# Subsea Methane Hydrates: Origin and Monitoring the Impacts of Global Warming
## 4 Full Waveform Inversion for Viscoelastic Media
### Frechet Derivative of the Operator \(B\)  
We now give a representation for the formal Frechet derivatives of the operator \(B\big{[}\lambda,\mu,\tau^{P},\tau^{S}\big{]}\) with respect to the parameters of the viscoelastic medium. We restrict ourselves to constructing a derivative only for the Lame parameter \(\lambda\); all other derivatives are constructed according to the same scheme. In order to avoid cumbersome calculations, we write down the system of Equation (4) in general form:  
\[L(\lambda,\mu,\tau^{P},\tau^{S})\vec{u}(x,z;x_{s},z_{s};\omega)=\vec{F}(x,z;x_ {s},z_{s};\omega) \tag{16}\]  
Let us now introduce the increment of the parameter \(\lambda\) and the corresponding increment of the wave field \(\partial\vec{u}\), and then, up to the small of the second order, we obtain the following relation connecting these increments:  
\[L(\lambda,\mu,\tau^{P},\tau^{S})\delta\vec{u}(x,z;x_{s},z_{s};\omega)=-\frac{ \partial L}{\partial\lambda}\delta\lambda\ \vec{u}(x,z;x_{s},z_{s};\omega) \tag{17}\]  
Here, \((x_{s},z_{s})\) are coordinates of the point source. Let us consider now [PERSON]'s matrix \(G(x,z;x_{s},z_{s};\omega)\), which describes, at the point \((x,z)\), a wave field from the source point of the oriented force at the point \((x_{s},z_{s})\). Let us rewrite Equation (17) as:  
\[\left\{\begin{array}{l}\omega^{2}\rho\delta u_{x}+\frac{\partial}{\partial x }\bigg{(}((\lambda+2\mu)(1+S\tau^{P})div\ \delta\vec{u}-2\mu(1+S\tau^{S})\frac{\partial\delta u_{z}}{\partial z}) \bigg{)}+\\ \\ \frac{\partial}{\partial z}\bigg{(}\mu(1+S\tau^{S})(\frac{\partial\delta u _{x}}{\partial z}+\frac{\partial\delta u_{z}}{\partial x})\bigg{)}=-\frac{ \partial}{\partial x}\Big{(}\delta\lambda(1+S\tau^{P})div\ \vec{u}\Big{)};\\ \\ \frac{\partial}{\partial z}\bigg{(}((\lambda+2\mu)(1+S\tau^{P})div\ \vec{u}-2\mu(1+S\tau^{S})\frac{\partial\delta u _{z}}{\partial z})\bigg{)}+\\ \\ \omega^{2}\rho\delta u_{z}+\frac{\partial}{\partial x}\bigg{(}\mu(1+S\tau^{S}) (\frac{\partial\delta u_{x}}{\partial z}+\frac{\partial\delta u_{z}}{\partial x })\bigg{)}=\\ \\ -\frac{\partial}{\partial z}\Big{(}\delta\lambda(1+S\tau^{P})div\ \vec{u}\Big{)}.\end{array}\right. \tag{18}\]  
We write down the solution of the resulting system in the form:  
\[\delta\vec{u}(x_{r},z_{r};x_{s},z_{s};\omega)=\int\limits_{D}G(x_{r},z_{r}; \xi,\zeta;\omega)\left[\begin{array}{l}\frac{\partial}{\partial x}\big{[} \delta\lambda(1+S\tau^{P})\ div\vec{u}(\zeta,zeta;x_{s},z_{s};\omega)\big{]} \\ \\ \frac{\partial}{\partial z}\big{[}\delta\lambda(1+S\tau^{P})\ div\vec{u}(\zeta,\zeta;x_{s},z_{s};\omega)\big{]}\end{array}\right]d\xi d\zeta \tag{19}\]  
Let us suppose that the function \(\delta\lambda\) is smooth and vanishes at the boundary of D:  
\[\delta\vec{u}(x_{r},z_{r},x_{s},z_{s};\omega)=-\int\limits_{D}\left[\begin{array} []{l}\frac{\partial G(x_{r},z_{r};\xi,\zeta;\omega)}{\partial x}\\ \\ \frac{\partial G(x_{r},z_{r};\xi,\zeta;\omega)}{\partial z}\end{array}\right] \delta\lambda(1+S\tau^{P})div\vec{u}()d\xi d\zeta \tag{20}\]Following the definition of a derivative, we conclude, that the solution of the system of Equation (18) defines the Frechet derivative with respect to the parameter \(\lambda\). All other derivatives with respect to the parameters of the viscoelastic medium are constructed in exactly the same way. To construct adjoint operators, we find the scalar product of both parts of (20) using an arbitrary function \(\phi(x_{r},z_{r},x_{s},z_{s},\omega)\) followed by rearrangement of the integral to extract the scalar product with the function \(\delta\lambda(\xi,\eta)\).