Improvement of Nakamura technique by singular spectrum analysis Roberto Carniel a , Fausto Barazza a , Paolo Pascolo b a Dipartimento di Georisorse e Territorio, Universita ` di Udine, Italy b Dipartimento di Ingegneria Civile, Universita ` di Udine, Italy Accepted 14 August 2005 Abstract In this work we investigate the application of the Singular Spectrum Analysis to improve Nakamura Technique. The SSA has a wide and multidisciplinary range of applications; it allows a time series to be decomposed into different components, e.g. the signal itself, as well as various noise components, which can be subsequently removed from the data. Removal of the minor component of the data can lead to significant improvements in the identification of the system. In this paper the use of SSA Technique allows to optimize the signal to noise ratio before computing the classical Nakamura spectral ratios. A number of typical applications are also presented. q 2005 Elsevier Ltd. All rights reserved. Keywords: Singular spectrum analysis; Nakamura spectral ratio; Site effects; De-noising; Seismic risk 1. Introduction The application of the Singular Spectrum Analysis (SSA) on Nakamura Technique can, in some cases, improve the results of the simple Nakamura spectral ratio. The SSA allows the time series to be decomposed into different components, e.g. the signal itself, as well as various noise components, which can be subsequently removed from the time series. As shown in Ref. [1], removal of the minor components of the data can lead to significant improvements in the identification of the system. The tremor is usually measured along two horizontal directions and one vertical direction, so we can obtain two Nakamura spectral ratios that in some cases show some difference. The cause of this difference may be the presence of noise. The SSA technique can reduce this effect. In addition the SSA technique allows to obtain various de- noising levels, so a level can be chosen such that the ‘matching’ between the two Nakamura spectral ratios is maximum, obtaining an improvement with respect to the simple application of Nakamura spectral ratio. 1.1. The Quarter wave law Let’s consider a sedimentary layer upon a bedrock layer. The sedimentary layer behaves like a filter, amplifying some frequencies and attenuating others. The frequencies for which the maximum amplifications are observed are said resonance frequencies and the knowledge of such frequencies can provide important information about the superficial layer. If the material is linear elastic, an hypothesis which is generally acceptable for small deformations, and homo- geneous, the resonance frequencies are given by the so called quarter wave law: f shear;k Z ð2k C 1Þ c S 4H ; f long;k Z ð2k C 1Þ c P 4H where c S and c P are, respectively, the propagation velocity of the S and P waves in the superficial layer and H is its thickness. For soft layers we have c P z(1.5–3)c S from which we obtain f 0 Z f shear;0 Z c S 4H ; f 1 Z f long;0 Z c P 4H xð1:5–3Þf 0 ; f 2 Z f shear;1 Z 3 c S 4H Z 3f 0 ; . (1) The frequency f 0 is said the fundamental frequency and its knowledge allows to derive a relationship between c S and H. We can associate to each eigen-frequency a corresponding eigen-mode (or eigen-shape); as an example the first three eigen-modes are represented in Fig. 1. It is noteworthy that in the bedrock the velocities are much higher than those in the soft layer, so that the quarter wave law gives frequencies much higher than f 0 , which can therefore be neglected. Soil Dynamics and Earthquake Engineering 26 (2006) 55–63 www.elsevier.com/locate/soildyn 0267-7261/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2005.08.005 E-mail address: roberto.carniel@uniud.it (R. Carniel).