ENSEMBLE FORECASTING – A NEW PARADIGM IN NUMERICAL WEATHER PREDICTION

ENSEMBLE FORECASTING – A NEW PARADIGM IN

NUMERICAL WEATHER PREDICTION

Zoltan Toth

Global Systems Division, ESRL/OAR/NOAA

Acknowledgements:

Isidora Jankov, Malaquias Pena,Yuanfu Xie, Paula McCaslin, Paul Schultz, Linda Wharton, Roman

Krysztofowicz, Yuejian Zhu, Andre Methot, Tom Hamill, Kathy Gilbert, et al.

INTRODUCTION• Share experience / perspective

– 19 yrs at NCEP/EMC• Operational needs

– Over a year at NOAA research lab (GSD)• Research opportunities

• Review host of NWP research / development issues pursued at NOAA– Some closer, others further from operational use

• Highlight collaborative opportunities– Research thrives on exchange of ideas

OUTLINE / SUMMARY• Why ensembles?

– Knowledge about uncertainty• Complete the forecast with probabilistic information• User needs for covariance / scenarios

• Capturing uncertainty in initial conditions – Focus on dynamical consistency in perturbations– Derive estimate of error variance in best analysis

• New NWP modeling paradigm– Focus on ensemble (not single value) forecasting– Stochastically represent effect of unresolved scales

• Choice of DA scheme– Variational or sequential?– Both can use ensemble-based covariances

• Challenges in data assimilation– Careful attention to many details related to

• Forward operators, control variables, DA/forecast cycle, moist constraints, dynamical consistency (initialization of forecast with imperfect models), etc

– Ensemble-based covariances – “hybrid” mehod?

DATA ASSIMILATION / ENSEMBLE FORECASTING PREAMBLE

• Objective– Probabilistic / ensemble estimate of current /

future state of system• Ensemble of gridded multivariate 3D fields

• Approach– Estimate initial state – Data assimilation

• Represent via ensemble

– Project initial ensemble into future – Ensemble forecasting

• Capture model related uncertainty

NUMERICAL WEATHER PREDICTION (NWP) BASICS

COMPONENTS OF NWP• Create initial condition reflecting state of the atmosphere, land, ocean• Create numerical model of atmosphere, land, ocean

ANALYSIS OF ERRORS• Errors present in both initial conditions and numerical models• Coupled atmosphere / land / ocean dynamical system is chaotic

– Any error amplifies exponentially until nonlinearly saturated– Error behavior is complex & depends on

• Nature of instabilities• Nonlinear saturation

IMPACT ON USERS• Analysis / forecast errors negatively impact users

– Impact is user specific (user cost / loss situation)• Information on expected forecast errors needed for rational decision making

– Spatial/temporal/cross-variable error covariance needed for many real life applications– How can we provide information on expected forecast errors?

WHAT INFORMATION USERS NEED• General characteristics of forecast users

– Each user affected in specific way by• Various weather elements at • Different points in time & • Space

• Requirements for optimal decision making for weather sensitive operation– Probability distributions for single variables

• Lack of information on cross-correlations– Covariances needed across

• Forecast variables, space, and time

• Format of weather forecasts– Joint probability distributions

• Provision of all joint distributions possibly needed by users is intractable– Encapsulate best forecast info into calibrated ensemble members

• Possible weather scenarios – 6-Dimensional Data-Cube (6DDC)

» 3 dimensions for space, 1 each for time, variable, and ensemble members

• Provision of weather information– Ensemble members for sophisticated users

• Other types of format derived from ensemble data– All forecast information fully consistent with calibrated ensemble data

HOW CAN WE REDUCE & ESTIMATE EXPECTED FORECAST ERRORS?

STATISTICAL APPROACH• Statistically assess errors in past unperturbed forecasts (eg, GFS, RUC)

– Can correct for systematic errors in expected value– Can create probabilistic forecast information – Eg, MOS PoP

• Limitation– Case dependent variations in skill not captured– Error covariance information practically not attainable

DYNAMICAL APPROACH – Ensemble forecasting• Sample initial & model error space - Monte Carlo approach

– Leverage DTC Ensemble Testbed (DET) efforts• Prepare multiple analyses / forecasts –

– Case dependent error estimates– Error covariance estimates

• Limitation– Ensemble formation imperfect – not all initial / model errors represented

DYNAMICAL-STATISTICAL APPROACH• Statistically post-process ensemble forecasts

– Good of both worlds– How can we do that?

AVIATION EXAMPLE• Recovery of a carrier from weather related disruptions

– Operational decisions depend on multitude of factors• Based on United / Hemispheres March 2009 article, p. 11-12

• Factors affecting operations– Weather – multiple parameters

• Over large region / CONUS during coming few days– Federal regulations / aircraft limitations

• Dispatchers / load planners– Aircraft availability

• Scheduling / flight planning– Maintenance

• Pre-location of spare parts & other assets where needed– Reservations

• Rebooking of passengers– Customer service

• Compensation of severely affected customers

• How to design economically most viable operations?– Given goals / requirements / metrics / constraints

SELECTION OF OPTIMAL USER PROCEDURES• Generate ensemble weather scenarios ei, i = 1, n• Assume weather is ei, define optimal operation procedures oi

• Assess cost/loss cij using oi over all weather scenarios ej

• Select oi with minimum expected (mean) cost/loss ci over e1,…en as optimum operation

COST/LOSS cij GIVEN ej

WEATHER & oi OPERATIONS

ENSEMBLE SCENARIOS

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o2 c21 c22 . c2n c2

. . . . . .

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DATA ASSIMILATION BASICS• Two distinct objectives

– Reproduce reality as faithfully as possible• May hinder NWP forecast application

– Create initial condition leading to best NWP forecast• Initial state must be consistent with model dynamics

• Technique– Probabilistic / ensemble estimate of current / future state of system

• Ensemble of gridded multivariate 3D fields– Bayesian combination of “prior” & observations

• Must have error estimates for both prior & observations

• Basic functionalities (steps) needed– Relate observations to model variables

• Forward operators– Combine information from various observing systems into “superob”

• For each model variable• Accurate error estimates needed

– Spread effect of observations across time/space/variables• Use dynamical constraints, ensemble-based covariances, etc

– Combine prior and observationally based analysis• Use error estimates

CHOICE OF DA SCHEME• Criteria

– Actual or expected quality of performance• Results / expectations

– Some indications that 4DVAR offers higher quality• Dynamically constraint increments not restricted to ensemble

space• 4Dvar with ensemble-based covariance superior

– Buehner et al – “hybrid” scheme

• How to estimate error variance in 4Dvar analysis?Two approaches• Run ensemble-based DA

– Very expensive– Not 4Dvar, but ensemble-based DA errors are

estimated• Estimates affected by DA methods/assumptions

• Alternative approach– Based on basic assumptions independent of DA

schemes– Under development / testing

ESTIMATING & REPRESENTING INITIAL CONDITION RELATED UNCERTAINTY

• Objective– Make initial perturbations consistent with

uncertainty in analysis

Two approaches available• Variational DA

– Estimate uncertainty in 3/4Dvar analysis– Initialize ensemble with estimated analysis error

variance• Ensemble-based DA

– Use one of several ensemble-based DA schemes

Accurate estimates of analysis error variances is critical for the proper initialization of ensembles. This variance is the initial uncertainty that the ensemble perturbations try to mimic. Because of large computational costs, not all DA schemes explicitly compute analysis error variances (e.g., GSI does not). Furthermore, estimates of analysis errors derived via DA schemes are influenced by the assumptions used to create the analysis fields, resulting in a scheme-dependent analysis error. For example, in regions where observations are scarce or the DA scheme gives low weights to them, the analysis errors will be highly correlated with the first guess errors, making it difficult to estimate the true errors (Simmons and Hollinsworth, 2002). A methodology for the estimation of analysis and forecast errors is introduced here. The method is based on a few simple assumptions that are independent of any data assimilation method and provides error estimates with a range of uncertainty.

A schematic showing an EDAS scheme where the ensemble (based on ET) produce a flow-dependent error covariance matrix that is combined with the static covariance matrix generated by a variational DA (GSI) and, in turn the analysis error-variance is used to initialize the Ensemble forecast scheme. The method introduced in this paper allows a scheme-independent estimation of error covariances to initialize the ensemble.

Estimating analysis error variances for ensemble initializationMalaquías Peña1, Isidora Jankov2 and Zoltan Toth3

1IMSG at EMC/NCEP/NOAA, 2CIRA at GSD, 3GSD/ESRL/NOAA

NOAA Earth SystemResearch Laboratory

))((2)()()( 2222 TATFTATFAFd

where d is the perceived root mean square error, F is the forecast, A is the

analysis, T is the true state and ρ is the correlation between true forecast error

and true analysis error.

Defining the true analysis error-variance: f02 ≡ (A-T)2

and the true forecast error-variance: flead2 ≡ (F-T)2,

the perceived forecast error variances measured on each lead time can be estimated via the following set of equations:

2. ConceptThe perceived error (forecast minus analysis at the verifying time) variance, is decomposed into the true analysis error variance and the true forecast error variance:

The following assumptions are used to simplify the problem:

1. Small initial errors grow exponentially and saturate following a logistic function. Therefore, the evolution of errors can be parameterized with a minimal number of parameters that can be obtained via observations of perceived errors. Departures from this evolution of errors will be attributed to model errors, which will be modeled with a continuous function.

2. At short lead times, errors are local. That is, we ignore advection of errors from neighboring gridpoints. At long lead times, errors at any gridpoint results from the influence of all surrounding gridpoints.

3. The correlation between analysis errors and forecast errors decreases on each analysis cycle at a power rate:

1. Introduction

The goal is to estimate the true analysis and forecast error-variance consistent with the perceived error variance observed on each lead time. This can be expressed as a minimization of the following J:

0612

02

626 2ˆ ffffd hhh

01222

02

122

12 2ˆ ffffd hhh

:

...,18,12,6),ˆmax( 122 hhhiwddJ iii }where the estimated perceived error variance,

2ˆid is given in (1).

3. Assumptions

ρm = (ρ1)m , m=2,..Mwhere ρ1 is the correlation at 6h lead time, ρ2 =(ρ1)2 is the correlation at 12h lead time, ρ3 = ρ1 ρ2 , is the correlation at 18h, etc. Only one parameter (ρ1) needs to be determined.

4. Application to Ensemble Data Assimilation Systems

5. Tests of the analysis error estimation method

A schematic showing an Ensemble Data Assimilation scheme where two assimilation schemes, one flow-dependent and the other static, are run and combined to produce a hybrid analysis. In this scheme, the analysis error estimation (derived from the EnKF scheme) is fed into the ensemble generation scheme but does not reflect the analysis error from the GSI.

Lagarias, J.L., J. A. Reeds, M.H. Wrights and P.E. Wright, 1998: Convergence properties of the Nelder-Mead Simplex Method in Low dimensions, SIAM J. Optim., 9, 112-147Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci. 20: 130–141. Simmons, A. J. and A. Hollingsworth, 2002: Some aspects of the improvement in skill of numerical weather prediction. Quart. J. Roy. Meteor. Soc., 128:647–677.

Top panel. In blue: True Error variance; in red: Perceived Error variance; in green: the modeled Perceived Error variance. Bottom panel. In black: Anomaly correlation of analysis errors and forecast errors at different leads as generated from the model; in magenta: correlation using (2).GFS 500hPa total energy error at two gridpoints

(2)

SIMULATED FORECASTS WITH THE LORENZ 40 VARS

References

We apply the method to estimate analysis and forecast errors in the 3-variables Lorenz model (Lorenz 1963) under a perfect model scenario. Synthetic data is generated from the control run (nature) plus a random value. A 3DVar scheme is used to assimilate the data.

Left: Estimate of true forecast error variance (f, blue)

and the analysis error (f0) as estimated by the method,

and the fitting curve (dhat) on the perceived error (d). Right panels: The diagnosed correlation and the fitting error.Note that the optimization procedure (the simplex method; Lagrarias et al., 1998) produces a very good fit with the observed perceived errors.

Point in the Extratropics

Point in the Tropics Same as above but for a point in the tropics. Compare the true forecast error variances on left panels and note that the true errors are underestimated in the tropics. Right panels: Compare the correlation function and note that analysis and forecasts are much more correlated in the tropics than in the extratropics

(1)

wi is a weighting function to ensure that the fitting is best at the initial time, where we have most confidence of the measurements.

PROPOSED HYBRID EDAS

GSI-ENKF HYBRID EDAS

A schematic showing an EDAS scheme where the ensemble (based on ET) produce a flow-dependent error covariance matrix that is combined with the static covariance matrix generated by a variational DA (GSI) and, in turn the analysis error-variance is used to initialize the Ensemble forecast scheme. The method introduced in this paper allows a scheme-independent estimation of error covariances to initialize the ensemble.

A schematic showing an Ensemble Data Assimilation scheme where two assimilation schemes, one flow-dependent and the other static, are run and combined to produce a hybrid analysis. In this scheme, the analysis error estimation (derived from the EnKF scheme) is fed into the ensemble generation scheme but does not reflect the analysis error from the GSI.

PROPOSED HYBRID EDAS

GSI-ENKF HYBRID EDAS

ENSEMBLE INITIALIZATION

• Objectives– Perturbation variance reflect analysis error variance– Covariance reflect error dynamics

• Minimize noise in ensemble

• Current approaches – Gaussian, linear– Ensemble DA (EnKF, ETKF, EnSRF, etc)

• Need to control filter divergence, spurious correlations• Inflate variance, localize covariances

– Noise added in process

– Ensemble Transform method• Dynamical consistency• Need good error variance estimate• Ad hoc localization of variance (Rescaling)

CONTROLING NOISE IN ENSEMBLE DATA ASSIMILATION

Malaquias Peña1 and Zoltan Toth

Environmental Modeling CenterNCEP/NWS/NOAA

1 SAIC at EMC/NCEP/NOAA

Acknowledgements:

Mozheng Wei, Takemasa Miyosi, & Roman Krzysztofowicz

Pena & Toth, NPG

Ensemble-based DAAnalysis ensemble

(Xa): Initial conditions and error covariance

(Pa)

Observation ( y ) with error (co)variance

Forecast ensemble (Xb) and error covariance

(Pb=B)

DYNAMICSModel projects initial state into the future

Sampling errors make B and Pa noisy leading to

a) Spurious long distance correlations

b) Filter divergence

Ad –Hoc Solutions:Negative impact of finite ensemble size:

Localization (e.g. Shur product)

• Inflation (multiplicative noise)• Additive noise

NO

ISE

STATISTICS • Define forecast error covariance B • Adjust B for DA applications• Merge Xb with observations (y)

Traditional mitigation efforts

Ad hoc noise is inserted in EnKF procedure to reduce negative effects of sampling error in B

…but forecasts from noisy initial states have sub-optimal performance.

ALTERNATIVE APPROACH• Dynamical cycling of ensemble perturbations

– Avoid addition of noise into ensemble forecasts• Do not feed back noise into ensemble forecasts• Preserve relevant info on dynamics of system

– Minimize statistical effects on forecasts• Manipulation of Pb should only minimally affect Pa

=> Pb reflects dynamics of system

Hybrid ET

The hybrid approach is a regularization strategy

NMC ETKF Hybrid

Spectrum flatter than ETKF alone

Eigenvalues of B

B-1 exists

After Hamill and Snyder, 2000; α=0.5Design 3

ETR

StatisticsAdd small (10%) value to diagonal of B

ET without regularization

ET with regularizationBy construction ET is rank deficient. Regularization allows B to be invertible

Retains flow-dependent covariance structure

Design 3

Forecast performance

ETR

3DVAR

Regular ETKF w / cycled noise

ETKF w / noise not cycled

Hybrid ETKF

• Ensemble-based forecast error covariance has sampling error• Addition of random noise, inflation, or localization

– Reduces rank deficiency (less ill-conditioned) BUT– Introduces noise wrt dynamics

• If cycled => Suboptimal covariance & forecast performance

• Alternatives tested: – Noise added to B to reduce rank deficiency is NOT cycled

• Two sets of first guesses – effective but expensive

– Ensemble Transform with Regularization of B (ETR) • Affects mainly the variance structure of Pa

– Minimal effect on covariance

• Superior performance with large ensemble– Effect is expected to be less with realistically small ensemble

• Particle Data Assimilation Coming …

WHAT WE LEARNT

ENSEMBLE INITIALIZATION – NEW APPROACHES

• Can rescaling method be revised?– To mimic background error reducing effect of DA?

• Work in progress• Still Gaussian / linear approach

• Quest for non-Gaussian ensemble formation– Need for highly nonlinear situations

• E.g, convective processes, Tropical Cyclone development– Positive impact expected both for

• Ensemble performance• Data assimilation (cleaner covariances)

MODEL RELATED UNCERTAINTIES

• Origin– Due to truncation / approximations

• Finite spatial resolution• Finite time steps• Approximations in physics

• Forecast impact– Random errors added at each time step

• Can be considered only stochastically

• Design of current generation models– Aimed at making best single (unperturbed) forecast

• Minimize RMS error– Intentionally ignores effect of unresolved processes

• Fine spatial scales• Fine time scales• Full physics

– No stochastic representation of unresolved processes

NEW NWP MODELING PARADIGM

• Ensemble application– Effect of unresolved processes must be represented

• Otherwise ensemble cloud misses reality

• Approach– Stochastically simulate (generate) variance equal to

error associated with each process truncated /approximated in model

• Major change in modeling approach– Focus on / test in ensemble

• Instead of single unperturbed forecast– Build stochastic element to represent model related

errors into each model component• Major effort needed

CURRENT METHODS TO REPRESENTMODEL RELATED ERRORS

• Multi-model/physics (Houtekamer et al)– Ad hoc and pragmatic approach– Unique & distinct solutions

• Cannot provide continuum of realizations– No scientific foundation

• Giving up ideal of capturing nature in cognizant manner

• Stochastic perturbations added (Buizza et al)– Formal (not fully informed) response to need– Random noise has very limited effect

• Structured noise used at NCEP (Hou et al)

• Stochastic physics (Teixeira et al)– Right approach?– Capture/simulate, not suppress effect of unresolved

processes

REPRESENTING MODEL RELATED UNCERTAINTY:

A STOCHASTIC PERTURBATION (SP) SCHEMEGeneral Approach: Add a stochastic forcing term into the tendencies of the model eqsStrategy: Generate the S terms from (random)

linear combinations of the conventional

perturbation tendencies.

Desired Properties of Forcing1. Applied to all variables2. Approximately balanced 3. Smoothly varying in space and time4. Flow dependent5. Quasi-orthogonal

Example of Combination Coefficients

Reduced number of excessive outliers

Reduced bias

Comparable RMSE

Increase Spread

Increased Spread

---- Operation---- Operation + SP---- Operation + optimal pp (upper limit)

ImprovedProbabilistic Performance

Goal: Represent effect of unresolved processes

Dingchen Hou

FINE SCALE ENSEMBLE EXPERIMENTSTO CAPTURE MODEL RELATED UNCERTAINTY

• Recognize importance of microphysics for moist processes

• Capture model related forecast uncertainty

• Hydrometeorological Testbed (HMT) ensemble

• Two model cores, various microphysics schemes

Ensemble Prediction System Development for Aviation and other Applications

Isidora Jankov1, Steve Albers1, Huiling Yuan3, Linda Wharton2, Zoltan Toth2, Tim

Schneider4, Allen White4 and Marty Ralph4

1Cooperative Institute for Research in the Atmosphere (CIRA),Colorado State University, Fort Collins, CO

Affiliated with NOAA/ESRL/ Global Systems Division

2NOAA/ESRL/Global Systems Division

3Cooperative Institute for Research in Environmental Sciences (CIRES)University of Colorado, Boulder, CO

Affiliated with NOAA/ESRL/Global Systems Division

4 NOAA/ESRL/Physical Sciences Division

BACKGROUND

• Objective• Develop fine scale ensemble forecast system

• Application areas• Aviation (SF airport)• Winter precipitation (CA & OR coasts)• Summer fire weather (CA)

• Potential user groups• Aviation industry, transportation, emergency and

ecosystem management, etc

EXPERIMENTAL DESIGN 2009-2010

Nested domain: • Outer/inner nest grid spacing 9 and 3 km, respectively.• 6-h cycles, 120hr forecasts foe the outer nest and 12hr forecasts for the inner nest • 9 members (listed in the following slide)• Mixed models, physics & perturbed boundary conditions from NCEP Global Ensemble

• 2010-2011 season everything stays the same except initial condition perturbations?

QPF

32

Example of 24-h QPF9-km resolution

9 members:ARW-TOM-GEP0ARW-FER-GEP1ARW-SCH-GEP2ARW-TOM-GEP3NMM-FER-GEP4ARW-FER-GEP5ARW-SCH-GEP6ARW-TOM-GEP7NMM-FER-GEP8

HMT QPF and PQPF24-hr PQPF

0.1 in.

1 in.

2 in.

48-hr forecast starting at 12 UTC, 18 January 2010

Reliability of 24-h PQPF

34 34http://esrl.noaa.gov/gsd/fab

OAR/ESRL/GSD/Forecast Applications Branch

Reliability diagrams of 24-h PQPF 9-km resolution Dec 2009 - Apr 2010

Observed frequency vs forecast probabilityOverforecast of PQPFSimilar performance for different lead times

Brier skill score (BSS):Reference brier score is Stage IV sample climatologyBSS is only skilful for 24-h lead time at all thresholds and for 0.01 inch/24-h beyond 24-h lead time.

CYCLING FINE SCALE PERTURBATIONS

• How to create dynamically conditioned fine scale perturbations consistent with forcing from global ensemble?

• Current approach• Interpolate global perturbed initial conditions• Fine scale motions missing initially

• Need to spin up• UKMet, Canadian, part of NCEP SREF etc ensembles

• Cycle LAM perturbations

Initial Perturbations for HMT-10/11“Cycling” GEFS (or SREF) perturbations

36

00Z 06Z

Global Model Analysis interpolated on LAM grid

LAM forecast driven by global analysis

Forecast Time12Z

Perturbations

37

Cloud CoverageJuly 30 2010 00UTC

00hr

03hr

06hr

LAPS CYC NOCYC

ON THE HORIZON:COUPLED DA – ENSEMBLE SYSTEMS

• Analysis / forecast error estimation independent of DA schemes• Poster results

• Rescaling of global perturbations consistent with DA

• Nonlinear / non-Gaussian initial perturbations via Bayesian particle filters• Coupled with 4-DVAR

• Bayesian particle filter for nonlinear / non-Gaussian DA / ensemble forecast system

CHALLENGES IN DA - RELATE OBSERVATIONS TO MODEL VARIABLES

• Critical step in relating reality to NWP model• Wide range of observing systems /

instruments / sensors• Construct “forward operators”

• Tedious but important work

• Requires detailed knowledge about observing system and model

• Examples - Relate• Radar reflectivity to

• Convective processes in model

• Radar radial wind to• 3-dimensional wind structure

Intensity: WRF Katrina forecast by STMAS

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

j

Track: WRF 20km Katrina forecast by STMAS

Best track:every 6 hours

WRF-ARW 72 hour fcst w/Ferrier physics:every 3 hours

CHALLENGES IN DAIMPROVE PRIOR / BACKGROUND FIELD

• Background must contain all information available• Prior to latest observations

• Ideally a short range NWP forecast from latest analysis• As used in global DA

• Only limited success with Limited Area Model (LAM) applications• Full LAM DA/forecast cycling attempts fail• Noise around boundary conditions amplify via cycling?

• Current approach• Periodically cold-start LAM analysis cycle from global analysis

• E.g., NAM, RUC at NCEP• Scientifically unsatisfactory, suboptimal performance

• Promising experiments at GSD• Use lateral boundary as constraint in LAM analysis• Bring in dynamically consistent fine scale info from LAM

background

Cycling Impact on STMAS analysis

Without Withcycling

STMAS-WRF ARW cycling Impact


CHALLENGES IN DAENSURE CONSISTENCY WITH MODEL DYNAMICS 1

• Dynamically inconsistent information is lost - insult• Quick transitional process introduces additional

errors – injury

• Possible approaches• Balance constraints – widely used• Digital filter – E.g., Rapid Refresh cycle at GSD• 4-DVAR – used in global DA• Additional constraints

• Local Analysis and Prediction System (LAPS) “Hot Start”• “Conceptual” relationships among moist & other variables

Local Analysis and Prediction System (LAPS)

Steve Albers, Dan Birkenheuer, Isidora Jankov

Paul Schultz, Zoltan Toth, Yuanfu Xie, Linda Wharton

http://esrl.noaa.gov/gsd/fabOAR/ESRL/GSD/Forecast Applications Branch

Data Ingest

Intermediatedata files

Traditional GSI

Data

STMAS3D

Trans

Trans

Post proc1 Post proc2 Post proc3

Model prep

WRF-ARW MM5 WRF-NMM

EnsembleForecast

ProbabilisticPost

Processing

ErrorCovariance

LAPS DA-Ensemble

System

47http://esrl.noaa.gov/gsd/fabOAR/ESRL/GSD/Forecast Applications Branch

Analysis Scheme

Three-Dimensional Cloud Analysis

METAR

LAPS HOT START INITIALIZATION

FH

FL

0ˆ T

sqq

c

+ FIRST GUESS

Cloud / Reflectivity / Precip Type (1km analysis)

DIA

Obstructions to visibility along approach paths

Analysis6-hr LAPS Diabatically initialized

WRF-ARW forecast

13 June 2002

50http://esrl.noaa.gov/gsd/fabOAR/ESRL/GSD/Forecast Applications Branch

Developing Squall Line Animation

Thresholded ReflectivityBias & ETS June 13 2002

2hr HOT FcstAnalysis 2hr NO-HOT Fcst

850 mb Analyzed and Simulated Reflectivity

16 June 2002


Mature Squall Line Animation

Initialized with LAPS Initialized with NAM

Thresholded ReflectivityBias & ETS June 16 2002


• Choice of control variable affects increments

• Integrated variables like streamfunction & velocity potential poor choice• To preserve integrated quantities, analysis must

introduce spurious fine scale fluctuations to compensate for observation related changes

• Use vorticity and divergence (or u,v)

Control Variable Issues

Xie et al 2002 has studied the analysis impact by selecting different control variables for wind, ψ-χ, u-v, or ζ-δ VARS.

Conclusions:• ζ-δ is preferable (long waves on OBS; short on

background);• u-v is neutral; • ψ-χ is not preferred: short on OBS; long on

background;


Non-physical error by ψ-χ VARS

• Ψ and χ are some

type of integral of

u-v. A correction to

the background toward the obs changes the integral of u. The background term adds opposite increment to the analysis for keeping the same integral as the background.


Responses of three type of VARS

For a single obs of u, e.g., ψ-χ, u-v, or ζ-δ VARS have different responses as shown here. A ψ-χ VAR may not show this clearly as many filters applied.



• Background error covariance info used to spread observational info across space/time/variables

• Problems with estimation & use of covariance abound• Poor estimates from small sample of lagged or

ensemble forecasts• Long distance correlation estimates on small scales can

be spurious

• Possible solution• Multiscale analysis by Yuanfu Xie et al• Space-Time Multiscale Analysis System (STMAS)

Space and Time Multiscale Analysis System (STMAS)

Yuanfu Xie, Steve Albers, Brad Beechler, Huiling Yuan

Contributors:

GSD: Dan Birkenheuer, Zoltan Toth, Steve Koch

UC Boulder: Tomoko Koyama

National Marine Data and Information Service (NMDIS): Wei Li, Zhongjie He, Dong Li

Central Weather Bureau (CWB): Wenho Wang, Jenny Hui

NCAR: Paul Beringer

Collaborators:

MIT/Lincoln Lab, CWB, NMDIS, NOAA/MDL and NOAA/ARL


What is STMAS?• STMAS is a multiscale analysis based on a multi-grid technique

• Starts analyzing large, then progressively finer scales

• Retrieves resolvable information from observations with model constraints on its coarse grid analysis

• Becomes a standard 4DVAR except with a narrow banded error covariance matrix at the finest scale gridded analysis.

• Features• Improved computational efficiency• Multi-scale features resolved• Expected to improve 4DVAR analysis given imperfect error covariance

estimation

• Can use ensemble-based background error covariances


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

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


• What is best approach to initialize imperfect models?• Systematic difference between reality and its model

• Traditional approach• Remove systematic model error from background

forecast

• Introduce new paradigm• “Map” observations onto model attractor• Assimilate data in model attractor space• Run numerical forecast• “Remap” forecast state back to space of reality

65

FORECAST INITIALIZATION FOR IMPERFECT NUMERICAL MODELS

Environmental Modeling Center

NOAA/NWS/NCEP USA

1SAIC at Environmental Modeling Center, NCEP/NWS

Acknowledgements: Dusanka Zupanski, Guocheng Yuan

http://wwwt.emc.ncep.noaa.gov/gmb/ens/index.html

Zoltan Toth

and

Malaquias Pena Mendez1

66

• CHAOTIC ERRORS– Statistical approach

• “NMC” method (differences between past short-range forecasts verifying at same time)

• Ensemble method (differences between past ensemble forecasts verifying at same time)

– Dynamical approach• 4DVAR – Norm dependent adjustments

• Ensemble-based DA (large ensemble of current forecasts) – Norm-independent adjustments

• MODEL-RELATED ERRORS– 3 approaches used to cope with model errors in DA:

• Assume model-related errors don’t differ from chaotic errors (Ignore problem)

– Inflation of background errors (ie, move analysis closer to observations)» Multiply background error covariance matrix in 3/4DVAR» Increase initial perturbation size in ensemble-based DA

• Assume model-related errors are stochastic with characteristics different from chaotic errors (D. Zupanski et al)

– Introduce additional (model) error covariance term (allow analysis to move closer to obs.)

– How statistics determined?

• Assume errors are systematic (Dee et al)

– Estimate systematic difference between analysis and background

– Before their use, move background by systematic difference closer to analysis

HOW TO TREAT FORECAST ERRORS IN DA

Move background toward nature

Move background toward nature

IS THIS THE RIGHT MOVE?Treat initial and model error the same way?

67

HOW TO USE MAPPING IN DA / NWP FORECASTING?

1. Estimate the mapping between nature and the model attractors

2. Map the observed state of nature into the space of the model attractor– Move obs. with mapping

vector

3. Analyze data4. Run the model from the

mapped initial condition5. “Remap” the analysis and

forecast back to the phase space of nature

Standard procedure

New

ste

pN

ew s

tep

Challenging step

68

MAPPING VECTOR

• Definition

– Vector that provides best remapped forecast performance

• Estimation

1. Difference between long term time means of forecast trajectory & nature – Evaluate at observation sites (H)

In practice, nature is not known• Use traditional analyses as proxy:

2. Adaptive technique• If systematic errors are regime dependent, or climate means are

not available• Details later

69

MODEL & DA DETAILS• Lorenz (1963) 3-variable model:

bzxydt

dz

xzyrxdt

dy

xydt

dx

)( “NATURE”

=10

b =8/3

r =28

“MODEL”

=9

z=z+2.5

• Three initialization schemes

– Perfect initial conditions– Replacement

• All 3 variables observed• Observational error = 2 (~5% natural variability)

– 3-DVAR:• 15 time step cycle length (~7 hrs in atmosphere)• Diagonal R, R=2• B based on independent forecast errors, empirically tuned

variance

Runge-Kutta numerical scheme with a time step of 0.01

))()(())()(()()(2 11 xHyMRxHyMxxBxxJ TfTf

70

RESULTS – CLIMATE MEAN MAPPINGPERFECT INITIAL CONDITIONS

67% error reduction

• Except for very short lead time, mapped forecast beats traditional forecast

• Remapped forecast beats traditional forecast at all leads

• Effect of bias correction negligible (except at short lead time)

• Drift-induced errors much reduced• Shadowing period extended 3-fold

71

CONCLUSIONS

• “Perfectly” known state is not best initial condition for imperfect models

• Intentionally moving initial condition away from nature, toward model attractor yields superior forecast

• Mapped forecasts used in DA yield superior analysis• Adaptive, regime dependent mapping vector

estimation needs less data and yields improved analysis/forecast performance

Taking a step back brings us closer to reality

• If, like a fly, attracted too close to the light you get burnt;

• By staying back, we can better understand / simulate nature

OPPORTUNITIES FOR COLLABORATION

Beyond ensemble topics

• Dynamical downscaling

• Variational moist data assimilation (“hot start”)

• NWP forecast initialization; cycling LAM DA/forecasts

• Control variables

• Satellite and GPS data assimilation• Chinese and Japanese Geostationary satellite

• Low level wind and solar radiation

OUTLINE / SUMMARY• Why ensembles?

– Knowledge about uncertainty• Complete the forecast with probabilistic information• User needs for covariance / scenarios

• Capturing uncertainty in initial conditions – Focus on dynamical consistency in perturbations– Derive estimate of error variance in best analysis

• New NWP modeling paradigm– Focus on ensemble (not single value) forecasting– Stochastically represent effect of unresolved scales

• Choice of DA scheme– Variational or sequential?– Both can use ensemble-based covariances

• Challenges in data assimilation– Careful attention to many details related to

• Forward operators, control variables, DA/forecast cycle, moist constraints, dynamical consistency (initialization of forecast with imperfect models), etc

– Ensemble-based covariances – “hybrid” mehod?

BACKGROUND

Noise in EnKF

Noise added

Forecast (Xb noisy)

Xa

Xa noisy Bnoisy

DA(xa)

Design 1. Cycles Xb noisy . Typical application

Design 2. Cycles Xb clean . New design

Noise added

Forecast (Xb noisy)

Xa noisy

Xa noisy

noise

Forecast (Xb noisy)

Xa

Xa noisy Bnoisy

Forecast (Xb clean)Bclean

Pa clean

DA(xa)

noise

Forecast (Xb noisy)

Xa clean

Xa noisy

Forecast (Xb clean)

Pa noisy

Too expensive to run in an operational environment

Noise in EnKF (Cont.)

Forecast (Xb)Xa B+Binvertible

DA(xa)

Design 3. Correlated noise in B

Forecast (Xb)Xa

Examples for Design 3:

A) Hybrid ensemble Kalman Filter (Hamill and Snyder, 2000)

• Bhybrid =(1-α)BEnKF+αB3DVar

• α is a weighting parameter between 0 (only flow-dependent) and 1 (only 3DVar)

• Variance and covariance of B equally affected

B) New approach: Regularization (Tikonov, 1943) procedure for B

• Brid = BEnKF+ λDiag(BEnKF)

• λ is a ridging parameter

• Variance is affected primarily (that is later scaled back)

• Covariance only minimally affected

Pa

ET with Regularization (ETR)• ET is an ensemble perturbation method (produces Xa (=XfT) )• Since B can be formed from Xf, and R-1 is known, the analysis

precision relationship:

can be added to ET for DA applications• When B-1 exists, ET plus the analysis precision relationship is

equivalent to ETKF (Bishop et al 2001)• By construction, however, ET produces a B that is rank

deficient, thus B-1 does not exist (ETKF framework does not constrain B to be invertible)

• A regularization procedure is implemented to reduce the rank deficiency problem in ET, thus ETR

111 RBA

Experimental Setting

• 3-DVar Minimizing the following cost function

)]()()()[()( HxyRHxyxxBxxxJ oTobb 11

2

1

• Lorenz 96 Model (with ’07 pars)

where F= 5.1 and m=1,..,21

• Perfect model scenario. One observation per grid-point. 6-hr assimilation cycle• Uncorrelated observational error: normally distributed random noise with unit variance (R=I)

• Ensemble Transform (perturbation method)

• B is computed using NMC method

TXX fa xxxxxxX

K211K

1,...,,where

fTf A XX 1)(and T is a function of eigenvectors and eigenvalues of

RMS analysis error

ETKF with cycling random noise

ETR with minimal regularization (.001)

ETR with scaled analysis amplitude

Evolution of analysis error in three gridpoints (different color lines) for ETKF and ETR; same start.

ETKF no cycling random noise

Time (days)

In this situation, ETR ≈ ET

Mae=0.42Nearly

identical

Mae=0.21

Mae=0.42

Mae=0.36

Design 1

Design 2 Design 3

Cycling noise

Snapshot of B and B-1

Well-sampled, stable, fixed (no “errors of the day”)

ETKF, no inflation

Noisy covariance due to sampling errors. Spurious long-distance correlations. Matrix ill-conditioned.Filter diverges

3DVAR – NMC method

After 4 cycles

3DVar - NMC method

ETKF with random perturbations added to each member before performing time integration. Inverse becomes stable Prevents filter divergence; However, noise cycled

ETKF, no inflation

Design 1

Two ensembles run with ETKF, one with (used for estimating B, then discarded), another without addition of noise (used for cycling covariance); Noise still impacts initial perturbations

NMC method ETKF, no inflation

Design 2

NMC method ETKF, no inflationC

ycle

d no

ise

Design 1 Design 2

Documents

ENSEMBLE FORECASTING – A NEW PARADIGM IN NUMERICAL WEATHER PREDICTION