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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