Space and Time Multiscale Analysis System A sequential variational approach

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

Outline

• Applications;• Sequential variational analysis approach;

• Multigrid implementation;• Numerical results;• Summary.


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


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.


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( )


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:


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.


Background

Observation Longer wave

Background

Observation Longer wave

Resolvable Information for a Given Observation Network

Difference on longer wave Difference on shorter wave


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.


An idealized multiscale caseLeft: Mesonet surface stations; Right: An analysis function


An idealized multiscale case (cont.)


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( )


=0.5 0.7 0.9

A single 3DVAR with different

These analyses tend toapproximate the truth:


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.


Comparison:

Single 3DVARWith =0.5 or 0.9

STMAS-RF


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

∑


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.


Multigrid TechniqueUsing the number of gridpoints to control

the base functions.


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.


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.


Different Implementations of STMAS

Recursive filter Wavelet Multigrid


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


FAA/MIT Boundary Detection Application


STMAS for frontal boundary detection


STMAS vs. HPC


STMAS Dennis analysis (850mb)

STMAS (V) Background (V)


STMAS Dennis analysis (850mb)STMAS (U) STMAS wind


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)


Intensity: WRF Katrina forecast by STMAS

Wind Barb, Windspeed image,Pressure contour at 950mb Surface pressure

j


KATRINA LAPS / HRS(2005)


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

9/22/2009 5th Oceanic D-A Workshop

00-01hr 800mb wind initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS)


00-01hr wind cross-section initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS)


00-01hr 800mb reflectivity initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS)


00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS)


00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, WRF forecast, RUC vs. STMAS


Future Modeling Considerations

• Resolution (horizontal & vertical)

• Microphysics Scheme

• Reflectivity Calculation


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.


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.


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.


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


STMAS/LSI Boundary Detection


Space and Time Multiscale Analysis System A sequential variational approach

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