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Diffuse wavefield and scattering process measured Diffuse wavefield and scattering process measured at surface and underground sites: seismological at surface and underground sites: seismological

observations on the equipartition of energyobservations on the equipartition of energy

M. La RoccaM. La Rocca11, D. Galluzzo, D. Galluzzo11, E. Del Pezzo, E. Del Pezzo11, L. Margerin, L. Margerin22, R. Scarpa, R. Scarpa33

1 – Istituto Nazionale di Geofisica e Vulcanologia - Osservatorio 1 – Istituto Nazionale di Geofisica e Vulcanologia - Osservatorio Vesuviano, Napoli, ItalyVesuviano, Napoli, Italy2 - Université de Tolouse, Tolouse, France 2 - Université de Tolouse, Tolouse, France 3 - Università degli Studi di Salerno, Salerno, Italy3 - Università degli Studi di Salerno, Salerno, Italy

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Wave Scattering and Diffusion

The propagation of waves in heterogeneous media modifies the wavefield through various processes. Scattering and intrinsic attenuation are the most important among them. For a given heterogeneous medium and wave frequency it is possible to define a “mean free path” and a corresponding “mean free time” that characterize the average scattering length and time. The wavefield observed after a time much greater than the “mean free time”, that is after many scattering events, is expected to be diffuse.

In a diffuse wavefield all wave modes are equally present. In other words, a large number of waves with the same amplitude but random phase and wave vector propagate through the observation point at any time. Consequences of this definition are that the diffuse wavefield has stationary amplitude and is incoherent.

The existence of diffuse wavefields is well known for Electromagnetic and acoustic waves. What about seismic waves?

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Scattering of Seismic Waves

The case of seismic waves is more complex than EM and acoustic waves because of the interaction between P and S waves. The propagation of a seismic wave through a discontinuity interface involves not only reflection and refraction, but also wave conversion. Near the free surface the propagation of surface waves further complicates the observed wavefield.

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Scattering of Seismic Waves

The amplitude of scattered waves strongly depends on the scattering angle.

(Figures from the paper by J.A. Turner, BSSA 1998)

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Multiple Scattering of Seismic Waves

The earthquake coda was first described by Aki and Chouet (1975) who introduced the single scattering model. More complex models were introduced later to better describe the envelope of coda wave energy. Among these, the multiple scattering model is particularly efficient to describe the propagation of seismic waves in heterogeneous media, the coda envelope, the diffusive regime, and other phenomena.

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Diffusive regime in a seismic wavefield

Theory predicts the existence of seismic wavefields in diffusive regime. For lapse time >> mean free time the seismic coda is expected to be a diffuse wavefield. That means:

1) waves coming from any directions with random phase; 2) signal mostly incoherent; 3) amplitude slowly changing with time; 4) equipartition of energy among the wave modes.

Seismic noise has many properties of diffuse wavefield. What about the earthquake coda? Are we able to observe a diffuse seismic wavefield?

Results are controversial because the very most of available data are recorded at or very near the free surface where the observed signals may be strongly affected by the local structure (site effects).

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The Energy Equipartition

Theory predicts the equipartition of energy among all wave modes in a diffusive regime. For body waves the equipartition principle states that the ratio between S and P energy density is given by

Experimental estimation of the energy equipartition is difficult because P and S waves can not be separated. Moreover, signals recorded at surface contains a lot of surface waves and may be strongly affected by site effects.

The H/V ratio (H/V = (N+E) / 2Z) can be computed easily. For body waves in diffusive regime we expect H/V = 1. Another measurable parameter is the Horizontal / Vertical energy ratio Eh/E

For body waves in diffusive regime we expect Eh/E = 0.67.

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Is the late coda a diffuse wavefield?

The direct P wave is expected to come from the source, carrying a strong source imprint, and spatially coherent. On the contrary, coda waves are expected to come from any directions, to be chaotic and incoherent. How to measure these properties of the wavefield? We need 3D seismic arrays, but we don't have any!In this work we use a 2D underground array.

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Analysis of coda waves recorded by an underground array

In this presentation we describe the results of detailed analysis performed on seismic data recorded by an array installed underground in the INFN Laboratory of Gran Sasso. We compare the underground recordings with signals recorded at surface by a broad band station.

FONTARI UNDERSEIS

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UNDERSEIS ARRAY was operating in the INFN Laboratory from 2003 to 2010. The array was composed by 20 short period three component stations. In 2009 U33 became broad band, and two broad band seismic stations were installed at surface in the Gran Sasso area. FON8 was very near the UND array.

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DATA SETMany tens of local and regional earthquakes were recorded at UND array and FON8. The 35 events characterized by the best signal to noise ratio were selected and analyzed in this work.

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Seismic Arrays

Seismic array: many stations installed in a homogeneous 2D configuration at distances comparable with the wavelength of the signal to study. With a dense seismic array we can sample the wavefield both in time and space.

Array data can be used for deep investigation of coda waves.

Methods of analysis: Array Methods

Array methods permit the estimation of apparent velocity and propagation direction of the seismic waves. We have used the Beam Forming in frequency domain and the Semblance in time domain.

Beam Forming power spectrum

Cross-spectral matrix

Semblance

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Methods of analysis: Spatial Coherence

The coherence measures the similarity of two signals fi(t), fj(t),

Taking advantage of seismic arrays, we can extend the computation of the coherence to N signals to measure their similarity among the array stations. This is a measure of the spatial coherence.

Coherence of M signals recorded at the same time at different places

Cross-spectral matrix computed for N signals with smoothing over M frequencies.

i, j = 1, … N

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Examples of seismograms and their coherence

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The coherence is frequency dependent: lower the frequency, higher the coherence. This is not an intrinsic feature of the seismic wavefield, but rather it is a resolution limit or our array. A reliable estimate of the coherence at low frequency requires a much larger array.

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Coherence of coda waves

How to identify a diffusive regime in the seismic wavefield? We look for signals characterized by travel time >> mean free time, and low coherence, but also a good SNR. These conditions can not be found in the same time window for the whole frequency range.

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We select different time windows along the coda that will be used to study the wavefield properties in different frequency bands.

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Energy partition ratio PEhr and H/V ratio have been computed in different time-frequency windows along the coda. Results are in good agreement with the values expected in case of energy equipartition for body waves.

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H/V and PEhr computed for the same earthquake recorded at surface and underground are significantly different. The theoretical value of H/V at surface is difficult to compute because it depends on the velocity structure, site effects, topography, …

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H/V ratio versus time has been computed for any stations of UND array in many different frequency bands. Values tend to 1 along the coda, in agreement with the hypothesis of diffuse wavefield.

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H/V ratio versus time has been computed at FON8 in the same frequency bands. Values are always greater than 1 along the coda, in strong contrast with the result observed at UND array.

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H/V and PEhr computed for regional and local earthquakes show comparable results at UND array. For local earthquakes the estimation is less stable because the coda is usually shorter, and the condition “lapse time >> mean free time” may not be satisfied.

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Spectral analysis of coda waves and seismic noise: comparison between underground and surface recordings. Results at surface are very oscillating along the coda.

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Results of array analysis applied to the three components separately indicate a broad distribution of backazimuth. This is in good agreement with the hypothesis that late coda consists of body waves in almost diffusive regime.

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Results of Semblance analysis are also compatible with a diffusive regime in the very late coda.

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Polarization analysis

Polarization analysis have been performed at all array stations along the coda in many different frequency bands. Results show very scattered polarization azimuth at most of the stations. This is in agreement with the hypothesis of diffuse wavefield.

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Numerical Simulations

Simulations of the Eh/E at different depth have been computed numerically by using two different velocity models, under the assumption of wavefield composed of body and surface waves.

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Numerical simulations

Results at low frequency show some qualitative agreement with the observed values. We believe that the main cause of discrepancy is due to the flat layer model that does not fit at all the area of Gran Sasso massif. In future simulations it will be necessary to include the local topography.

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Resume

We have analyzed local and regional earthquakes recorded by an underground seismic array. We performed the following analysis and focused our attention on coda waves:

Spatial coherence of the seismic wavefield;

Estimation of the H/V amplitude ratio vs frequency and vs time;

Estimation of the Eh/E vs frequency;

Comparison of observed H/V and Eh/E with simulated values;

Array analysis;

Polarization analysis.

Our results proved that the use of an underground seismic array is very useful to investigate the late coda of earthquakes finding diffuse wavefield and energy equipartition.

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Conclusions

Results of our analysis (Coherence, H/V, Eh/E, propagation parameters, polarization) indicate that the late coda of local and regional earthquakes recorded at about 1.4 km depth is composed of body waves in a diffusive regime.

As the lapse time increases along the coda the features of diffuse wavefield become more evident. This is particularly evident for regional earthquakes. For local earthquakes the observations are limited by the coda duration.

The energy equipartition in the late coda is inferred by the results of Eh/E ratio. However, a final proof of the energy equipartition would be achievable only by using a 3D underground array. That would allow for a complete separation of longitudinal and shear waves.

We believe that the main cause of discrepancy between observed and simulated results is due to the flat layer model that does not fit the area of Gran Sasso massif. In future simulations it will be necessary to include the local topography.

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Useful readings

Egle, JASA 1981; Weaver, JASA 1982; Weaver, JASA 1985; Aki, BSSA 1992; Papanicolau et al., BSSA 1996; Margerin et al., GJI 1998;Turner, BSSA 1998; Shapiro et al., 2000, BSSA 2000;Hennino et al., PRL 2001; Margerin et al., BSSA 2001;Wegler, GJI 2005; Sanchez-Sesma et al., WM 2008; Margerin et al., GJI 2009; Weaver, ES 2010; Yamamoto & Sato, JGR 2010; Nakahara & Margerin, BSSA 2011;Sanchez-Sesma et al., JGE 2011; Margerin, GJI 2013; La Rocca et al., GJI 2013.