First-Arrival Traveltime Tomography Based on the Adjoint State Method with Independence of Surface Normal Vectors

Preconditioned transmission + reflection joint traveltime tomography with adjoint‐state method for subsurface velocity model building

“…According to the surface normal vectors–independent adjoint‐state traveltime tomography method (Han et al, 2019), the ASJT objective function can be defined as $$$$\begin{eqnarray} J(\mathbf{v}) &=& {\omega}_{1}{J}_{T}(\mathbf{v})+{\omega}_{2}{J}_{R}(\mathbf{v})=\frac{{\omega}_{1}}{2}\underset{\mathrm{\Omega}}{\int}{\left({t}_{T}\left(\mathbf{x}\right)-{t}_{T}^{\textit{obs}}\left(\mathbf{x}\right)\right)}^{2}{\delta}_{\mathbf{x},\mathbf{g}}\nonumber\\ &&+\,\frac{{\omega}_{2}}{2}\underset{\mathrm{\Omega}}{\int}{\left({t}_{R}\left(\mathbf{x}\right)-{t}_{R}^{\textit{obs}}\left(\mathbf{x}\right)\right)}^{2}{\delta}_{\mathbf{x},\mathbf{g}}, \end{eqnarray}$$$$…”

“…The corresponding adjoint Equation () for the calculation of the adjoint‐state variables $${\lambda}_{T}$$ and $${\lambda}_{R}$$ is an extension from adjoint‐state transmission traveltime inversion (Han et al., 2019), which can be derived by the following perturbation theory (Nocedal & Wright, 2006): $$$$\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{c}\nabla \cdot \left({\lambda}_{T}\left(\mathbf{x}\right)\nabla {t}_{T}\right)=\left({t}_{T}\left(\mathbf{x}\right)-{t}_{T}^{\textit{obs}}\left(\mathbf{x}\right)\right){\delta}_{\mathbf{x},\mathbf{g}}\\ \def\eqcellsep{&}\begin{array}{c}{\nabla \cdot \left({\lambda}_{R}^{g}\left(\mathbf{x}\right)\nabla {t}_{R}\right)=\left({t}_{R}\left(\mathbf{x}\right)-{t}_{R}^{\textit{obs}}\left(\mathbf{x}\right)\right){\delta}_{\mathbf{x},\mathbf{g}}}\\ {\nabla \cdot \left({\lambda}_{R}^{s}\left(\mathbf{x}\right)\nabla {t}_{T}\right)={\lambda}_{R}^{g}\left(\mathbf{x}\right){\delta}_{\mathbf{x},{\mathbf{x}}_{\mathbf{r}}}}\end{array} \end{array} \right.. \end{equation}$$$$…”

“…According to the surface normal vectors-independent adjointstate traveltime tomography method (Han et al, 2019), the ASJT objective function can be defined as…”

“…The corresponding adjoint Equation ( 6) for the calculation of the adjoint-state variables 𝜆 𝑇 and 𝜆 𝑅 is an extension from adjoint-state transmission traveltime inversion (Han et al, 2019), which can be derived by the following perturbation theory (Nocedal & Wright, 2006):…”

“…It has relatively minimal memory requirements and is particularly suitable for parallel computing (Benaichouche et al., 2015; Noble et al., 2010). Due to these advantages, the AST method recently has gained much attention in velocity model building in both seismic exploration and the Earth internal structure study (Huang & Bellefleur, 2012; Han et al., 2019; Li et al., 2014; Li et al., 2017; Tong, 2021; Waheed et al., 2016a, 2016b).…”