Forward problem statement

The computational domain (catchment) \(\Omega \in \mathbb{R}^2\) is a 2D spatial domain and \(t > 0\) denotes the physical time. A lattice \(\mathcal{T}_{\Omega}\) covers \(\Omega\) and the number of active cells within a catchment \(\Omega\) is denoted \(N_x\). Let \(\mathcal{D}_\Omega(x), \forall x \in \Omega\) be the drainage plan obtained from terrain elevation processing considering river network.

The hydrological model is a dynamic operator \(\mathcal{M}\) mapping observed input fields of rainfall, evapotranspiration and eventually snow and temperature \(\boldsymbol{P}(x, t'), \; \boldsymbol{E}(x, t'), \; \boldsymbol{S}(x, t'), \; \boldsymbol{T}(x, t'), \; \forall (x, t') \in \Omega \times [0, t]\) onto discharge field \(Q(x, t)\) such that:

(1)\[ Q\left(x,t\right)=\mathcal{M}\left[\boldsymbol{P}\left(x,t'\right),\boldsymbol{E}\left(x,t'\right),\boldsymbol{h}\left(x,0\right),\boldsymbol{\theta}\left(x\right)\right], \forall (x, t') \in \Omega \times \left[0,t\right]\]

with \(\boldsymbol{h}(x, t)\) the \(N_s\)-dimensional vector of model states 2D fields and \(\boldsymbol{\theta}\) the \(N_{\theta}\)-dimensional vector of model parameters 2D fields.

Note that a pre-regionalization function \(\mathcal{F}_{R}\) can be considered in the forward model to link model parameters to physiographic descriptors \(\boldsymbol{D}\) (see section Regionalization operators) such that:

(2)\[ \boldsymbol{\theta}(x)=\mathcal{F}_{R}(\boldsymbol{D}(x),\boldsymbol{\rho}(x)), \forall x \in \Omega\]

with \(\boldsymbol{\rho}\) the tunable parameters that is regionalization control vector. In that case the forward model is a composed function \(\mathcal{M}\left(\mathcal{F}_{R}\left(\rho\right)\right)\) depending on \(\boldsymbol{\rho}\).

Remark that all operators of the forward model can contain trainable neural networks.