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Elastic Least-Squares Imaging in Tilted Transversely Isotropic Media for Multicomponent Land and Pressure Marine Data

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Abstract

Traditional elastic reverse-time migration (RTM) involves P-/S-wave separation for the source and receiver wavefields, followed by applying the zero-lag cross-correlation imaging condition to produce PP and PS images. In anisotropic media, P-/S-wave decomposition requires a higher memory and computational cost than that in isotropic media. In addition, finite acquisition apertures and band-limited source functions result in unsatisfactory resolutions and amplitudes. To mitigate these problems, we present an elastic least-squares imaging method for tilted transversely isotropic media and apply it to land multicomponent and marine pressure data. Unlike traditional RTM, we use the relative perturbations to the product of density and squared axial (compressional/shear) velocities as reflectivity models (\(\Delta \ln{C}_{33}\) and \(\Delta \ln{C}_{55}\)), and estimate them by solving a linear inverse problem. Numerical experiments illustrate that subsurface reflectors can be well resolved in adjoint images for land multicomponent data, because of the presence of both P- and S-waves in seismograms. Least-squares migration helps to further improve spatial resolution and image amplitudes. Since there are no direct S-waves in marine streamer data, adjoint RTM images of \(\Delta \ln{C}_{55}\) are mainly resolved with the converted S-waves and are not as good as those in \(\Delta \ln{C}_{33}\) images. By approximating the Hessian inverse, least-squares migration allows us to take advantage of the weak converted P–S–P-waves and improve the \(\Delta \ln{C}_{55}\) image quality. Numerical experiments for synthetic and field data demonstrate the feasibility and advantage of the proposed least-squares TTI RTM compared with wave-mode separation-based elastic RTM. In field data experiments, we observe that since there are no strong P–S–P converted waves in streamer pressure records from the marine survey, the reflectors in \(\Delta \ln{C}_{55}\) image might be mainly imaged from P-waves due to the amplitude versus offset (AVO) effects.

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Acknowledgements

This research is supported by TOTAL E&P USA Inc and the Ph.D. program of Department of Geoscience at the University of Texas at Dallas. We appreciate the comments and suggestions from editor Jeffrey Gu and reviewers Alexey Stovas and Igor Ravve. The co-author GM was supported by the sponsors of the UT-Dallas Geophysical Consortium. We are grateful to the Texas Advanced Computing Center (TACC) for providing computational resources. This paper is Contribution No. 1359 of the Department of Geosciences at the University of Texas at Dallas.

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Correspondence to Jidong Yang.

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Appendices

Appendix 1: Detailed Derivations of the Adjoint Wave Equation and Sensitivity Kernels

Taking the variation of the augmented misfit function in Eq. 15 and neglecting the high-order terms, we obtain

$$\begin{aligned} \Delta \chi ({\mathbf {v}},\varvec{\upsigma },{\mathbf {w}},\varvec{\uptau })&= \int _0^{t_{\rm max}}\int _{\varOmega }[d_{cal}({\mathbf {x}},t)-d_{\rm obs}({\mathbf {x}},t)]\Delta {d}_{cal}({\mathbf {x}})\delta ({\mathbf {x}}-{\mathbf {x}}_r) {\rm d}{\mathbf {x}}^3{\rm d}t \nonumber \\&\quad + \int _0^{t_{\rm max}}\int _{\varOmega }\Big \{{\mathbf {w}}\cdot \left( \rho _0\partial _{t}\Delta {\mathbf {v}} -{\mathbf {P}}\Delta \varvec{\upsigma }\right) \nonumber \\&\quad + \varvec{\uptau }\cdot \left( \partial _{t}\Delta \varvec{\upsigma } -\Delta {\mathbf {D}}{\mathbf {P}}^{\rm T}{\mathbf {v}} -{\mathbf {D}}{\mathbf {P}}^{\rm T}\Delta {\mathbf {v}} \right) \Big \}{\rm d}{\mathbf {x}}^3{\rm d}t, \end{aligned}$$
(26)

where \(\delta ({\mathbf {x}})\) is the Kronecker delta function and \(\Delta {\mathbf {v}}\) and \(\Delta \varvec{\upsigma }\) are the perturbed particle velocity and stress wavefields. \(\Delta {\mathbf {D}}\) is the perturbed stiffness matrix and can be expressed as

$$\begin{aligned} \Delta {D}={\mathbf {M}}\frac{\partial {\mathbf {C}}}{\partial {C}_{33}}{\mathbf {M}}^{\rm T}{C}^0_{33}\Delta \ln {C}_{33} +{\mathbf {M}}\frac{\partial {\mathbf {C}}}{\partial {C}_{55}}{\mathbf {M}}^{\rm T}{C}^0_{55}\Delta \ln {C}_{55}. \end{aligned}$$
(27)

Using integration by parts, we have the following equalities

$$\begin{aligned} \int _0^{t_{\rm max}}{\mathbf {w}}\cdot \partial _{t}\Delta {\mathbf {v}}{\rm d}t= & {} \left[ {\mathbf {w}}\cdot {\mathbf {v}}\right] |_0^{t_{\rm max}}-\int _0^{t_{\rm max}}\Delta {\mathbf {v}}\cdot\partial _{t}{\mathbf {w}}{\rm d}t, \quad \nonumber \\ \int _0^{t_{\rm max}}\varvec{\uptau }\cdot \partial _{t}\Delta \varvec{\upsigma }{\rm d}t= & {} \left[ \varvec{\uptau }\cdot \varvec{\upsigma }\right] |_0^{t_{\rm max}}-\int _0^{t_{\rm max}}\Delta \varvec{\upsigma }\cdot\partial _{t}\varvec{\uptau }{\rm d}t. \end{aligned}$$
(28)

Because of the initial and end conditions for \({\mathbf {v}}\) and \(\varvec{\upsigma }\)

$$\begin{aligned} {\mathbf {v}}({\mathbf {x}},0)&= {} 0, \partial _{t}{\mathbf {v}}({\mathbf {x}},0)=0, \varvec{\upsigma }({\mathbf {x}},0)=0, \partial _{t}\varvec{\upsigma }({\mathbf {x}},0)=0, \nonumber \\ {\mathbf {w}}({\mathbf {x}},{t_{\rm max}})&= {} 0, \partial _{t}{\mathbf {w}}({\mathbf {x}},{t_{\rm max}})=0, \varvec{\uptau }({\mathbf {x}},{t_{\rm max}})=0, \partial _{t}\varvec{\uptau }({\mathbf {x}},{t_{\rm max}})=0, \end{aligned}$$
(29)

Eq. 28 is reduced to

$$\begin{aligned} \int _0^{t_{\rm max}}{\mathbf {w}}\cdot \partial _{t}\Delta {\mathbf {v}}{\rm d}t=-\int _0^{t_{\rm max}}\Delta {\mathbf {v}}\cdot\partial _{t}{\mathbf {w}}{\rm d}t, \quad \int _0^{t_{\rm max}}\varvec{\uptau }\cdot \partial _{t}\Delta \varvec{\upsigma }{\rm d}t=-\int _0^{t_{\rm max}}\Delta \varvec{\upsigma }\cdot\partial _{t}\varvec{\uptau }{\rm d}t. \end{aligned}$$
(30)

Similarly, the terms involving the spatial derivatives can be simplified to

$$\begin{aligned}&\int _{\varOmega }{\mathbf {w}}\cdot \left( {\mathbf {P}}\Delta \varvec{\upsigma }\right) {\rm d}{\mathbf {x}}^3=-\int _{\varOmega }\Delta \varvec{\upsigma }\cdot \left( {\mathbf {P}}^{\rm T}{\mathbf {w}}\right) {\rm d}{\mathbf {x}}^3, \nonumber \\&\int _{\varOmega }\varvec{\uptau }\cdot \left( {\mathbf {D}}{\mathbf {P}}^{\rm T}\Delta {\mathbf {v}}\right) {\rm d}{\mathbf {x}}^3=-\int _{\varOmega }\Delta {\mathbf {v}}\cdot \left( {\mathbf {P}}{\mathbf {D}}^{\rm T}\varvec{\uptau }\right) {\rm d}{\mathbf {x}}^3. \end{aligned}$$
(31)

Substituting Eqs. 27, 30 and 31 into Eq. 26 yields

$$\begin{aligned} \Delta \chi ({\mathbf {v}},\varvec{\upsigma },{\mathbf {w}},\varvec{\uptau })&= \int _0^{t_{\rm max}}\int _{\varOmega }[d_{cal}({\mathbf {x}},t)-d_{\rm obs}({\mathbf {x}},t)]\Delta {d}_{cal}({\mathbf {x}})\delta ({\mathbf {x}}-{\mathbf {x}}_r) {\rm d}{\mathbf {x}}^3{\rm d}t \nonumber \\&\quad +\, \int _0^{t_{\rm max}}\int _{\varOmega }\Big \{ -\Delta {\mathbf {v}}\cdot \left[ \rho _0\partial _{t}{\mathbf {w}} - {\mathbf {P}}{\mathbf {D}}^{\rm T}\varvec{\uptau }\right] \nonumber \\&\quad -\Delta \varvec{\upsigma }\cdot \left[ \partial _{t}\varvec{\uptau }-{\mathbf {P}}^{\rm T}{\mathbf {w}} \right] \nonumber \\&\quad -\Delta \ln {C}_{33}\left[ {C}^0_{33}{\mathbf {M}}\frac{\partial {\mathbf {C}}}{\partial {C}_{33}}{\mathbf {M}}^{\rm T}{\mathbf {P}}^{\rm T}{\mathbf {v}}\cdot \varvec{\uptau }\right] \nonumber \\&\quad -\Delta \ln {C}_{55}\left[ {C}^0_{55}{\mathbf {M}}\frac{\partial {\mathbf {C}}}{\partial {C}_{55}}{\mathbf {M}}^{\rm T}{\mathbf {P}}^{\rm T}{\mathbf {v}}\cdot \varvec{\uptau }\right] \Big \}{\rm d}{\mathbf {x}}^3{\rm d}t. \end{aligned}$$
(32)

Provided the Lagrange multiplier wavefields satisfy

$$\begin{aligned} \rho _0\partial _{t}{\mathbf {w}} - {\mathbf {P}}{\mathbf {D}}^{\rm T}\varvec{\uptau }=\Delta {\mathbf {d}}_{mul}({\mathbf {x}}_r,t), \quad \partial _{t}\varvec{\uptau }-{\mathbf {P}}^{\rm T}{\mathbf {w}}=\Delta {\mathbf {d}}_{p}({\mathbf {x}}_r,t), \end{aligned}$$
(33)

the variation of the misfit function reduces to

$$\begin{aligned} \Delta \chi = \int _{\varOmega } \left[ K_{33}\Delta \ln {C}_{33} + K_{55}\Delta \ln {C}_{55} \right] {\rm d}{\mathbf {x}}^3 , \end{aligned}$$
(34)

where \(\Delta {\mathbf {d}}\) is data residual and subscripts mul and p denote multicomponent and pressure data, respectively. \(K_{33}\) and \(K_{55}\) are the sensitivity kernels

$$\begin{aligned} K_{33}({\mathbf {x}})= & {-} \int _0^{t_{\rm max}}\left[ C^0_{33}({\mathbf {x}}){\mathbf {M}}({\mathbf {x}})\frac{\partial {\mathbf {C}}({\mathbf {x}})}{\partial {C}_{33}}{\mathbf {M}}^{\rm T}({\mathbf {x}}){\mathbf {P}}^{\rm T}{\mathbf {v}}({\mathbf {x}},t) \cdot \varvec{\tau }({\mathbf {x}},t)\right] {\rm d}t, \nonumber \\ K_{55}({\mathbf {x}})= & {-} \int _0^{t_{\rm max}}\left[ C^0_{55}({\mathbf {x}}){\mathbf {M}}({\mathbf {x}})\frac{\partial {\mathbf {C}}({\mathbf {x}})}{\partial {C}_{55}}{\mathbf {M}}^{\rm T}({\mathbf {x}}){\mathbf {P}}^{\rm T}{\mathbf {v}}({\mathbf {x}},t) \cdot \varvec{\tau }({\mathbf {x}},t)\right] {\rm d}t. \end{aligned}$$
(35)

Equation 34 illustrates the relation between the changes in the data misfit and the model parameter perturbations, in terms of the forward wavefields \({\mathbf {v}}({\mathbf {x}},t)\) and \(\varvec{\sigma }({\mathbf {x}},t)\) as well as the multiplier wavefields \({\mathbf {w}}({\mathbf {x}},t)\) and \(\varvec{\tau }({\mathbf {x}},t)\).

Appendix 2: Detailed Expressions for the 2D Born Modeling and Adjoint Migration Operators

In the 2D Born modeling, the background wavefields \(v_i^0\) and \(\sigma _{ij}^0\) (\(i,j=x,z\)) satisfy

$$\begin{aligned}&\rho \partial _{t} v^0_{x} - \left( \partial _{x}\sigma ^0_{xx} +\partial _{z} \sigma ^0_{xz}\right) = 0, \nonumber \\&\rho \partial _{t} v^0_{z} - \left( \partial _{x}\sigma ^0_{xz} +\partial _{z} \sigma ^0_{zz}\right) = 0, \nonumber \\&\partial _{t} \sigma ^0_{xx} - D^0_{11}\partial _{x}v^0_x -D^0_{13}\partial _{z}v^0_z - D^0_{15}\left( \partial _{x}v^0_{z} +\partial _{z} v^0_{x}\right) = f_{xx}, \nonumber \\&\partial _{t} \sigma ^0_{zz} - D^0_{13}\partial _{x}v^0_x -D^0_{33}\partial _{z}v^0_z - D^0_{35}\left( \partial _{x}v^0_{z} +\partial _{z} v^0_{x}\right) = f_{zz}, \nonumber \\&\partial _{t} \sigma ^0_{xz} - D^0_{15}\partial _{x}v^0_x -D^0_{35}\partial _{z}v^0_z - D^0_{55}\left( \partial _{x}v^0_{z} +\partial _{z} v^0_{x}\right) = f_{xz}, \end{aligned}$$
(36)

and the perturbed wavefields \(\Delta {v}_i\) and \(\Delta {\sigma }_{ij}\) satisfy

$$\begin{aligned}&\rho \partial _{t} \Delta {v}_{x} - \left( \partial _{x}\Delta \sigma _{xx} +\partial _{z} \Delta \sigma _{xz}\right) = 0, \nonumber \\&\rho \partial _{t} \Delta {v}_{z} - \left( \partial _{x}\Delta \sigma _{xz} +\partial _{z} \Delta \sigma _{zz}\right) = 0, \nonumber \\&\partial _{t} \Delta \sigma _{xx} - D^0_{11}\partial _{x}\Delta {v}_x -D^0_{13}\partial _{z}\Delta {v}_z - D^0_{15}\left( \partial _{x}\Delta {v}_{z} +\partial _{z} \Delta {v}_{x}\right) = f^{vir}_{xx}, \nonumber \\&\partial _{t} \Delta \sigma _{zz} - D^0_{13}\partial _{x}\Delta {v}_x -D^0_{33}\partial _{z}\Delta {v}_z - D^0_{35}\left( \partial _{x}\Delta {v}_{z} +\partial _{z} \Delta {v}_{x}\right) = f^{vir}_{zz}, \nonumber \\&\partial _{t} \Delta \sigma _{xz} - D^0_{15}\partial _{x}\Delta {v}_x -D^0_{35}\partial _{z}\Delta {v}_z - D^0_{55}\left( \partial _{x}\Delta {v}_{z} +\partial _{z} \Delta {v}_{x}\right) = f^{vir}_{xz}. \end{aligned}$$
(37)

\(f^{vir}_{ij}\) (\(i,j=x,z\)) is the virtual source and can be written as

$$\begin{aligned} f^{vir}_{xx}&= \left[ p_{11}\partial _{x}v^0_{x} + p_{13}\partial _{z}v^0_{z} + p_{15}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{33}\Delta \ln {C}_{33} \nonumber \\&\quad + \left[ q_{11}\partial _{x}v^0_{x} + q_{13}\partial _{z}v^0_{z} + q_{15}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{55}\Delta \ln {C}_{55}, \nonumber \\ f^{vir}_{zz}&= \left[ p_{13}\partial _{x}v^0_{x} + p_{33}\partial _{z}v^0_{z} + p_{35}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{33}\Delta \ln {C}_{33} \nonumber \\&\quad + \left[ q_{13}\partial _{x}v^0_{x} + q_{33}\partial _{z}v^0_{z} + q_{35}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{55}\Delta \ln {C}_{55}, \nonumber \\ f^{vir}_{xz}&= \left[ p_{15}\partial _{x}v^0_{x} + p_{35}\partial _{z}v^0_{z} + p_{55}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{33}\Delta \ln {C}_{33} \nonumber \\&\quad + \left[ q_{15}\partial _{x}v^0_{x} + q_{35}\partial _{z}v^0_{z} + q_{55}\left( \partial _{x}v^0_{z}+\partial _{z}v^0_{x}\right) \right] C^0_{55}\Delta \ln {C}_{55}, \end{aligned}$$
(38)

with

$$\begin{aligned} p_{11}= & {} 1+2\epsilon \cos ^2\theta -2(1+\epsilon -r_p)\sin ^2\theta \cos ^2\theta , q_{11} = 2(r+2)\sin ^2\theta \cos ^2\theta ,\nonumber \\ p_{13}= & {} r_p+2(1+\epsilon -r_p)\sin ^2\theta \cos ^2\theta , q_{13} = r_s-2(r_s+2)\sin ^2\theta \cos ^2\theta ,\nonumber \\ p_{33}= & {} 1+2\epsilon \sin ^2\theta -2(1+\epsilon -r_p)\sin ^2\theta \cos ^2\theta , q_{33} = 2(r_s+2)\sin ^2\theta \cos ^2\theta ,\nonumber \\ p_{15}= & {} \frac{1}{2}\sin 2\theta \left[ \epsilon +(1+\epsilon -r_p)\cos 2\theta \right] , q_{15} = -\frac{1}{2}(2+r_s)\sin 2\theta \cos 2\theta ,\nonumber \\ p_{35}= & {} \frac{1}{2}\sin 2\theta \left[ \epsilon -(1+\epsilon -r_p)\cos 2\theta \right] , q_{35} = \frac{1}{2}(2+r_s)\sin 2\theta \cos 2\theta ,\nonumber \\ p_{55}= & {} 2(1+\epsilon -r_p)\sin ^2\theta \cos ^2\theta , q_{55} = 1-2(2+r_s)\sin ^2\theta \cos ^2\theta , \end{aligned}$$
(39)

where \(r_p=({f+\delta (1+f)})/{\sqrt{f(f+2\delta )}}\), \(r_s=-\left[ 1+{(f+\delta)}/{\sqrt{f(f+2\delta )}}\right]\) and \(f=1-{v^2_s}/{v^2_p}\). For 2D problems, the adjoint wave equation can be written as

$$\begin{aligned}&\partial _{t} \varepsilon ^{\dagger }_{xx}({\mathbf {x}},t) - \partial _{x}v^{\dagger }_{x}({\mathbf {x}},t) = \Delta {d}_p({\mathbf {x}}_r,T-t), \nonumber \\&\partial _{t} \varepsilon ^{\dagger }_{zz}({\mathbf {x}},t) - \partial _{z}v^{\dagger }_{z}({\mathbf {x}},t) = \Delta {d}_p({\mathbf {x}}_r,T-t), \nonumber \\&\partial _{t} \varepsilon ^{\dagger }_{xz}({\mathbf {x}},t) - \left( \partial _{z}v^{\dagger }_{x}({\mathbf {x}},t)+\partial _{x}v^{\dagger }_z({\mathbf {x}},t)\right) = 0, \nonumber \\&\rho _0\partial _{t}v^{\dagger }_{x}({\mathbf {x}},t) - \partial _{x}\left( D_{11}\varepsilon _{xx}({\mathbf {x}},t)+D_{13}\varepsilon _{zz}({\mathbf {x}},t)+D_{15}\varepsilon _{xz}({\mathbf {x}},t)\right) \nonumber \\&\quad -\partial _{z}\left( D_{15}\varepsilon _{xx}({\mathbf {x}},t)+D_{35}\varepsilon _{zz}({\mathbf {x}},t)+D_{55}\varepsilon _{xz}({\mathbf {x}},t)\right) =\Delta {d}_x({\mathbf {x}}_r,T-t), \nonumber \\&\rho _0\partial _{t}v^{\dagger }_{z}({\mathbf {x}},t) - \partial _{x}\left( D_{15}\varepsilon _{xx}({\mathbf {x}},t)+D_{35}\varepsilon _{zz}({\mathbf {x}},t)+D_{55}\varepsilon _{xz}({\mathbf {x}},t)\right) \nonumber \\&\quad -\partial _{z}\left( D_{13}\varepsilon _{xx}({\mathbf {x}},t)+D_{33}\varepsilon _{zz}({\mathbf {x}},t)+D_{35}\varepsilon _{xz}({\mathbf {x}},t)\right) =\Delta {d}_z({\mathbf {x}}_r,T-t). \end{aligned}$$
(40)

\(\Delta {d}_{p}\) is the data residuals for pressure records, and \(\Delta {d}_x\) and \(\Delta {d}_{z}\) are data residuals for multicomponent data. The corresponding sensitivity kernels have the following forms

$$\begin{aligned} K_{33}({\mathbf {x}})&=\int _0^{t_{\rm max}}\Big \{\left[ p_{11}\partial _{x}v_x({\mathbf {x}},t)+p_{13}\partial _{z}v_{z}({\mathbf {x}},t)+p_{15}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{xx}({\mathbf {x}},t) \nonumber \\&\quad \left[ p_{13}\partial _{x}v_x({\mathbf {x}},t)+p_{33}\partial _{z}v_{z}({\mathbf {x}},t)+p_{35}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{zz}({\mathbf {x}},t) \nonumber \\&\quad \left[ p_{15}\partial _{x}v_x({\mathbf {x}},t)+p_{35}\partial _{z}v_{z}({\mathbf {x}},t)+p_{55}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{xz}({\mathbf {x}},t) \Big \}{\rm d}t, \nonumber \\ K_{55}({\mathbf {x}})&=\int _0^{t_{\rm max}}\Big \{\left[ q_{11}\partial _{x}v_x({\mathbf {x}},t)+q_{13}\partial _{z}v_{z}({\mathbf {x}},t)+q_{15}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{xx}({\mathbf {x}},t) \nonumber \\&\quad \left[ q_{13}\partial _{x}v_x({\mathbf {x}},t)+q_{33}\partial _{z}v_{z}({\mathbf {x}},t)+q_{35}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{zz}({\mathbf {x}},t) \nonumber \\&\quad \left[ q_{15}\partial _{x}v_x({\mathbf {x}},t)+q_{35}\partial _{z}v_{z}({\mathbf {x}},t)+q_{55}\left( \partial _{z}v_{x}({\mathbf {x}},t)+\partial _{x}v_{z}({\mathbf {x}},t)\right) \right] \varepsilon ^{\dagger }_{xz}({\mathbf {x}},t) \Big \}{\rm d}t, \nonumber \\ \end{aligned}$$
(41)

where \(p_{ij}\) and \(q_{ij}\) (\(i,j=1,3,5\)) have the same definitions as in Eq. 39.

Appendix 3: An Elastic TTI LSM Experiment for a 3D Four-Layer Model

We present a small 3D numerical experiment in Fig. 19 to illustrate the proposed 3D elastic TTI LSM method. The model parameters are shown in Fig. 19a and Table 1. The first layer is water, and the other three layers are TTI solids. The model is discretized using a grid of \(251\times 101\times 201\), with 20 m horizontal spacing and 10 m vertical spacing. There are thirteen shots in the inline direction and five shots in the crossline direction. Each shot is recorded by an array of \(251\times 101\) hydrophones for pressure data (Fig. 19b). LSM results for the first and thirteenth iterations are shown in Fig. 19c–f. Migration models are built by smoothing only the solid interfaces. Similar to the 2D examples, the \(\Delta \ln {C}_{33}\) and \(\Delta \ln {C}_{55}\) images in the first iteration have limited illumination and resolution (Fig. 19c, d). By iteratively approximating the Hessian inverse, LSM enlarges the subsurface illumination range and improves the spatial resolution (Fig. 19e, f).

Fig. 19
figure 19

An elastic TTI LSM experiment for the pressure data from a 3D four-layer model (a). Model parameters are given in Table 1. b One shot gather with the source located at inline = 2.5 km and crossline = 1.0 km. c, d\(\Delta \ln {C}_{33}\) and \(\Delta \ln {C}_{55}\) images at the first iteration. e, f Corresponding images at the thirteenth iteration

Table 1 Model parameters for the 3D four-layer model shown in Fig. 19

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Yang, J., Hua, B., Williamson, P. et al. Elastic Least-Squares Imaging in Tilted Transversely Isotropic Media for Multicomponent Land and Pressure Marine Data. Surv Geophys 41, 805–833 (2020). https://doi.org/10.1007/s10712-020-09588-3

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