The budget equation

To relate the surface flux to measurements at height $z_{M}$ we have to specify all relevant sources and flux contributions in continuity equation 2.8, and integrate it over a virtual box $V$, which is situated directly above the surface patch of our interest. The measurement position is placed somewhere on the upper boundary of the box. The contact area between the virtual box and the Earth's surface will be called $B_{0}$. The side boundary $B_{s}$ of the box consists of lines in vertical direction. The top surface of the box is taken parallel to ground surface $B_{0}$, and is called $B_{t}$. To achieve a stable estimate, we take the time average of the integrated continuity equation. The integrated continuity equation, which is called the budget equation of an arbitrary quantity with concentration $\xi$ is:

$$0 = \underbrace{\phantom{\sum_{i}\hspace{-0.5cm}} \frac{1}{\Delta t}... ...} - \underbrace{\sum_{i} \int_{V} \overline{S}_{i}\mbox{d}V}_{\mbox{creation}}$$

$$+ \underbrace{\int_{B_{t}} \left\lgroup \overline{\xi \vec{u}} \... ...t \vec{n}_{\mbox{\scriptsize in}} \mbox{d}B_{0}}_{\mbox{emission from ground}}$$

$$+ \underbrace{\sum_{j} \int_{B_{t}} \!\!\! \left\lgroup \overlin... ...criptsize in}} \right \rgroup \mbox{d}B_{0}}_{\mbox{other fluxes from ground}}$$ (2.17)

where  $\vec{n}_{\mbox{\scriptsize out}}$ is a unit vector, perpendicular to the surface of box $V$, and vector  $\vec{n}_{\mbox{\scriptsize in}}$ is a unit vector, which for reasons of convenience points from the Earth's surface perpendicularly into virtual box $V$. Velocity  $\vec{u}_{\xi}$ is the velocity of the gas-component, which carries $\xi$ through the ground/atmosphere interface (most significant candidate is often evaporating water). At all other segments of the boundary of $V$ (i.e. on $B_{s}$ and $B_{t}$) we assume that diffusion is insignificant, and hence take  $\vec{u}_{\xi} = \vec{u}$.