest coefficients (or, as we will describe later, the “smallest nodes”) that result from the analysis.

The tremor is measured along two orthogonal directions and a vertical one, allowing one thus to compute two spectral ratios. The presence of differences between the two ratios can be due to lateral anisotropies or, more often, to the presence of directional noise (for instance anthropogenic noise). An algorithm, wavelet analysis for site effects estimation (WASEE), is presented here which tries to reconstruct the two signals using, when possible, the most characteristic nodes that have almost the same position in the wavelet packet tree.

Section snippets

The quarter wave law

Let us consider the representation of a sedimentary layer upon a bedrock layer as in Fig. 1. When seismic waves pass through the sedimentary layer they will be modified according to the characteristics of the layer. If we suppose that the ideal soft layer is homogeneous and elastic, it is possible to evaluate the fundamental resonance frequency through the quarter wave law $f_{0} = \frac{4 C_{s}}{H},$ where $C_{s}$ is the velocity of the S waves and $H$ is the depth of the soft layer.

The Nakamura technique

The Nakamura technique [1], [2], also known as quasi transfer spectrum (QTS) or H/V technique, allows one to evaluate the fundamental frequency of a soft layer by measuring only the tremor at the surface. If we take again into consideration the typical geological structure of a sedimentary basin (Fig. 1), the tremor is expected to be composed by surface waves and body waves. The latter will be modified by the filter action of the soft layer. It is therefore possible to define the spectra of the

Multi-resolution wavelet analysis

Through the wavelet analysis it is possible to decompose any square-integrable function in $L^{2}$ as a sum of bases functions that characterise two orthogonal complementary subspaces $V_{j}$ and $W_{j}$ (with $V_{j} = V_{j - 1} \oplus W_{j - 1}$ ). The orthonormal space $V_{0}$ is defined by the family ${φ (x - k), k \in Z}$ . Because $V_{0} \subset V_{1}$ it is possible to write any function in $V_{0}$ as a linear combination $φ (x) = \sqrt{2} \sum_{k = - \infty}^{\infty} h_{k} φ (2 x - k),$ where ${h_{k}}$ is a square-summable sequence of coefficients which define a linear operator.

The space $W_{0}$ is defined by the

Discrete wavelet transform (DWT)

Every function in $L^{2}$ can be approximated by using the space $V_{J}$ where $J$ is a sufficiently large number, so the space $V_{J}$ tends to $L^{2}$ and $W_{J}$ tends to ${0}$ . In this way every function $f (x) \approx f_{J} (x) \in V_{J}$ can be decomposed as the sum of two functions that belong, respectively, to $V_{J - 1}$ and $W_{J - 1}$ , i.e by using (3), (4): $f_{J} (x) = \sum_{k = - \infty}^{\infty} a_{J, k} φ_{J, k} (x) = \sum_{k = - \infty}^{\infty} a_{J - 1, k} φ_{J - 1, k} (x) + \sum_{k = - \infty}^{\infty} d_{J - 1, k} ψ_{i - 1, k} (x) = A_{1} (x) + D_{1} (x) = \sum_{k = - \infty}^{\infty} a_{J - 2, k} φ_{J - 2, k} (x) + \sum_{k = - \infty}^{\infty} d_{J - 2, k} ψ_{i - 2, k} (x) + \sum_{k = - \infty}^{\infty} d_{J - 1, k} ψ_{i - 1, k} (x) = A_{2} (x) + D_{2} (x) + D_{1} (x) = \dots .$ The functions $A_{i} (x)$ and $D_{i} (x$

Wavelet packet transform (WPT)

In Fig. 5 an example is shown of wavelet packet decomposition; such representation is also called wavelet packet tree. The spaces $W_{j}^{p}$ of the decomposition can be called “nodes” of the tree; so the general space $W_{j}^{p}$ refers to a node at depth (or level) $j$ in the tree and at position $p + 1$ in the level $j$ .

Let us consider a detail space $W_{j}^{p}$ and its orthonormal basis $ψ_{j}^{p} (t - 2^{j} k)$ .

For the Theorem 8.1 of Coifman, Meyer and Wickerhouser, presented in [9], it is possible to divide the detail space $W_{j}^{p}$ as the

De-noising via threshold

Once the DWT is computed it is possible to de-noise a signal by discarding some coefficients, usually the smallest ones. The threshold idea consists in removing all the coefficients whose values are smaller then a suitably chosen threshold value $τ$ . The most used definition for $τ$ is given in [10] $τ = σ \sqrt{\ln N},$ where $σ$ is the standard deviation of the coefficients and $N$ is the length of the signal. There are two kinds of threshold; the soft-threshold for which after the thresholding the new coefficients

Cost function

Once the wavelet packet tree is computed, it is possible to apply to every node a cost function and then calculate the best tree by selecting the nodes that cover all the tree and, at the same time, have the minimum cost value. The most common cost function is the Shannon entropy, defined by Coifman and Wickerhauser [11].

Let us consider a non-negative sequence $p = {p_{i}}$ with $\sum_{i} p_{i} = 1$ ; the entropy value of the sequence is defined as $H (p) = - \sum_{i} p_{i} \log_{2} p_{i},$ where conventionally we assume $0 \log_{2} 0 = 0$ .

Let ${x_{i}}$ be

Algorithm

As we have seen, by applying the WPT to a signal, it is possible to compute the wavelet coefficients that characterise every node and every level of the WPT. Then it is possible to apply to every node a cost function, which for this work is the Shannon's entropy function, defined by Coifman and Wickerhauser [11]. The main idea of the original algorithm WASEE is to reconstruct the analysed signals by using only some characteristics nodes of a level.

The principle scheme of the algorithm is the

Discussions and conclusion

We have seen that through the wavelet analysis it is possible to estimate the fundamental frequency of a soft layer, independently from the seismic wave-field that creates the ambient vibrations. If the simple Nakamura technique is used, it could be difficult to identify the peak; therefore some further tools have been developed in order to try to improve Nakamura technique. One of these applications has been recently presented by Carniel et al. [12]. They have used the singular spectrum

Acknowledgements

The authors wish to thank the seismology group of the University of Udine, coordinated by M. Riuscetti; the functional mechanics group of the University of Udine, coordinated by P.B. Pascolo; Marta Tárraga and Josh Jones for useful discussions. The comments of Amara Graps and Maria José Gonzalez Fuentes substantially improved the quality of the paper.

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