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Downscaling Climate VariablesDownscaling:
Inferring climate variations on smaller spatial/temporal scales
than resolution of climate model/forecast
1Marina Timofeyeva, 2David Unger and 3Cecile Penland1UCAR and NWS/NOAA
2NWS/NOAA
3OAR/NOAA
Contributors: Robert Livezey and Rachael Craig
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Outline
• Introduction: Local Climate Variables• Downscaling Seasonal Temperature
Forecasts • Downscaling Seasonal Precipitation
Forecast• Temporal Downscaling• Summary
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Introduction: Definitions
Downscaling to a Local Climate Variable:• Downscaling – inferring climate variations on smaller
spatial/temporal scales than resolution of climate model/forecast
• Local – points, station, small grid, etc. Key: higher resolution than the original variable used for downscaling
• Climate – mean daily, weekly, monthly, seasonal (3-4 month) temperature, precipitation, wind fields, etc.
• Variable – main object of interest: observation or forecast. Note climate variable is often considered in form of parameters of distribution
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Introduction: Climate Variables
Slide courtesy: P.SardeshmukhNOAA NWS and OARNOAA NWS and OAR
Standard Deviation of 500mb Geopotential height Anomalies in JFM
Legend:
Contours are every 10 m- > 45 m
- > 75 m
Introduction (cont.)
Downscaling Methods:• Dynamical – applications are on meteorological
scale, climate variables are estimated as averages of continuous model runs
• Statistical – variable can be modeled at defined temporal scale, e.g. monthly, weekly, seasonal, etc, if predictability (deviation from observational noise and/or forecast skill) at such scale exists
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Introduction (cont.)
Downscaling requirements:
• Model Simplicity
• Validity of Distribution
• Existence of potential predictability
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Introduction: Assumptions
Assumptions must be appropriate for the dynamical system being downscaled.
Example: If the amplitude of a Rossby wave is normally distributed, the energy in that wave cannot be normally distributed. (In fact, it would be chi-squared.)
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Introduction: Source of Predictability
0
0.2
0.4
0.6
0.8
1
1.2
0.001 0.01 0.1 1 10
20 days
( = 2 / r )
200020000 200 2
~ 10 dayseasonal~ 6 yr~ 60 yr daily
anthropogenic forcing ?
ENSOeffect
synoptic broadening
red noisebackground
Idealized spectrum of extratropical height variability
P
log
Time Averages
Periods
Slide courtesy: P.SardeshmukhSlide courtesy: P.Sardeshmukh
NOAA NWS and OARNOAA NWS and OAR
0
0.2
0.4
0.6
0.8
1
1.2
0.001 0.01 0.1 1 10
20 days
( = 2 / r )
200020000 200 2
~ 10 dayseasonal~ 6 yr~ 60 yr daily
anthropogenic forcing ?
ENSOeffect
synoptic broadening
red noisebackground
Idealized spectrum of extratropical height variability
P
log
Time Averages
Periods
0
0.2
0.4
0.6
0.8
1
1.2
0.001 0.01 0.1 1 10
20 days
( = 2 / r )
200020000 200 2
~ 10 dayseasonal~ 6 yr~ 60 yr daily
anthropogenic forcing ?
ENSOeffect
synoptic broadening
red noisebackground
Idealized spectrum of extratropical height variability
P
log
Time Averages
Periods
Downscaling Temperature Forecasts
Source for Downscaling: CPC forecasts
Questions to be answered:
• Why downscale?
• What distribution is appropriate?
• Is there potential predictability?
• How do we do it?
• What is the outcome?
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Downscaling Temperature Forecasts
• Why downscale?
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Downscaling Temperature Forecasts
When there is a climate signal, CPC has a reason to change the odds from climatological distribution
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One way dynamics affects probability:A temperature equation with cooling and
heating:
Also, let’s say that the heating Q has a Gaussian white noise component to it:
Q = Qo + Q
dTdt
T Q
Justification for Temperature PDF
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Example:
The pdf f(T) is described by the following equation:
where is essentially the variance of Q.
This is the equation for a Gaussian distribution.
Thus, Gaussian systems are equivalent in probability to linear dynamical systems.
f (T )
t
T
T Qo f (T ) 1
2
2
T 2 f (T )
Justification for Temperature PDF
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Downscaling Temperature ForecastsPredictability of
The Downscaling Source :– Moderate to high national-scale
skill confined to Fall/Winter strong ENSO years at short to medium leads
– Otherwise, skill is primarily modest and level with lead (derived from biased climatologies, i.e. long-term trend)
– Worst forecasts are for• Fall/Winter at short to medium
leads in the absence of strong-ENSO
• Summer/Fall at medium to long leads even for strong ENSOs: No remedy except to advance the science
-10
0
10
20
30
40
50
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5
All:DJF,JFM,FMA ENSO:DJF,JFM,FMA
Other:DJF,JFM,FMA
-10
0
10
20
30
40
50
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5
All:FMA,MAM,AMJ ENSO:FMA,MAM,AMJ
Other:FMA,MAM,AMJ
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He
idke
Ski
ll S
core
He
idke
Ski
ll S
core
Lead (month)
Lead (month)
Downscaling Temperature Forecasts
Predictability of the Downscaling Source – Map
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Downscaling Temperature Forecasts
Forecasted Temperature (°F)
PO
F (
%)
The CPC POE outlooks for each CD are used as downscaling source for station specific outlooks.Historical NCDC data (1959 to present) for station and CD are used in developing downscaling relations that, together with CPC operational forecasts, are used for station POE outlooks
Observed T
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Downscaling Temperature ForecastsHow CPC adjusts CD forecast distribution back towards climatology depending upon forecast skill.
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1. CPC fits a normal distribution consistent with the forecasted tercile
probabilities to get TCD ,which is the mean of the forecasted CD pdf.
2. Adjusted CD distribution forecast then will have
mean and std :
T^
CD TCD; ^
CD CD 1 2
where
TCD isthedeterministic forecast (from1971 - 2000),
CD the1971 - 2000 std,and
thecorrelation skill for theTCD forecast
NOTE : Low skill pushes the forecast towards theclimatology
Downscaling Temperature Forecasts
yclimatologthetowardondistributithepushesncorrelatioLowNOTE
CDandstationbetweentcoefficienncorrelatiotheisr
anddeviationdardtansstationtheiswhere
rTraTbaT
regressionbyestimatedarestdandmeanondistributiStation
i
i
iiiCDiCD
iiCDiii
:
20001971
;1;
:.3
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yclimatologstationtowardndistibutioforecastpush
CDandstationwbncorrelatiolowandskillforecastCDlowBothNOTE
rr
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stdandmeanhasondistributiforecaststationtheandCombining
iiiCD
CD
ii
CDiii
/:
111
;
:,32.4
222
2^^
^^
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Downscaling Temperature Forecasts
y = 1.2658x - 9.5544R2 = 0.9104
y = -0.0711x + 29.528R2 = 0.0009
y = 1.1707x - 5.1282R2 = 0.9335
15
20
25
30
35
40
45
50
15 20 25 30 35 40 45 50
CD Temperature
Sta
tion
Tem
pera
ture
1458
130
9181
Linear (1458)
Linear (9181)
Linear (130)
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Downscaling Temperature Forecasts
Adjustment of Intercept (ai) for local trend at the station is needed IF the trend over last 10 years is statistically significant:
306.2%95)10('
10
10
)20001971(
10
'
islevelConfidenceformembersofsampleforcutoffsStudent
yearsofnumbertheisn
sdifferencetheofdeviationdardtansyearlasttheiss
sdifferencetheofmeanicallogatolimctheisX
etemperaturCDandstationbetweensdifferencetheofmeanyearlasttheisx
ondistributitsStudentforcutoff
n
s
Xxabs
x
x
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Downscaling Temperature Forecasts
2
4
6
8
10
12
1961 1971 1981 1991 2001
SLC-CD83 Trend
)(
10,*1
21*
1
2
20001971,
1,,
formeanyeariadji
yearyearCDyearSTyear
aa
yearsNwhereN
TTN
NOAA NWS and OARNOAA NWS and OAR
Δ (
°F)
Downscaling Temperature Forecasts
ri – Station/CD Correlation
ρ (CD fcst/obs corr)
Sp
read
of
Sta
tion
Fore
cast
0.5
0.6
0.7
0.8
0.9
1.0
0.5 0.6 0.7 0.8 0.9 1
0.5
0.7
0.8
0.9
1
Climatological Spread
Confident Prediction
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Downscaling Temperature Forecasts
Outcome – NWS Local Climate Product:
Outlook Graphics are dynamically generated for every location (1,141 sites; about 10 sites per WFO CWA)
Text interpretation of probability information for general public avoids use of very technical terms
Intuitive navigating options
Clickable maps for changing locations
Main menu and interactive (clickable) map and graphs
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Downscaling Precipitation Forecasts
Source for Downscaling: CPC forecasts
Questions to be answered:• Why downscale? –discussed in previous section• What distribution is appropriate?• Is there potential predictability?• How do we do it?• What is the outcome? – discussed in previous
section
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Downscaling Precipitation Forecasts
Mean = 0.30St. Dev.= 0.38Median = 0.19Mode = 0.01Skewness = 3.11Kurtosis = 14.67
Mean = 60.7St.Dev.= 13.6Median = 59.5Mode = 52.0Skewness = 0.225Kurt = -0.526
Temperature is a normally distributed variable, therefore the downscaling method based on regression can provide good estimates
Precipitation (right chart) is too skewed for normal distribution. The regression would require a transformation of this variable. Compositing can be used for Precipitation forecasts because it does not employ regression analysis.
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Downscaling Precipitation Forecast
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Distributions of seasonal precipitation totals are too skewed
0
0.1
0.2
0.3
1 2 3 4 5 6 7
Precipitation amount bins
Rel
ativ
e F
req
uen
cy
Station CD
Downscaling Precipitation Forecast
Is there Potential Predictability in CPC Precipitation Forecasts?– Useable national-scale
skill entirely confined to Fall/Winter strong ENSO years in short to medium leads
– Otherwise skill is statistically indistinguishable from zero
-10
-5
0
5
10
15
20
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5
All:FMA,MAM,AMJ ENSO:FMA,MAM,AMJ
Other:FMA,MAM,AMJ
-10
-5
0
5
10
15
20
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5
All:DJF,JFM,FMA ENSO:DJF,JFM,FMA
Other:DJF,JFM,FMA
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He
idke
Ski
ll S
core
He
idke
Ski
ll S
core
Lead (month)
Lead (month)
Downscaling Precipitation Forecast
Predictability of CPC Precipitation Forecasts
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Downscaling Precipitation Forecast
• Which distribution is an appropriate assumption for precipitation?– Data: 1960 – 2003 3 month (DJF, …OND)
total precipitation for 87 locations in NWS WR– Kolmogorov-Smirnoff GOF test of
Distributions: Normal, Lognormal and Gamma– Mapping CPC forecast potential predictability
on fit of an assumed distribution
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Downscaling Precipitation Forecast
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Which distribution is an appropriate assumption for precipitation?
0%
20%
40%
60%
80%
100%
120%
FMA MAM AMJ MJJ JJA JAS ASO SON OND NDJ DJF JFM
Season
Pe
rce
nta
ge
of
No
n-V
iab
le S
tati
on
s f
or
DS
us
ing
re
gre
ss
ion
Normal Lognormal Gamma
Downscaling Precipitation Forecast
• What does it mean?– Linear regression cannot be used because
distribution assumptions, used by regression tests, are not met in many cases
– Several alternatives:• Variable transformation, e.g. sqrt, ln, etc.• Normal Quantile transformation• Special Case, zero precipitation amounts, require
the use of two model forecast systems: 1. forecast probability of precipitation chance and 2. forecast probability of precipitation amount
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Downscaling Precipitation Forecast
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Warning : To apply a nonlinear transformation we must ensure a straightforward procedure to transform the downscaled predictions back to physical units.
For example, log transformation has a relationship between parameters in transformed (α,β) and untransformed (μ,σ) domains (Aitchison and Brown, 1957):
221 e )1(
2222 ee
Downscaling Precipitation Forecast
0
1
2
3
4
5
6
7
-3 -2 -1 0 1 2 3
Quantiles of Standard Normal
Pre
cip
itat
ion
Station CD
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Parameters of the linear regression are quantiles of standard normal distribution
y = 0.9x + 0.007R2 = 0.83
-3
-2
-1
0
1
2
3
-4 -2 0 2 4
CD Q tranformed dataS
tati
on
Q t
ran
sfo
rmed
dat
a
Disaggregation - Seasonal to Monthly
• Regression and Average of 3 estimates• Simultaneous spatial and temporal downscaling possible• Tm- = bs- Ts- + as- ; S- = m-2,m-1,m, R=Lower• Tm0 = bs0 Ts0 + as0 ; S0 = m-1,m,m+1, R=Best • Tm+ = bs+ Ts+ + as+ ; S+ = m ,m+1,m+2, R=Lower
Tm= (Tm- + Tm0 + Tm+ )/3
M =3
MAM FMA JFM
Temporal Downscaling
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+ +
.031
.023
.028 .019
.040 .036
.026 .030
.094 .103
.074 .090
.035 .030
.012 .015
1-Mo
FD CD3-Mo
CRPS Skill Scores: Temperature
-.009 .002
-.006 -.008
.002 .001
.011 .004
.044 .038
.050 .047
.013 .016
.027 .026
.055 .059
.055 .058
.027 .029
.026 .023
.020 .021
.024 .024
.051 .045
.041 .034
.065 .055
.042 .035
High
Moderate
Low
None
Skill
.10
.05
.01
1-Month Lead, All initial times
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Downscaling Other than Seasonal Climate Variables
• Alternative – Statistical downscaling of variables representing stochastic structure of climate variables at finer than seasonal scale.
• Example - statistical downscaling model is linked with a GCM by using most predictable fields (e.g., SST, Wind fields) as forcing. Downscaling model is a correlation model between variables derived from the GCM fields and variables representing stochastic structure of local climate variables
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Downscaling Other than Seasonal Climate Variables
• Stochastic structure variables of temperature – Insolation term (T, A and phase), AR terms (Φ) and white noise term (ε):
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45
1 181 361 541 721 901 1081
0
100
200
300
400
500
600
Observed T Fitted T Insolation
PHASETMN
Temperature (ºC)
Insolation (Watts/m2)
α
β
days
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i
n
kkikphaseii zIATT
1
_
*
Lessons learned
• Keep your model simple and your assumptions in mind
• To have good downscaling results, the original prediction skills must be good.
• The statistics between large and small scales must be robust and appropriate.
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Additional Thoughts• Models which don’t represent the current climate well cannot be
credibly downscaled statistically– for even the current climate with methods based only on
observations– for the current climate with methods based on model
corrections if either (a) the model is missing important variability or (b) observational data are limited
• Models of future climate downscaled statistically is problematic because climate change is inherently a non-stationary process
• Nested or linked model downscaling implies major technical challenges as well as assumptions about scale interactions if attempted for future climates (possible solution is global high-resolution models)
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