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[SSS06-P07] Synthesis of dynamic strain envelopes in two-dimensional random heterogeneous media based on multiple forward scattering approximation
Keywords:DAS, Envelope, Strain, Small-scale Heterogeneity
The physical quantity of the output record of the DAS observation is the strain or strain rate. These are the spatial derivative of the displacement or velocity recorded by conventional seismometers. The spatial derivative enhances the sensitivity to the small-scale medium heterogeneity. Therefore, observed DAS waveforms may be much different from each other even at neighboring measurement points. In the present study, we quantitatively evaluate such effect of spatial derivation and present a theoretical method to calculate the envelope of strain waveform in a short-wavelength fluctuated media based on the Markov approximation which is a multiple forward scattering approximation.
We assume that random seismic velocity fluctuation can be characterized by a Gaussian type auto-correlation function. When the characteristic scale of the random fluctuation is larger than the wavelength, the forward scattering is dominant and the P to S or S to P wave conversions are negligible. We consider that a plane P wavelet vertically enters the random medium from the bottom along the z-axis. We expand the scalar potential by plane waves. When the forward scattering is dominant, the amplitude of the plane wave satisfies the parabolic type wave equation. To calculate the envelope, we introduce the two-frequency mutual coherence function (TFMCF). The TFMCF obeys the parabolic type equation. For the Gaussian type random medium, we derive the expression of the strain envelope by using the TFMCF. The envelope of xx-component strain is the inverse Fourier transform of the fourth derivative of the TFMCF with respect to the transverse coordinate. Similarly, that of the zz-component strain can be described by the inverse Fourier transform of the sum of the TFMCF and its fourth derivative. The xz-component strain envelope is the inverse Fourier transform of the second derivative of the TFMCF the same expression as the x-component of the velocity envelope. The time scale of the strain envelope is scaled by the characteristic time as well as the velocity envelope. The characteristic time is proportional to “ε2 z2 / a”, where ε is the RMS fractional fluctuation, a is the characteristic scale of the random heterogeneity and z is the propagation distance.
Comparison of the zz and xx components of the strain envelopes with the z and x components of the velocity envelopes indicates that the excitation of the xx-component relative to the zz component is smaller than that of the x component relative to the z component. The peak delay times of the strain envelopes of both components are about 4 times larger than that of the velocity envelopes.
We compare the obtained theoretical envelopes with the envelopes calculated by using the finite-difference simulation of the seismic wave propagation in two-dimensional random media. Theoretical envelopes well fit the finite-difference envelopes except for the coda, at the same level as the case of the velocity envelopes.
We assume that random seismic velocity fluctuation can be characterized by a Gaussian type auto-correlation function. When the characteristic scale of the random fluctuation is larger than the wavelength, the forward scattering is dominant and the P to S or S to P wave conversions are negligible. We consider that a plane P wavelet vertically enters the random medium from the bottom along the z-axis. We expand the scalar potential by plane waves. When the forward scattering is dominant, the amplitude of the plane wave satisfies the parabolic type wave equation. To calculate the envelope, we introduce the two-frequency mutual coherence function (TFMCF). The TFMCF obeys the parabolic type equation. For the Gaussian type random medium, we derive the expression of the strain envelope by using the TFMCF. The envelope of xx-component strain is the inverse Fourier transform of the fourth derivative of the TFMCF with respect to the transverse coordinate. Similarly, that of the zz-component strain can be described by the inverse Fourier transform of the sum of the TFMCF and its fourth derivative. The xz-component strain envelope is the inverse Fourier transform of the second derivative of the TFMCF the same expression as the x-component of the velocity envelope. The time scale of the strain envelope is scaled by the characteristic time as well as the velocity envelope. The characteristic time is proportional to “ε2 z2 / a”, where ε is the RMS fractional fluctuation, a is the characteristic scale of the random heterogeneity and z is the propagation distance.
Comparison of the zz and xx components of the strain envelopes with the z and x components of the velocity envelopes indicates that the excitation of the xx-component relative to the zz component is smaller than that of the x component relative to the z component. The peak delay times of the strain envelopes of both components are about 4 times larger than that of the velocity envelopes.
We compare the obtained theoretical envelopes with the envelopes calculated by using the finite-difference simulation of the seismic wave propagation in two-dimensional random media. Theoretical envelopes well fit the finite-difference envelopes except for the coda, at the same level as the case of the velocity envelopes.