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?