Smoothing the Q(h) relation

[SouthEndMusic]

In the grand scheme of boosting performance by making the whole RHS of the Ribasim ODE problem C1 smooth, doing this for TabulatedRatingCurve is one of the steps. Currently the $Q(h)$ relation is given by piecewise linear interpolation of the input data. To make this more smooth, we can use a cubic polynomial that rounds out the corners. As proposed by @Huite, this is easiest to understand by hydrologist by taking some fixed small interval centered around each data point where the linear interpolation is replaced by a polynomial (some care must be taken half this interval is bigger than the distance between points in the $x$ direction).

For the nicest integration this could be incorporated into DataInterpolations.jl. For some inspiration see here, which uses an interval around the data points whose size is relative to the distance between points instead of absolute as proposed here.

[Huite]

Commenting this for completeness, this paper

Model smoothing strategies to remove microscale discontinuities and spurious secondary optima in objective functions in hydrological calibration
Dmitri Kavetski, George Kuczera
First published: 08 March 2007 https://doi.org/10.1029/2006WR005195

primarily suggest using logistic smoothing. This has the upside of being better than C2, but I'd say with a significant cost since our Qh tables contain quite a lot of rows.

Basically, you need a logistic function for each jump of dQ/dh. For integration, to keep it continous, you need to sum as many integrals (a softplus) as there are jumps (because the function is only 0 or 1 in the limit as x to ±infinity). For e.g. 20 points, that means 20 softplus evaluations.