AN INVESTIGATION OF STRONG AND WEAK

CONSTRAINTS TO IMPROVE VARIATIONAL SURFACE

ANALYSES

Daniel Paul Tyndall

4 March 2010

Department of Atmospheric Sciences

University of Utah

Salt 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 fields

Error correlation to metrological parameter or topography field

Fundamental 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 constraintAttempting 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 χAnalyzing constraint impacts on mesoscale

analysesFound 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 Ny 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 localization

2. Vectorization and parallelization

3. 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 sparseRequires covariance localization (300 km) to make it

sparsePbHT reduction in memory: 187 MB → 83 MB

Optimal 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 parallelizationfor 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 changes

2. 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 executablesExecutables can be run in UNIX, Windows,

or Mac OSComputer 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?