## Inverse Modelling

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Notebook

Introduction to Inverse Modelling applied to Atmospheric Science

### Initial concepts

We consider that a physical system is described by a set of variables named state variables and grouped in a state vector, x however those variables are not directly observables or are difficult to measure and indirect methods are used to assess the state of the system. These other variables are grouped in the observation vector y. The relation between both is given by the forward model F by $\mathbf{y}=\mathbf{F}(\mathbf{x},\mathbf{b})+\mathbf{\epsilon}$

where b is named parameter vector, and the $$\mathbf{\epsilon}$$ is named observational error due to uncertainties in the observations, the forward model and model parameters.

The inversion modelling give us methods to estimate the values of x given the values of y and our forward model.

In general we design the foward model according to state vector and observation vector, in the case of remote sensing of the atmosphere, we rely on the radiative transfer theory and the model F will be close related with the radiation (our observations) that arrives to a sensor. The inverse modelling however can be applied to estimate surface-fluxes given concentratons or used in data assimilation.