3d/4d-var methods liang xu (nrl) jcsda summer colloquium on satellite da 1 3d-var/4d-var solution...

3D/4D-Var Methods Liang Xu (NRL)JCSDA Summer Colloquium on Satellite DA 1

3D-Var/4D-Var Solution Methods3D-Var/4D-Var Solution Methods

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…

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…


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


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


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


TerminologyTerminology


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.


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.


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).


Minimization algorithmsMinimization algorithms


Minimization algorithms …Minimization algorithms …


Good 3D/4D-Var lecturesGood 3D/4D-Var lectures

Elias Valur Holm

• http://old.ecmwf.int/newsevents/training/lecture_notes/pdf_files/ASSIM/Ass_algs.pdf

Mike Fisher

• http://old.ecmwf.int/archive/newsevents/training/meteorological_presentations/2013/DA2013/Fisher/TC_lecture_1.pdf




Yannick Tremolet

• http://old.ecmwf.int/archive/newsevents/training/meteorological_presentations/pdf/DA/Weak4DVar.pdf


NRL Observation Space 3D/4D-Var NAVDAS1/NAVDAS-AR2/COAMPS-AR3

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)


A generalized cost functionA 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.


Two special casesTwo special cases


Example: derivation of 3D-Var solutionExample: derivation of 3D-Var solution

Hessian matrix

Sherman-Morrison-Woodbury

Primal

Dual


NAVDAS vs. NAVDAS-AR NAVDAS vs. NAVDAS-AR

NAVDAS-AR(4D-Var)

NAVDAS(3D-Var)obs1

obs2

obs3

obs4

obs1

obs4

obs2

obs3Major

DifferencesNAVDAS(3D-Var)

NAVDAS-AR(4D-Var)

Time information

in observations

Not preserved Preserved

Use of forecast model to constrain analysis

No Yes

Computation cost

Proportional to square of number of

observations

Minimal cost for additional observations on top of a fixed cost


Euler-Lagrange (E-L) equationEuler-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.


Solution to the linear problemSolution to the linear problem


A key matrix/vector multiplicationA key matrix/vector multiplication


A key matrix/vector multiplication …A key matrix/vector multiplication …


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.


Summary of linear solution (inner loop)Summary of linear solution (inner loop)


Flow chart of NAVDAS-ARFlow chart of NAVDAS-AR


Observation sensitivity equation 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


Test to validating adjoint of ARTest to validating adjoint of AR

Innovation

Adjoint sensitivityto observations

Adjoint sensitivity to the Initialcondition at timestep “m”

Analysis Incrementsat timestep “m”

timestepm

lhs(ob space)

rhs(model space)

lhs/rhs

6 56.5270671075137443 56.5267786303384554 1.00000506135851186


NAVDAS-AR data assimilation systemNAVDAS-AR data assimilation system

Observation(y) NAVDAS-AR Forecast

ModelForecast

(xf)

Gradient ofCost Function

J: (J/ xf)

Background(xb)

Analysis(xa)

Adjoint of theForecast Model

Tangent Propagator

ObservationSensitivity

(J/ y)

BackgroundSensitivity

(J/ xb)

AnalysisSensitivity

(J/ xa)

Observation Impact<y-H(xb)> (J/ y)

Adjoint of NAVDAS-

AR

Ob Error SensitivityJ/ eob

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?


Special characters of NAVDAS-ARSpecial 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 1st 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.


Discussion 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.


Discussion … 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 34

Thank YouThank You


Typical procedures in 3D/4D-VarTypical 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.


Basic components of NAVDAS-ARBasic 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.


Two implicit cost functions in ARTwo implicit cost functions in AR

4DVar cost function to be minimized. It is not explicitly calculated in AR.

4D-Rep cost function, , associated with the FCG “SOLVER”. It is not explicitly calculated in AR either.

• Calculated from a 6 h data assimilation window centered at 2009041506.

• The number of assimilated observations was 812,277.

• is the inner loop iteration number.

• The plot of ‘omega’ values vs iteration number is not shown here.

(from Chua and Xu 2008)

Both cost functions shown convergence when the regular stopping criteria in AR.

Solving to minimize a cost function

At the minimum

=

3d/4d-var methods liang xu (nrl) jcsda summer colloquium on satellite da 1 3d-var/4d-var solution...

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