Modeling of six-component seismogram envelopes using Radiative Transfer Theory Peter Gaebler1,2, Christoph Sens-Schönfelder1 and Michael Korn2 1
GFZ Potsdam, Section 2.4, Seismology, Potsdam
2
University Leipzig, Institute of Geophysics and Geology, Leipzig
Contact:
[email protected]
1 Abstract
4 MC-RTT vs FD simulations of six component envelopes
Measurements of rotational motions of the seismic wave-field have recently shown a high amount of rotational energy also in the coda of seismic events. Especially the rotational motion in the P-wave coda bears interesting information as it can only be excited by scattering of the wave-field at 3D heterogeneities. It indicates the conversion from P to S energy in the Earth’s subsurface and consequently the scattering of high-frequency seismic waves. A suitable method to describe the propagation and scattering of high-frequency waves is the Radiative Transfer Theory (RTT). RTT describes the spatial and temporal distribution of seismic energy emitted from a seismic source. It considers scattering and mode conversions between P, SV and SH polarizations. It also includes the angular dependent scattering pattern derived from the Born approximation. The RTT does not contain any phase information of the seismic waves. Therefore the energy of superimposing waves is considered additive and no interferences between the seismic signals are treated. This makes the RTT a very useful tool for the description of scattered waves. It has recently been shown that the RTT is a powerful method to compute seismogram envelopes for the three translational components of the seismic wave-field. In this study we extend the capabilities of the method to model six-component seismogram envelopes in an elastic medium with randomly distributed velocity and density perturbations. The approach implements Monte Carlo (MC) solutions to the radiative transfer equation. The three additional rotational components can provide independent information about the Earth’s structure and the seismic source. They can for example be used to further constrain scattering properties and thus help to discriminate between intrinsic and scattering attenuation. The results of the MC simulations are verified by comparison with 3D full wave-field finite difference simulations. Six-component seismogram envelopes from the two different approaches are compared. We obtain reasonable agreement not only for the translational components but also for the rotational energy. In conclusion, the RTT is a useful approach to model the six-component seismogram envelopes of highfrequency wave-fields from the initial onset of the direct P-wave to the later part of the S-wave coda in random elastic media.
The results of the Monte Carlo Simulations are verified by comparison with 3D finite difference simulations using a modified version of the finite difference modeling software FDMPI [Bohlen,2002]. The simulations are similiar to the ones decribed by Przybilla and Korn, 2008. We use a model size of 76×76×76km with a compressional isotropic source placed at the the center of the model. The dominant source frequency is 6Hz, mean P and S velocities are 6.0km/s and 3.46km/s. Random velocity fluctuations with an exponential autocorrelation function (eACF) are added to the model. The eACF is described by the fractional fluctuation and by the correlation distance a. We record translational and rotational components of the wave-field at receivers placed on spheres at distances of 10, 20 and 30km from the source (see figure 3a). Figure 3b and 3c show single traces of translational and rotational motions rotated in the LQT-system recorded at 30km distance in a exponential random medium with = 0.1 and a = 1. L
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Figure 1: Top three traces show ground velocities, bottom trace shows rotation rate around the vertical axis. P-wave arrival is marked in orange.
3 RTT - Radiative Transfer Theory RTT describes the the energy flux from a radiating surface into a unit solid angle around direction n, this quantity is called specific intensity I(n, r). It depends on position r, as well as on the direction of the energy flux n. Phenomenologically the transport problem can be regarded as a conservation of energy flux in a direction n and at position r. Figure 2 illustrates the conservation of energy flux in the radiative transfer equations.
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To compare the six-component envelopes from the FD and MC simulations following data processing steps were done:
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Figure 3: a) Receivers placed at 10, 20, 30km distance b) Radial and transversal translational motions recorded at 30km distance c) Rotations around the radial and the
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Figure 1 shows translational and vertical rotational ground motion induced by the March 2011 Tohoku earthquake recorded at the Fundamentalstation Wettzell, Germany. Rotational motions around a vertical axis can only be excited by horizontally polarized shear waves. The high amount of energy visible in the vertical rotation rate at the time of the P-wave arrival therefore clearly indicates a conversion from P to S energy. This can be achieved by scattering of the high-frequency seismic wave-field at local or regional small scale heterogeneities.
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Additionally to the three components of translational motion the seismic wave-field is described by six components of strain and three components of rotational motion [Aki and Richards, 2002]. Due to the steady increase in sensor sensitivity it is nowadays possible to measure rotational motions in the seismic wave field induced by earthquakes and even by ambient seismic noise with very high accuracy. There are several applications using the records of rotational motions, for example:
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2 Rotational Motions in Seismology
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Comparison between MC-RTT and FD 10 10 envelopes show a very good agreement from the initial P-wave onset until late parts of the S-wave coda. Modelling of 10 10 translational motions agrees with previous 4 14 4 14 0 2 6 8 10 12 16 18 0 2 6 8 10 12 16 18 tim e [s ] tim e [s ] results by Przybilla and Korn, 2008. Figure 4: Comparison between MC-RTT and FD simulations The results of this study clearly demonstrate that also the three rotational components of motion can be modeled using RTT. We therefore conclude, that the RTT approach can be used to analyze six-component-envelopes of the high-frequency seismic wave-field. 1
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5 Real data application Figure 5b shows stacked hilbert envelopes for 11 earthquakes at 10 stations (see figure 5a). The earthquake station distance ranges from 100 to 160km. Additionally to the three translational envelopes (black) we measure a vertical rotation envelope (red) using data from the ringlaser instrument located at the Fundamentalstation Wettzell. These four component records will be inverted for following medium parameters using MC-RTT simulations: • fractional fluctuation • correlation length a • quality factor for intrinsic S-wave attenuation QS
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Figure 2: Energy conservation in the radiative transfer equations. Illustrated is the energy flux from a radiating surface (RS) into a unit solid angle (USA) around a direction
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The Radiative Transfer Equations are solved using Monte Carlo techniques. In this study a so called Particle Counting Method is used. The specific intensity I(n, r) is modeled by a number density of particles N (n, r) moving into direction n and loctated a position r. Particles are emitted from a source following a fixed radiation pattern and are described by following parameters:
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During the course of the simulation run particles can undergo scattering processes at medium heterogeneities following the Born approximation. This processes include mode conversion and a change of propagation direction. To account for attenuation particles loose energy (I P , IθS , IφS ) as they are propagating through the medium. When no scattering events occur particles move through the medium according to ray theory. This includes the interaction with interfaces (reflection, transmission, mode conversions) or the propagation in depth-dependent velocity structures. Snapshots of the energy field are stored in an four-dimensional-array consisting of three spatial and one temporal dimension.
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Figure 5: (a) Triangles indicate location of Gräfenberg array stations plus the location of the Fundamentalstation Wettzell. Circles show locations of the August 2011 Vogtland swarm earthquakes greater magnitude 3. (b) Stacked envelopes from 11 earthquakes for the three translational components of the shown stations (black). Rotation rate around the vertical axis measured with the ringlaser in Wettzell is shown in red. The blue and orange line indicate the expected arrival of the P respectively the S-wave, assuming vP = 5.6km/s and vS = 3.4km/s.
6 References • Aki and Richards. 2002. Quantitative Seismology. University Science Books. • Bohlen. 2002. Parallel 3-D viscoelastic finite difference seismic modelling. Computers and Geosciences. • Przybilla and Korn. 2008. Monte Carlo simulation of radiative energy transfer in continuous elastic random media-three-component envelopes and numerical validation. Geophysical Journal International. • Sens-Schönfelder, Margerin and Campillo. 2009. Laterally heterogeneous scattering explains Lg blockage in the Pyrenees. Journal of Geophysical Research.
