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Forecasts & their Errrors

This study explores forecast errors in meteorological models, analyzing the factors that contribute to these errors and their implications. The research highlights the importance of probabilistic forecasting, model evaluation, and data assimilation for improving forecast accuracy.

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Forecasts & their Errrors

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  1. Forecasts & their Errrors Thanks to: Stefano Migliorini (NCEO), Mark Dixon (MetO), Mike Cullen (MetO), Roger Brugge (NCEO) Horiz. winds and pressure, at 5.5 km Met Office North Atlantic/European LAM Ross Bannister National Centre for Earth Observation (the Data Assimilation Research Centre) Forecast Possible error in forecast

  2. Forecast errors • Forecast errors (from a numerical model): • are a fact of life! • depend upon the model formulation, • synoptic situation (‘flow dependent’), • model’s initial conditions, • length of the forecast. • are impossible to calculate in reality, δx = xf - xt. • Of interest: • forecast error statistics - the probability density fn. of xt, Pf(xt). • Applications: • probabilistic forecasting. • model evaluation/monitoring. • state estimation (data assimilation).

  3. Seminar structure • Probability density functions (PDFs) of the state, Pf(xt). • The use of Pf(xt) in data assimilation problems. • Measuring Pf(xt). • Modelling Pf(xt) for large-scale data assimilation. • Refining Pf(xt) for large-scale data assimilation. • Challenges for small-scale Meteorology.

  4. σ = √var(δx) PDF of state, Pf(xt) Forecast comprising a single number Pf(xt) xf = xt + δx 0 xt xf Probable state Impossible state Possible but unlikely state

  5. σx2 = √var(δx2) σx1 = √var(δx1) cov(δx1,δx2) Two-component state vector Forecast comprising two numbers xf = xt + δx Pf(xt) 0

  6. u –– v –– p –– T –– q xf = Geophysical error covariances – B • The B-matrix • specifies the PDF of errors in xf(Gaussianity assumed) • describes the uncertainty of each component of xf and • how errors of elements in xf are correlated • is important in data assimilation problems 107 – 108 elements δu δv δp δT δq B = δu δv δp δT δq 107 – 108 elements structure function associated (e.g.) with pressure at a location

  7. Example standard deviations (square-root of variances) From Ingleby (2001)

  8. Example geophysical structure functions (covariances with a fixed point) Univariate structure function Multivariate structure functions

  9. t = 0 1.0 0.9 0.8 0.7 t > 0 1.0 0.9 0.8 0.7 Covariances are time dependent Structure function for tracer in simple transport model

  10. Pf(xt) \ B in data assimilation • Data assimilation combines the PDFs of • forecast(s) from a dynamical model, Pf(xt)and • measurements, Pob(y|xt) • to allow an ‘optimal estimate’ to be found (Bayes’ Theorem). • Maximum likelihood solution (Gaussian PDFs) PDF of combination of forecast and observational information forecast = prior knowledge Solved e.g. by direct inversion or by variational methods

  11. x(0) initial conditions y(t1) y(t2) y(t3) y(t4) y(t5) Tracer assimilation –– sources/ sinks Tracer + source/sink assimilation Data assimilation example(for inferred quantities) T, q, O3satellite radiances Initial conditionsinferred frommeasurements made at a later time Sources/sinks of tracer, rmeasurements of tracer r 30-day assimilation Pseudo satellite tracks

  12. Dangers of misspecifying Pf(xt) \ B in data assimilation? Example 1: Anomalous correlations of moisture across an interface Example 2: Anomalous separability of structure functions around tilted structures Normally dry air Normally moist air

  13. Measuring Pf(xt) \ B Forecast errors are impossible to measure in reality, δx = xf - xt. All proxy methods require a data assimilation system. Analysis of innovations Differences between varying length forecasts x √2 δx t Canadian ‘quick covs’ Ensembles x x √2 δx t t

  14. PDF in model variables Transform to new variables that are assumed to be univariate 107 – 108 elements δu δv δp δT δq (multivariate) model variable (univariate) control variable control variable transform δu δv δp δT δq 107 – 108 elements Modelling Pf(xt) \ Bwith transforms for data assimilation

  15. Ideas of ‘balance’ to formulate K (and hencePf(xt) \ B) these are not the same (clash of notation!) ← streamfunction (rot. wind) pert. (assume ‘balanced’) ← velocity potential (div. wind) pert. (assume ‘unbalanced’) ← residual pressure pert. (assume ‘unbalanced’) Implied f/c error covariance matrix H geostrophic balance operator (δψ → δpb) T hydrostatic balance operator (written in terms of temperature) Approach used at the ECMWF, Met Office, Meteo France, NCEP, MSC(SMC), HIRLAM, JMA, NCAR, CIRA Idea goes back to Parrish & Derber (1992)

  16. Assumptions • This formulation makes many assumptions e.g.: • That forecast errors projected onto balanced variables are uncorrelated with those projected onto unbalanced variables. • The rotational wind is wholly a ‘balanced’ variable (i.e. large Bu regime). • That geostrophic and hydrostatic balances are appropriate for the motion being modelled (e.g. small Ro regime).

  17. A. ‘Non-correlation’ test vertical model level latitude

  18. Modified transform B. Rotational wind is not wholly balanced Could there be an unbalanced component of δψ? Standard transform H geostrophic balance operator T hydrostatic balance operator H anti-geostrophic balance operator

  19. Modified transform Non-correlation test for refined model vertical model level latitude

  20. from Berre, 2000 C. Are geostrophic and hydrostatic balances always appropriate? E.g. test for geostrophic balance

  21. What next? Hi-resolution forecasts need hi-resolution Pf(xt) \ B • The Reading/MetO HRAA Collaboration • www.met.rdg.ac.uk/~hraa • Can forecast error covariances at hi-resolution be successfully modelled with the transform approach? • What is an appropriate transform at hi-resolution? • At what scales do hydrostatic and geostrophic balance become inappropriate? • There is little known theory to guide us at hi-res. • → What is the structure of forecast error covariances in such cases? High impact weather!

  22. Hi-resolution ensembles Early results from Met Office 1.5 km LAM (a MOGREPS-like system) Thanks to Mark Dixon (MetO), Stefano Migliorini (NCEO), Roger Brugge (NCEO)

  23. Summary • All measurements are inaccurate and all forecasts are wrong! • Accurate knowledge of forecast uncertainty (PDF) is useful: » to allow range of possible outcomes to be predicted, » to give allowed ways that a forecast can be modified by observations (data assimilation). • For synoptic/large scales the forecast error PDF is modelled with a change of variables and balance relations. • For hi-res (convective scales) the forecast error PDF is still important but there is no formal theory to guide PDF modelling: » hydrostatic/geostrophic balance less appropriate, » non-linearity/dynamic tendencies may be more important.

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