FORMULATION METHODS FOR STATISTICAL APPLICATION IN WEATHER FORECASTING

1. FORMULATION METHODS FOR STATISTICAL APPLICATION IN WEATHER FORECASTING BPKC : CLM (STATS)

2. FORMULATION METHODS FOR STATISTICAL APPLICATION IN WEATHER FORECASTING • Classical , PPM , MOS Methods • Of the three, only PPM and MOS use NWP model output, while the Classical Method uses only observations. BPKC : CLM (STATS)

3. CLASSICAL METHODS • Similar to subjective methods, used to predict weather, based on current observations or recent weather conditions. • This approach does not use numerical model forecasts and relies purely on observed data. • Before Dynamical Models, statistical systems were limited to Classical Approach. • To develop equations with Classical Method, observations of the initial and resultant weather conditions are needed. BPKC : CLM (STATS)

4. CLASSICAL METHODS • The resultant weather conditions are the ‘Predictands’, while the initial conditions are the ‘Predictors’. • For Example, for forecasting Max Temperature for tomorrow, input would consist only of observational data available at the time that the forecast was to be made. • This situation can be expressed as:- BPKC : CLM (STATS)

5. CLASSICAL METHODS • Here is estimate (forecast) of ‘Predictand’ (dependent variable) ‘Y’ at time ‘t’ and ‘X0’ is a vector of observational data (independent variables) at initial time ‘0’. • Observations are not necessarily made at initial time, but must be available at that time. • Accuracy of any given forecast based on this approach is strongly dependent on significant changes occur in atmosphere between times of predictor observations and time of validity of resultant forecast. BPKC : CLM (STATS)

6. CLASSICAL METHODS • Greater the forecast projection, greater will be the chance of significant changes. • Hence, this approach is good for short-range forecasts, but its skill falls sharply beyond a few hours. Practically no skill in medium range. • Work best when the weather tends to be persistent (low variability). • This approach is also used for very long-range seasonal forecasts, where there is little skill in numerical model predictions. BPKC : CLM (STATS)

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8. PERFECT PROGNOSIS METHOD • The need to accurately predict surface weather elements, led to the development of Perfect Prognosis Method (PPM) (Klien et al., 1959). • This is an objective method, in which, a concurrent relation is developed between the parameter to be predicted and the observed circulation around the location of interest, using several years of data. • PPM technique is based on the assumption that numerical model forecasts are “Perfect”. BPKC : CLM (STATS)

9. 1st Statistical approach for dealing with forecasts from NWP • Perfect Prog (Klein et al. 1959) • Takes NWP model forecasts for future atmosphere assuming them to be perfect • Perfect prog regression equations are similar to classical regression equations except they do not incorporate any time lag. Example: equations specifying tomorrows predictands are developed using tomorrow’s predictor values. If the NWP forecasts for tomorrow’s predictors really are perfect, the perfect-prog regression equations should provide very good forecasts.

10. PERFECT PROGNOSIS METHOD

11. PERFECT PROGNOSIS METHOD • Despite numerical models not being perfect, this approach gives an estimate of what to expect, if numerical models are correct • It does not require numerical model data for development, but uses numerical output when equations are applied operationally • Hence, it is important to make sure that variables used as ‘Predictors’ in development of Perfect Prognosis Equations will be available from NWP models, so that equations can be applied operationally BPKC : CLM (STATS)

12. PERFECT PROGNOSIS METHOD • Extensive sample of upper air observations and surface weather reports is collected • Multiple Linear Regression equations are then derived • In applying PPM equations, Dynamical Model output for a specific forecast projection is substituted for the developmental observations to give a forecast for the appropriate valid time BPKC : CLM (STATS)

13. PERFECT PROGNOSIS METHOD • These specification equations can be used to generate guidance from any dynamical model as long as the model produces the required predictors and the original time relationships among predictors and predictand are preserved • The PPM approach can be expressed numerically as:- BPKC : CLM (STATS)

14. PERFECT PROGNOSIS METHOD • Here is estimate of ‘Predictand’ ‘Y’ at time ‘0’ and ‘X0’ is a vector of observations of variables that can be predicted by numerical models • Even though is an estimate, it is not a forecast, it is only a specification • In application, is inserted into equation (2) to provide a forecast as:-  BPKC : CLM (STATS)

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16. MODEL OUTPUT STATISTICS • Like PPM, the MOS approach is also an objective type of forecasting technique (Glahn & Lowry, 1972). • In contrast to Classical Approach and Perfect Prognosis Approach (which uses numerical model output only in operational application of the equations), the Model Output Statistics (MOS) approach uses numerical model forecasts for both the development and the operational application of the equations BPKC : CLM (STATS)

17. 2nd Statistical approach • Model Output Statistics (MOS) • Preferred because it can include directly in the regression eqns the influences of specific characteristics of different NWP models at different projections into the future. • To get MOS forecast eqns you need a developmental data set with historical records of predictand, and records of the forecasts by NWP model. • Separate MOS forecast equations must by made for different forecast projections. Example: Predictand is tomorrow’s 1000-800mb thickness as forecast today by a certain NWP model.

18. MODEL OUTPUT STATISTICS

19. MODEL OUTPUT STATISTICS • MOS Approach requires an archive of dynamical model forecasts : usually 2-3 years of data are needed • MOS Approach relates a weather observation (Predictand) to variables forecast by a numerical model (Predictors) • The Predictors from the numerical model are usually forecasts that are valid at about the same time as the Predictand BPKC : CLM (STATS)

20. MODEL OUTPUT STATISTICS • Some statistical method (usually Multiple Linear Regression) is then used to determine relationships among various observed weather elements and numerical model output variables at projections near (before, at, or amer) the specific valid time of the Predictand. • To make operational forecasts, MOS equations are usually applied to the same dynamical model that provided the developmental sample. BPKC : CLM (STATS)

21. MODEL OUTPUT STATISTICS • Mathematically MOS can be expressed as:- • where is estimate of Predictand ‘Y’ at time ‘t’ and is a vector of forecasts from numerical models. • The numerical model predictions need not be limited at time ‘t’; however, the projection times of the different variables will usually be grouped around ‘t’. BPKC : CLM (STATS)

22. MODEL OUTPUT STATISTICS • The equations (4) account for some of the bias and systematic errors found in the dynamical model. • Local topographic and environmental conditions of a location are also automatically accounted for in the forecast system. • MOS technique also recognises the predictability of the model variables by selecting those variables that provide useful forecast information. BPKC : CLM (STATS)

23. MODEL OUTPUT STATISTICS • Finally, as the dynamical model predictions deteriorate with increasing time, this approach produces forecasts that tend towards the mean of the predictand in the developmental sample. • Drawback of this technique is that a sufficient sample of Model Output is required in order to derive a stable relation. Hence, it cannot be applied immediately when a new model is made operational. • Also, if model undergoes a major change, MOS relations will have to be developed again. BPKC : CLM (STATS)

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25. COMPARISON • Classical Approach is most useful for very short range (0-12 hour), or if NWP model forecast data is not available. • It is relatively simple to use, since it requires only that data, which are usually available without special archiving procedures. • PPM and MOS approaches require NWP model forecast data in operational application of the equations. BPKC : CLM (STATS)

26. COMPARISON • However, since PPM equations are derived from observed data only, there is usually a longer sample of data available from which to derive the equations. • In contrast, MOS equations need to be developed from a stable sample of numerical model forecast data. • Because most NWP models are constantly evolving, developmental sample for MOS equations is likely to be relatively short. BPKC : CLM (STATS)

27. COMPARISON • When a model is changed significantly, forecast produced by MOS has less skill and accuracy and equations will have to be redeveloped. In contrast, PPM forecasts usually do not deteriorate with change in model. • Generally an improved numerical model will lead to improved forecasts. • If model is not modified and sufficient sample of stable numerical model output is available, MOS approach will produce the best results. BPKC : CLM (STATS)

28. COMPARISON BPKC : CLM (STATS)

29. COMPARISON BPKC : CLM (STATS)

30. Advantages and Disadvantages of Perfect-Prog and MOS:

31. Perfect Prog • Advantages: • Large developmental sample (fit using historical climate data) • Equations developed without NWP info, so changes to NWP models don’t require changes in regression equations • Improving NWP models will improve forecasts • Same equations can be used with any NWP models • Disadvantages: • Potential predictors must be well forecast by the NWP model

32. MOS • Advantages: • Model-calculated, but un-observed quantities can be predictors • Systematic errors in the NWP model are accounted for • Different MOS equations required for different projection times • Method of choice when practical • Disadvantages: • Requires archived records (several years) of forecast from NWP model to develop, and models regularly undergo changes. • Different MOS equations required for different NWP models

33. ANY QUESTION ? BPKC : CLM (STATS)

34. MOS (Additional Information)

35. REVISION • MOS relates observations of the weather element to be predicted (PREDICTANDS) to appropriate variables (PREDICTORS) via a statistical method Predictors can include: • NWP model output interpolated to observing site • Prior observations • Geoclimatic data – terrain, normals, lat/long, etc. Current statistical method: Multiple Linear Regression (forward selection)

39. Temperature • Dry bulb temperature (every 3 h) • Dew point (every 3 h) • Daytime maximum temperature [0700 – 1900 LST] (every 24 h) • Nighttime minimum temperature [1900 – 0800 LST] (every 24 h) • Wind • U- and V- wind components (every 3 h) • Wind speed (every 3 h) • Sky Cover • Clear, few, scattered, broken, overcast [binary/MECE] (every 3 h)

40. PoP/QPF • PoP: accumulation of 0.01” of liquid-equivalent precipitation in a {6/12/24} h period [binary] • QPF: accumulation of {0.10”/0.25”/0.50”/1.00”/2.00”*} CONDITIONAL on accumulation of 0.01” [binary/conditional] • 6 h and 12 h guidance every 6 h; 24 h guidance every 12 h • 2.00” category not available for 6 h guidance • Thunderstorms • 1+ lightning strike in gridbox [binary] • Severe • 1+ severe weather report in gridbox [binary]

41. Ceiling Height • CH < 200 m, 200-400 m, 500-900 m, 1000-1900 m, 2000-3000 m, 3100-6500 m, 6600-12000 m, > 12000 m [binary/MECE] • Visibility • Visibility < ½ km, < 1 km, < 2 kms, < 3 kms, < 5 kms, < 6 kms[binary] • Obstruction to Vision • Observed fog (fog w/ vis < 1 km), mist (fog w/ vis>1-2 km), haze (includes smoke and dust), blowing phenomena, or none [binary]

42. Precipitation Type • Pure snow (S); freezing rain/drizzle, ice pellets, or anything mixed with these (Z); pure rain/drizzle or rain mixed with snow (R) • Conditional on precipitation occurring • Precipitation Characteristics (PoPC) • Observed drizzle, steady precip, or showery precip • Conditional on precipitation occurring • Precipitation Occurrence (PoPO) • Observed precipitation on the hour – does NOT have to accumulate

43. Stratification

44. Pooling Data

45. MOS equations are multivariate of the form: • Y = c0 + c1*X1 + c2*X2 + … + cN*XN • C’s are constants, X’s are predictors • N is the number of predictors in the equation and is specified when the equations are developed.

46. MOS Development Strategy • Forward Selection ensures that the “best” or most STATISTICALLY IMPORTANT predictors are chosen first. • First predictor selected accounts for greatest reduction of variance (RV) • Subsequent predictors chosen give greatest RV in conjunction with predictors already selected • STOP selection when max # of terms reached, or when no remaining predictor will reduce variance by a pre-determined amount

47. Meteorological consistencies – SOME checks • T > Td; min T < T < max T; dir = 0 if wind speed = 0 • Truncation (no probabilities < 0, > 1) • Monotonicityenforced (for elements like QPF) • BUT temporal coherence is only partially checked • Generation of “best categories”

48. Unconditional Probabilities from Conditional • If event B is conditioned upon A occurring: • Prob(B|A)=Prob(B)/Prob(A) • Prob(B) = Prob(A) × Prob(B|A) • Example: • Let A = event of >10 mm., and B = event of > .25 mm., then if: • Prob (A) = .70, and • Prob (B|A) = .35, then • Prob (B) = .70 × .35 = .245 B A U

49. Truncating Probabilities • 0 <Prob (A) < 1.0 • Applied to QPFand thunderstorm probabilities • If Prob(A) < 0, Probadj (A)=0 • If Prob(A) > 1, Probadj (A)=1.

50. Normalizing MECE Probabilities • Sum of probabilities for exclusive and exhaustive categories must equal 1.0 • If Prob (A) < 0, then sum of Prob (B) and Prob (C) = D, and is > 1.0. • Set: Probadj (A) = 0, • Probadj (B) = Prob (B) / D, • Probadj (C) = Prob (C) / D