space and time multiscale analysis system a sequential variational approach yuanfu xie, steven koch...
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Space and Time Multiscale Analysis System
A sequential variational approach
Yuanfu Xie, Steven Koch
Steve Albers and Huiling Yuan
Global Systems Division
Earth System Research Laboratory
5th Oceanic D-A Workshop9/22/2009
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Outline
• Applications;• Sequential variational analysis approach;
• Multigrid implementation;• Numerical results;• Summary.
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LAPS III Configuration
Data Ingest
Intermediatedata files
LAPS GSI
Data
STMAS3D
Trans
Trans
Post proc1 Post proc2 Post proc3
Model prep
WRF-ARW MM5 WRF-NMM
ForecastVerification
ErrorCovariance
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STMAS Applications
• FAA/MIT boundary detection;• MDL boundary detection/nowcasting;• Storm Prediction Center nowcasting;• Central Weather Bureau reanalysis;• AOML/ESRL hurricane data assimilation.• National Marine Data Information Service for
SST and height analysis;• Tornado applications.
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It is derived from a statistical analysis assuming the error’s distribution is Gaussian. It solves a variational problem:
subject to a model constraint for 4DVAR
Single 3-4DVAR approach
€
min1
2u− ub( )
TB−1 u− ub( ) +
1
2Hu− y( )
TO−1 Hu − y( )
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A Lognormal 1DVAR
€
Pa (u)∝ Po(u− uo)Pb (u− ub )
Po(u− uo)∝ e−
ln(u−uo )−τ o( )2
2σ o2
, Pb (u− ub )∝ e−
ln(u−ub )−τ b( )2
2σ b2
minln(u− uo) − τ o( )
2
2σ o2
+ln(u− ub ) − τ b( )
2
2σ b2
PDF:
Cost function:
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Error Covariance
The critical issue is the accuracy of the background error covariance matrix B.
It is hard to obtain it. It usually has a size of about 106 x 106 /2. It is time, location, and flow dependent. Model forecast differences are usually used
to construct the covariance but model bias is missed from the covariance.
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Background
Observation Longer wave
Background
Observation Longer wave
Resolvable Information for a Given Observation Network
Difference on longer wave Difference on shorter wave
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STMASSTMAS is implemented in two steps. 1. It retrieves the resolvable observation
information. 2. After the resolvable information retrieved,
STMAS is reduced to a standard statistical variational analysis
With long waves retrieved, STMAS deals with a localized error covariance, a banded matrix, at its last phase of analysis.
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An idealized multiscale caseLeft: Mesonet surface stations; Right: An analysis function
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An idealized multiscale case (cont.)
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A Recursive Filter 3DVAR
A single 3DVAR
And B is approximated by a recursive filter (Hayden and Purser 95):
€
ui' =αui−1
' + 1−α( )ui left pass
ui" =αui+1
" + 1−α( )ui' right pass
€
min1
2u− ub( )
TB−1 u− ub( ) +
1
2Hu− y( )
TO−1 Hu − y( )
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=0.5 0.7 0.9
A single 3DVAR with different
These analyses tend toapproximate the truth:
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Recursive filter version of STMAS A sequential variational analysis implemented
through a recursive filter.
1. Solve the VAR with large , e.g. 0.999;2. Subtract the analysis from observation
values used in previous VAR analysis;3. Reduce by a fraction, say in (0.5,1);4. Return to 1 if it is necessary;5. Add the previous analyses together.
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Comparison:
Single 3DVARWith =0.5 or 0.9
STMAS-RF
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Approximation Theory
Any smooth function can be decomposed by a series of base functions that is complete.
where k can be Fourier based functions, wavelets, or any smooth function with decreasing scales with k.
STMAS uses a sequence of truncated series
to approximate the the information contained in obs and background
€
f = α kφkk= 0
∞
∑
€
f = α kφkk=M
M +N
∑
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Advantages STMAS is an iterative variational analysis;
It retrieves longer wave information that can be resolved by the observation network;
It is a variational generalization of a single 3-4DVAR and it can handle advanced data and observation errors globally;
Various balances can be applied at different STMAS levels, e.g., geostrophic on large scale analysis; hydrostatic over small ones.
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Multigrid TechniqueUsing the number of gridpoints to control
the base functions.
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STMAS Multigrid Implementation
The number of grid points over a given domain determines the shortest wavelength allowed. A multigrid uses the number of grid points to control the wavelength.
STMAS solves its variational problem over the coarsest grid and obtains observation information for longest waves. By gradually increasing the number of grid points, STMAS multigrid gains shorter waves by each iterations.
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An efficient analysis system Since the multigrid determines the wavelength, there
is no correlation involved in STMAS variational analysis over a given grid.
Only computation for the cost function is simple interpolations.
An STMAS 5 km surface analysis of 6 state variables over eastern US (two third of CONUS) using recursive filter spends 15 minutes; A multigrid STMAS analysis could take about 40 seconds.
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Different Implementations of STMAS
Recursive filter Wavelet Multigrid
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Real time analysis
STMAS runs a real time analysis every 15 minutes over a domain over the east of US. It provides wind, temperature, pressure, dew-point, and many other analyzed fields.
http://laps.noaa.gov/request/nph-laps.cgi
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FAA/MIT Boundary Detection Application
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STMAS for frontal boundary detection
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STMAS vs. HPC
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STMAS Dennis analysis (850mb)
STMAS (V) Background (V)
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STMAS Dennis analysis (850mb)STMAS (U) STMAS wind
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STMAS: A typhoon test
Analysis ofradial+derived wind
Substract (-)Derived wind
(where it is available)
u v
Analysis of Derived wind Substract (-) Derived wind
(where it is available)
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Intensity: WRF Katrina forecast by STMAS
Wind Barb, Windspeed image,Pressure contour at 950mb Surface pressure
j
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KATRINA LAPS / HRS(2005)
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Windsor tornado case, 22 May 2008
• Tornado touched down at Windsor, Colorado around 17:40 UTC, 22 May 2008
• STMAS initialization 1.67 km 301 x 313 background model: RUC 13km, 17 UTC hot start (cloud analysis)• Boundary conditions: RUC 13km, 3-h RUC forecast (initialized at 15 UTC)• WRF-ARW 1.67 km, 1-h forecast Thompson microphysics
• Postprocessing: Reflectivity
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00-01hr 800mb wind initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS)
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00-01hr wind cross-section initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS)
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00-01hr 800mb reflectivity initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS)
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00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS)
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00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, WRF forecast, RUC vs. STMAS
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Future Modeling Considerations
• Resolution (horizontal & vertical)
• Microphysics Scheme
• Reflectivity Calculation
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Future of Multigrid STMAS
• At each multigrid level, STMAS solves a single 4DVAR just like other 4DVARs;
• At coarser grids, its adjoint of the 4DVARs exists and the adjoint is used;
• At finer grid where there is no adjoint, a non-differentiable optimization method will be used;
• At its finest grid, an EnKF can be applied.
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Summary
STMAS-MG (multigrid) is an efficient, multiscale analysis system;
It can use all possible data sources, including radar, satellite, SFMR and so on;
It can also impose balances or constraints to its analysis directly;
STMAS will become a sequential 4DVAR.
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Comparison of the responsesComparison of RF and Barnes
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81
wavelength (dimensionless)
Response
RF-0.5
RF-0.7
RF-0.9
BN-0.5
BN-0.7
BN-0.9
For each responsefunction of a single3DVAR, a can befound such thatits response is closeto the 3DVAR one.
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Composite of two corrections
The responses of two corrections of both Barnes and 3DVAR:
Successive responses
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81
wavelength (dimensionless)
Responsemulti-rf
multi-bn
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STMAS/LSI Boundary Detection
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