AN INVESTIGATION OF STRONG AND WEAK

CONSTRAINTS TO IMPROVE VARIATIONAL SURFACE

ANALYSES

Daniel Paul Tyndall4 March 2010

Department of Atmospheric SciencesUniversity of UtahSalt Lake City, UT

Outline Introduction Literature Review

2DVar/3DVar Analysis MethodologiesStrong and Weak Constraints

Current ProgressAnalysis Equation SolutionModifications to 2DVar analysis systemComputer Independent Analysis SystemComparison to INCA

Research Goals Research Timeline

Introduction High resolution analysis needs:

Operational weather forecastingWildfire managementRoad maintenance operationsAir pollution management

Typical data assimilation techniques:Cressman method2D variational (2DVar) and 3D variational

(3DVar) methods4D variational (4DVar) and ensemble methods

Literature Review

Data Assimilation 2DVar/3DVar ingredients

ObservationsBackground fieldBackground and observation error covariance

matrices Typical undersampling problem

Observation to grid point ratios:○ 1.5:100 for Real-Time Mesoscale Analysis (RTMA;

de Pondeca 2007)○ 1.7:1000 for Integrated Nowcasting through

Comprehensive Analysis (INCA)

The Cost Function 2DVar and 3DVar analyses depend on

the cost function:

Expanded to:2 ( ) b oJ J J ax

1 12 ( ) ( ) ( ) [ ( ) ] [ ( ) ]J a a b a b a o a ox x x x x x y x yT Tb oP H P H

background observations

Constraints Goal: adding data to undersampled

analysis equation Understood balances or correlations

between meteorological fields can help constrain the analysis equation

Constraints can be formulated as:Weak constraintsStrong constraints

Weak Constraints Implemented as 3rd term in cost function:

Usually takes form:

Does not force analysis to fit constraintSometimes constraint is an approximation

Multiple constraints can be combined into a single term

Makes solution of analysis equation more complicated

1( ) ( )cJ a c a cx x x xT

cP

2 ( ) b o cJ J J J ax

Strong Constraints Implemented into cost function through:

Modification of Pb

Modification of background field Assumes constraint is perfect May add:

Balanced coupling between 2 assimilated fieldsError correlation to metrological parameter or

topography fieldFundamental law or impose limit to analysis

Strong Constraint Implementations Protat and Zawadzki (1999)

Utilized continuity equation as strong constraintTrying to form 3D wind field through assimilation

of Doppler velocities from multiple radar receivers

Gustafsson et al. (2001)Geostrophic approximation as a strong

constraint in new version of HIRLAM modelNew version believed to out perform old version

because of constraints

Strong Constraint Implementations (continued) Žagar et al. (2004); Žagar et al. (2005)

Implemented shallow water equation model as strong constraint

Attempting to assimilate wind information in tropics

Weak Constraint Implementations Protat and Zawadzki (1999)

Also used Doppler velocities from receivers as weak constraint (in addition to continuity equation strong constraint)

Analysis problem would become oversampled otherwise

Analysis method resulted in unrepresentative wind velocities○ Probably due to integration technique of

strong constraint

Weak Constraint Implementations (continued) Xie et al. (2002)

Tested geostrophic constraints between u and v wind components and ψ and χ

Analyzing constraint impacts on mesoscale analyses

Found that constraint helped u and v wind assimilation, but degraded mesoscale features when using ψ and χ assimilation

Literature Review Conclusions Poorly implemented constraints can degrade

analysis Where is all the research on mesoscale

constraints? Xie et al. (2002) and Protat and Zawadzki (1999)

only ones here to look at mesoscale problemsOther mesoscale research looks at radar

assimilation, but not conventional surface observation assimilation

Doesn’t seem to be a lot of research on this particular topic

Current Progress

Solving the Analysis Equation Analysis space (used by Tyndall 2008, local

analysis system [LSA])

Observation space (Lorenc 1986, da Silva et al. 1995, to be used in this research)

1 1( ) [ ( )] o bν y xT T T T Tb b o b b oP +P H P HP P H P H

a bx x νbP

1( ) ( ) o by x ηTb oH HP H P

a bx x ηTbP H

x xN Nx xN N x yN N

x xN N

y yN N y yN N

x yN N

Modified 2DVar Analysis System Modified analysis system written in

MATLAB Like Tyndall (2008), uses Generalized

Minimum Residual (GMRES) method to solve analysis equation

Why MATLAB?Easy parallelizationEasy vectorizationEasy post processing of graphicsIntuitive debugger

Analysis System Improvements

1. Sparse matrices/covariance localization2. Vectorization and parallelization3. Precomputation of pbht for data denial

experiments

Sparse Matrices and Covariance Localization Using built-in sparse matrix data type Test domain of 39,817 grid points and 588

observations (5-km resolution) H is mathematically sparse

Reduction in memory: 187 MB → 0.3 MB Pb is not mathematically sparse

Requires covariance localization (300 km) to make it sparse

PbHT reduction in memory: 187 MB → 83 MBOptimal computation time when PbHT is converted to

sparse after computation

Vectorization and Parallelization Vectorization adds an order of magnitude

increase in computation speed MATLAB has easy for loop parallelization

for k=1:numxb; pb_row = zeros(1,numxb); dx = radius .* cos(pi .* xb_lat ./180.) .* pi .* .. (xb_lon - xb_lon(k)) ./ 180.; dy = radius .* pi .* (xb_lat - xb_lat(k)) ./ 180.; dz = xb_felv - xb_felv(k); r2 = dx .* dx + dy .* dy; z2 = (dz .* dz); pb_row(1,:) = sigb .* (exp(-r2./rad2).*exp(-z2/radz2)); pbht(k,:) = pb_row * ht;end;

Pre-computation of pbht pbht does not need to be recomputed

unless:1. Matrix Pb changes2. Observation locations change

Optimizations decreased pbht computation time: 7 h → 7 min on 6 2-GHz cores

Data denial data set easily created by: Single observation innovation = 0 Particular observation error = 109

Operating System Independent Analysis System MATLAB can create compiled executables

Executables can be run in UNIX, Windows, or Mac OS

Computer running executables does not need MATLAB license

Analysis system easily ported to this framework when GUI is completed

Is it worth it?Kochanski seminar – analyses too complex

Analysis Domain Proposing to investigate impacts of

constraints over Austria Why Austria?

High resolution background fields already computed and used for different analysis system (INCA)

Approximate spatially uniform observation dataset

Can compare 2DVar analyses to INCA analyses as a baseline

Comparison to INCA

Date/Time

2DVar RMSE

INCA RMSE

Bkg. RMSE

2007111918 2.07 1.92 2.692007111919 2.19 2.06 2.822007111920 2.28 2.19 2.942007111921 2.30 2.19 3.012007111922 2.31 2.23 3.052007111923 2.43 2.34 3.142007112000 2.44 2.41 3.042007112001 2.52 2.50 3.102007112002 2.58 2.57 3.182007112003 2.65 2.65 3.28

2DVar and INCA temperature analyses tested during 4 day Föhn period

Period selected because of high INCA errors

2DVar found to have similar RMSE to INCA (0.1-0.2°C agreement)

Difference between 2DVar and INCA Temperature Analyses (0500 UTC 21 November 2007)

2DVar Analysis Increments (0500 UTC 21 November 2007)

2DVar Integrated Data Influence

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.8

0.9

1.0

Larger Differences between 2DVar and INCA… Certain times where

2DVar does poorly compared to INCA

Why is this the case?

Date/Time

2DVar RMSE

INCA RMSE

Bkg. RMSE

2007112211 2.62 2.43 3.092007112212 2.64 2.48 3.172007112213 2.70 2.55 3.342007112309 2.87 2.68 3.172007112310 2.78 2.48 3.042007112311 2.59 2.28 2.872007112312 2.69 2.39 2.972007112313 2.69 2.41 2.932007112314 2.62 2.37 2.752007112315 2.42 2.21 2.60

Cross Validation Results1100 UTC 23 November 2007

Difference between 2DVar and INCA Temperature Analyses1100 UTC 23 November 2007

Research Goals and Timeline

Research Goals Test various strong and weak analysis constraints Current hypotheses:

Specifying Pb using both spatial distances and potential temperature gradients will improve 2-m temperature analyses

10-m wind analyses can be improved by added terrain-channeling constraint

Need accurate estimates of background error correlation Using method by Lönnberg and Hollingsworth (1986); also

used by Tyndall (2008) Test hypotheses through data denial experiments and

RMSE and sensitivity statistics (see Tyndall 2008)

Research Timeline Project will be composed of two journal

publications First publication to be submitted summer

2010Comparison between INCA and 2DVar

systems Second publication to be submitted

summer 2011Investigation of strong and weak constraints

on surface variational analyses

Questions?