# Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed
## 3 Final Time Wave Inversion Problem for Variable Sound Speed
### Proof of Theorem 1  
Let \(c\) be a non-trapping sound speed. We first prove two crucial Lemmas, from which Theorem 1 follows rather quickly. **Lemma 1**. : _Assume that \(\mathbf{W}_{T}f=0\), then \(f=0\). That is, \(\mathbf{W}_{T}\) is injective._  
Proof. : Let \(p\) denote the solution of (1)-(3). We will construct a solution \(\bar{p}\) of the wave equation which is periodic in time with period \(4T\) such that \(\bar{p}=p\) on \(\mathbb{R}^{3}\times[0,T]\). Once this is done, we obtain \(f=\bar{p}(\,\cdot\,,0)=\bar{p}(\,\cdot\,,4^{n}T)\) for any \(n\). Using (10), we arrive at  
\[\forall\mathbf{x}\in\mathbb{R}^{3}\colon\quad f(\mathbf{x})=\lim_{n\to\infty}p (\mathbf{x},4^{n}T)=0\,.\]  
It remains to construct the above-mentioned solution \(\bar{p}\) of the wave equation. The idea is to properly reflect the solution \(p\) in the time variable \(t\) through the time moments \(t=T,2T,\ldots\), as follows. We first construct \(\bar{p}\) on \([0,2T]\) by the odd reflection of \(p\) through the moment \(t=T\): \(\bar{p}(\,\cdot\,,T)=p(\,\cdot\,,T)\) for \(t\in[0,T]\) and \(\bar{p}(\,\cdot\,,T)=-p(\cdot\,,2T-t)\) for all \(t\in[T,2T]\). Since \(p(\,\cdot\,,T)=0\) on \(\mathbb{R}^{3}\), we obtain that \(\bar{p}\) and \(\bar{p}_{t}\) are continuous at \(t=T\). Therefore, \(p\) is continuous on \([0,2T]\) and solves the wave equation on that interval. Next note that \(\bar{p}_{t}(\,\cdot\,,2T)=-\bar{p}_{t}(\,\cdot\,,0)=0\) on \(\mathbb{R}^{3}\). By the even reflection through \(t=2T\): \(\bar{p}(\,\cdot\,,T)=\bar{p}(\,\cdot\,,4T-t)\) for all \(t\in[2T,4T]\), we obtain that \(\bar{p}\) is a solution of the wave equation in \([0,4T]\). Finally, we extend the solution by periodicity with period \(4T\). Noting that \(\bar{p}(\,\cdot\,,0)=\bar{p}(\,\cdot\,,4T)\) and \(\bar{p}_{t}(\,\cdot\,,0)=\bar{p}_{t}(\,\cdot\,,4T)=0\), we obtain that \(\bar{p}\) and \(\bar{p}_{t}\) are continuous for all time and \(\bar{p}\) satisfies the wave equation in \(\mathbb{R}^{3}\times\mathbb{R}_{+}\). This finishes our proof. **Lemma 2**. : _There is a constant \(C\) and a pseudo-differential operator \(\mathbf{K}_{T}\) of order at most \(-1\) such that_  
\[\forall f\in L^{2}(B(0,R))\,\colon\quad\left\|f\right\|_{L^{2}(\Omega)}\leq 2 \Big{(}\left\|\mathbf{W}_{T}f\right\|_{L^{2}(\mathbb{R}^{3})}+\left\|\mathbf{K}_ {T}f\right\|_{L^{2}(B(0,R))}\Big{)}\,.\]  
Proof. : Let \(p\) denote the solution of wave Equations (1)-(3) and recall the parametrix formula \(p(\mathbf{y},t)=\frac{1}{(2\pi)^{3}}\sum_{\sigma=\pm}\int_{\mathbb{R}^{3}}a_{ \sigma}(\mathbf{y},t,\xi)e^{i\phi_{\pm}(\mathbf{y},T,\xi)}\hat{f}(\xi)\,d\xi= \sum_{\sigma=\pm}p_{\sigma}(\mathbf{y},t)\); see [47]. Here, the phase function \(\phi_{\pm}\) solves the eikonal equation \(\partial_{t}\phi_{\pm}(\mathbf{y},t,\xi)\pm c(\mathbf{y})\left|\
abla_{\mathbf{ y}}\phi_{\pm}(\mathbf{y},t,\xi)\right|=0\) for \((\mathbf{y},t)\in\mathbb{R}^{3}\times\mathbb{R}_{+}\) with the initial condition \(\phi_{\pm}(\mathbf{x},0,\xi)=\mathbf{x}\cdot\xi\). The amplitude function is a classical symbol \(a_{\pm}(\mathbf{y},t,\xi)=\sum_{k=0}^{\infty}a_{-k,\pm}(\mathbf{y},t,\xi)\), where \(a_{-k}\) is homogeneous of order \(-k\) in \(\xi\). Its leading term \(a_{0,\pm}\) satisfies the transport equation  
\[\Big{(}\partial_{t}\phi_{\pm}(\mathbf{y},t,\xi)\partial_{t}-c^{2}(\mathbf{y}) \
abla_{\mathbf{y}}\phi_{\pm}(\mathbf{y},t,\xi)\cdot\
abla_{\mathbf{y}}+C_{\pm }(\mathbf{y},t,\xi)\Big{)}a_{0,\pm}(\mathbf{y},t,\xi)=0\,, \tag{12}\]  
with the initial condition \(a_{\pm,0}(\mathbf{x},0,\xi)=1/2\), see [41]. Here, \(C(\mathbf{y},\xi,t)\) only depends on the sound speed \(c\) and the phase function \(\phi_{\pm}\). Let us denote by \(\gamma_{\mathbf{x},\xi}\) the unit speed geodesics originating at \(\mathbf{x}\) along the direction \(\xi\). Then, \(\gamma_{\mathbf{x},\xi}\) is a characteristics curve of the above transport equation; that is, (12) reduces to a homogeneous ODE on each geodesic curve.