\section{Transport adjoint} \def\A{{S}} \def\B{{D}} \def\V{{\cal V}} \def\ta{t_s} \def\tb{t_d} \def\M{M} \def\Mc{M^{\mathsf{ex.}}} \def\ME{\mathcal{M}} \def\MEc{\mathcal{M}^{\mathsf{ex.}}} \def\q{c} \def\m{q} \def\S{\xi} \def\aaa{{(\bf A)}} \def\bbb{{(\bf B)}} \def\C{\epsilon} \def\cbar{\bar{c}} \def\exbar{\bar{\epsilon}} \def\exrbar{\bar{\epsilon}^*} \def\grad{\vec{\mbox{grad}}} \def\der#1\def\ex{\epsilon} \def\ex{\epsilon} \def\exr{\epsilon^*} \def\xe{$^{133}$Xe } \def\ba{${ ^{140}}$Ba } \def\dix#1{10$^{#1}$} \def\microbq{$\mu$Bq } \def\m#1{$^{-#1}$} % #2{\frac{\partial #1}{\partial #2}} \def\dep#1{\left(#1\right)} \def\depb#1{\left[#1\right]} \def\dem{1/2} \def\eq#1{Eq.~\ref{eq:#1}} \def\fg#1{Fig.~\ref{fg:#1}} \def\sec#1{Section~\ref{sec:#1}} \def\dq{{\dep{\delta q}}} \def\S{S} \def\D{D} \def\ts{t_s} \def\td{t_d} \def\vs{V_s} \def\vd{V_d} \def\ms{M_s} \def\md{M_d} \def\M{M} \def\Mc{M^{\mathsf{ex.}}} \def\ME{\mathcal{M}} \def\MEc{\mathcal{M}^{\mathsf{ex.}}} \def\c{c} \def\q{q} \def\dt#1{\frac{\partial #1}{\partial t}} \def\C{\epsilon} \def\k{\kappa} \def\l{\lambda} \def\g{\gamma} \def\se{\sigma} \def\pe{\pi} \def\div#1{\mbox{div}\dep{#1}} \def\divrhov#1{\div\dep\rho\vec{V}#1} \def\drhodt#1{\frac{\partial \rho #1}{\partial t}} \subsection{Adjoint derivation} \def\cout{J} We are now going to use the adjoint method in order to derive the results of the previous section. The measurement performed in the detector which, for any solution $c$ of \eq{direct}, can be written as \begin{equation}\label{eq:cout} \cout=\int_{\Omega\times\tau} \rho \mu c \ \dxdt \end{equation} depends linearly on the concentration $c$, which itself depends linearly not only on the emission $\sigma$ but also on the possible lateral inflow of tracer. That inflow occurs along the {\em inflow boundary} $\domegai$ of the physical domain $\Omega$ under consideration, \ie\ along the part of the boundary $\domega$ along which the velocity $\V$ is directed towards the interior of $\Omega$ ($\V.\n<0$ where $\n$ is the unit outward normal vector). The adjoint method defines a general approach to explicitly determine the link between any observable quantity (here $J$) and any input parameter (here the source and lateral inflow of tracers). It is implemented here as follows. Equation~(\ref{eq:direct}) is introduced in \eq{cout} as \begin{eqnarray} \cout&=&\int_{\Omega\times\tau} \rho \mu c \ \dxdt\nonumber \\\label{eq:adjun} &-&\int_{\Omega\times\tau} \rho c^* \depb{\dt{c}+\V.\grad c +\lambda c-\sigma}\ \dxdt \end{eqnarray} where the function $c^*\dep{\x,t}$ (which is basically a Lagrange multiplier) is to be defined. Let us transform through integration by parts the advective part in \eq{adjun}: \begin{eqnarray}\label{eq:adjdeux} I&=&\int_{\Omega\times\tau} \rho c^* \depb{\dt{c}+\V.\grad c} \dxdt\\ &=&\int_{\Omega}\depb{\rho c^* c}_{t_i}^{t_f} \dx\nonumber +\int_\tau \depb{\rho\V c^* c.\n}_{\domega} dt\\ &-& \int_{\Omega\times\tau} c \depb{\dt{\rho c^*}+\div{\rho\V c^*}}\dxdt \end{eqnarray} One recognizes here the conservative flux form of the continuity equation for $c^*$ which can be transformed into its advective form by using the continuity equation \begin{equation}\label{eq:conserv} \dt{\rho}+\div{\rho \V}=0 \end{equation} so that, after rearangement of terms, \begin{eqnarray}\label{eq:adjtrois} \cout&=&\int_{\Omega\times\tau} \rho c^* \sigma \ \dxdt\nonumber\\ &-&\int_{\Omega}\depb{\rho c^* c}_{t_i}^{t_f} \dx -\int_\tau \depb{\rho\V c^* c.\n}_{\domega} dt\nonumber\\ &+&\int_{\Omega\times\tau} \rho c \depb{\dt{c^*}+\V.\grad c^*-\lambda c^*+\mu} \ \dxdt \end{eqnarray} By taking for $c^*$ the solution of \eq{retro} defined by the condition that $c^*=0$ at time $t_f$ and along the outflow boundary $\domegao$ ($\V.\n>0$), we finally obtain \begin{equation}\label{eq:adjquatre} \cout= \int_{\Omega} {\rho c^* c}_{|_{t_i}} \dx -\int_\tau \depb{\rho\V c^* c.\n}_{{\domega}_i} dt +\int_{\Omega\times\tau} \rho c^* \sigma \ \dxdt \end{equation} which explicitly expresses $\cout$ as a function of the initial condition $c$ at $t_i$, of the tracer inflow $\rho\V c.\n_{|\domegai}$ and of the tracer emission $\sigma$. The corresponding multiplicative factors depend linearly on the adjoint variable $c^*$, which turns out to be identical to the {\em retro-tracer} concentration introduced in the previous section. In the more general case of a nonlinear evolution equation and/or observable, it would have been necessary \cite[as shown for instance by ][]{Tala:87} to consider first order perturbations $\delta c$, $\delta\sigma$ and $\delta J$ in place of $c$, $\sigma$ and $\cout$. The analogue of \eq{adjquatre} would then provide local sensitivities of $J$ with respect to input parameters. Those sensitivities would still depend linearly on the adjoint variable. Coming back to our original problem, with a tracer injected at time $\ts$ (with $c=0$ before time $\ts$) and no lateral inflow of tracer, \eq{adjquatre} reduces to \begin{equation} \cout=\int_{\Omega\times\tau}\rho\mu c\dxdt= \int_{\Omega\times\tau}\rho \sigma c^*\dxdt \end{equation} which completes the mathematical proof or the reciprocity principle (\ref{eq:princip1}). Note that with exactly the same algebra, it can be shown that for a conservative tracer ($\lambda=0$ and $\depb{\rho\V c.\n}_{{\domega}_i}=0$), and for times at which no emission nor measurement occurs, \begin{equation} \frac{d}{dt} \int_{\Omega} \rho c c^* \dx=0 \end{equation} At any time, $\int_{\Omega} \rho c c^* \dxdt$ is an evaluation of the measurement as a combination of the distribution of tracer coming from the source $S$ and of the distribution of the air which will be sampled later on at time $\td$ in the detector. It is seen that the advective equation \begin{equation} \frac{\partial c}{\partial t}+\V.\grad\ c=0 \end{equation} is identical with its own adjoint with respect to the air-weighted scalar product~(\ref{eq:scalarproduct}). That property directly results from the conservation of the mass of air. It was mentioned by \cite{Tala:87} that, if a linear evolution equation conserves a scalar product in time, then that equation is identical with its adjoint with respect to the conserved scalar product. The proof is as follows. \def\scalg#1{\left[#1\right]} Let us consider an equation of the form \begin{equation}\label{eq:directg} \frac{d c }{dt}=Lc \end{equation} where $c$ is a state vector and $L$ a linear operator. The adjoint equation of \eq{directg} with respect to a given scalar product $\scalg{ , }$ reads \begin{equation}\label{eq:adjointg} -\frac{d c^*}{dt}=L^*c^* \end{equation} Let us assume that the scalar product is conserved by \eq{directg} meaning that for any solution $c$ of \eq{directg}, the quantity $\scalg{c,c}$ is constant in time, so that \begin{eqnarray} 0 &=&\frac{d}{dt}\scalg{c,c}\\ &=&\scalg{\frac{dc}{dt},c}+\scalg{c,\frac{dc}{dt}}\\ &=&\scalg{Lc,c}+\scalg{c,Lc}\\ &=&\scalg{\dep{L+L^*}c,c} \end{eqnarray} Thus, being true for any $c$, implies $L+L^*=0$, which shows that \eq{directg} and \eq{adjointg} are identical. In the case of pure advection, the tracer massic concentration $c$ is conserved for any air mass element $dm=\rho\dx$. This means that the quantity $\int\rho c^2\dx$ where the integral is taken over a given mass of air, is conserved in time. It results that the advection equation is self adjoint with respect to the scalar product (\ref{eq:scalarproduct}). In the adjoint derivation above, it is the presence of the density $\rho$ in the second integral in \eq{adjun} which allows to take advantage of the conservation of mass (\ref{eq:conserv}) and to obtain the exact symmetry between \eq{retro} and \eq{direct}.