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,

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 colleagues… 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 colleagues…

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 2 The 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 Outline Introduction Terminology ML, MAP, and MV (3D/4D-Var) estimate Gaussian pdf  ML  MAP  3D/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 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:  1 st 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 Terminology

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 6 Terminology …

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 10 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-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 algorithms

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 13 Minimization algorithms …

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 14 Minimization algorithms …

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 15 Good 3D/4D-Var lectures Elias Valur Holm http://old.ecmwf.int/newsevents/training/lecture_notes/pdf_f iles/ASSIM/Ass_algs.pdf Mike Fisher http://old.ecmwf.int/archive/newsevents/training/meteorolog ical_presentations/2013/DA2013/Fisher/TC_lecture_1.pdf http://old.ecmwf.int/archive/newsevents/training/meteorolog ical_presentations/2013/DA2013/Fisher/TC_lecture_2.pdf http://old.ecmwf.int/archive/newsevents/training/meteorolog ical_presentations/2013/DA2013/Fisher/TC_lecture_3.pdf http://old.ecmwf.int/archive/newsevents/training/meteorolog ical_presentations/2013/DA2013/Fisher/TC_lecture_4.pdf Yannick Tremolet http://old.ecmwf.int/archive/newsevents/training/meteorolog ical_presentations/pdf/DA/Weak4DVar.pdf

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 16 NRL Observation Space 3D/4D-Var NAVDAS 1 /NAVDAS-AR 2 /COAMPS-AR 3 1 NRL Atmospheric Variational Data Assimilation System (3D-Var) 2 NRL Atmospheric Variational Data Assimilation System – Accelerated Representer (Global 4D-Var) 3 Coupled Ocean/Atmosphere Mesoscale Prediction System– Accelerated Representer (Regional Mesoscale 4D-Var)

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 17 A generalized cost function A measurement of misfit in the initial background state. A measurement of misfit in the NWP model (including misfit in lateral boundary conditions in the case of limited area models). A measurement of misfit in the observations and observation operator.

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 18 Two special cases

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 19 Example: derivation of 3D-Var solution Hessian matrix Sherman-Morrison-Woodbury Primal Dual

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 20 NAVDAS vs. NAVDAS-AR NAVDAS-AR (4D-Var) NAVDAS (3D-Var) obs1obs2obs3obs4obs1obs4obs2obs3 Major Differences NAVDAS (3D-Var) NAVDAS-AR (4D-Var) Time information in observations Not preservedPreserved Use of forecast model to constrain analysis NoYes Computation cost Proportional to square of number of observations Minimal cost for additional observations on top of a fixed cost

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 21 Euler-Lagrange (E-L) equation 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 problem

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 23 A key matrix/vector multiplication

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 24 A key matrix/vector multiplication …

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 25 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=P b H T z 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)

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 27 Flow chart of NAVDAS-AR

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 28 Observation sensitivity equation The results of targeted observing field programs can be interpreted by extending the adjoint sensitivity vector into observation space – Roger Daley The adjoint of NAVDAS-AR was obtained by simply changing the order of subroutine calls to the forward problem – Xu et al (2006) Adjoint of NAVDAS-AR

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 29 Test to validating adjoint of AR Innovation Adjoint sensitivity to observations Adjoint sensitivity to the Initial condition at timestep “m” Analysis Increments at timestep “m” timestep m lhs (ob space) rhs (model space) lhs/rhs 656.527067107513744356.52677863033845541.00000506135851186

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 30 NAVDAS-AR data assimilation system Observation (y) NAVDAS-ARForecast Model Forecast (x f ) Gradient of Cost Function J: (  J/  x f ) Background (x b ) Analysis (x a ) Adjoint of the Forecast Model Tangent Propagator Observation Sensitivity (  J/  y) Background Sensitivity (  J/  x b ) Analysis Sensitivity (  J/  x a ) Observation Impact (  J/  y) Adjoint of NAVDAS- AR Ob Error Sensitivity  J/  ob What is the impact of the observations on the forecast accuracy? How to adjust the specified observation and background errors to improve the NWP forecast?

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 31 Special characters of NAVDAS-AR It searches for the minimum in the observation space. The size of the control variables equals the number of observations to be assimilated for both strong and weak constraint. The gradient of the 4D-Var cost function is not explicitly calculated in NAVDAS-AR (see extra slide at the end). A solution to a set of linear equations is sought instead. The observation error is used as the 1 st level of preconditioning during minimization. The strong constraint (perfect model assumption) is only a special case, where model error variance is set to zero. The coding of the adjoint of NAVDAS-AR is very simple.

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 32 Discussion Recap the underlying assumptions used in 3D/4D-Var All errors are normally distributed (Gaussian pdfs) need to properly screen data before getting into the DA system. The dynamical process is weakly non-linear need to have a very good 3D/4D background information, such that DA only adding small analysis increments. No bias in all the errors (current 3D/4D–Var are bias-blind) need to remove biased observations before assimilation. Biases in background/model remain. Observation errors are not correlated spatially and temporally data thinning, super-ob, and increasing observation error. Errors from different sources are not uncorrelated problem remains.

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 33 Discussion … Pros and Cons of 3D/4D-Var Cons: Needs ADJ (TLM) of NWP and observation operator, respectively Requires Gaussian pdf with zero bias. Needs to be weakly non-linear Doesn’t automatically provide a posteriori pdf Pros: Performs generally better than most of the other DA schemes Solves a full rank problem (no need for “localization”) Allows the use of outer loops to account for nonlinearity Easy to add dynamical constraints Ensemble of 3D/4D-Var now is a common practice Long window weak constraint 4D-Var can now parallel in time

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 34 Thank You

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 35 Typical procedures in 3D/4D-Var Form a cost function that properly measures all the misfits in a least-square sense Chose a proper algorithm to minimize the cost function, such as model or observation space algorithm, respectively. The choices of algorithms are often based on what resources are available. In addition to proper data selection, certain variable transformation may be needed to ensure all the fundamental assumptions valid. Some levels of preconditioning to speed up the convergence of typical iterative algorithm are often employed. Adjoint models (NWP or observation operator) are often used to efficiently calculate the gradient of the cost function.

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 36 Basic components of NAVDAS-AR Nonlinear NAVGEM : Calculate innovation and provide low resolution basic state trajectory required in TLM/Adjoint models. Observation pre-process and VarBC: Provide quality-controlled and bias-corrected observations to be used in the Solver. Adjoint and tangent linear models: Calculate special Matrix/Vector multiplication in both the Solver & Post-multiplier. Observation operators and associated Jacobians (CRTM): Calculate innovation vector and to provide gradient information in both the Solver and Post-multiplier. A Flexible Conjugate Gradient (FCG) Solver: Solve a set of high dimensional linear equations. Error covariance models: Specify observation, background, model error covariance.

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 37 Two implicit cost functions in AR (from Chua and Xu 2008)

3D/4D-Var Methods Liang Xu (NRL) JCSDA Summer Colloquium on Satellite DA 38

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 3D-Var/4D-Var Solution Methods Liang Xu Naval Research Laboratory, Monterey,

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Presentation on theme: "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,"— Presentation transcript:

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