Treating Model Error in Variational

Data Assimilation

Nancy Nichols Data Assimilation Research Centre

University of Reading

Treating Model Error in Variational

Data Assimilation

Nancy Nichols Data Assimilation Research Centre

University of Reading

Without Assimilation

Treating Model Error in Variational

Data Assimilation

Without Assimilation

Nancy Nichols Data Assimilation Research Centre

University of Reading

With assimilation

Treating Model Error in Variational

Data Assimilation

With Assimilation Without Assimilation

Nancy Nichols Data Assimilation Research Centre

University of Reading

With Model Error Correction

1.

Weak Constraint 4DVar

System Equations

Prior:

Model:

Observations:

where

and errors are uncorrelated in time

subject to

Variational Assimilation with

Model Error

Can solve using the adjoint technique as

before. Now the adjoints are increased

by an additional set of adjoint variables

giving the sensitivity of the objective

function with respect to each of the

model error variables .

In practice this is too expensive for real time

forecasting, but simplifications can be used.

Adjoint Method

2.

Systematic Model Errors

Many sources of model error are systematic

and also correlated in time.

Sources of model error include:

• Limited resolution

• Wrong forcing

• Inaccurate parameters

• Errors at boundaries

• Discretization errors

• Random disturbances

Model Errors

Augmented Method

To treat systematic errors we augment the

dynamic equations with a simple model for

the dynamics of the errors. Then we only

need to estimate the initial error . The

additional adjoints can then be calculated

efficiently. If it is assumed that the error is

a constant ‘bias’ error then the gradients

can be found directly from the previous

adjoint equations.

0

Variational Assimilation with

Systematic Model Error

Minimize with respect to x0 and e0

subject to

Variational Assimilation with

Systematic Model Error

Minimize with respect to x0 and e0

subject to Constant

Bias

Error

Example - Lorenz 63

Alternative simple error models

Example - Linear Advection Equation

Model: Linear Advection 1-D Upwind Scheme

Initial conditions: Square wave

Boundary conditions: Periodic

Stepsize: t = 1/80 x = 1/40

Observations: Exact solution to ut + ux = 0 at

20 unevenly spaced points at each time step

Solid = Truth, Dotted = Background, + = Observation, Red = With Assimilation

Solid = Truth, Dotted = Background, + = Observation, Red = With Assimilation

Evolving Error Model

Augmented Method

Details can be found in:

A.K. Griffith and N.K. Nichols, Adjoint techniques in data assimilation

for treating systematic model error, J. of Flow, Turbulence and

Combustion, 65, 2000, 469 - 488.

N.K. Nichols, Data Assimilation: Making Sense of Observations,

(eds. W Lahoz, B Khattatov and R Menard), Springer, 2010, pp 13 –

40. (doi: 10.1007/978-3-540-74703-1)

M.J. Bell, M.J. Martin and N.K. Nichols, Assimilation of data into an

ocean model with systematic errors near the equator, Quarterly Journal

of the Royal Meteorological Society, 130, 2004, 873-894.

5.

Conclusions

4D Variational Data Assimilation is a powerful

technique for estimating and predicting the

states of very large environmental systems.

It is used in major operational forecasting

centres. The method can be adapted to a

wide variety of problems and can be simplified

by using approximations in the procedure.

Conclusions

Many challenges left!

Example – 1D Nonlinear Shallow Water Equations