.. _math_num_documentation.inverse.inverse_problem_statement:

=========================
Inverse problem statement
=========================

.. warning::
   Section in development

Hydrological optimization problems generally consist in fitting catchment model response to the available observations, that is potentially multi-source and multi-site heterogeneous data used to inform the model. The misfit between observations and model response is measured with a cost function :math:`J` (see section :ref:`Cost functions <math_num_documentation.signal_analysis.cost_functions>`) that depends on hydrological parameters :math:`\theta` through the forward hydrological model :math:`\mathcal{M}` composed of hydrological operators that can include a pre-regionalization (see section :ref:`Forward problem statement <math_num_documentation.forward.forward_problem_statement>`). The inverse problem consists in searching an optimal parameter set :math:`\hat{\boldsymbol{\theta}}` such that:

.. math::
   :name: eq:inverse-prl
   
   \hat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}J\left(\boldsymbol{\theta}\right).

.. **TODO**. extend/generalize vs proba cost func, bayesian, mcmc, etc.; add control vectors hyperparam...