What does an inverse model do?'Forward' model
'Inverse' model
Examples of inverse modelling ...
... and any situation in where:

Parameter Estimation by Maximum Likelihood (Method of Least Squares)Gaussian error characteristics (one variable)(two or more variables) 
Ingredients (for an inversion)
The forward model (strong constraint)How will 1,2,3 combine to give the most likely set of parameters?Bayes' Theorem:
Maximum likelihood = minimum penalty

Notes on the cost function
For unknown parameters in , and observations in ,
Why "least squares"?
Eg. if is nonlinear ...

Methods of Inverting1. Cressman Analysis
2. Best Linear Unbiased Estimator (BLUE)
3. Variational Analysis (4dVar)
4. Kalman filter

Example with BLUE
1 unknown parameter, 1 observation, 1 initial estimate
BLUE formulae:

Example with BLUE(Astronomy  Inverting Kepler's Equation) Want to determine orbital parameters:
Physics of the forward model:

Inversion results

4Dimensional Variational Data AssimilationLeith, 1993:... the atmosphere "is a chaotic system in which errors introduced into the system can grow with time ... As a consequence, data assimilation is a struggle between chaotic destruction of knowledge and its restoration by new observations."

Schematic limb radiance operator

Adjoint Variables and Adjoint Operators
The adjoint of an operator propagates the adjoint variables in the reverse sense (this is just the chain rule generalised to many variables)

Example of '4d'Var. with a simple chaotic systemThe double pendulum

Other GFD applicationsSources/sinks determinationForward model (tracer transport Eq.):
What are the sources/sinks, given observations of ?
Cost function:
Gradient w.r.t. :

Other bonuses of doing inverse modelling / DA
Some difficulties with inverse modelling / DA

SummaryInverse methods:
