# Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed
## 3 Final Time Wave Inversion Problem for Variable Sound Speed
### Proof of Theorem 1
We then write  
\[\mathbf{W}_{T}(f)(\mathbf{y}) = \frac{1}{(2\pi)^{3}}\sum_{\sigma=\pm}\int_{\mathbb{R}^{3}}a_{ \sigma}(\mathbf{y},T,\xi)e^{i\phi_{\pm}(\mathbf{y},T,\xi)}\hat{f}(\xi)\,d\xi= \sum_{\sigma=\pm}\mathbf{W}_{\sigma}(f)(\mathbf{y}).\]  
Each operator \(\mathbf{W}_{\pm}\) is a Fourier integral operator (FIO) with the canonical relation given by the pairs \((\mathbf{y}_{\pm},\lambda\eta_{\pm};\mathbf{x},\lambda\xi)\) for any \(\lambda\in\mathbb{R}\), \(\xi,\eta\) unit vectors, \(\mathbf{y}_{\pm}=\gamma_{\mathbf{x},\xi}(\pm T)\), and \(\eta_{\pm}=\hat{\gamma}_{\mathbf{x},\xi}(\pm T)\). Let \(\mathbb{R}^{3}\) be equipped with the metrics \(c^{-2}(\mathbf{x})\,\,d\mathbf{x}^{2}\). Then, \((\mathbf{y}_{\pm},\eta_{\pm})\) is obtained by translating \((\mathbf{x},\xi)\) on the geodesic \(\gamma_{\mathbf{x},\pm\xi}\) by the distance \(T\). From the initial condition of \(\phi_{\pm}\) and \(a_{0,\pm}\) we see that, up to lower order terms,  
\[p_{-}(\mathbf{x},0)=p_{+}(\mathbf{x},0)=\frac{1}{2}\,f(\mathbf{x})\,. \tag{13}\]  
Heuristically, under Equations (1)-(3), each singularity of \(f\) at \((\mathbf{x},\xi)\) is broken into two equal parts. They propagate along the geodesic \(\gamma_{\mathbf{x},\xi}\) in the opposite directions \(\pm\xi\) to generate a singularity of \(\mathbf{W}_{T}(f)\) at \((\mathbf{y}_{\pm},\eta_{\pm})\). From the standard theory of FIOs (see [50]), the adjoint \(\mathbf{W}_{\pm}^{*}\) translates \((\mathbf{y}_{\pm},\eta_{\pm})\) back to \((\mathbf{x},\xi)\) and \(\mathbf{W}_{\pm}^{*}\mathbf{W}_{\pm}\) is a pseudo differential operator. On the other hand, \(\mathbf{W}_{\mp}^{*}\mathbf{W}_{\pm}\) is a FIO whose canonical relation consists of the pairs \((\mathbf{y},\eta;\mathbf{x},\xi)\) given by \(\mathbf{y}=\gamma_{\mathbf{x},\xi}(\pm 2T)\), and \(\eta=\hat{\gamma}_{\mathbf{x},\xi}(\pm 2T)\). That is, \(\mathbf{W}_{\pm}^{*}\mathbf{W}_{\mp}\) is an infinitely smoothing operator on \(B\). Therefore, microlocally, we can write \(\mathbf{W}_{T}^{*}\mathbf{W}_{T}f=\mathbf{W}_{+}^{*}\mathbf{W}_{+}(f)+\mathbf{ W}_{-}^{*}\mathbf{W}_{-}(f)\). We will show that the principal symbol \(\theta_{\pm}(\mathbf{x},\xi)\) of \(\mathbf{W}_{\pm}^{*}\mathbf{W}_{\pm}\) satisfies \(\theta_{\pm}(\mathbf{x},\xi)=1/4\). This result can be intuitively understood as follows. Let us consider \(\mathbf{W}_{+}^{*}\mathbf{W}_{+}\) and a singularity of \(f\) at \((\mathbf{x},\xi)\). Under Equations (1)-(3), half of this singularity propagates into the direction \(\xi\) (corresponding to the function \(p_{+}\)). At the moment \(t=T\), it is transformed to a singularity of \(\mathbf{W}_{+}(f)=p_{+}(T)\) at \((\mathbf{y}_{+},\eta_{+})\). Under the adjoint Equation (15), half of this singularity propagates back to \((\mathbf{x},\xi)\) at \(t=0\) to generate a singularity of \(\mathbf{W}_{+}^{*}\mathbf{W}_{+}(f)\). It is natural to believe that this recovered singularity is \(1/4\) of the original singularity of \(f\) (due to twice splitting, as described). The proof below verifies this intuition. Indeed, denote by \(q_{+}\) the solution of the time-reversed wave equation, e.g., Equation (15), with the initial condition given by \(g_{+}=\mathbf{W}_{+}(f)\). Then, by definition (see Theorem 2) \(\mathbf{W}_{T}^{*}\mathcal{S}_{+}=q_{+}(\,\cdot\,,0)|_{B}\). The solution \(q_{+}\) can be decomposed into the sum \(q_{+}=q_{0}+q_{1}\). Here, \(q_{0},q_{1}\), up to smooth terms, are solutions of the wave equations in \(\mathbb{R}^{3}\times(0,T)\) and satisfy \(q_{0}(\,\cdot\,,0)=\mathbf{W}_{+}^{*}(g_{+})\), \(q_{1}(\,\cdot\,,0)=\mathbf{W}_{-}^{*}(g_{+})\). We are only concerned with \(q_{0}\) since it defines \(\mathbf{W}_{+}^{*}\mathbf{W}_{+}f=q_{0}(\,\cdot\,,0)\). We can write  
\[q_{0}(\mathbf{y},t)=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}b(\mathbf{y},t, \xi)e^{i\phi_{+}(\mathbf{y},t,\xi)}\hat{f}(\xi)\,d\xi. \tag{14}\]  
Let \(b_{0}\) be the principal part of \(b\). Then, the principal symbol \(\theta_{+}\) of \(\mathbf{W}_{+}^{*}\mathbf{W}_{+}\) is given by \(\theta_{+}(\mathbf{x},\xi)=b_{0}(\mathbf{x},0,\xi)\).