## 3 Theory
### Inverse Problem
\[\frac{\partial E}{\partial\pi_{k}}=-\int_{0}^{T}\left(\frac{ \partial s_{1}^{\ddagger}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}},T-t^{\prime\prime}\right)}{\partial x_{1}^{\prime\prime}}+\frac{ \partial s_{2}^{\ddagger}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}},T-t^{\prime\prime}\right)}{\partial x_{2}^{\prime\prime}}\right) \cdot\left(\frac{\partial s_{0}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}}^{\prime},t\right)}{\partial x_{1}^{\prime\prime}}+\frac{\partial s _{0}\left(\stackrel{{\rightarrow}}{{x^{\prime}}}^{\prime\prime},t \right)}{\partial x_{2}^{\prime\prime}}\right)dt^{\
u}\Delta x^{\prime\prime 3} \tag{13}\] \[\frac{\partial E}{\partial\pi_{\mu_{k}}}=-\int_{0}^{T}\left[- \left(\frac{\partial s_{1}^{\ddagger}\left(\stackrel{{\rightarrow }}{{x^{\prime}}},T-t^{\prime\prime}\right)}{\partial x_{2}^{\prime\prime}}+ \frac{\partial s_{2}^{\ddagger}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}}^{\prime\prime},T-t^{\prime\prime}\right)}{\partial x_{1}^{\prime}} \right)\cdot\left(\frac{\partial s_{0}\left(\stackrel{{\rightarrow }}{{x^{\prime}}}^{\prime\prime},t^{\prime\prime}\right)}{\partial x_{2}^{ \prime}}+\frac{\partial s_{0}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}}^{\prime\prime},t^{\prime\prime}\right)}{\partial x_{1}^{\prime}}\right)\right.\] (14) \[\frac{\partial E}{\partial\rho_{k}}=\int_{0}^{T}\left(\frac{ \partial s_{1}^{\ddagger}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}}^{\prime\prime},T-t^{\prime\prime}\right)}{\partial t^{\prime\prime }}\frac{\partial s_{0}\left(\stackrel{{\rightarrow}}{{x^{ \prime}}}^{\prime\prime},t\right)}{\partial t^{\prime\prime}}+\frac{\partial s _{2}^{\ddagger}\left(\stackrel{{\rightarrow}}{{x^{\prime}}}^{ \prime\prime},T-t^{\prime\prime}\right)}{\partial t^{\prime\prime}}\frac{ \partial s_{0}\left(\stackrel{{\rightarrow}}{{x^{\prime}}}^{ \prime\prime},t\right)}{\partial t^{\prime\prime}}\right)dt^{\
u}\Delta x^{ \prime\prime}3 \tag{15}\]  
The parametrization considered in this study is \((\rho,V_{p},V_{s})\). The gradients are calculated for these parameters using the chain rule. To change the parametrization from the parameters \((\rho,\tau_{u},\mu_{u})\) to \((\rho,V_{p},V_{s})\) one can apply the chain rule according to the relations of unrelaxed moduli with the unrelaxed Lame parameters \((\rho^{\prime},\lambda_{u}^{\prime},\mu_{u}^{\prime})\) and seismic velocity parameters respectively (Equations (16) and (20)). \[\rho=\rho^{\prime},\pi_{u}=\lambda_{u}^{\prime}+2\mu_{u}^{\prime},\text{ and }\mu_{u}=\mu_{u}^{\prime} \tag{16}\]  
The gradients for density and Lame parameters can be expressed by  
\[\frac{\partial E}{\partial\rho^{\prime}}=\frac{\partial E}{\partial\rho} \frac{\partial\rho}{\partial\rho^{\prime}}+\frac{\partial E}{\partial\pi_{u}} \frac{\partial\pi_{u}}{\partial\rho^{\prime}}+\frac{\partial E}{\partial\mu_{u}} \frac{\partial\mu_{u}}{\partial\rho^{\prime}}=\frac{\partial E}{\partial\rho} \tag{17}\]  
\[\delta c_{n}=\delta m_{n}+\beta_{n}\delta c_{n-1} \tag{9}\]0\[ \begin{array}{l}\frac{\partial E}{\partial t_{u}^{\prime}}=\frac{\partial E}{ \partial\pi_{u}}\frac{\partial\pi_{u}}{\partial t_{u}^{\prime}}\frac{\partial E }{\partial\lambda}+\frac{\partial E}{\partial\mu_{u}}\frac{\partial\pi_{u}}{ \partial t_{u}^{\prime}}+\frac{\partial E}{\partial\rho}\frac{\partial\rho}{ \partial\mu_{u}^{\prime}}=2\frac{\partial E}{\partial\pi_{u}}+\frac{\partial E }{\partial\mu_{u}}\\ =-\int_{0}^{\Gamma}[(\frac{\partial\xi_{u}^{\prime}}{\partial x_{1}^{\prime }}+\frac{\partial\xi_{u}^{\prime}}{\partial x_{1}^{\prime}}).(\frac{\partial 80 _{1}}{\partial x_{2}^{\prime}}+\frac{\partial 80_{2}}{\partial x_{1}^{\prime}})+2( \frac{\partial\xi_{u}^{\prime}}{\partial x_{1}^{\prime}}\frac{\partial s0_{ 1}}{\partial x_{1}^{\prime}}\\ +\frac{\partial\xi_{u}^{\prime}}{\partial x_{2}^{\prime}}\frac{\partial s0_{ 2}}{\partial x_{2}^{\prime}})]dt^{\prime\prime}\Delta{x^{\prime\prime}}^{3} \end{array}