Transcript Slide 1

Daniel Paul Tyndall
4 March 2010
Department of Atmospheric Sciences
University of Utah
Salt Lake City, UT
Outline


Introduction
Literature Review
 2DVar/3DVar Analysis Methodologies
 Strong and Weak Constraints

Current Progress






Analysis Equation Solution
Modifications to 2DVar analysis system
Computer Independent Analysis System
Comparison to INCA
Research Goals
Research Timeline
Introduction

High resolution analysis needs:
 Operational weather forecasting
 Wildfire management
 Road maintenance operations
 Air pollution management

Typical data assimilation techniques:
 Cressman method
 2D variational (2DVar) and 3D variational
(3DVar) methods
 4D variational (4DVar) and ensemble methods
Data Assimilation

2DVar/3DVar ingredients
 Observations
 Background field
 Background 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:
2J ( xa )  Jb  J o

Expanded to:
2J ( xa )  ( xa  xb )T Pb1 ( xa  xb )  [H( xa )  yo ]T Po1[H( xa )  yo ]
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 constraints
 Strong constraints
Weak Constraints

Implemented as 3rd term in cost function:
2 J ( xa )  Jb  J o  J c

Usually takes form:
Jc  ( xa  xc )T Pc1 ( xa  xc )

Does not force analysis to fit constraint
 Sometimes constraint is an approximation
Multiple constraints can be combined into a
single term
 Makes solution of analysis equation more
complicated

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 constraint
 Trying 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 model
 New 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
problems
 Other 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
Solving the Analysis Equation

Analysis space (used by Tyndall 2008, local
analysis system [LSA])
(PbT +PbTHTPo1HPb )ν  PbTHTPo1[ yo  H( xb )]
Nx  Nx
Nx  Nx
Nx  N y
xa  xb  Pbν
Nx  Nx

Observation space (Lorenc 1986, da Silva et al.
1995, to be used in this research)
yo  H( xb )  (HPbHT  Po1 )η
Ny  Ny
xa  xb  PbHT η
Nx  N y
Ny  Ny
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 parallelization
Easy vectorization
Easy post processing of graphics
Intuitive debugger
Analysis System Improvements
1.
2.
3.
Sparse matrices/covariance localization
Vectorization and parallelization
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 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 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 changes
2. Observation locations change
Optimizations decreased pbht
computation time: 7 h → 7 min on 6 2GHz 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
2007111918
2007111919
2007111920
2007111921
2007111922
2007111923
2007112000
2007112001
2007112002
2007112003
2DVar INCA Bkg.
RMSE RMSE RMSE
2.07
2.19
2.28
2.30
2.31
2.43
2.44
2.52
2.58
2.65
1.92
2.06
2.19
2.19
2.23
2.34
2.41
2.50
2.57
2.65
2.69
2.82
2.94
3.01
3.05
3.14
3.04
3.10
3.18
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
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Larger Differences between
2DVar and INCA…


Certain times where
2DVar does poorly
compared to INCA
Why is this the
case?
Date/
Time
2007112211
2007112212
2007112213
2007112309
2007112310
2007112311
2007112312
2007112313
2007112314
2007112315
2DVar INCA Bkg.
RMSE RMSE RMSE
2.62
2.64
2.70
2.87
2.78
2.59
2.69
2.69
2.62
2.42
2.43
2.48
2.55
2.68
2.48
2.28
2.39
2.41
2.37
2.21
3.09
3.17
3.34
3.17
3.04
2.87
2.97
2.93
2.75
2.60
Cross Validation Results
1100 UTC 23 November 2007
Difference between 2DVar and
INCA Temperature Analyses
1100 UTC 23 November 2007
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 terrainchanneling 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
2010

 Comparison between INCA and 2DVar
systems

Second publication to be submitted
summer 2011
 Investigation of strong and weak constraints
on surface variational analyses