Robin HoganRobin HoganEwan O’Connor, Anthony IllingworthEwan O’Connor, Anthony Illingworth

University of Reading, UKUniversity of Reading, UK

Chris Ferro, Ian Jolliffe, David StephensonChris Ferro, Ian Jolliffe, David Stephenson

University of Exeter, UKUniversity of Exeter, UK

Verifying cloud forecasts:Verifying cloud forecasts:What is the “half-life” of a cloud forecast?What is the “half-life” of a cloud forecast?

Is the Equitable Threat Score really equitable?Is the Equitable Threat Score really equitable?

How skillful is a forecast?

• Most model evaluations of clouds test the cloud climatology– What about individual forecasts?

• Standard measure shows ECMWF forecast “half-life” of ~6 days in 1980 and ~9 days in 2000 – But virtually insensitive to clouds!

ECMWF 500-hPa geopotential anomaly correlation

Overview• The “Cloudnet” processing of ground-based radar and lidar

observations– Continuous evaluation of the climatology of clouds in models– Evaluation of the diurnal cycle of boundary-layer clouds

• Desirable properties of verification measures (skill scores)– Usefulness for rare events: the Symmetric Extreme Dependency

Score– Equitability: is the “Equitable Threat Score” equitable?

• Testing the skill of cloud forecasts from seven models– Skill versus cloud fraction, height, scale, forecast lead time, season...– Estimating the forecast “half life”

• Testing the skill of cloud forecasts from space– Evaluation of ECMWF model with ICESat/GLAS lidar

• Most results taken from these papers:– Hogan, O’Connor & Illingworth (QJ 2009)– Hogan, Ferro, Jolliffe & Stephenson (WAF, in press)

Project

• Aim: to retrieve and evaluate the crucial cloud variables in forecast and climate models– 8+ models: global, mesoscale and high-resolution forecast models– Variables: cloud fraction, LWC, IWC, plus a number of others– Sites: 4 across Europe plus worldwide ARM sites– Period: several years to avoid unrepresentative case studies

• Current status– Funded by US Department of Energy Climate Change Prediction

Program to apply to ARM data worldwide

Level 1b

• Minimum instrument requirements at each site– Cloud radar, lidar, microwave radiometer, rain gauge, model or sondes

Radar

Lidar

Level 1c

Ice

LiquidRain

Aerosol

• Instrument Synergy product– Example of target classification and data quality fields:

Level 2a/2b

• Cloud products on (L2a) observational and (L2b) model grid– Water content and cloud fraction

L2a IWC on radar/lidar grid

L2b Cloud fraction on model grid

ChilboltonObservations

Met OfficeMesoscale

Model

ECMWFGlobal Model

Meteo-FranceARPEGE Model

KNMIRACMO Model

Swedish RCA model

Cloud fraction

Cloud fraction in 7 models• Mean & PDF for 2004 for Chilbolton, Paris and Cabauw

Illingworth et al. (BAMS 2007)

0-7 km

– All models except DWD underestimate mid-level cloud– Some have separate “radiatively inactive” snow (ECMWF, DWD); Met

Office has combined ice and snow but still underestimates cloud fraction

– Wide range of low cloud amounts in models– Not enough overcast boxes, particularly in Met Office model

Diurnal cycle composite of clouds

Barrett, Hogan & O’Connor (GRL 2009)

Meteo-France:Local mixing scheme: too little entrainment

SMHI:Prognostic TKE scheme: no diurnal evolution

All other models have a non-local mixing scheme in unstable conditions and an explicit formulation for entrainment at cloud top: better performance over the diurnal cycle

Radar and lidar provide cloud boundaries and cloud properties above site

Joint PDFs of cloud fraction

• Raw (1 hr) resolution– 1 year from Murgtal– DWD COSMO model

• 6-hr averaging

ab

cd

…or use a simple contingency table

a = 7194 b = 4098

c = 4502 d = 41062

DWD model, Murgtal

Model cloud

Model clear-sky

a: Cloud hit b: False alarm

c: Miss d: Clear-sky hit

Contingency tables

For given set of observed events, only 2 degrees of freedom in all possible forecasts (e.g. a & b), because 2 quantities fixed: - Number of events that occurred n =a +b +c +d - Base rate (observed frequency of occurrence) p =(a +c)/n

Observed cloud Observed clear-sky

Skill-Bias diagrams

Positiveskill

Randomforecast

Negativeskill

Best possible forecast

Worst possible forecast

Under-prediction No bias Over-prediction

Random unbiased forecast

Constant forecast of non-occurrence

Constant forecast of occurrence

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Reality (n=16, p=1/4)

Forecast

-

5 desirable properties of verification measures

1. “Equitable”: all random forecasts receive expected score zero– Constant forecasts of occurrence or non-occurrence also score

zero– Note that forecasting the right cloud climatology versus height

but with no other skill should also score zero

2. Difficult to “hedge”– Some measures reward under- or over-prediction

3. Useful for rare events– Almost all measures are “degenerate” in that they asymptote to 0

or 1 for vanishingly rare events

4. Dependence on full joint PDF, not just 2x2 contingency table– Difference between cloud fraction of 0.9 and 1 is as important for

radiation as a difference between 0 and 0.1– Difficult to achieve with other desirable properties: won’t be

studied much today...

5. “Linear”: so that can fit an inverse exponential for half-life– Some measures (e.g. Odds Ratio Skill Score) are very non-linear

Hedging“Issuing a forecast

that differs from your true belief in order to improve your score”

(e.g. Jolliffe 2008)

• Hit rate H=a/(a+c)– Fraction of events

correctly forecast– Easily hedged by

randomly changing some forecasts of non-occurrence to occurrence

H=0.5

H=0.75

H=1

EquitabilityDefined by Gandin and Murphy (1992):• Requirement 1: An equitable verification measure awards all

random forecasting systems, including those that always forecast the same value, the same expected score– Inequitable measures rank some random forecasts above skillful

ones

• Requirement 2: An equitable verification measure S must be expressible as the linear weighted sum of the elements of the contingency table, i.e. S = (Saa +Sbb +Scc +Sdd) / n– This can safely be discarded: it is incompatible with other

desirable properties, e.g. usefulness for rare events

• Gandin and Murphy reported that only the Peirce Skill Score and linear transforms of it is equitable by their requirements – PSS = Hit Rate minus False Alarm Rate = a/(a+c) – b/(b+d)– What about all the other measures reported to be equitable?

Some reportedly equitable measures

HSS = [x-E(x)] / [n-E(x)]; x = a+d ETS = [a-E(a)] / [a+b+c-E(a)]

LOR = ln[ad/bc] ORSS = [ad/bc – 1] / [ad/bc + 1]

E(a) = (a+b)(a+c)/n is the expected value of a for an unbiased random forecasting system

Random and constant forecasts all score zero, so these measures are all equitable, right?

Simple attempts to hedge will fail for all these measures

Skill versus cloud-fraction threshold

• Consider 7 models evaluated over 3 European sites in 2003-2004

LOR implies skill increases for larger

cloud-fraction thresholdHSS implies skill decreases

significantly for larger cloud-fraction threshold

LORHSS

Extreme dependency score• Stephenson et al. (2008) explained this behavior:

– Almost all scores have a meaningless limit as “base rate” p 0– HSS tends to zero and LOR tends to infinity

• They proposed the Extreme Dependency Score:

– where n = a + b + c + d

• It can be shown that this score tends to a meaningful limit:– Rewrite in terms of hit rate H =a/(a +c) and base rate p =(a +c)/n :

– Then assume a power-law dependence of H on p as p 0:– In the limit p 0 we find

– This is useful because random forecasts have Hit rate converging to zero at the same rate as base rate: =1 so EDS=0

– Perfect forecasts have constant Hit rate with base rate: =0 so EDS=1

Symmetric extreme dependency score

• EDS problems:– Easy to hedge (unless

calibrated)– Not equitable

• Solved by defining a symmetric version:– All the benefits of EDS,

none of the drawbacks!

Hogan, O’Connor and Illingworth (2009 QJRMS)

Skill versus cloud-fraction threshold

SEDS has much flatter behaviour for all models (except for Met Office which underestimates high cloud occurrence significantly)

LORHSS SEDS

Skill versus height– Most scores not reliable

near the tropopause because cloud fraction tends to zero

LORHSS

LBSS

SEDS

• New score reveals:– Skill tends to slowly

decrease at tropopause

– Mid-level clouds (4-5 km) most skilfully predicted, particularly by Met Office

– Boundary-layer clouds least skilfully predicted

EDS

A surprise?• Is mid-level cloud well forecast???

– Frequency of occurrence of these clouds is commonly too low (e.g. from Cloudnet: Illingworth et al. 2007)

– Specification of cloud phase cited as a problem– Higher skill could be because large-scale ascent has largest

amplitude here, so cloud response to large-scale dynamics most clear at mid levels

– Higher skill for Met Office models (global and mesoscale) because they have the arguably most sophisticated microphysics, with separate liquid and ice water content (Wilson and Ballard 1999)?

• Low skill for boundary-layer cloud is not a surprise!– Well known problem for forecasting (Martin et al. 2000) – Occurrence and height a subtle function of subsidence rate,

stability, free-troposphere humidity, surface fluxes, entrainment rate...

Key properties for estimating ½ life

• We wish to model the score S versus forecast lead time t as:

– where 1/2 is forecast “half-life”

• We need linearity– Some measures “saturate” at high skill

end (e.g. Yule’s Q / ORSS)– Leads to misleadingly long half-life

• ...and equitability– The formula above assumes that score tends to zero for very long

forecasts, which will only occur if the measure is equitable

2/1/0

/0 2)( tt SeStS

• Expected values of a–d for a random forecasting system may score zero:– S[E(a), E(b), E(c), E(d)] = 0

• But expected score may not be zero!

– E[S(a,b,c,d)] = P(a,b,c,d)S(a,b,c,d)

• Width of random probability distribution decreases for larger sample size n– A measure is only equitable if positive

and negative scores cancel

Which measures are equitable?

ETS & ORSS are asymmetric

n = 16 n = 80

Asyptotic equitability • Consider first unbiased forecasts of events that occur with

probability p = ½

– Expected value of “Equitable Threat Score” by a random forecasting system decreases below 0.01 only when n > 30

– This behaviour we term asymptotic equitability

– Other measures are never equitable, e.g. Critical Success Index CSI = a/(a+b+c), also known as Threat Score

What about rarer events?• “Equitable Threat Score” still virtually equitable for n > 30

• ORSS, EDS and SEDS approach zero much more slowly with n – For events that occur 2% of the time (e.g. Finley’s tornado

forecasts), need n > 25,000 before magnitude of expected score is less than 0.01

– But these measures are supposed to be useful for rare events!

Possible solutions1. Ensure n is large enough that E(a) > 102. Inequitable scores can be scaled to make them equitable:

– This opens the way to a new class of non-linear equitable measures

),|E()max(

),|E(

s

sequit qpSS

qpSSS

3. Report confidence intervals and “p-values” (the probability of a score being achieved by chance)

What is the origin of the term “ETS”?

• First use of “Equitable Threat Score”: Mesinger & Black (1992)– A modification of the “Threat Score” a/(a+b+c)– They cited Gandin and Murphy’s equitability requirement that

constant forecasts score zero (which ETS does) although it doesn’t satisfy requirement that non-constant random forecasts have expected score 0

– ETS now one of most widely used verification measures in meteorology

• An example of rediscovery– Gilbert (1884) discussed a/(a+b+c) as a possible verification

measure in the context of Finley’s (1884) tornado forecasts– Gilbert noted deficiencies of this and also proposed exactly the

same formula as ETS, 108 years before!

• Suggest that ETS is referred to as the Gilbert Skill Score (GSS)– Or use the Heidke Skill Score, which is unconditionally equitable

and is uniquely related to ETS = HSS / (2 – HSS)

Hogan, Ferro, Jolliffe and Stephenson (WAF, in press)

• Truly equitable

• Asymptotically equitable

• Not equitable

Properties of various measures

Skill versus lead time

• Only possible for UK Met Office 12-km model and German DWD 7-km model– Steady decrease of skill with lead time– Both models appear to improve between 2004 and 2007

• Generally, UK model best over UK, German best over Germany– An exception is Murgtal in 2007 (Met Office model wins)

2004 2007

Forecast “half life”

• Fit an inverse-exponential:– S0 is the initial score and 1/2 is the half-life

• Noticeably longer half-life fitted after 36 hours– Same thing found for Met Office rainfall forecast (Roberts 2008)– First timescale due to data assimilation and convective events– Second due to more predictable large-scale weather systems

2004 20072.6 days

2.9 days2.9 days2.7 days2.9 days

2.7 days

2.7 days3.1 days

2.4 days

4.0 days4.3 days4.3 days

3.0 d

3.2 d

3.1 d

Met Office DWD

2/1/0 2)( tStS

• Different spatial scales? Convection?– Average temporally before calculating skill scores:

– Absolute score and half-life increase with number of hours averaged

Why is half-life less for clouds than pressure?

• Cloud is noisier than geopotential height Z because it is separated by around two orders of differentiation:

– Cloud ~ vertical wind ~ relative vorticity ~ 2streamfunction ~ 2pressure– Suggests cloud observations should be used routinely to evaluate models

Geopotential height anomaly Vertical velocity

Satellite observations: IceSAT• Cloud observations from IceSAT 0.5-micron

lidar (first data Feb 2004)• Global coverage but lidar attenuated by thick

clouds: direct model comparison difficult

Optically thick liquid cloud obscures view of any clouds beneath

Solution: forward-model the measurements (including attenuation) using the ECMWF variables

Lidar apparent backscatter coefficient (m-1 sr-1)

Latitude

Global cloud fraction comparison

ECMWF raw cloud fraction ECMWF processed cloud fraction

IceSAT cloud fraction

Wilkinson, Hogan, Illingworth and Benedetti (MWR 2008)

• Results for October 2003– Tropical convection peaks too

high– Too much polar cloud– Elsewhere agreement is good

• Results can be ambiguous– An apparent low cloud

underestimate could be a real error, or could be due to high cloud above being too thick

Testing the model skill from space

Clearly need to apply SEDS to cloud estimated from lidar & radar!

Unreliable region

Lowest skill: tropical boundary-layer clouds

Tropical skill appears to peak at mid-levels but cloud very infrequent

here

Highest skill in north mid-latitude and polar upper

troposphere

Is some of reduction of skill at low levels because of lidar

attenuation?

Wilkinson, Hogan, Illingworth and Benedetti (MWR 2008)

CCPP project• US Dept of Energy Climate Change Prediction Program recently

funded 5-year consortium project centred at Brookhaven, NY– Implement updated Cloudnet processing system at Atmospheric

Radiation Measurement (ARM) radar-lidar sites worldwide– Ingests ARM’s cloud boundary diagnosis, but uses Cloudnet for

stats– New diagnostics being tested

• Testing of NWP models– NCEP, ECMWF, Met Office, Meteo-France... – Over a decade of data at several sites: have cloud forecasts

improved over this time?

• Single-column model testbed– SCM versions of many GCMs will be run over ARM sites by Roel

Neggers – Different parameterization schemes tested– Verification measures can be used to judge improvements

US Southern Great Plains 2004

Winter2004

Summer2004

Summary and outlook• Model comparisons reveal:

– Half-life of a cloud forecast is between 2.5 and 4 days, much less than ~9 days for ECMWF 500-hPa geopotential height forecast

– In Europe, higher skill for mid-level cloud and lower for boundary-layer cloud, but larger seasonal contrast in Southern US

• Findings applicable to other verification problems:– “Symmetric Extreme Dependency Score” is a reliable measure of

skill for both common and rare events (given we have large enough sample)

– Many measures regarded as equitable are only so for very large samples, including the “Equitable Threat Score”, but they can be rescaled

• Future work (in addition to CCPP):– CloudSat & Calipso: what is the skill of cloud forecasts globally?– What is half-life of ECMWF cloud forecasts? (Need more data!) – Near-real-time evaluation for rapid feedback to NWP centres?– Dept of Meteorology Lunchtime Seminar, 1pm Tuesday 3rd Nov:

“Faster and more accurate representation of clouds and gases in GCM radiation schemes”

Monthly skill versus time• Measure of the skill of forecasting cloud fraction>0.05

– Comparing models using similar forecast lead time– Compared with the persistence forecast (yesterday’s

measurements)

• Lower skill in summer convective events

Statistics from AMF

• Murgtal, Germany, 2007– 140-day comparison

with Met Office 12-km model

• Dataset released to the COPS community– Includes German

DWD model at multiple resolutions and forecast lead times

Possible skill scoresContingency

tableObserved

cloud Observed clear

sky

Modeled cloud

ahit

b false alarm

Modeled clear sky

cmiss

d correct negative

DWD model

a = 7194 b = 4098

c = 4502 d = 41062

Perfect forecast

ap = 11696 bp = 0

cp = 0 dp = 45160

Random forecast

ar = 2581 br = 8711

cr = 9115 dr = 36449

• To ensure equitability and linearity, we can use the concept of the “generalized skill score” = (x-xrandom)/(xperfect-xrandom)– Where “x ” is any number derived from the joint PDF– Resulting scores vary linearly from random=0 to perfect=1

• Simplest example: Heidke skill score (HSS) uses x=a+d– We will use this as a reference to test other scores

• Brier skill score uses x=mean squared cloud-fraction difference, Linear Brier skill score (LBSS) uses x=mean absolute difference– Sensitive to errors in model for all values of cloud fraction

“Cloud” deemed to occur when cloud fraction f is larger than

some threshold fthresh

Alternative approach• How valid is it to estimate 3D cloud fraction from 2D slice?

– Henderson and Pincus (2009) imply that it is reasonable, although presumably not in convective conditions

• Alternative: treat cloud fraction as a probability forecast– Each time the model forecasts a particular cloud fraction, calculate

the fraction of time that cloud was observed instantaneously over the site

– Leads to a Reliability Diagram:

Jakob et al. (2004)

Perfect

No skillNo resolution

Simulate lidar backscatter:– Create subcolumns with max-rand

overlap– Forward-model lidar backscatter from

ECMWF water content & particle size– Remove signals below lidar sensitivity

ECMWF raw cloud fraction

ECMWF cloud fraction after processing

IceSAT cloud fraction

Testing the model climatology

Reduction in model due to lidar attenuation

Error due to uncertain extinction-to-backscatter ratio