Cost functions#

Let \(J\) be the cost function to minimize and \(\theta\in\Omega\subset\mathbb{R}^{N_{p}}\) be the \(N_p\)-dimensional model parameters. Then, the problem is to find an optimal solution \(\hat{\boldsymbol{\theta}}\) minimizing the cost function:

\[\hat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}J\left(\boldsymbol{\theta}\right).\]

As classically done in VDA, we assume here that \(J\) be a convex and differentiable cost function such that:

\[J(\theta) = \alpha J_{obs}(\theta) + \beta J_{reg}(\theta)\]

with \(J_{obs}\) measuring the misfit to observations, \(J_{reg}\) being a regularization term, \(\alpha\) the misfit to observations weight and \(\beta\) the regularization weight.

Firstly, the observation term \(J_{obs}\) can be generally expressed as follows for multi-gauge observation:

\begin{eqnarray} &J_{obs}& &=& &\sum_{k=1} ^ {N_o} \phi_k J_{obs,k}& \\ &J_{obs}& &=& &\text{med}(J_{obs,k})& \end{eqnarray}

with \(J_{obs,k}\) and \(\phi_{k}\) being respectively the constrained term and weight associated to each gauge \(k\in\left[1..N_{o}\right]\). The weighting is such that \(\sum_{k=1}^{N_{o}}\phi_{k}=1\). Note that \(N_{o}=1\) is the classical “mono-gauge” downstream calibration case.

Then each term \(J_{obs,k}\) can be broken down into “classical” objective function (COF) and “signatures-based” objective function (SOF) as follows:

\[J_{obs,k} = w_d j_{k,d} + \sum_{i=1} ^ {N_s} w_{s,i} j_{k,s,i}\]

with \(j_{k,d}\) being the COF calculated at gauge \(k\), which is the essential criteria for the optimization, \(j_{k,s,i}\) being the SOF calculating the relative error between simulation and observation associated to signature type \(i\in\left[1..N_{s}\right]\) at gauge \(k\), and \(w_d,w_{s,i}\) being the corresponding optimization weight.

COF can be one of:

  • nse

\[j_{k,d} = j_{k,\text{nse}} = \frac{\sum_{t=t^*} ^ {T} \left[ Q(k,t) - Q_o(k,t) \right] ^ 2}{\sum_{t=t^*} ^ {T} \left[ Q_o(k,t) - \overline{Q_o}(k,t) \right] ^ 2}\]

  • kge

\[\begin{split}\begin{eqnarray} j_{k,d} = \; &j_{k,\text{kge}}& &=& &\sqrt{(r - 1) ^ 2 + (\alpha - 1) ^ 2 + (\beta - 1) ^ 2}& \\ &r& &=& &\frac{\text{Cov} \left[ Q(k), \; Q_o(k) \right]}{\sqrt{\text{Var} \left[Q(k) \right]} \; \sqrt{\text{Var} \left[Q_o(k) \right]}}& \\ &\alpha& &=& &\frac{\overline{Q(k)}}{\overline{Q_o(k)}}& \\ &\beta& &=& &\frac{\sqrt{\text{Var}\left[Q(k) \right]}}{\sqrt{\text{Var}\left[Q_o(k) \right]}} \end{eqnarray}\end{split}\]

  • kge2

\[j_{k,d} = j_{k, \text{kge2}} = \left( j_{k, \text{kge}} \right) ^ 2\]

  • se

\[j_{k,d} = j_{k, \text{se}} = \sum_{t=t^*} ^ {T} \left[ Q(k,t) - Q_o(k,t) \right] ^ 2\]

  • rmse

\[j_{k,d} = j_{k, \text{rmse}} = \sqrt{\frac{j_{k, \text{se}}}{T - t^* + 1}}\]

  • logarithmic

\[j_{k,d} = j_{k, \text{logarithmic}} = \sum_{t=t^*} ^ {T} \left[ Q_o(k,t) \left( \ln \frac{Q(k,t)}{Q_o(k,t)} \right) ^ 2 \right]\]

Now, denote \(S_{i}^{o}\) and \(S_{i}^{s}\) are observed and simulated signature type \(i\) respectively. These signatures are defined and calculated as depicted in hydrological signatures section. Then, for each signature type \(i\), the corresponding SOF is computed depending on if the signature is:

  • continuous signature:

\[j_{k,s,i} = \left|\frac{S_{i}^{s}(k)}{S_{i}^{o}(k)}-1\right|\]

  • flood event signature:

\[j_{k,s,i} = \frac{1}{N_E}\sum_{e=1}^{N_{E}}\left|\frac{S_{i,e}^{s}(k)}{S_{i,e}^{o}(k)}-1\right|\]

where \(S_{i,e}^{s},S_{i,e}^{o}\) are the simulated and observed signature of event number \(e\in\left[1..N_{E}\right]\).