Some Fundamentals of
INVERSE MODELLING

Ross Bannister
Room 2L49, Dept. of Meteorology
D.A.R.C.

'Sky and Water I', M.C. Escher, 1938

# What does an inverse model do?

## 'Forward' model

 Model parameters: and

## Examples of inverse modelling ...

 Exact inversion techniques: Matrix inversion PV inversion Abel's integration equation Newton-Raphson method Inexact inversion applications: Data assimilation Satellite retrieval Medical Imaging Geology Astronomy Solar physics Missile interception

## ... and any situation in where:

• observations are noisy,
• observations are incomplete and irregular,
• parameters cannot be measured directly.

# Parameter Estimation by Maximum Likelihood (Method of Least Squares)

## Ingredients (for an inversion)

 1. Observations 2. A-priori (background) 3. Constraints ...

## 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 non-linear ...

## 1. Cressman Analysis

 Cheap. Easy to implement. A-priori in data-poor regions. No account of errors. Not dynamically consistent. Direct observations only. No forward model used.

## 2. Best Linear Unbiased Estimator (BLUE)

 Account taken of errors. Can use indirect obs. A-priori in data poor regions. Difficult to know . Difficult to use with non-linear operators. Expensive for large Nos. of degrees of freedom.

## 3. Variational Analysis (4d-Var)

• Account taken of errors.
• Can use indirect obs. and non-linear operators.
• Suitable for large Nos. of degrees of freedom.
• 4d-Var is dynamically consistent.
• A-priori in data poor regions.
• Difficult to know .
• Expensive.
• Difficult to implement and use.
• Needs preconditioning.

## 4. Kalman filter

• Account taken of errors.
• Can use indirect obs.
• A-priori in data poor regions.
• Evolves in time.
• Difficult to use with non-linear operators.
• Very expensive.
• Difficult to use practically.

# Example with BLUE

1 unknown parameter, 1 observation, 1 initial estimate

 Value Uncertainty Prior estimate Observation Forward model -------

BLUE formulae:

# Example with BLUE

(Astronomy - Inverting Kepler's Equation)

# 4-Dimensional Variational Data Assimilation

## Leith, 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."

The adjoint of an operator propagates the adjoint variables in the reverse sense

(this is just the chain rule generalised to many variables)

# Other GFD applications

## Sources/sinks determination

Forward model (tracer transport Eq.):

What are the sources/sinks, given observations of ?

Cost function:

## Other bonuses of doing inverse modelling / DA

• Model performance
• Observation quality

## Some difficulties with inverse modelling / DA

• Non-linearity (errors, parametrisations)
• Model budget disruption by obs.
• The 'initialization problem'
• Treatment of model error
• Null space
• Error characterization, esp. multivariate
• Artefacts from unrealistic

# Summary

Inverse methods:

• are an integral part of science
• infer information about model parameters using:
• noisy, irregular, and indirect measurements
• how the system behaves
• require expertise in:
• forward modelling
• inverse techniques
• dealing with large volumes of information
• make use of a number of methods and assumptions, for DA:
• Gaussian error characteristics
• method of least squares
• B.L.U.E. / 3d/4d Var. / Kalman filter
• can help assess:
• model performance
• observation quality
• becomes difficult esp:
• non-linear models
• large No. of degrees of freedom
• multivariate
• suffer potential problems:
• representing B
• artefacts
• null space