3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 1 3D-Var/4D-Var Solution Methods Liang Xu Naval Research Laboratory, Monterey, CA JCSDA Summer Colloquium on Satellite Data Assimilation CIRA, CSU, Fort Collins, CO 27 July 2015 Thanks to: Roger Daley, Andrew Bennett, Yoshi Sasaki Tom Rosmond, Ron Errico, and so many other
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3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 1 3D-Var/4D-Var Solution Methods Liang Xu Naval Research Laboratory, Monterey,
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3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 1
JCSDA Summer Colloquium on Satellite Data AssimilationCIRA, CSU, Fort Collins, CO
27 July 2015
Thanks to: Roger Daley, Andrew Bennett, Yoshi SasakiTom Rosmond, Ron Errico, and so many other colleagues…
Liang XuNaval Research Laboratory, Monterey, CA
JCSDA Summer Colloquium on Satellite Data AssimilationCIRA, CSU, Fort Collins, CO
27 July 2015
Thanks to: Roger Daley, Andrew Bennett, Yoshi SasakiTom Rosmond, Ron Errico, and so many other colleagues…
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 2
The big picture of 3D/4D-VarThe big picture of 3D/4D-Var
• Scientific aspect:
form a quadratic cost function in a weighted least-square sense
• Computational aspect:
find a 3D/4D analysis (an optimal 3D/4D state) by solving a series of linearized minimization problems
while (number of outer loop)
minimize the cost function(s) using calculus of variations (inner loop(s))
endwhile
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 3
OutlineOutline
• Introduction
• Terminology
• ML, MAP, and MV (3D/4D-Var) estimate
• Gaussian pdf MLMAP3D/4D-Var
• Key assumptions used in 3D/4D-Var
• Minimization algorithms
• An observation space 4D-Var example
• Discussions
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 4
Introduction Introduction
• Most of major operational centers use either 3D-Var or 4D-Var for their atmospheric data assimilation
• Different flavors of 3D/4D-Var Examples: primal-, dual-, model space-, observation space-,
incremental-, PSAS, 4DPSAS, representer, S4D-Var, W4D-Var, saddle point, etc.
• Examples of operational variational atmospheric data assimilation systems: 1st 4D-Var papers (Le Dimet & Talagrand 1986; Lewis & Derber
1985) 1st 3D-Var (primal, analysis space) - NMC in June 1991 1st 4D-Var (primal, model space, incremental, S4D-Var) - ECMWF
in November 1997 1st weak constraint 4D-Var (dual, observation space, W4D-Var,
accelerated representer, 4DPSAS) - NRL in August 2009
• Only a narrow view of 3D/4D-Var is provided here
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 5
TerminologyTerminology
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 6
Terminology …Terminology …
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 7
Terminology …Terminology …
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 8
Terminology …Terminology …
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 9
Terminology …Terminology …
State vector: n,
Observation vector: k,
Model forecast: M
Initial condition:
Observations: H,
where, M is the NWP model, H is the observation operator. , and are the error vectors, for the initial condition, model error at time step n, and observations, respectively.
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 10
ML, MAP, and MV (3D/4D-Var)ML, MAP, and MV (3D/4D-Var)
• Minimum variance (MV) estimate
MEAN – find an optimal state that minimizes the variances of the loss function of conditional mean.
• Maximum likelihood (ML) estimate & Maximum a posteriori (MAP)
MODE – find an optimal state that maximizes the posterior pdf.
For Gaussian pdf, MV, ML, and MAP estimates are identical and are equivalent to 3D/4D-Var.
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 11
Key assumptions used in 3D/4D-VarKey assumptions used in 3D/4D-Var
• Common assumptions used in 3D/4D-Var: Errors in background, observation, and observation operator
are normally distributed (Gaussian pdf) with zero mean (unbiased).
Errors in background, observation, and observation operator are not mutually correlated (uncorrelated).
Errors in observation is not correlated spatially and temporally.
Observation operator can be linearized (observation operator is weakly nonlinear).
• Assumptions special to 4D-Var: Model error is normally distributed and unbiased. Model errors are uncorrelated to other types of errors. Model can be linearized (model is weakly nonlinear). Model can be used to constraint the analysis either strongly
(perfect model) or weakly (imperfect model).
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 12
Minimization algorithmsMinimization algorithms
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 13
• The analyzed 4D state, where the generalized cost function is minimum, satisfies the following E-L equation.
• Notice that adjoint of NWP model and observation operator are resulted from taking the derivative of the cost function.
• The E-L equation is a coupled two point values problem and is generally not easy to be solved.
• It can be decoupled using the “representer method” when both the model and operator can be linearized.
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 22
Solution to the linear problemSolution to the linear problem
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 23
A key matrix/vector multiplicationA key matrix/vector multiplication
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 24
A key matrix/vector multiplication …A key matrix/vector multiplication …
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 25
The backward/forward SWEEP The backward/forward SWEEP
Input: an observation space vector
(4D) -- z
Initial background
error covariance -
smoothes adjoint fields
at the beginning of DA window.
Output: a model space vector (4D) --
g=PbHTz
Data Assimilation Window
forward TLM
backward ADJ
OB contribution
impact of model error
Based on Amerault
The ‘SWEEP’ is the engine of AR framework and is used for different applications, the FCG solver, post- & pre- multiplication for forward and adjoint of AR, respectively.
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 26
Summary of linear solution (inner loop)Summary of linear solution (inner loop)
3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 27
Flow chart of NAVDAS-ARFlow chart of NAVDAS-AR
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