Phase statistics of seismic coda waves

arXiv:0809.3556v2 [physics.geo-ph] 9 Jan 2009 Phase statistics of seismic coda waves D. Anache-M´enier and B. A. van Tiggelen Laboratoire de Physique et de Mod´elisation des Milieux Condens´es, Universit´e Joseph Fourier/CNRS, BP 166, 38042 Grenoble, France L. Margerin Centre Europ´een de Recherche et d’Enseignement des G´eosciences de l’Environnement, Universit´e Aix Marseille, CNRS, Aix en Provence, France We report the analysis of the statistics of the phase fluctuations in the coda of earthquakes recorded during a temporary experiment deployed at Pinyon Flats Observatory, California. The ob- served distributions of the first, second and third derivatives of the phase in the seismic coda exhibit universal power-law decays whose exponents agree accurately with circular Gaussian statistics. The correlation function of the spatial phase derivative is measured and used to estimate the mean free path of Rayleigh waves. PACS numbers: 46.65.+g, 91.30.Ab, 46.40.Cd In the short-period band (> 1 Hz) , ballistic arrivals of seismic waves are often masked by scattered waves due to small-scale heterogeneities in the lithosphere. The scat- tered elastic waves form the pronounced tail of seismo- grams known as the seismic coda [1, 2]. Even when scat- tering is prominent, it is still possible to define the phase of the seismic record by introducing the complex analytic signal ψ(t, r) = A(t, r)eiφ(t,r), with A the amplitude and φ the phase. In the past, many studies have focused on the modeling of the mean field intensity I(t) = ⟨A(t)2⟩ [see 3, for review]. The goal of the present paper is to study the statistics of the phase field in the coda. In the coda, the measured displacements result from the super- position of many partial waves which have propagated along different paths between the source and the receiver. Each path consists of a sequence of scattering events that affect the phase of the corresponding partial wave in a random way. For narrow-band signals, the phase field can therefore be written as φ(t, r) = ωt + δφ(t, r), where ω is the central frequency, and δφ denotes the random fluctu- ations. The trivial cyclic phase ωt cancels when a spatial phase difference is considered between two neighbouring points. Spatially resolved measurements are facilitated by dense arrays of seismometers that have been set up occasionally. We note that the phase of coda waves has not been given much attention so far. The advantage of phase is that it is not affected by the earthquake magni- tude, and that it contains pure information on scattering, not blurred by absorption effects. For the statistical anal- ysis of amplitude and phase fluctuations of direct arrivals, we refer the reader to e.g. Zheng and Wu [4]. I. PHASE DISTRIBUTIONS We study data sets from a temporary experiment de- ployed at Pinyon Flat Observatory (PFO), California, in 1990 by an IRIS program. This site exhibits a high level of regional seismic activity. The array (see Fig. 1) contained 58 3-components L-22 sensors (2Hz) and was −100 0 100 200 300 −250 −200 −150 −100 −50 0 50 x (m) y (m) FIG. 1: Geometry of the seismic array configured as a grid and two orthogonal arms with sensor spacings of 7 meters within the grid and 21 meters on the arms [5]. We selected 8 earthquakes of magnitude greater than 2 with good signal to noise ratio in the coda. Typ- ically, epicentral distances are less than 110 km and the coda lasts more than 30 seconds after the direct arrivals. To perform the statistical analysis, we filtered the sig- nal in a narrow frequency band centered around 7 Hz (±5%) and selected a 15 s time window starting around 5 seconds after the direct arrivals. In this time window, the signal is believed to be dominated by multiple scattering and is highly coherent along the array [6]. We evaluate the Hilbert transform of the vertical displacement which yields the imaginary part of the complex analytic signal ψ(t, r) = A(t, r)eiφ(t,r). From the complex field, two defi- nitions of the phase can be given: (1) The wrapped phase φ is defined as the argument of the complex field ψ in the range (−π : π]. (2) The unwrapped phase φu is obtained by correcting for the 2π jumps – occurring when φ goes through the value ±π - to obtain a continuous function with values in R. The φ distribution is flat [7]. How- ever, more information can be extracted by considering 2 higher-order statistics of the phase. For this purpose we consider the spatial derivative of the phase, which can be estimated in two different ways. (1) The first measurement relies on the difference of the wrapped phases ∆φ between two seismometers sepa- rate by a distance δ. Applying the simple finite difference formula φ′ ≈∆φ/δ an estimate of the spatial derivative is obtained. Note that the phase difference ∆φ takes val- ues between −2π and +2π which does not allow a precise estimate for the distribution of the derivative for values roughly larger than π/δ. Beyond this value our mea- surements will be dominated by finite diff

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