Reduce the number of states by combining the infiltration and evaporation states

[SouthEndMusic]

Say we combine these forcings into outward_forcing. Then the volume update over a timestep of size $\Delta t$ for this combined state for one basin is, with evapotranspiration and infiltration:

$$ V = \int_t^{t + \Delta t} \phi(\tau) \left[A(\tau)ET + I\right]\text{d}\tau $$

where $\phi(t)$ is the reduction factor and $ET$ and $I$ are constant over the timestep. We of course want to split this back into the evaporation update $VET$ and the infiltration update $VI$ such that $V = VET + VI$, but we don't know how to do that exactly. Say we approximate these with a trapezoid integration:

$$ VET^* = \Delta t ET\frac{\phi(t)A(t) + \phi(t + \Delta t)A(t + \Delta t)}{2} $$

$$ VI^* = \Delta t I \frac{\phi(t) + \phi(t + \Delta t)}{2} $$

We want however that our approximated evaporation and infiltration volumes agree with the ODE result $V$ to maintain the water balance. Therefore we scale these results to obtain the final approximation:

$$ VET' = \frac{VET^*}{VET^* + VI^*}V $$

$$ VI' = \frac{VI^*}{VET^* + VI^*}V $$

so that $VET' + VI' = V$.