## 3 Inversion
### Estimation of Amplitude Scaling Factors \(\alpha\) and \(\beta\)
This leads to \[h^{\varsigma}_{U}(\beta)=\left[\int_{-\infty}^{\infty}\big{[}\{ \mathbf{k}^{\varsigma}_{U2}\}^{t}(\tau^{\prime})\mathbf{k}^{\varsigma}_{U2}(\tau^ {\prime})-\{\mathbf{k}^{\varsigma}_{U1}(\tau^{\prime})\}^{t}\mathbf{k}^{\varsigma }_{U1}(\tau^{\prime})\big{]}\mathrm{d}\tau^{\prime}\right]\beta^{2} \tag{27}\] \[+\bigg{[}2\int_{-\infty}^{\infty}\big{[}\{\mathbf{k}^{\varsigma}_ {U2}(\tau^{\prime})\}^{t}\mathbf{l}^{\varsigma}_{U2}(\tau^{\prime})-\{\mathbf{ k}^{\varsigma}_{U1}(\tau^{\prime})\}^{t}\mathbf{l}^{\varsigma}_{U1}(\tau^{ \prime})\big{]}\mathrm{d}\tau^{\prime}-S(0)\bigg{]}\beta\] \[+\bigg{[}\int_{-\infty}^{\infty}\big{[}\{\mathbf{l}^{\varsigma}_ {U2}(\tau^{\prime})\}^{t}\mathbf{l}^{\varsigma}_{U2}(\tau^{\prime})-\{\mathbf{ l}^{\varsigma}_{U1}(\tau^{\prime})\}^{t}\mathbf{l}^{\varsigma}_{U1}(\tau^{ \prime})\big{]}\mathrm{d}\tau^{\prime}\bigg{]}=0.\]  
Two more expressions can be obtained by enforcing the energy conservation of the focusing function \(\mathbf{F}_{L}\). We find, akin to Equation (25), that  
\[\int_{-\infty}^{\infty}\big{[}\{\mathbf{F}^{-}_{L}(\tau+\tau^{\prime})\}^{t} \mathbf{F}^{-}_{L}(\tau^{\prime})-\{\mathbf{F}^{+}_{L}(\tau+\tau^{\prime})\}^ {t}\mathbf{F}^{+}_{L}(\tau^{\prime})\big{]}\mathrm{d}\tau^{\prime}=\mathbf{I}S( \tau). \tag{28}\]  
When we substitute the expressions for \(\mathbf{F}^{\pm}_{L}(\tau,\alpha,\beta)\) from Equation (24) into this result and repeat the abovementioned steps, we arrive at expressions for \(h^{\varepsilon}_{U}(\alpha)\) and \(h^{\varsigma}_{L}(\beta)\) (which are equivalent to Equations (26) and (27) with the subscript \(U\) replaced by \(L\)). The scaling factors \(\alpha\) and \(\beta\) could be found by evaluating the roots of \(h^{\varepsilon}_{U}(\alpha)\), \(h^{\varepsilon}_{U}(\alpha)\), \(h^{\varsigma}_{U}(\beta)\) and \(h^{\varsigma}_{L}(\beta)\). However, we have chosen an alternative approach based on minimizing the cost functions  
\[J^{\varepsilon}(\alpha)=\{h^{\varepsilon}_{U}(\alpha)\}^{2}+\{h^{\varepsilon }_{L}(\alpha)\}^{2}, \tag{29}\]  
and  
\[J^{\varsigma}(\beta)=\{h^{\varsigma}_{U}(\beta)\}^{2}+\{h^{\varsigma}_{L}( \beta)\}^{2}. \tag{30}\]  
In Figure 4, we show both cost functions as computed from the numerical data of our running example. We can use Matlab's _fminbnd_ routine to minimize these functions, yielding the estimates \(\alpha\approx 0.4716\) and \(\beta\approx 0.7946\) (their true values being \(0.4694\) and \(0.7934\), respectively). \[h^{p}_{U}(\alpha)=\biggl{[}\int_{-\infty}^{\infty}\bigl{[}\{ \mathbf{k}^{p}_{U2}\big{(}\tau^{\prime}\big{)}\}^{\dagger}\mathbf{k}^{p}_{U2 }\big{(}\tau^{\prime}\big{)}-\{\mathbf{k}^{p}_{U1}\big{(}\tau^{\prime}\big{)} \}^{\dagger}\mathbf{k}^{p}_{U1}\big{(}\tau^{\prime}\big{)}\big{]}\mathrm{d} \tau^{\prime}\biggr{]}\alpha^{2} \tag{26}\]0 Cost functions (**a**) \(J^{\varepsilon}(\alpha)\) and (**b**) \(J^{\varsigma}(\beta)\) for our numerical data. The dashed magenta lines denote the minima, \(0.4716\) (for \(\alpha\)) and \(0.7946\) (for \(\beta\)), that were found using Matlab’s _fminbnd_ routine. The solid cyan lines denote the exact values \(0.4694\) (for \(\alpha\)) and \(0.7934\) (for \(\beta\)), as extracted from the reference data.