Probability Forecasts from Ensembles and their Application at the SPC David Bright NOAA/NWS/Storm...
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Probability Forecasts Probability Forecasts from Ensembles and their from Ensembles and their
Application at the SPC Application at the SPC
David BrightNOAA/NWS/Storm Prediction Center
Norman, OK
AMS Short Course on Probabilistic Forecasting
January 9, 2005San Diego, CA
Where Americas Climate and Weather Services Begin
Outline
• Motivation for ensemble forecasting
• Ensemble products and applications– Emphasis on probabilistic products
• Ensemble calibration (verification)
• Decision making using ensembles
Outline
• Motivation for ensemble forecasting
• Ensemble products and applications– Emphasis on probabilistic products
• Ensemble calibration (verification)
• Decision making using ensembles
• Daily weather forecasts begin as an initial-value problem on large supercomputers
• To produce a skillful weather forecast requires:– An accurate initial state of the atmosphere to
begin the model forecast– Computer models that realistically represent the
evolution of the atmosphere (in a timely manner)• With a reasonably accurate initial analysis of the
atmosphere, the state of the atmosphere at any subsequent time can be determined by a super-mathematician." (Bjerknes 1919)
Example: DeterminismExample: Determinism
60h Eta Forecast valid 00 UTC 27 Dec 2004PMSL (solid); 10m Wind; 1000-500 mb thickness (dashed)
60h Eta Forecast valid 00 UTC 27 Dec 2004PMSL (solid); 10m Wind; 1000-500 mb thickness (dashed)
Precip amount (in) and type (blue=snow; green=rain)
Example: DeterminismExample: Determinism
60h Eta Forecast valid 00 UTC 27 Dec 2004
“Truth” 00 UTC 27 Dec 2004
Example: DeterminismExample: Determinism
60h Eta Forecast valid 00 UTC 27 Dec 2004
“Truth” 00 UTC 27 Dec 2004
• Ignores forecast uncertainty• Potentially misleading• Oversells forecast capability
??Example: DeterminismExample: Determinism
Ensemble forecasting can be traced back to the discovery of the "Butterfly Effect" (Lorenz 1963, 1965)…
Atmo a non-linear, non-periodic, dynamical system causes even tiny errors to grow upscale ... resulting in forecast uncertainty and eventually chaos
The Butterfly EffectThe Butterfly Effect
The Butterfly EffectThe Butterfly Effect
• Discovery of the “butterfly effect” (Lorenz 1963)
• Simplified climate model… When the integration was restarted with 3 (vs 6) digit accuracy, everything was going fine until…
Time
• Solutions began to diverge
Solutions diverge
Time
The Butterfly EffectThe Butterfly Effect
• Soon, two “similar” but clearly unique solutions
Solutions diverge
Time
The Butterfly EffectThe Butterfly Effect
• Eventually, results revealed two uncorrelated and completely different solutions (i.e., chaos)
Solutions diverge
Time
Chaos
The Butterfly EffectThe Butterfly Effect
• Ensembles can be used to provide information on forecast uncertainty
• Information from the ensemble typically consists of…
(1) Mean(2) Spread(3) Probability
Ensembles useful in this range!
Solutions diverge
Time
Chaos
The Butterfly EffectThe Butterfly Effect
• Ensembles extend predictability…
• A deterministic solution is no longer skillful when its error variance exceeds climatic variance • An ensemble remains skillful until error saturation (i.e., until chaos occurs)
Solutions diverge
Chaos
Time
Ensembles extend predictability
Ensembles especially useful in this range!
The Butterfly EffectThe Butterfly Effect
- NWP models...- Doubling time of small
initial errors ~ 1 to 2 days
- Maximum large-scale (synoptic to planetary) predictability ~10 to 14 days
It’s hard to get it right the first time!
Example: Synoptic Scale VariabilityExample: Synoptic Scale Variability7 day forecast – NCEP MREF 500 MB Height7 day forecast – NCEP MREF 500 MB Height
GFS “Control” Forecast GFS -12h “Control”
GFS Pert European Model and Start
• Reveals forecast uncertainty, e.g., se U.S. precip• Sensible weather often mesoscale dominated
Example: Mesoscale VariabilityExample: Mesoscale Variability1.5 day forecast – NCEP SREF Precipitation1.5 day forecast – NCEP SREF Precipitation
500 mb Hght (Dec. 2004; Greater U.S. Area)
Climate SD
1.41 x Climate SDGFS
Ens Means
Limit of deterministic skill ~7.5 days
Limit of ensemble skill ~10.5 days
1 2 3 4 5Days
7 8 9 10 11 12
RMSE
20 m
40 m
80 m
100 m
120 m
Error Growth with Time: GFSError Growth with Time: GFS
Determinism
Ensemble
Ensembles vs. DeterminismEnsembles vs. Determinism
Evaluating Weather Forecasts
Outline
• Motivation for ensemble forecasting
• Ensemble products and applications
–Emphasis on probabilistic products• Ensemble calibration (verification)
• Decision making using ensembles
SPC Approach to EnsemblesSPC Approach to Ensembles
• Develop customized products based on a particular application (severe, fire wx, etc.)
• Design operational guidance products that…– Help blend deterministic and ensemble approaches– Facilitate transition toward probabilistic thinking– Aid in critical decision making
• Increase confidence • Alert for rare but significant events
F15 SREF MEAN500 MB HGHT,TEMP,WIND
Ensemble MeansEnsemble Means
Synoptic-Statistical RelationshipsSynoptic-Statistical RelationshipsMean + SpreadMean + Spread
• Examples of simple relationships between dispersion patterns and synoptic interpretation can be defined.
• Obtain a quick overview of range of weather situations from ensemble statistics.
Amplitude
Location
F15 SREF MEAN/SD500 MB HGHT
Ensemble Mean + SpreadEnsemble Mean + Spread
F000 F048
F096 F144
500 mb Mean Height (solid) and Standard Deviation (dashed/filled)
Increased spread Less predictability Less forecast confidence
Ensemble Mean + SpreadEnsemble Mean + Spread
F000 F048
F096 F144
500 mb Mean Height and Normalized VarianceNormalize the ensemble variance by climatic variance
Values approaching 2 (dark color fill) => Ensemble variance saturated based on climo
2
Ensemble Mean + Normalized SpreadEnsemble Mean + Normalized Spread
F000 F048
F096 F144
500 mb Member Height “Spaghetti” - 5640 meter contour
Another way to view uncertainty:Another way to view uncertainty:SpaghettiSpaghetti
F63 SREF POSTAGE STAMP VIEW: PMSL, HURRICANE FRANCES F63 SREF POSTAGE STAMP VIEW: PMSL, HURRICANE FRANCES
Red = EtaBMJ
Yellow= Yellow= EtaKFEtaKF
Blue = RSM
White = White = OpEtaOpEta
SREF Member
F15 SREF MEDIAN/RANGECAPE
At least 1 memberhas >= 500 J/kg
All 16 membershave >=500 J/kg CAPE
Median
Spatial Variability: Median + RangeSpatial Variability: Median + Range
• Arithmetic mean…– Easy to compute and understand– Tends to increase coverage of light
pcpn and decrease max values.
3-hr Total Pcpn NCEP SREFF63 Valid 09 Oct 2003 00 UTC
Ways to view central value: MeanWays to view central value: Mean
• Median…– If the majority of members
don’t precip, will show large areas of no precip. Thus, often limited in areal extent.
3-hr Total Pcpn NCEP SREF
Ways to view central value: MedianWays to view central value: Median
• The blending of two PDFs, when one provides better spatial representation [e.g., ensemble mean QPF] and the other greater accuracy [e.g., QPF from all members]. See Ebert (MWR 2001) for more info.
Rank Ens Mean Rank Member QPF 1 1 2 16 3 32
Ways to view central value: Ways to view central value: Probability MatchingProbability Matching
• Probability matching…– Ebert (2001)
Found to be the best ensemble averaged QPF
– Max values restored; pattern from ens mean
3-hr Total Pcpn NCEP SREF
Ways to view central value: Ways to view central value: Probability MatchingProbability Matching
Probability 144h 2 meter Td <= 25 degF
Probabilistic Output of Basic Products: Probabilistic Output of Basic Products: 2 m Dewpoint2 m Dewpoint
Probability 144h Haines Index > 5
Probabilistic Output of Derived Probabilistic Output of Derived Products: Haines IndexProducts: Haines Index
Probability Convective Pcpn Probability Convective Pcpn >> .01” .01”
Prob Conv Pcpn > .01”Valid 00 UTC 20 Sept 2003
Prob Conv Pcpn > .01”Valid 00 UTC 20 Sept 2003
Probability Convective Pcpn Probability Convective Pcpn >> .01” .01”
Pcpn probs due to physics - No EtaBMJ members?!
Red = EtaBMJYellow = EtaKFBlue = RSM
Spaghetti: Different physicsSpaghetti: Different physics
Note clustering by model
Spaghetti: OutliersSpaghetti: Outliers
F39 SREF SPAGHETTI F39 SREF SPAGHETTI (1000 J/KG)(1000 J/KG)
Red = EtaBMJYellow = EtaKFBlue = RSMWhite solid = 12 KM OpEta (12 UTC)
12 UTC operational Eta clearlyan outlier from 09 UTC SREF - Is this the result of ICs or resolution? - Is this a better fcst (updated info) or an outlier
F15 SREF MINIMUMMINIMUM 2 METER RH
F15 SREF MAXIMUMMAXIMUMFOSBERG FIREWX INDEX
• Any member can contribute to the max or min value at a grid point
Extreme ValuesExtreme Values
Combined Probability ChartsCombined Probability Charts
• Probability surface CAPE >= 1000 J/kg– Generally
low in this case
– Ensemble mean < 1000 J/kg (no gold dashed line)
CAPE (J/kg)Green solid= Percent Members >= 1000 J/kg; Shading >= 50%
Gold dashed = Ensemble mean (1000 J/kg)F036: Valid 21 UTC 28 May 2003
• Probability deep layer shear >= 30 kts– Strong mid
level jet through Iowa
10 m – 6 km Shear (kts)Green solid= Percent Members >= 30 kts; Shading >= 50%
Gold dashed = Ensemble mean (30 kts)F036: Valid 21 UTC 28 May 2003
Combined Probability ChartsCombined Probability Charts
• Convection likely WI/IL/IN– Will the
convection become severe?
3 Hour Convective Precipitation >= 0.01 (in)Green solid= Percent Members >= 0.01 in; Shading >= 50%
Gold dashed = Ensemble mean (0.01 in)F036: Valid 21 UTC 28 May 2003
Combined Probability ChartsCombined Probability Charts
• Combined probabilities very useful
• Quick way to determine juxtaposition of key parameters
• Not a true probability– Not
independent– Different
members contribute
Prob Cape >= 1000 X Prob Shear >= 30 kts X Prob Conv Pcpn >= .01” F036: Valid 21 UTC 28 May 2003
Combined Probability ChartsCombined Probability Charts
Severe ReportsRed=Tor; Blue=Wind; Green=Hail
Prob Cape >= 1000 X Prob Shear >= 30 kts X Prob Conv Pcpn >= .01” F036: Valid 21 UTC 28 May 2003
• Combined probabilities a quick way to determine juxtaposition of key parameters
• Not a true probability
– Not independent– Different
members contribute
• Fosters an ingredients-based approach on-the-fly
Combined Probability ChartsCombined Probability Charts
F15 SREF PROBABILITYP12I x RH x WIND x TMPF(< .01” x < 10% x > 30 mph x > 60 F)
Ingredients for extreme fire weatherconditions over the Great Basin
Combined or Joint Probabilities - Not a true probability - An ingredients-based, probabilistic approach - Useful for identifying key areas
Combined Probability ChartsCombined Probability Charts
F15 SREF PROBABILITYTPCP x RH x WIND x TMPF(< .01” x < 10% x > 30 mph x > 60 F)
Ingredients for extreme fire weatherconditions over the Great Basin
Combined Probability ChartsCombined Probability Charts
Elevated Instability – General ThunderElevated Instability – General ThunderNCEP SREF 30 Sept 2003 09 UTC F12
Mean MUCAPE/CIN (Sfc to 500 mb AGL) Mean LPL (Sfc to 500 mb AGL)
Parcel Equilibrium LevelParcel Equilibrium Level NCEP SREF 30 Sept 2003 09 UTC F12
Mean Temp (degC) MUEquilLvl (Sfc to 500 mb AGL)
Prob Temp MUEquilLvl < -20 degC (Sfc to 500 mb AGL)
Lightning VerificationLightning Verification
Gridded Lightning Strikes 18-21 UTC 30 Sept 2003(40 km grid boxes)
Microphysical ExampleMicrophysical ExampleProbability cloud top temps > -8 degC Probability cloud top temps < -12 degC
Ice Crystals Unlikely Ice Crystals Likely
NCEP SREF 7 Oct 2003 21 UTC F15
Mode Mode
Most Common Precip Type (Snow = Blue); Mean Precip (in); Mean 32o F Isotherm
F015 SREF Valid: 00 UTC 21 December 2004
Probability Dendritic Layer Probability Dendritic Layer >> 50 mb 50 mb
F015 SREF Valid: 00 UTC 21 December 2004
F015 SREF Valid: 00 UTC 21 December 2004
Probability of Banded PrecipitationProbability of Banded PrecipitationPotentialPotential
Probability MPV < .05 PVU (saturated; 900 to 650 mb layer) x Probability Deep Layer FG > 1
Probability Omega <= -3 microbar/sProbability Omega <= -3 microbar/s
F015 SREF Valid: 00 UTC 21 December 2004
Probability 6h Precip >= .25”Probability 6h Precip >= .25”
F015 SREF Valid: 00 UTC 21 December 2004
Outline
• Motivation for ensemble forecasting
• Ensemble products and applications– Emphasis on probabilistic products
• Ensemble calibration (verification)• Decision making using ensembles
Combine Thunderstorm Ingredients Combine Thunderstorm Ingredients into Single Parameterinto Single Parameter
• Three first-order ingredients (readily available from NWP models):– Lifting condensation level > -10o C– Sufficient CAPE in the 0o to -20o C layer – Equilibrium level temperature < -20o C
• Cloud Physics Thunder Parameter (CPTP) CPTP = (-19oC – Tel)(CAPE-20 – K) K
where K = 100 Jkg-1 and CAPE-20 is MUCAPE in the 0o C to -20o C layer
Example CPTP: One MemberExample CPTP: One Member
18h Eta Forecast Valid 03 UTC 4 June 2003
Plan view chart showing where grid point soundings support lightning (given a convective updraft)
SREF Probability CPTP SREF Probability CPTP >> 1 1
15h Forecast Ending: 00 UTC 01 Sept 2004Uncalibrated probability: Solid/Filled; Mean CPTP = 1 (Thick dashed)
3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
SREF Probability Precip SREF Probability Precip >> .01” .01”
15h Forecast Ending: 00 UTC 01 Sept 2004Uncalibrated probability: Solid/Filled; Mean precip = 0.01” (Thick dashed)
3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
Joint Probability (Assumed Independence)Joint Probability (Assumed Independence)
15h Forecast Ending: 00 UTC 01 Sept 2004Uncalibrated probability: Solid/Filled
P(CPTP > 1) x P(Precip > .01”)3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
Perfect Forecast
No Skill
Climatology
P(CPTP > 1) x P(P03I > .01”)
Uncalibrated ReliabilityUncalibrated Reliability (5 Aug to 5 Nov 2004)(5 Aug to 5 Nov 2004)
Frequency[0%, 5%, …, 100%]
Adjusting ProbabilitiesAdjusting Probabilities
1) Calibrate based on the observed frequency of occurrence
– Very useful, but may not provide information concerning rare or extreme (i.e., low probability) events
2) Use statistical techniques to estimate probabilities in the tails of the distribution (e.g., Hamill and Colucci 1998; Stensrud and Yussouf 2003)
Ensemble CalibrationEnsemble Calibration1) Bin separately P(CPTP > 1) and P(P03M > 0.01”) into
11 bins (0-5%; 5-15%; …; 85-95%; 95-100%)2) Combine the two binned probabilistic forecasts into one
of 121 possible combinations (0%,0%); (0%,10%); … (100%,100%)
3) Use NLDN CG data over the previous 60 days to calculate the frequency of occurrence of CG strikes for each of the 121 binned combinations
4) Bin ensemble forecasts as described in steps 1 and 2 and assign the observed CG frequency (step 3) as the calibrated probability of a CG strike
5) Calibration is performed for each forecast cycle (09 and 21 UTC) and each forecast hour; domain is entire U.S. on 40 km grid
Before Calibration
Joint Probability (Assumed Independence)Joint Probability (Assumed Independence)
P(CPTP > 1) x P(Precip > .01”)3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
15h Forecast Ending: 00 UTC 01 Sept 2004Uncorrected probability: Solid/Filled
After Calibration
Calibrated Ensemble Thunder Probability Calibrated Ensemble Thunder Probability
15h Forecast Ending: 00 UTC 01 Sept 2004Calibrated probability: Solid/Filled
3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
Calibrated Ensemble Thunder ProbabilityCalibrated Ensemble Thunder Probability
15h Forecast Ending: 00 UTC 01 Sept 2004Calibrated probability: Solid/Filled; NLDN CG Strikes (Yellow +)
3 hr valid period: 21 UTC 31 Aug to 00 UTC 01 Sept 2004
Perfect Forecast
No Skill
Perfect Forecast
No Skill
Calibrated ReliabilityCalibrated Reliability (5 Aug to 5 Nov 2004)(5 Aug to 5 Nov 2004)
Calibrated Thunder Probability
Climatology
Frequency[0%, 5%, …, 100%]
Adjusting ProbabilitiesAdjusting Probabilities
1) Calibrate based on the observed frequency of occurrence
– Very useful, but may not provide information concerning extreme (i.e., low probability) events
2) Use statistical techniques to estimate probabilities in the “tails” of the distribution (e.g., Hamill and Colucci 1998; Stensrud and Yussouf 2003)
• Consider 2 meter temperature prediction from NCEP SREF– Construct a “rank histogram” of the ensemble
members (also called Talagrand diagram)• Rank individual members from lowest to highest• Find the verifying rank position of “truth” (RUC 2
meter analysis temperature) • Record the frequency with which truth falls in that
position (for a 15 member ensemble there are 16 rankings)
Adjusting ProbabilitiesAdjusting Probabilities
Adjusting ProbabilitiesAdjusting ProbabilitiesUncorrected Talagrand Diagram
Warm bias in 15h fcst of 12 UTC NCEP SREF
Uniform Distribution
2m temperature ending 27 December 2004
Under-dispersive
Truth is colder than all SREF membersa disproportionate amount of time
Clustering by model
Use 14-day bias to account for bias in forecastUse 14-day bias to account for bias in forecast
Members 1 through 15 of NCEP SREF
Adjusting ProbabilitiesAdjusting ProbabilitiesBias Adjusted Talagrand Diagram
Near neutral bias in 15h fcst of 12 UTC NCEP SREF
Large bias eliminated butremains under-dispersive
Uniform Distribution
2m temperature ending 27 December 2004
• Build the pdf by using observed data to fit a statistical distribution (Gamma, Gumbel, or Gaussian) to the tails
• This produces a calibrated pdf based on past performance– “Past performance does not guarantee future
results.”
Adjusting ProbabilitiesAdjusting Probabilities
Adjusting ProbabilitiesAdjusting ProbabilitiesCorrected Talagrand Diagram
~Uniform distribution in 15h fcst of 12 Z SREF
Uniform Distribution
SREF probabilities now reflect expectedoccurrence of event even in the “tails”
2m temperature ending 27 December 2004
Adjusted Temperature FcstAdjusted Temperature Fcst
Max temp (50%) valid 12 UTC 5 Jan to 00 UTC 6 Jan 2004
Probabilistic Temperature ForecastProbabilistic Temperature ForecastNorman, OK (95% Confidence)Norman, OK (95% Confidence)
50.0%
2.5%
2.5%
Dec 27 Dec 28 Dec 29
Norman, OK Temp Forecast from SREF
Actual mins & maxes indicated by red dots
Temp (degF)
Local Time 4 AM 6 PMMid Mid
Probabilistic MeteogramProbabilistic MeteogramProbability of severe thunderstorm ingredients: OUN; Runtime: 09 UTC 21 April
•Information on how ingredients are evolving•Viewing ingredients via probabilistic thinking
Probabilistic MeteogramProbabilistic MeteogramProbability of severe thunderstorm ingredients: OUN; Runtime: 09 UTC 21 April
•Information on how ingredients are evolving•Viewing ingredients via probabilistic thinking
Outline
• Motivation for ensemble forecasting
• Ensemble products and applications– Emphasis on probabilistic products
• Ensemble calibration (verification)
• Decision making using ensembles
Decision MakingDecision Making
• Probabilities from an uncalibrated, under-dispersive ensemble system are still useful in quantifying uncertainty
• Trends in probabilities (dprog/dt) may indicate less spread among members as t 0
12h Prob Thunder 12h Prob Severe
Prob Thunder Prob Severe
Day 6
Day 5
Day 4
Day 3 Day 2
•Increased probabilistic resolution as event approaches
• Run-to-run consistency
•Time-lagged members (weighted) add continuity to forecast
Trend over 5 daysTrend over 5 daysfrom NCEP MREF from NCEP MREF (Valid: 22 Dec 2004)(Valid: 22 Dec 2004)
Results…Results…
Decision MakingDecision Making
• Probabilities from an un-calibrated, under-dispersive ensemble system are still useful to quantify uncertainty
• Trends in probabilities (dprog/dt) may indicate less spread among members as t 0
• Decision theory can be used with or without separate calibration
Decision Theory ExampleDecision Theory Example
• Consider the calibrated thunderstorm forecasts presented earlier [see Mylne (2002) for C/L model]…
User: Electronics storeCritical Event: Lightning strike/surgeCost to protect: $300Expense of a Loss: $10,000
Yes No
Yes Hit
$300
F.A.
$300
No Miss
$10,000
C.R.
$0
Observed
Forecast
a = F.A. – C.R. Miss + F.A. – Hit – C.R.
C/L = a = 0.03
If no forecast information is available,user will always protect if a < o, andnever protect if a > o, where o is climatological frequency of the event
Decision Theory ExampleDecision Theory Example• If the calibration were perfect, then user would
seek protective action whenever forecasted probability is > a.
But, forecast is not perfectlyreliable…
• Apply a cost-loss model to assist in the decision (prior calibration is unnecessary)
• Define a cost-loss model as in Murphy (1977); Legg and Mylne (2004); Mylne (2002)– This can be done without
probabilistic calibration as the technique implicitly calibrates based on past performance
• V = Eclimate - Eforecast
Eclimate – Eperfect
Decision Theory ExampleDecision Theory Example
Decision Theory ExampleDecision Theory Example
V = Eclimate - Eforecast
Eclimate - Eperfect
V = a general assessment of forecast valuerelative to the perfect forecast (i.e., basically a skill score).
V = 1 indicates a perfect forecast system (i.e., action is taken only when necessary)
V < 0 indicates a system of equal or lesser value than climatology
V = Eclimate - Eforecast
Eclimate - Eperfect
Eclimate = min[ (1-o)F.A. + oHit; (1-o)C.R. + oMiss ]
Eperfect = oHit
Eforecast= hHit + mMiss + fF.A.+ rC.R.
o = climatological freq = h + m
Yes No
Yes Hit
$300
F.A.
$300
No Miss
$10,000
C.R.
$0
Yes No
Yes Hit
(h)
F.A.
(f)
No Miss
(m)
C.R.
(r)
Decision Theory ExampleDecision Theory ExampleObserved
Observed
Forecast
Forecast
Costs:
Performance:
Decision Theory ExampleDecision Theory Example
Never Protect
Always Protect
10%
Action probability for a = .03 is 7% with V = .64
PotentialValue
0.0
0.5
0.001 0.01 0.10
a = Cost/Loss Ratio (log scale)1.00
1.0Maximum Potential Value of the Forecast and its Associated Probability
.008 .14
SummarySummary
• Ensembles provide information on mean, spread, and forecast uncertainty (probabilities)
• Derived products viewed in probability space have proven useful at the SPC
• Combined or joint probabilities very useful• When necessary, ensembles can be
calibrated to provide reliable estimates of probability and/or aid in decision making
SPC SREF Products on WEBSPC SREF Products on WEB
http://www.spc.noaa.gov/exper/sref/
ReferencesReferencesBright, D.R., M. Wandishin, R. Jewell, and S. Weiss, 2005: A physically based parameter for
lightning prediction and its calibration in ensemble forecasts. Preprints, Conference on Meteor. Appl. of Lightning Data, AMS, San Diego, CA (CD-ROM 4.3)
Cheung, K.K.W., 2001: A review of ensemble forecasting techniques with a focus on tropical cyclone forecasting. Meteor. Appl., 8, 315-332.
Ebert, E.E., 2001: Ability of a poor man's ensemble to predict the probability and distribution of precipitation. Mon. Wea. Rev., 129, 2461-2480.
Hamill, T.M. and S.J. Colucci, 1998: Evaluation of Eta-RSM ensemble probabilistic precipitation forecasts. Mon. Wea. Rev., 126, 711-724.
Legg, T.P. and K.R. Mylne, 2004: Early warnings of severe weather from ensemble forecast information. Wea. Forecasting, 19, 891-906.
Mylne, K.R. 2002: Decision-making from probability forecasts based on forecast value. Meteor. Appl., 9, 307-315.
Sivillo, J.K. and J.E. Ahlquist, 1997: An ensemble forecasting primer. Wea. Forecasting, 12, 809-818.
Stensrud, D.J. and N. Yussouf, 2003: Short-range ensemble predictions of 2-m temperature and dewpoint temperature over New England. Mon. Wea. Rev., 131, 2510-2524.
Wilks, D.S., 1995: Statistical Methods in the Atmospheric Sciences. International Geophysics Series, Vol. 59, Academic Press, 467 pp.