Two perspectives on equipartition in diffuse elastic fields in three dimensions

Perton, M.; Sánchez‐Sesma, Francisco J.; Rodríguez-Castellanos, Alejandro; Campillo, Míchel; Weaver, Richard L.

doi:10.1121/1.3177262

Cited by 58 publications

(56 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, we find for all sites that the scaling coefficient for only the vertical component is 15-30 per cent lower compared to the horizontal scaling coefficient, probably due to the more heterogeneous vertical composition. From a theoretical point a view, this might indicate equal energies for the transverse and longitudinal waves with the independent modes of vibration in a full 3-D space (Perton et al 2009). …”

Section: Discussionmentioning

confidence: 99%

Statistical properties of the seismic noise field: influence of soil heterogeneities

Pilz¹,

Parolai²

2014

Geophysical Journal International

View full text Add to dashboard Cite

S U M M A R YSeismic noise is generally considered as a reproducible and temporarily stationary natural source of energy. We present a study on the statistical features of the soil motion due to the seismic noise wavefield and the dependencies on the near-surface geology. We have investigated the variations of the 3-D average squared soil displacement over different timescales. The results clearly indicate ballistic behaviour for short timescales being indicative for the properties of the shallow material. Differences in the structural heterogeneity of the subsoil produce different scattering properties, changing the character of motion from nearly ballistic to diffusive on frequency-dependent timescales for all materials. Although in a strict sense the seismic noise wavefield is not completely isotropic, an ultimate pre-condition for a diffusive wavefield, the deviations compared to a uniform distribution are rather small. This means that the emergence of the Green's function is effective for all network sites after a sufficient self-averaging process that is provided by the scattering and the random spatial-temporal noise source distribution.

show abstract

Section: Discussionmentioning

confidence: 99%

Statistical properties of the seismic noise field: influence of soil heterogeneities

Pilz¹,

Parolai²

2014

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…The cross-terms disappear when the noise sources are uncorrelated; hence, a single crosscorrelation of noise observations at x A and x B gives, analogous to equation 7, ͕G͑x B ,x A ,t͒ ‫ם‬ G͑x B ,x A ,‫מ‬t͖͒ ‫ء‬ S N ͑t͒, where S N ͑t͒ is the autocorrelation of the noise ͑see Figure 6f͒. Note that the symmetry of the responses in Figure 6e and f relies again on the isotropic illumination of the receivers, i.e., on the net power flux of the illuminating wavefield being ͑close to͒ zero ͑van Tiggelen, 2003; Malcolm et al, 2004;Sánchez-Sesma et al, 2006;Snieder et al, 2007;Perton et al, 2009;Weaver et al, 2009;Yao et al, 2009͒. Of course, what is demonstrated here for a 2D distribution of sources also holds for a 3D source distribution. In that case, all sources in Fresnel volumes rather than Fresnel zones contribute to the retrieval of the direct wave between x A and x B .…”

Section: Tutorial On Interferometry: Part 1 75a199mentioning

confidence: 99%

Tutorial on seismic interferometry: Part 1 — Basic principles and applications

et al. 2010

View full text Add to dashboard Cite

Seismic interferometry involves the crosscorrelation of responses at different receivers to obtain the Green's function between these receivers. For the simple situation of an impulsive plane wave propagating along the x-axis, the crosscorrelation of the responses at two receivers along the x-axis gives the Green's function of the direct wave between these receivers. When the source function of the plane wave is a transient ͑as in exploration seismology͒ or a noise signal ͑as in passive seismology͒, then the crosscorrelation gives the Green's function, convolved with the autocorrelation of the source function. Direct-wave interferometry also holds for 2D and 3D situations, assuming the receivers are surrounded by a uniform distribution of sources. In this case, the main contributions to the retrieved direct wave between the receivers come from sources in Fresnel zones around stationary points. The main application of direct-wave interferometry is the retrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surfacewave velocity distribution of the subsurface. Seismic interferometry is not restricted to retrieving direct waves between receivers. In a classic paper, Claerbout shows that the autocorrelation of the transmission response of a layered medium gives the plane-wave reflection response of that medium. This is essentially 1D reflected-wave interferometry. Similarly, the crosscorrelation of the transmission responses, observed at two receivers, of an arbitrary inhomogeneous medium gives the 3D reflection response of that medium. One of the main applications of reflected-wave interferometry is retrieving the seismic reflection response from ambient noise and imaging of the reflectors in the subsurface. A common aspect of direct-and reflected-wave interferometry is that virtual sources are created at positions where there are only receivers without requiring knowledge of the subsurface medium parameters or of the positions of the actual sources.

show abstract

“…Weaver (1985) studied diffuse fields at the free surface of half-space and found the contributions of P, SV, SH and Rayleigh waves to surface energy densities. Sánchez-Sesma et al (2008) and Perton et al (2009) also included the free surface to have a half-space in 2D and 3D, respectively, and studied the transition between deep space and the free surface. In fact, the energy density per unit area associated with Rayleigh waves is: Perton et al (2009) computed the energy densities against depth in an elastic half-space.…”

Section: Energy Densitiesmentioning

confidence: 99%

“…On the other hand, equipartition means that the available energy in the phase space goes with equal average amounts, to all the possible states. Extending the results by Weaver (1982, 1985) Sánchez-Sesma et al (2008 and Perton et al (2009) pointed out that in addition to modes the equipartition allows description with degrees of freedom. These two aspects of equipartition lead to the interpretation of microtremor H/V spectral ratios or MHVSR in terms of the imaginary part of Green's functions (Sánchez-Sesma et al 2011b) using diffuse field assumption (DFA).…”

Section: Introductionmentioning

confidence: 99%

Modeling and inversion of the microtremor H/V spectral ratio: physical basis behind the diffuse field approach

Sánchez‐Sesma

2017

Earth Planets Space

View full text Add to dashboard Cite

Microtremor H/V spectral ratio (MHVSR) has gained popularity to assess the dominant frequency of soil sites. It requires measurement of ground motion due to seismic ambient noise at a site and a relatively simple processing. Theory asserts that the ensemble average of the autocorrelation of motion components belonging to a diffuse field at a given receiver gives the directional energy densities (DEDs) which are proportional to the imaginary parts of the Green's function components when both source and receiver are the same point and the directions of force and response coincide. Therefore, the MHVSR can be modeled as the square root of 2 × ImG 11 /ImG 33 , where ImG 11 and ImG 33 are the imaginary parts of Green's functions at the load point for the horizontal (sub-index 1) and vertical (subindex 3) components, respectively. This connection has physical implications that emerge from the duality DED force and allows understanding the behavior of the MHVSR. For a given model, the imaginary parts of the Green's functions are integrals along a radial wavenumber. To deal with these integrals, we have used either the popular discrete wavenumber method or the Cauchy's residue theorem at the poles that account for surface waves normal modes giving the contributions due to Rayleigh and Love waves. For the retrieval of the velocity structure, one can minimize the weighted differences between observations and calculated values using the strategy of an inversion scheme. In this research, we used simulated annealing but other optimization techniques can be used as well. This last approach allows computing separately the contributions of different wave types. An example is presented for the mouth of Andarax River at Almería, Spain.

show abstract

Two perspectives on equipartition in diffuse elastic fields in three dimensions

Cited by 58 publications

References 38 publications

Statistical properties of the seismic noise field: influence of soil heterogeneities

Statistical properties of the seismic noise field: influence of soil heterogeneities

Tutorial on seismic interferometry: Part 1 — Basic principles and applications

Modeling and inversion of the microtremor H/V spectral ratio: physical basis behind the diffuse field approach

Contact Info

Product

Resources

About