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Data Sensitivity Kernels

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Full-3D Seismic Waveform Inversion

Part of the book series: Springer Geophysics ((SPRINGERGEOPHYS))

Abstract

The discrepancies between synthetic and observed waveforms can be quantified in various ways. The waveform difference (i.e., subtracting the synthetic waveform from the observed waveform and minimizing the energy of the difference) is one approach that has been widely adopted in the active-source full-wave inversion community. In passive-source large-scale (e.g., global- and continental-scale) tomography using transmitted waves, discrepancies are often quantified using phase-delay time and/or group-delay time. The theoretical formulation derived in Chap. 3 therefore needs to be extended to arbitrary misfit measures, which is formalized using data functionals in Sect. 4.1 of this chapter. In Sect. 4.2, I will introduce the concept of the wavefield perturbation kernel (WPK) and give a general approach for deriving Fréchet kernels of arbitrary data functionals. For data functionals defined in terms of the complex phase of the waveform, I will draw connections with the Rytov transform and analyze the validity of the Rytov approximation. In Sect. 4.3, I generalize the formulation to different parameterization of the source and structural models. In particular, I will derive Fréchet kernels for anisotropy and anelastic attenuation. In Sect. 4.4, I will give tutorials about how to compute the Fréchet kernels using the forward wavefield from the source and the RGT.

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Notes

  1. 1.

    Here I have dropped the \(\tilde {}\) on top of the Green’s tensor and the forward wavefield used in Eqs. 3.413 and 3.414 to reduce clutter.

  2. 2.

    An increase in \(\Delta \tau _{q}\) leads to a decrease in waveform amplitude and \(\Delta \tau _{q}\) has the unit of time, therefore it is named the amplitude-reduction time.

  3. 3.

    This identity can be verified by expressing \(\nabla ^{2}\left [a(\mathbf {x})b(\mathbf {x})\right ]=\nabla \cdot \left [\nabla \left (ab\right )\right ]=\nabla \cdot \left (a\nabla b+b\nabla a\right )=\nabla \cdot \left (a\nabla b\right )+\nabla \cdot \left (b\nabla a\right )=a\nabla ^{2}b+\nabla a\cdot \nabla b+\nabla b\cdot \nabla a+b\nabla ^{2}a=a\nabla ^{2}b+b\nabla ^{2}a+2\nabla a\cdot \nabla b.\)

  4. 4.

    Note that the Green’s tensor \(\mathbf {G}\) used in Eq. 4.185 here is the RGT, i.e., the Green’s tensor from the receiver to the scatterer, while the one used in Eqs. (16)–(17) in (Zhao et al. 2005) is the Green’s tensor from the scatterer to the receiver. Therefore no transpose operation is applied on the RGT in Eq. 4.185. The kernel given in Eq. 4.185 is for the relative perturbation \(\delta \alpha /\alpha \), while the kernels in Eqs. (16)–(17) in (Zhao et al. 2005) are for the absolute perturbation \(\delta \alpha \). Therefore Eq. 4.185 has the factor \(\alpha ^{2}(\mathbf {x})\), while Eqs. (16)–(17) in (Zhao et al. 2005) have the factor \(\alpha (\mathbf {x})\).

  5. 5.

    Note that no transpose operation is needed on the RGT \(\mathbf {G}\) in Eq. 4.186 here, while the Green’s tensor used in Eqs. (18)–(19) in (Zhao et al. 2005), which is from the scatterer to the receiver, needs to be transposed. The kernel given in Eq. 4.186 here is for the relative perturbation \(\delta \beta /\beta \), while those in Eqs. (18)–(19) in (Zhao et al. 2005) are for the absolute perturbation \(\delta \beta \). Therefore Eq. 4.186 here has the factor \(\beta ^{2}(\mathbf {x})\), while Eqs. (18)–(19) in (Zhao et al. 2005) have the factor \(\beta (\mathbf {x})\). Equation 4.186 here and Eqs. (18)–(19) in (Zhao et al. 2005) differ by a minus sign.

  6. 6.

    Note that the expressions for \(J_{s},K_{s},M_{s},G_{s},B_{s},H_{s},D_{s},E_{s}\) given here differ from the corresponding equations in (Chen and Tromp 2007) by a minus sign, which is due to differences in the coordinate system. The coordinate system used here is the A&R coordinate, which is consistent with the one used in Sect. 2.5.3 of (Babuska and Cara 1991), therefore expressions for \(G_{s},B_{s},H_{s},E_{s}\) given here are identical to the corresponding equations given in Sect. 2.5.3 of (Babuska and Cara 1991).

  7. 7.

    The COO format stores a list of (row, column, value) tuples.

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  1. Department of Geology & Geophysics, University of Wyoming, Laramie, Wyoming, USA

    Po Chen

  2. Department of Earth Sciences, National Cheng Kung University, Tainan, Taiwan

    En-Jui Lee

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Chen, P., Lee, EJ. (2015). Data Sensitivity Kernels. In: Full-3D Seismic Waveform Inversion. Springer Geophysics. Springer, Cham. https://doi.org/10.1007/978-3-319-16604-9_4

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