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**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.

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**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

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**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?

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**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

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**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?

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**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

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**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 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?

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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 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

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**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 e1 e2 . en o1 c11 c12 cn c1 o2 c21 c22 c2n c2 on cn1 cn2 cnn EXPECTED COST PROCEDURES OPERATION

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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

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**Criteria CHOICE OF DA SCHEME Results / expectations**

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

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**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

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**Estimating analysis error variances for ensemble initialization**

Malaquías Peña1, Isidora Jankov2 and Zoltan Toth3 1IMSG at EMC/NCEP/NOAA, 2CIRA at GSD, 3GSD/ESRL/NOAA NOAA Earth System Research Laboratory 1. Introduction 4. Application to Ensemble Data Assimilation Systems 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. GSI-ENKF HYBRID EDAS 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 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. 2. Concept The perceived error (forecast minus analysis at the verifying time) variance, is decomposed into the true analysis error variance and the true forecast error variance: 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: f ≡ (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: 5. Tests of the analysis error estimation method SIMULATED FORECASTS WITH THE LORENZ 40 VARS 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. } (1) : 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). 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: where the estimated perceived error variance, is given in (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. GFS 500hPa total energy error at two gridpoints Point in the Extratropics 3. Assumptions 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. 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: 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 ρm = (ρ1)m , m=2,..M (2) where 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. References 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, Lorenz, 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.

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**GSI-ENKF HYBRID EDAS PROPOSED HYBRID EDAS**

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 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.

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**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)

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**CONTROLING NOISE IN ENSEMBLE DATA ASSIMILATION**

Malaquias Peña1 and Zoltan Toth Environmental Modeling Center NCEP/NWS/NOAA 1 SAIC at EMC/NCEP/NOAA Acknowledgements: Mozheng Wei, Takemasa Miyosi, & Roman Krzysztofowicz Pena & Toth, NPG

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**Ensemble-based DA Negative impact of finite ensemble size:**

Analysis ensemble (Xa): Initial conditions and error covariance (Pa) Forecast ensemble (Xb) and error covariance (Pb=B) Observation ( y ) with error (co)variance DYNAMICS Model projects initial state into the future STATISTICS Define forecast error covariance B Adjust B for DA applications Merge Xb with observations (y) Negative impact of finite ensemble size: Ad –Hoc Solutions: Sampling errors make B and Pa noisy leading to a) Spurious long distance correlations b) Filter divergence Localization (e.g. Shur product) NOISE Inflation (multiplicative noise) Additive noise

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**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

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**Hybrid ET The hybrid approach is a regularization strategy Design 3**

After Hamill and Snyder, 2000; α=0.5 B-1 exists NMC ETKF Hybrid Eigenvalues of B Spectrum flatter than ETKF alone The hybrid approach is a regularization strategy

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**ETR Design 3 Statistics Add small (10%) value to diagonal of B**

ET without regularization ET with regularization By construction ET is rank deficient. Regularization allows B to be invertible Retains flow-dependent covariance structure

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**Forecast performance Regular ETKF w / cycled noise Hybrid ETKF 3DVAR**

ETKF w / noise not cycled ETR

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WHAT WE LEARNT 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 …

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**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)

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**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

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**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

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**CURRENT METHODS TO REPRESENT MODEL 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

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**Goal: Represent effect of unresolved processes**

REPRESENTING MODEL RELATED UNCERTAINTY: A STOCHASTIC PERTURBATION (SP) SCHEME General Approach: Add a stochastic forcing term into the tendencies of the model eqs Strategy: Generate the S terms from (random) linear combinations of the conventional perturbation tendencies. Goal: Represent effect of unresolved processes Comparable RMSE Desired Properties of Forcing 1. Applied to all variables 2. Approximately balanced 3. Smoothly varying in space and time 4. Flow dependent 5. Quasi-orthogonal Increased Spread Reduced bias ---- Operation ---- Operation + SP ---- Operation + optimal pp (upper limit) Example of Combination Coefficients Improved Probabilistic Performance Increase Spread Dingchen Hou Reduced number of excessive outliers

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**FINE SCALE ENSEMBLE EXPERIMENTS TO 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

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**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

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**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 Source: Marty Ralph ESRL MAIN MESSAGE: HMT brings forecasters and researchers together to improve precipitation forecasting (amount and type), which has been one of the most challenging meteorological problems to date. Skill scores remain very low (0.29 threat score for 24 h 1 inch QPF) and difficult to improve. Concerted effort is required to advance this, and to achieve the associated benefits in many sensitive sectors from transportation to flood control to water supply. Background: Aimed at accelerating the development, testing and infusion of new technologies, models, and scientific results from the research community into daily forecasting operations of the National Weather Service (NWS) and its River Forecast Centers (RFCs). Interagency planning conducted through the USWRP recommended implementing HMT as a major approach to improving QPF. An HMT proof of concept test was conducted in Current Status: HMT-West planning identified the American River of California as the first test watershed based on several factors (hydrologic model testing was moving to mountains and needed to address snow, Sacramento is at great risk for catastrophic urban inundation due to undersized flood control system, USWRP Planning meeting recommended the area-see BAMS article Nov 2005). HMT-West began its field phase in 2005 and will last through 2009 with variations in activity and emphasis from year to year. HMT is guided by a full-time Project Manager at ESRL (Tim Schneider) and by a cross-NOAA HMT Advisory Panel. Lessons learned about observing systems, models and forecast systems will provide solid scientific and practical basis for investments in long-term infrastructure enhancements. Next Steps: HMT will be implemented incrementally in different regions of the United States. Phase I: HMT West, Cool Season; Phase-II: HMT East, all seasons; Phase-III: HMT Central, warm season. The current plan is provided above, but will be adjusted based on funding levels. Potential Benefits: The HMT Testbed is expected to accelerate the improvements in QPF and QPE, and provide state-of-the-art precipitation data for evaluation of candidate next generation streamflow prediction systems through the “Distributed Model Intercomparison Project-II (DMIP-II). As an example of societal benefits, HMT-West has engaged local decision makers in flood control and reservoir operations to develop next generation QPF that is accurate enough to consider “forecast-based” reservoir operations. This approach could set the stage for major advances in flood control and water supply as the science and practice of precipitation moves from the past era where accuracies were too low to base major reservoir operations on, to an era where advanced QPF can mitigate flood risk while not risking loss of water supply etc.

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**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 season everything stays the same except initial condition perturbations? WRF-ARW and WRF-NMM WRF-ARW runs: Ferrier, Schultz, Thompson microphysics WRF-NMM run: Ferrier microphysics 8 GFS ensemble members provide LBCs for the mixed-model, mixed-physics ensemble One additional member uses WRF-ARW with Thompson microphysics and GFS deterministic run provides LBCs,

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**QPF Example of 24-h QPF 9-km resolution 9 members: ARW-TOM-GEP0**

ARW-FER-GEP1 ARW-SCH-GEP2 ARW-TOM-GEP3 NMM-FER-GEP4 ARW-FER-GEP5 ARW-SCH-GEP6 ARW-TOM-GEP7 NMM-FER-GEP8

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**HMT QPF and PQPF 24-hr PQPF**

48-hr forecast starting at 12 UTC, 18 January 2010 0.1 in. 1 in. 2 in.

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**Reliability of 24-h PQPF Reliability diagrams of 24-h PQPF**

9-km resolution Dec Apr 2010 Observed frequency vs forecast probability Overforecast of PQPF Similar performance for different lead times Brier skill score (BSS): Reference brier score is Stage IV sample climatology BSS is only skilful for 24-h lead time at all thresholds and for 0.01 inch/24-h beyond 24-h lead time. OAR/ESRL/GSD/Forecast Applications Branch 34

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**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

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**interpolated on LAM grid**

Initial Perturbations for HMT-10/11 “Cycling” GEFS (or SREF) perturbations Perturbations LAM forecast driven by global analysis Global Model Analysis interpolated on LAM grid 00Z 06Z 12Z Forecast Time

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Cloud Coverage July UTC LAPS CYC NOCYC 00hr 03hr 06hr

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**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

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**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

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**j Intensity: WRF Katrina forecast by STMAS Wind Barb, Windspeed image,**

Pressure contour at 950mb Surface pressure j

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**Best track: every 6 hours Track: WRF 20km Katrina forecast by STMAS**

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

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**CHALLENGES IN DA IMPROVE 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

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**Cycling Impact on STMAS analysis**

With Without cycling

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**STMAS-WRF ARW cycling Impact**

OAR/ESRL/GSD/Forecast Applications Branch

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**CHALLENGES IN DA ENSURE 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

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**Local Analysis and Prediction System (LAPS)**

Steve Albers, Dan Birkenheuer, Isidora Jankov Paul Schultz, Zoltan Toth, Yuanfu Xie, Linda Wharton OAR/ESRL/GSD/Forecast Applications Branch

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**LAPS DA-Ensemble System**

Data Data Ingest Intermediate data files Error Covariance Trans Traditional GSI Analysis Scheme STMAS3D Trans Post proc1 Post proc2 Post proc3 Model prep WRF-ARW MM5 WRF-NMM Probabilistic Post Processing Ensemble Forecast OAR/ESRL/GSD/Forecast Applications Branch 47 47 47

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**LAPS HOT START INITIALIZATION**

Three-Dimensional Cloud Analysis F H F L METAR + FIRST GUESS

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**Cloud / Reflectivity / Precip Type (1km analysis)**

Obstructions to visibility along approach paths DIA

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**6-hr LAPS Diabatically initialized**

WRF-ARW forecast Analysis 13 June 2002 Developing Squall Line Animation OAR/ESRL/GSD/Forecast Applications Branch 50

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**Thresholded Reflectivity**

Bias & ETS June

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**850 mb Analyzed and Simulated Reflectivity**

Analysis 2hr HOT Fcst 2hr NO-HOT Fcst 16 June 2002 Initialized with LAPS Initialized with NAM Mature Squall Line Animation OAR/ESRL/GSD/Forecast Applications Branch

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**Thresholded Reflectivity**

Bias & ETS June

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**CHALLENGES IN DA ENSURE CONSISTENCY WITH MODEL DYNAMICS 2**

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)

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**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; OAR/ESRL/GSD/Forecast Applications Branch

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**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. OAR/ESRL/GSD/Forecast Applications Branch

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**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. OAR/ESRL/GSD/Forecast Applications Branch

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**CHALLENGES IN DA ENSURE CONSISTENCY WITH MODEL DYNAMICS 2**

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)

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**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 OAR/ESRL/GSD/Forecast Applications Branch

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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 OAR/ESRL/GSD/Forecast Applications Branch

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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 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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**CHALLENGES IN DA ENSURE CONSISTENCY WITH MODEL DYNAMICS 3**

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

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**FORECAST INITIALIZATION FOR IMPERFECT NUMERICAL MODELS**

Zoltan Toth and Malaquias Pena Mendez1 Environmental Modeling Center NOAA/NWS/NCEP USA 1SAIC at Environmental Modeling Center, NCEP/NWS Acknowledgements: Dusanka Zupanski, Guocheng Yuan

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**HOW TO TREAT FORECAST ERRORS IN DA**

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 Move background toward nature IS THIS THE RIGHT MOVE? Treat initial and model error the same way? Move background toward nature

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**HOW TO USE MAPPING IN DA / NWP FORECASTING?**

Challenging step Estimate the mapping between nature and the model attractors Map the observed state of nature into the space of the model attractor Move obs. with mapping vector Analyze data Run the model from the mapped initial condition “Remap” the analysis and forecast back to the phase space of nature New step Standard procedure New step

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**MAPPING VECTOR Vector that provides best remapped forecast performance**

Definition Vector that provides best remapped forecast performance Estimation 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
**Runge-Kutta numerical scheme with a time step of 0.01**

MODEL & DA DETAILS Lorenz (1963) 3-variable model: “NATURE” =10 b =8/3 r =28 “MODEL” =9 z=z+2.5 Runge-Kutta numerical scheme with a time step of 0.01 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

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**RESULTS – CLIMATE MEAN MAPPING**

PERFECT 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

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**Mapped forecasts used in DA yield superior analysis **

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

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**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

73
**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?

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BACKGROUND

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**Noise in EnKF Design 1. Cycles Xb noisy . Typical application**

Forecast (Xb noisy) Xa noisy DA (xa) Forecast (Xb noisy) Noise added Xa noisy Bnoisy Xa noisy Noise added Pa noisy Xa Design 2. Cycles Xb clean . New design Forecast (Xb noisy) Forecast (Xb noisy) Xa noisy Bnoisy Xa noisy noise DA (xa) noise Forecast (Xb clean) Forecast (Xb clean) Xa Bclean Xa clean Pa clean Too expensive to run in an operational environment

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**Noise in EnKF (Cont.) Design 3. Correlated noise in B DA (xa)**

Forecast (Xb) Forecast (Xb) Xa B+Binvertible Xa Pa 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

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**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

78
**Lorenz 96 Model (with ’07 pars)**

Experimental Setting 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) 3-DVar Minimizing the following cost function B is computed using NMC method Ensemble Transform (perturbation method) where and T is a function of eigenvectors and eigenvalues of

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**RMS analysis error Design 1 Design 2 Design 3 Mae=0.42 Mae=0.42**

Evolution of analysis error in three gridpoints (different color lines) for ETKF and ETR; same start. Design 1 ETR with minimal regularization (.001) ETKF with cycling random noise Cycling noise In this situation, ETR ≈ ET Nearly identical Mae=0.42 Mae=0.42 Time (days) Design 2 Design 3 ETKF no cycling random noise ETR with scaled analysis amplitude Mae=0.36 Mae=0.21

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**Snapshot of B and B-1 3DVAR – NMC method ETKF, no inflation**

After 4 cycles 3DVAR – NMC method ETKF, no inflation Well-sampled, stable, fixed (no “errors of the day”) Noisy covariance due to sampling errors. Spurious long-distance correlations. Matrix ill-conditioned. Filter diverges

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**3DVar - NMC method ETKF, no inflation Design 1**

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

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**NMC method ETKF, no inflation Design 2**

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

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NMC method ETKF, no inflation Design 1 Design 2 Cycled noise

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