## 3 Inversion
### Estimation of Amplitude Scaling Factors \(\alpha\) and \(\beta\)  
In Appendix C, we show that the focusing function matrix \(\mathbf{F}\) can be written as an explicit function of \(\tau\), \(\alpha\) and \(\beta\), according to  
\[\underbrace{ \begin{pmatrix}\mathbf{F}^{-}_{U}(\tau,\alpha,\beta)\\ 2\mathbf{F}^{\mathrm{t}}_{LU}(\tau,\alpha,\beta)\\ \mathbf{F}^{-}_{L}(\tau,\alpha,\beta)\\ 2\mathbf{F}^{-}_{L}(\tau,\alpha,\beta)\end{pmatrix} }_{\mathbf{F}(\tau,\beta, \beta)}=\underbrace{ \begin{pmatrix}\mathbf{k}^{p}_{U1}(\tau)&\mathbf{k}^{ \mathrm{s}}_{U1}(\tau)\\ \mathbf{k}^{\mathrm{t}}_{U2}(\tau)&\mathbf{k}^{\mathrm{s}}_{U2}(\tau)\\ \mathbf{k}^{\mathrm{t}}_{U1}(\tau)&\mathbf{k}^{\mathrm{s}}_{U1}(\tau)\\ \mathbf{k}^{\mathrm{t}}_{U2}(\tau)&\mathbf{k}^{\mathrm{s}}_{U2}(\tau)\\ \mathbf{k}^{\mathrm{t}}_{U2}(\tau)&\mathbf{k}^{\mathrm{s}}_{U2}(\tau)\\ \end{pmatrix} }_{\mathbf{K}(\tau)} \begin{pmatrix}\alpha&0\\ 0&\beta\end{pmatrix} ^{\frac{1}{2}}+\underbrace{ \begin{pmatrix}\mathbf{I}^{p}_{ U1}(\tau)&\mathbf{I}^{\mathrm{s}}_{U1}(\tau)\\ \mathbf{I}^{p}_{U2}(\tau)&\mathbf{I}^{\mathrm{s}}_{U2}(\tau)\\ \mathbf{I}^{\mathrm{t}}_{U1}(\tau)&\mathbf{I}^{\mathrm{s}}_{U1}(\tau)\\ \mathbf{I}^{\mathrm{t}}_{U2}(\tau)&\mathbf{I}^{\mathrm{s}}_{U2}(\tau)\\ \end{pmatrix} }_{\mathbf{L}(\tau)} \begin{pmatrix}\alpha&0\\ 0&\beta\end{pmatrix} ^{-\frac{1}{2}}. \tag{24}\]  
In this formulation, \(\mathbf{k}(\tau)\) and \(\mathbf{l}(\tau)\) represent \(2\times 1\) vectors, which can be explicitly computed for each value of \(\tau\) from the recorded data (see Appendix C for details). To estimate the amplitude scaling factors \(\alpha\) and \(\beta\), we make use of a relation for energy conservation [42; 43], which has been used earlier for the estimation of amplitude scaling factors in acoustic media [44]. This criterion can be written as  
\[\int_{-\infty}^{\infty}\bigl{[}\{\mathbf{F}^{+}_{U}\big{(}\tau+\tau^{\prime} \big{)}\}^{\dagger}\mathbf{F}^{+}_{U}\big{(}\tau^{\prime}\big{)}-\{\mathbf{F} ^{-}_{U}\big{(}\tau+\tau^{\prime}\big{)}\}^{\dagger}\mathbf{F}^{-}_{U}\big{(} \tau^{\prime}\big{)}\big{]}\mathrm{d}\tau^{\prime}=\mathbf{I}S(\tau). \tag{25}\]  
When we substitute the expressions for \(\mathbf{F}^{\pm}_{U}(\tau,\alpha,\beta)\) from (24) into (25), subtract \(\mathbf{I}S(\tau)\) on both sides and evaluate the result at \(\tau=0\), we find four expressions for \(\alpha\) and \(\beta\) (i.e., the four entries of the \(2\times 2\) matrix equation). The first of these expressions (corresponding to the first diagonal entry of the matrix) is independent of \(\beta\). We multiply this expression by \(\alpha\) and define the left-hand side of the result as \(h^{p}_{U}(\alpha)\). We find that  
\[h^{p}_{U}(\alpha)=\biggl{[}\int_{-\infty}^{\infty}\bigl{[}\{ \mathbf{k}^{p}_{U2}\big{(}\tau^{\prime}\big{)}\}^{\dagger}\mathbf{k}^{p}_{U2 }\big{(}\tau^{\prime}\big{)}-\{\mathbf{k}^{p}_{U1}\big{(}\tau^{\prime}\big{)} \}^{\dagger}\mathbf{k}^{p}_{U1}\big{(}\tau^{\prime}\big{)}\big{]}\mathrm{d} \tau^{\prime}\biggr{]}\alpha^{2} \tag{26}\] \[+\biggl{[}2\int_{-\infty}^{\infty}\bigl{[}\{\mathbf{k}^{p}_{U2} \big{(}\tau^{\prime}\big{)}\}^{\dagger}\mathbf{I}^{p}_{U2}\big{(}\tau^{\prime} \big{)}-\{\mathbf{k}^{p}_{U1}\big{(}\tau^{\prime}\big{)}\}^{\dagger}\mathbf{I }^{p}_{U1}\big{(}\tau^{\prime}\big{)}\big{]}\mathrm{d}\tau^{\prime}-S(0)\biggr{]}\alpha\] \[+\biggl{[}\int_{-\infty}^{\infty}\bigl{[}\{\mathbf{l}^{p}_{U2}\}^{ \dagger}\big{(}\tau^{\prime}\big{)}\mathbf{l}^{p}_{U2}\big{(}\tau^{\prime} \big{)}-\{\mathbf{l}^{p}_{U1}\big{(}\tau^{\prime}\big{)}\}^{\dagger}\mathbf{l }^{p}_{U1}\big{(}\tau^{\prime}\big{)}\big{]}\mathrm{d}\tau^{\prime}\biggr{]}=0.\]  
In a similar way, the last of our four expressions (corresponding to the last diagonal entry of the matrix) is independent of \(\alpha\). We multiply this expression by \(\beta\) and define the left-hand side of the result as \(h^{s}_{U1}(\beta)\).