# Subsea Methane Hydrates: Origin and Monitoring the Impacts of Global Warming
## 4 Full Waveform Inversion for Viscoelastic Media  
Let us now consider the solution of a multi-parameter inverse problem for a viscoelastic medium. Let us assume that, for a given acquisition, we know the solution of the system of Equation (4) for the frequency range \((\omega_{1},\omega_{2})\). We treat this wave field solution as the result of the action of the nonlinear operator B on the parameters of the viscoelastic medium: the Lame coefficients \(\lambda\), \(\mu\) and the relaxation times \(\tau^{P}\), \(\tau^{S}\). Thus, the multi-parameter inverse problem for viscoelastic media is simply the inversion of the nonlinear operator B implicitly introduced by (4):  
\[\vec{u}(x_{r},z_{r};x_{s},z_{s};\omega)=B[\lambda,\mu,\tau^{P},\tau^{S}],\quad \omega_{1}\leq\omega\leq\omega_{2} \tag{5}\]  
Let us search for the solution of this Equation (5) by nonlinear least squares minimizing the objective functional, characterizing the discrepancy between the measured and synthetic data:  
\[\Phi[\lambda,\mu,\tau^{P},\tau^{S}]=\|\vec{u}(x_{r},z_{r};x_{s},z_{s};\omega)- B[\lambda,\mu,\tau^{P},\tau^{S}]\|^{2} \tag{6}\]  
To solve such a problem, researchers typically use local minimization techniques that reduce to computations of the gradient (5). In particular, it is worth mentioning the conjugate gradient method and its various modifications, (see [40,41]) or quasi-Newtonian methods (see Chapter 6 in the book [42]). The kernel of gradient-type methods is the construction of an iterative process that minimizes the objective functional (6) and updates the resulting solution. Some of the simplest and most popular approaches for this are the steepest descent, conjugate gradient and quasi-Newton methods. The key idea here is the search of the next approximation along a certain direction, whose construction is based on the knowledge of the current gradient. The simplest technique here is the steepest descent in which the next value of the  
Figure 5: Comparison of three seismic traces on the \(z=0\) line. Black corresponds to a perfectly elastic medium, red corresponds to a target layer with a Q factor of 20, and blue corresponds to a target layer with a Q factor of 10.  
parameters is sought along the direction of the antigradient providing the fastest decrease in the objective functional:  
\[[\lambda_{k+1},\mu_{k+1},\tau_{k+1}^{p},\tau_{k+1}^{S}]=[\lambda_{k},\mu_{k},\tau_ {k}^{p},\tau_{k}^{S}]-\alpha_{k}\
abla\Phi[\lambda_{k},\mu_{k},\tau_{k}^{p},\tau _{k}^{S}]. \tag{7}\]  
The step length \(\alpha_{k}\) is found as a result of one-dimensional minimization:  
\[\alpha_{k}=arg\min\Phi\Big{[}\Big{[}\lambda_{k},\mu_{k},\tau_{k}^{p},\tau_{k} ^{S}\Big{]}-\alpha\
abla\Phi\Big{]}. \tag{8}\]  
However, as we have already mentioned, the steepest descent method has a number of disadvantages that significantly slow down and sometimes stop its convergence for high-dimensional problems and complex functionals. That is why various modifications are often used instead, such as various implementations of the conjugate gradient method and quasi-Newtonian methods. In this paper, we search for a minimum point using the conjugate gradient method. Let us now describe numerical experiments on the application of full wave form inversion in a viscoelastic medium. To do this, we use the model presented in Figure 4. Let us start by obtaining the ratios for calculating the gradient. Gradient of the Objective Functional in the Problem of Full Waveform Inversion in Viscoelastic Media  
Let us start with the derivation of relations that determine the gradient of the objective functional \(\Phi\) in accordance with the representation (6).