Cost Function

This section follows on from the Cost function section introduced in the forward problem. The aim of this section is to present how is constructed the observation \(J_{obs}\) and regularization \(J_{reg}\) terms and how the regularization weighting coefficient \(\alpha\) can be automatically estimated.

Remember the expression for the cost function \(J\)

\[J = J_{obs} + \alpha J_{reg}\]

Observation term

First, we can expressed the observation term at a gauge \(J_{obs, g}\) such as:

\[J_{obs, g} = \sum_{c=1}^{N_c} w_c j_c \;\;\; \forall g \in [1, N_g]\]

with \(j_c\) and \(w_c\) being respectively any efficiency metric or signature (see sections Efficiency & Error Metric and Hydrological Signature) and associated weight.

Then, the observation terms \(J_{obs, g}\) for each gauge \(g\) are aggregated to give the final observation term \(J_{obs}\) such as:

\[J_{obs} = \sum_{g=1}^{N_g} w_g J_{obs, g}\]

with \(J_{obs, g}\) and \(w_g\) being respectively the observation term and weight associated to each gauge \(g\in[1, N_g]\).

Another less standard method of aggregating observation terms at each gauge \(J_{obs, g}\) is as follows:

\[J_{obs} = \text{quant}\left(J_{obs, 1}, .., J_{obs, N_g}, \; q\right)\]

with \(\text{quant}\) the quantile function and \(q\) the associated quantile value.

Note

\(q\) can take the following values 0.25, 0.5 and 0.75 which are respectively the lower quartile, the median and the upper quartile.

Regularization term

The regularization term \(J_{reg}\) can be expressed as follows:

\[J_{reg} = \sum_{c=1}^{N_c} w_c j_c\]

with \(j_c\) and \(w_c\) being respectively any regularization function (see section Regularization Function) and associated weight.

Regularization weighting coefficient

Typically, the value of the weighting coefficient \(\alpha\) is set by the user according to the weight that is to be placed on the observation \(J_{obs}\) and regularization \(J_{reg}\) term. Certain methods can be used to determine a more appropriate coefficient.

fast

This method consists of a single optimisation iteration to determine the coefficient \(\alpha\):

\[\alpha = \frac{J_{obs, 0} - J_{obs}}{J_{reg}}\]

with \(J_{obs, 0}\) the observation term before optimizing and \(J_{obs}\), \(J_{reg}\) respectively the observation and regularization term after optimization.

l-curve

This method consists of a series of 6 optimisation iterations to determine the coefficient \(\alpha\) based on L-Curve:

Warning

Section in development