Computing the Sensitivity Kernels for 2.5-D Seismic Waveform Inversion in Heterogeneous, Anisotropic Media

Abstract

2.5-D modeling and inversion techniques are much closer to reality than the simple and traditional 2-D seismic wave modeling and inversion. The sensitivity kernels required in full waveform seismic tomographic inversion are the Fréchet derivatives of the displacement vector with respect to the independent anisotropic model parameters of the subsurface. They give the sensitivity of the seismograms to changes in the model parameters. This paper applies two methods, called ‘the perturbation method’ and ‘the matrix method’, to derive the sensitivity kernels for 2.5-D seismic waveform inversion. We show that the two methods yield the same explicit expressions for the Fréchet derivatives using a constant-block model parameterization, and are available for both the line-source (2-D) and the point-source (2.5-D) cases. The method involves two Green’s function vectors and their gradients, as well as the derivatives of the elastic modulus tensor with respect to the independent model parameters. The two Green’s function vectors are the responses of the displacement vector to the two directed unit vectors located at the source and geophone positions, respectively; they can be generally obtained by numerical methods. The gradients of the Green’s function vectors may be approximated in the same manner as the differential computations in the forward modeling. The derivatives of the elastic modulus tensor with respect to the independent model parameters can be obtained analytically, dependent on the class of medium anisotropy. Explicit expressions are given for two special cases—isotropic and tilted transversely isotropic (TTI) media. Numerical examples are given for the latter case, which involves five independent elastic moduli (or Thomsen parameters) plus one angle defining the symmetry axis.

Appendices

Appendix A: Self-adjoint and Anti-self-adjoint Operators

Substituting Eq. 2 for Eq. 6, one has

$$ \begin{aligned} \int\limits_{\Upomega } {{\mathbf{v}} \cdot {\mathbf{D}}_{1} {\mathbf{u}}\,{\text{d}}\Upomega } & = \int\limits_{\Upomega } {\left. {v_{j} {\frac{\partial }{{\partial x_{i} }}}\left( {c_{ijkl} {\frac{{\partial u_{k} }}{{\partial x_{l} }}}} \right) + \left( {\rho \omega^{2} \delta_{jk} - c_{2jk2} k_{y}^{2} } \right)v_{j} u_{k} } \right]} \\ & = \int\limits_{\Upomega } {{\frac{\partial }{{\partial x_{i} }}}\left( {c_{ijkl} v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}}} \right)\,{\text{d}}\Upomega - \int\limits_{\Upomega } {\left[ {c_{ijkl} \frac{{\partial v_{j} }}{{\partial x_{i} }}\frac{{\partial u_{k} }}{{\partial x_{l} }} - \left( {\rho \omega^{2} \delta_{jk} - c_{2jk2} k_{y}^{2} } \right)v_{j} u_{k} } \right]} } \,{\text{d}}\Upomega \\ & = \oint\limits_{\Upgamma } {(c_{3jkl} \,{\text{d}}x - c_{1jkl} \,{\text{d}}z)v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}}\,{\text{d}}\Upgamma } - \int\limits_{\Upomega } {\left[ {c_{ijkl} \frac{{\partial v_{j} }}{{\partial x_{i} }}\frac{{\partial u_{k} }}{{\partial x_{l} }} - \left( {\rho \omega^{2} \delta_{jk} - c_{2jk2} k_{y}^{2} } \right)v_{j} u_{k} } \right]} \,{\text{d}}\Upomega \\ & = - \int\limits_{\Upomega } {\left[ {c_{ijkl} \frac{{\partial v_{j} }}{{\partial x_{i} }}\frac{{\partial u_{k} }}{{\partial x_{l} }} - \left( {\rho \omega^{2} \delta_{jk} - c_{2jk2} k_{y}^{2} } \right)v_{j} u_{k} } \right]} \,{\text{d}}\Upomega = \int\limits_{\Upomega } {{\mathbf{u}} \cdot {\mathbf{D}}_{1} {\mathbf{v}}\,{\text{d}}\Upomega } . \\ \end{aligned} $$

(42)

Here, Stoke’s theorem, the natural boundary condition v|_Γ→∞ = u|_Γ→∞ = 0 and the symmetric property of the elastic moduli c _2jk2 = c _2kj2 are applied to Eq. 42, where Γ stands for the boundary of the domain Ω. Equation 42 shows the operator D ₁(m) is self-adjoint.

Substituting Eq. 2 into Eq. 7, we obtain

$$ \begin{aligned} \int\limits_{\Upomega } {{\mathbf{v}} \cdot {\mathbf{D}}_{2} {\mathbf{u}}\,{\text{d}}\Upomega } & = ik_{y} \int\limits_{\Upomega } {\left. {v_{j} {\frac{\partial }{{\partial x_{i} }}}(c_{ijk2} u_{k} ) + c_{2jkl} v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}}} \right]\,{\text{d}}\Upomega } \\ & = ik_{y} \left[ {\int\limits_{\Upomega } {{\frac{\partial }{{\partial x_{i} }}}(c_{ijk2} v_{j} u_{k} )\,{\text{d}}\Upomega + \left( {\int\limits_{\Upomega } {c_{2jkl} v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}} - } c_{ijk2} u_{k} {\frac{{\partial v_{j} }}{{\partial x_{i} }}}} \right)} \,{\text{d}}\Upomega } \right] \\ & = ik_{y} \left[ {\oint\limits_{\Upgamma } {(c_{3jk2} \,{\text{d}}x - c_{1jk2} \,{\text{d}}z)v_{j} u_{k} } + \left( {\int\limits_{\Upomega } {c_{2jkl} v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}} - } c_{ijk2} u_{k} {\frac{{\partial v_{j} }}{{\partial x_{i} }}}} \right)\,{\text{d}}\Upomega } \right. \\ & = ik_{y} \left[ {\int\limits_{\Upomega } {\left( {c_{2jkl} v_{j} {\frac{{\partial u_{k} }}{{\partial x_{l} }}} - c_{ijk2} u_{k} {\frac{{\partial v_{j} }}{{\partial x_{i} }}}} \right)\,} {\text{d}}\Upomega } \right] = - \int\limits_{\Upomega } {{\mathbf{u}} \cdot {\mathbf{D}}_{2} {\mathbf{v}}\,{\text{d}}\Upomega } . \\ \end{aligned} $$

(43)

Here, we once again apply Stoke’s theorem, the natural boundary condition and the symmetric property of c _2kji  = c _ijk2 to Eq. 43, which shows the operator D ₂(m) to be an anti-self-adjoint one.

Appendix B: Derivatives of the Elastic Modulus Tensor

Applying Eq. 33, we obtain the derivatives of the thirteen non-zero elastic moduli (c ₁₁, c ₁₂, c ₁₃, c ₁₅, c ₂₂, c ₂₃, c ₂₅, c ₃₃, c ₃₅, c ₄₄, c ₄₅, c ₅₅, c ₆₆) with respect to the five independent parameters $(c_{{1^{\prime}1^{\prime}}} ,\;c_{{1^{\prime}3^{\prime}}} ,\;c_{{3^{\prime}3^{\prime}}} ,\;c_{{4^{\prime}4^{\prime}}} ,\;c_{{6^{\prime}6^{\prime}}} )$ and the dip angle of the axis of symmetry θ ₀:

$$ \begin{aligned} {\frac{{\partial c_{11} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}}& = \cos^{4} \theta_{0} ,\quad {\frac{{\partial c_{11} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = {\frac{{\sin^{2} 2\theta_{0} }}{2}},\quad {\frac{{\partial c_{11}}}{{\partial c_{{3^{\prime}3^{\prime}}} }}} = \sin^{4} \theta_{0} ,\quad{\frac{{\partial c_{11} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}}= \sin^{2} 2\theta_{0} , \\ {\frac{{\partial c_{11} }}{{\partial \theta_{0} }}}& = 2\sin 2\theta_{0} (c_{{3^{\prime}3^{\prime}}} \sin^{2} \theta_{0} - c_{{1^{\prime}1^{\prime}}} \cos^{2} \theta_{0} ) + (c_{{1^{\prime}3^{\prime}}} + 2c_{{4^{\prime}4^{\prime}}} )\sin 4\theta_{0} , \\ {\frac{{\partial c_{12} }} {{\partial c_{{1^{\prime}1^{\prime}}} }}} & = \cos^{2} \theta_{0} ,\quad {\frac{{\partial c_{12} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \sin^{2} \theta_{0} ,\quad {\frac{{\partial c_{12} }}{{\partial c_{{6^{\prime} 6^{\prime}}} }}} = - 2\cos^{2} \theta_{0} , \\ {\frac{{\partial c_{12} }}{{\partial \theta_{0} }}} & = \sin 2\theta_{0} (c_{{1^{\prime}3^{\prime}}} - c_{{1^{\prime}1^{\prime}}} + 2c_{{6^{\prime}6^{\prime}}} ), \\ {\frac{{\partial c_{13} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = {\frac{{\sin^{2} 2\theta_{0} }}{4}},\quad {\frac{{\partial c_{13} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \cos^{4} \theta_{0} + \sin^{4} \theta_{0} ,\quad {\frac{{\partial c_{13} }}{{\partial c_{{3^{\prime} 3^{\prime}}} }}} = {\frac{{\sin^{2} 2\theta_{0} }}{4}},\quad \; \begin{array}{*{20}c} {{\frac{{\partial c_{13} }}{{\partial c_{{4^{\prime} 4^{\prime}}} }}} = - \sin^{2} 2\theta_{0} } \\ \end{array} , \\ {\frac{{\partial c_{13} }}{{\partial \theta_{0} }}} & = [0.5(c_{{1^{\prime}1^{\prime}}} + c_{{3^{\prime}3^{\prime}}} - 4c_{{4^{\prime}4^{\prime}}} ) - c_{{1^{\prime}3^{\prime}}} ]\sin 4\theta_{0} , \\ {\frac{{\partial c_{15} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} &= \pm [- \cos^{3} \theta_{0} \sin \theta_{0}],\quad {\frac{{\partial c_{15} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \pm{\frac{{\sin 4\theta_{0} }}{4}},\quad {\frac{{\partial c_{15} }} {{\partial c_{{3^{\prime}3^{\prime}}} }}} =\pm \sin^{3} \theta_{0} \cos \theta_{0} ,\quad {\frac{{\partial c_{15} }}{{\partial c_{{4^{\prime} 4^{\prime}}} }}} = \pm{\frac{{ \sin 4\theta_{0} }}{2}}, \\ {\frac{{\partial c_{15} }}{{\partial \theta_{0} }}} &=\pm [(2c_{{4^{\prime}4^{\prime}}} + c_{{1^{\prime}3^{\prime}}} - c_{{1^{\prime}1^{\prime}}} )\left( {\cos^{4} \theta_{0} - \frac{3}{4}\sin^{2} 2\theta_{0} } \right) + (c_{{3^{\prime}3^{\prime}}} - 2c_{{4^{\prime}4^{\prime}}} - c_{{1^{\prime}3^{\prime}}} )\left( {\frac{3}{4}\sin^{2} 2\theta_{0} - \sin^{4} \theta_{0} } \right) ], \\ {\frac{{\partial c_{22} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = 1, \\ {\frac{{\partial c_{23} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = \sin^{2} \theta_{0} ,\quad {\frac{{\partial c_{23} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \cos^{2} \theta_{0} ,\quad {\frac{{\partial c_{23} }}{{\partial c_{{6^{\prime}6^{\prime}}} }}} = - 2\sin^{2} \theta_{0} , \\ {\frac{{\partial c_{23} }}{{\partial \theta_{0} }}} & = (c_{{1^{\prime}1^{\prime}}} - 2c_{{6^{\prime}6^{\prime}}} - c_{{1^{\prime}3^{\prime}}} )\sin 2\theta_{0} , \\ {\frac{{\partial c_{25} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = \pm [- 0.5\sin 2\theta_{0}] ,\quad {\frac{{\partial c_{25} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \pm 0.5\sin 2\theta_{0} ,\quad {\frac{{\partial c_{25} }}{{\partial c_{{6^{\prime}6^{\prime}}} }}} =\pm \sin 2\theta_{0} , \\ {\frac{{\partial c_{25} }}{{\partial \theta_{0} }}} & =\pm (2c_{{6^{\prime}6^{\prime}}} + c_{{1^{\prime}3^{\prime}}} - c_{{1^{\prime}1^{\prime}}} )\cos 2\theta_{0} , \\ {\frac{{\partial c_{33} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = \sin^{4} \theta_{0} ,\quad {\frac{{\partial c_{33} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = 0.5\sin^{2} 2\theta_{0} ,\quad {\frac{{\partial c_{33} }}{{\partial c_{{3^{\prime}3^{\prime}}} }}} = \cos^{4} \theta_{0} ,\quad {\frac{{\partial c_{33} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}} = \sin^{2} 2\theta_{0} , \\ {\frac{{\partial c_{33} }}{{\partial \theta_{0} }}} & = 2\sin 2\theta_{0} (c_{{1^{\prime}1^{\prime}}} \sin^{2} \theta_{0} - c_{{3^{\prime}3^{\prime}}} \cos^{2} \theta_{0} ) + (c_{{1^{\prime} 3^{\prime}}} + 2c_{{4^{\prime}4^{\prime}}} )\sin 4\theta_{0} , \\ {\frac{{\partial c_{35} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = \pm[- \sin^{3} \theta_{0} \cos \theta_{0}] ,\quad {\frac{{\partial c_{35} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = \pm [- 0.25\sin 4\theta_{0}] ,\quad {\frac{{\partial c_{35} }}{{\partial c_{{3^{\prime} 3^{\prime}}} }}} = \pm\cos^{3} \theta_{0} \sin \theta_{0} ,\quad {\frac{{\partial c_{35} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}} = \pm [- 0.5\sin 4\theta_{0}] , \\ {\frac{{\partial c_{35} }} {{\partial \theta_{0} }}} & = \pm \left\{(c_{{1^{\prime}3^{\prime}}} + 2c_{{4^{\prime}4^{\prime}}} - c_{{1^{\prime}1^{\prime}}} ) \left( {\frac{3}{4}\sin^{2} 2\theta_{0}} - \sin^{4} \theta_{0}\right) + [c_{{3^{\prime}3^{\prime}}} - (c_{{1^{\prime}3^{\prime}}}+ 2c_{{4^{\prime\prime}4^{\prime}}} )]\left( {\cos^{4} \theta_{0} -\frac{3}{4}\sin^{2} 2\theta_{0} } \right)\right\}, \\{\frac{{\partial c_{44} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}}& = \cos^{2} \theta_{0} ,\quad {\frac{{\partial c_{44} }}{{\partial c_{{6^{\prime}6^{\prime}}} }}} = \sin^{2} \theta_{0} ,\quad{\frac{{\partial c_{44} }}{{\partial \theta_{0} }}} =(c_{{6^{\prime}6^{\prime}}} - c_{{4^{\prime}4^{\prime}}} )\sin2\theta_{0} , \\ {\frac{{\partial c_{46} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}} & = \pm 0.5\sin 2\theta_{0} ,\quad{\frac{{\partial c_{46} }}{{\partial c_{{6^{\prime}6^{\prime}}} }}}= \pm[ - 0.5\sin 2\theta_{0}] ,\quad {\frac{{\partial c_{46}}}{{\partial \theta_{0} }}} = \pm (c_{{4^{\prime}4^{\prime}}} -c_{{6^{\prime}6^{\prime}}} )\cos 2\theta_{0} , \\ {\frac{{\partial c_{55} }}{{\partial c_{{1^{\prime}1^{\prime}}} }}} & = 0.25\sin^{2}2\theta_{0} ,\quad {\frac{{\partial c_{55} }}{{\partial c_{{1^{\prime}3^{\prime}}} }}} = - 0.25\sin^{2} 2\theta_{0} ,\quad{\frac{{\partial c_{55} }}{{\partial c_{{3^{\prime}3^{\prime}}} }}}= 0.25\sin^{2} 2\theta_{0} ,\quad {\frac{{\partial c_{55}}}{{\partial c_{{4^{\prime}4^{\prime}}} }}} = \cos^{2} 2\theta_{0} ,\\ {\frac{{\partial c_{55} }}{{\partial \theta_{0} }}} & =[0.5(c_{{1^{\prime}1^{\prime}}} + c_{{3^{\prime}3^{\prime}}} -2c_{{1^{\prime}3^{\prime}}} ) - 2c_{{4^{\prime}4^{\prime}}} ]\sin4\theta_{0} , \\ {\frac{{\partial c_{66} }}{{\partial c_{{4^{\prime}4^{\prime}}} }}} & = \sin^{2} \theta_{0} ,\quad{\frac{{\partial c_{66} }}{{\partial c_{{6^{\prime}6^{\prime}}} }}}= \cos^{2} \theta_{0} ,\quad {\frac{{\partial c_{66} }}{{\partial\theta_{0} }}} = (c_{{4^{\prime}4^{\prime}}} -c_{{6^{\prime}6^{\prime}}} )\sin 2\theta_{0} . \\ \end{aligned}$$

(44)

According to the relationship between the elastic moduli $\{ c_{{1^{\prime}1^{\prime}}} ,c_{{1^{\prime}3^{\prime}}} ,c_{{3^{\prime}3^{\prime}}} ,c_{{4^{\prime}4^{\prime}}} ,c_{{6^{\prime}6^{\prime}}} \}$ and the Thomsen parameters {ρ, α ₀, β ₀, ɛ, δ ^*, γ}(Thomsen, 1986):

$$ \begin{aligned} c_{{1^{\prime}1^{\prime}}} & = \rho (1 + 2\varepsilon )\alpha_{0}^{2} , \\ c_{{1^{\prime}3^{\prime}}} & = \rho \{ \alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (1 + \varepsilon ) - \beta_{0}^{2} ]\}^{1/2} - \rho \beta_{0}^{2} , \\ c_{{3^{\prime}3^{\prime}}} & = \rho \alpha_{0}^{2} , \\ c_{{4^{\prime}4^{\prime}}} & = \rho \beta_{0}^{2} , \\ c_{{6^{\prime}6^{\prime}}} & = \rho (1 + 2\gamma )\beta_{0}^{2} , \\ \end{aligned} $$

(45)

we have

$$ \begin{aligned} \frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{\partial \rho } &= (2\varepsilon + 1)\alpha_{0}^{2} , \quad {{\frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{{\partial \alpha_{0} }}} = 2\rho \alpha_{0} (2\varepsilon + 1),} \quad {{\frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{{\partial \beta_{0} }}} = 0,} \\ {\frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{\partial \varepsilon }}& = 2\rho \alpha_{0}^{2} , \quad{{\frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{{\partial \delta^{*} }}} = 0,}\quad {{\frac{{\partial c_{{1^{\prime}1^{\prime}}} }}{\partial \gamma }} = 0;} \\ \end{aligned} $$

$$ \begin{aligned} {\frac{{\partial c_{{1^{\prime}3^{\prime}}} }}{\partial \rho }} & = [\alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (\varepsilon + 1) - \beta_{0}^{2} ]^{1/2} - \beta_{0}^{2} , \\ {\frac{{\partial c_{{1^{\prime}3^{\prime}}} }}{{\partial \alpha_{0} }}} & = {\frac{{\rho \alpha_{0} [2\alpha_{0}^{2} (\delta^{*} + \varepsilon + 1) - \beta_{0}^{2} (\varepsilon + 2)]}}{{[\alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (\varepsilon + 1) - \beta_{0}^{2} ]^{1/2} }}}, \\ {\frac{{\partial c_{{1^{\prime}3^{\prime}}} }}{{\partial \beta_{0} }}} & = \left\{ {{\frac{{[2\beta_{0}^{2} - \alpha_{0}^{2} (\varepsilon + 2)]}}{{[\alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (\varepsilon + 1) - \beta_{0}^{2} ]^{1/2} }}} - 2} \right\}\rho \beta_{0} , \\ {\frac{{\partial c_{{1^{\prime}3^{\prime}}} }}{\partial \varepsilon }} & = {\frac{{\rho \alpha_{0}^{2} (\alpha_{0}^{2} - \beta_{0}^{2} )}}{{2[\alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (\varepsilon + 1) - \beta_{0}^{2} ]^{1/2} }}}, \\ {\frac{{\partial c_{{1^{\prime}3^{\prime}}} }}{{\partial \delta^{*} }}} & = {\frac{{\rho \alpha_{0}^{4} }}{{2[\alpha_{0}^{4} \delta^{*} + (\alpha_{0}^{2} - \beta_{0}^{2} )[\alpha_{0}^{2} (\varepsilon + 1) - \beta_{0}^{2} ]^{1/2} }}}, \\ \end{aligned} $$

$$ \begin{aligned}{\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{\partial \rho }} &= \alpha_{0}^{2} , \quad {\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{{\partial \alpha_{0} }}} = 2\rho \alpha_{0} , \quad {\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{{\partial \beta_{0} }}} = 0, \\ {\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{\partial \varepsilon }} &= 0, \quad {\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{{\partial \delta^{*} }}} = 0, \quad {{\frac{{\partial c_{{3^{\prime}3^{\prime}}} }}{\partial \gamma }} = 0,} \\ {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{\partial \rho }} &= \beta_{0}^{2} , {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{{\partial \alpha_{0} }}} = 0, \quad {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{{\partial \beta_{0} }}} = 2\rho \beta_{0} , \\ {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{\partial \varepsilon }} &= 0, \quad {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{{\partial \delta^{*} }}} = 0, \quad {\frac{{\partial c_{{4^{\prime}4^{\prime}}} }}{\partial \gamma }} = 0; \\ {\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{\partial \rho }} &= (2\gamma + 1)\beta_{0}^{2} , \quad {\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{{\partial \alpha_{0} }}} = 0, \quad {{\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{{\partial \beta_{0} }}} = 2\rho \beta_{0} (2\gamma + 1),} \\ {\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{\partial \varepsilon }} &= 0, \quad {\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{{\partial \delta^{*} }}} = 0, \quad{{\frac{{\partial c_{{6^{\prime}6^{\prime}}} }}{\partial \gamma }} = 2\rho \beta_{0}^{2} ,} \\ \end{aligned} $$

(46)