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Energy-norm imaging condition

The proposed method is well-suited for applications to elastic imaging and inversion using multi-component seismic data. In this section, we derive the energy-norm imaging condition using analytical wavefield, and show its connection to wave-mode decomposition.

The energy norm with respect to a SPD matrix $ \mathbf{M}$ can be defined as:

$\displaystyle e^2(\mathbf{u}) = \vert\vert\mathbf{u}\vert\vert _{\mathbf{M}} = \mathbf{u}^\intercal \mathbf{M}\mathbf{u}\;.$ (34)

In the acoustic case, the imaging condition based on the energy-norm is also referred to as the impedance sensitivity kernal (Zhu et al., 2009) or the inverse scattering imaging condition (Whitmore and Crawley, 2012):

$\displaystyle I_a = \rho \mathbf{u}_t^\intercal \mathbf{w}_t + (v \nabla \mathbf{u})^\intercal (v \nabla \mathbf{w}) \;.$ (35)

For general elastic media, the energy norm imaging condition can be expressed as (Kiyashchenko et al., 2007; Rocha et al., 2016)

$\displaystyle I_e = \rho \mathbf{u}_t^\intercal \mathbf{w}_t + \mathbf{u}^\intercal \mathbf{D}\mathbf{C}\mathbf{D}^\intercal \mathbf{w}\;.$ (36)

Using the analytic wavefield, equation 36 becomes

$\displaystyle I_e = \rho ( \mathbf{u}_t^\intercal \mathbf{w}_t + \mathbf{u}^\in...
...hscr{R}\left[ \rho (\Phi \hat{\mathbf{u}})^*(\Phi \hat{\mathbf{w}}) \right] \;,$ (37)

where $ \mathscr{R}$ represents the operator of taking the real part. In a locally homogeneous medium, equation 37 can be expressed as

$\displaystyle I_e = \mathscr{R}\left[ \mathscr{F}^{-1} \left[ \rho \hat{\mathbf{u}}^* \mathbf{A}\hat{\mathbf{w}} \right] \right] \;,$ (38)

where $ \mathscr{F}^{-1}$ represents inverse spatial Fourier transform. $ \mathbf{A}$ is SPD and has the eigenvalue decomposition as equation 13, and the part in the wavenumber domain of equation 38 can be further expanded as
$\displaystyle \rho \hat{\mathbf{u}}^* \mathbf{A}\hat{\mathbf{w}} =$   $\displaystyle \rho \left( \mathbf{Q}^\intercal \hat{\mathbf{u}} \right)^* \mathbf{V} \mathbf{Q}^\intercal \hat{\mathbf{w}}$  
$\displaystyle =$   $\displaystyle \begin{bmatrix}\hat{\mathbf{u}}_p^* & \hat{\mathbf{u}}_{s1}^* & \...
...t{\mathbf{w}}_p \\ \hat{\mathbf{w}}_{s1} \\ \hat{\mathbf{w}}_{s2} \end{bmatrix}$  
$\displaystyle =$   $\displaystyle \sum\limits_{i=p,s1,s2} v_i^2 k^2 \hat{\mathbf{u}}_i^* \hat{\mathbf{w}}_i \;.$ (39)

Equation 39 corresponds to the summation of the image produced by three pure wave-mode reflections (P-P, S1-S1 and S2-S2 images) using the formula prescribed in equations 35. Therefore, the proposed framework provides the possibility of individually accessing the contribution of each wave-mode in the elastic energy-norm imaging condition, in addition to outputing their summation.

The imaging condition prescribed in equation 38 is the cross-correlation of two positive-frequency analytical wavefields. This corresponds to the back-scattering part of the wavefield. If one instead performs cross-correlation between two wavefield with opposite signs of frequency, e.g. $ \mathbf{u}^*$ only contains negative frequency and $ \mathbf{w}$ only contains positive frequency, it will lead to a different forward-scattering imaging condition

$\displaystyle I_t = \mathscr{R}\left[ \mathscr{F}^{-1} \left[ \rho \hat{\mathbf...{u}_t^\intercal \mathbf{w}_t + \mathbf{u}^\intercal \mathbf{A}\mathbf{w}) \;.$ (40)

The forward-scattering imaging condition corresponds to the tomographic correlation between two wavefields. It is conventionally treated as low-frequency noise in RTM, but is what FWI needs for performing low-frequency updates in the velocity gradient Ramos-Martinez et al. (2016); Díaz and Sava (2012,2013). Equation 40 can also be expanded in a similar fashion as equation 39 to access individual contributions from each wave mode.

It is important to note that the inverse- and forward-scattering imaging conditions using scalar and vector analytical wavefields discussed in this section are not restricted to one-step wave extrapolation. There are different ways of obtaining an analytical wavefield using conventional finite-different or pseudo-spectral wave extrapolation, for example, by separately propagating a wavefield using a Hilbert-transformed source wavelet (Hu et al., 2016; Shen and Albertin, 2015) and use it as the imaginary part of the analytical wavefield.

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Next: Numerical examples Up: Theory Previous: Low-rank approximation