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Downscaling Climate Variables
Downscaling:
Inferring climate variations on smaller spatial/temporal scales
than resolution of climate model/forecast
1Marina
Timofeyeva, 2David Unger and 3Cecile Penland
1UCAR
and NWS/NOAA
2NWS/NOAA
3OAR/NOAA
Contributors: Robert Livezey and Rachael Craig
NOAA NWS and OAR
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
Standard Deviation of 500mb Geopotential height Anomalies in JFM
Legend:
Contours are every 10 m
- > 45 m
- > 75 m
Slide courtesy: P.Sardeshmukh
NOAA NWS and OAR
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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Idealized spectrum of extratropical
height variability
1.2
Introduction: Source of Predictability
1.2
1
1
0.8
P
P
Idealized spectrum of extratropicalsynoptic
broadening
height variability
synoptic
broadening
0.8
1.2
0.6
0.61
0.4
0.8
P
0.4
0.2
0.6
anthropogenic
forcing ?
anthropogenic
forcing ?
0.2
0
0.4
0.001
0.01
0 anthropogenic
?
0.001 forcing 0.01
0.2
ENSO
effect
ENSO
effect
ENSO 0.1
effect
log 0.1

synoptic
broadening
red noise
background
red noise
background
1
10
red1noise
background
10
20 days
1
2
10
( =20
2 
/ r)
days
2
log  
Periods
0
20000
0.001
2000
0.01
Periods
20000
2000
0.1
200
log200

( = 2 
/ r)
Time
Periods
Averages
Time
Averages
Time
~ 20000
60 yr
~ 2000
6 yr
200
seasonal
~ 10 day
20 days
~ 60 yr
~ 6 yr
seasonal
~ 10 day
( = 2 / r )
NOAA
NWS
and OAR
~ 6 yr
seasonal
2
daily
daily
Slide courtesy: P.Sardeshmukh
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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Justification for Temperature PDF
Example:
One way dynamics affects probability:
A temperature equation with cooling and
heating:
dT
 T  Q
dt
Also, let’s say that the heating Q has a
Gaussian white noise component to it:
Q = Qo +  Q
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Justification for Temperature PDF
The pdf f(T) is described by the following
equation:
f (T )

1 2
T  Qo  f (T )  

 f (T )
2
t
T
2 T
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.
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Downscaling Temperature Forecasts
– Otherwise, skill is primarily
modest and level with lead
(derived from biased
climatologies, i.e. long-term
trend)
50
40
30
20
10
0
-10
Heidke Skill Score
Predictability of
The Downscaling Source :
– Moderate to high national-scale
skill confined to Fall/Winter
strong ENSO years at short to
medium leads
0.5
1.5
2.5
3.5
4.5
5.5
6.5
9.5
10.5 11.5 12.5
ENSO:FMA,MAM,AMJ
Heidke Skill Score
– 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
8.5
Lead (month)
All:FMA,MAM,AMJ
Other:FMA,MAM,AMJ
50
40
30
20
10
0
-10
7.5
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
Lead (month)
All:DJF,JFM,FMA
Other:DJF,JFM,FMA
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ENSO:DJF,JFM,FMA
Downscaling Temperature Forecasts
Predictability of the Downscaling Source – Map
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Downscaling Temperature Forecasts
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
100
Observed T
POF (%)
CD96 forecast
Phoenix forecast
80
CD96 climatology
60
40
20
0
82
84
86
88
90
92
94
96
98
Forecasted Temperature (°F)
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Downscaling Temperature Forecasts
How CPC adjusts CD forecast distribution back towards
climatology depending upon forecast skill.
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 isthe deterministic forecast (from1971 - 2000),
 CD the1971 - 2000 std,and
 the correlation skill for theTCD forecast
NOTE : Low skill pushes the forecast towards the climatology
NOAA NWS and OAR
Downscaling Temperature Forecasts
3. Station distributi on mean and std are estimated by regression :
*
T i  ai  bi TCD
*
i
2
 ai 
ri TCD ;  i   i 1  ri ;
 CD
where  i is the station1971  2000 standard deviation and
ri is the correlatio n coefficient between station and CD
NOTE : Low correlatio n pushes the distributi on toward the climatology
4. Combining 2 and 3, the station forecast distribution has mean and std :
^
^
T i  ai  bi T CD ;
^
 i  i
^

  CD
1  1 
  CD

2

 2
2 2
r


1


ri
i
i


NOTE : Both low CD forecast skill and low correlatio n b / w station and CD
push forecast distibutio n toward station climatology
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Downscaling Temperature Forecasts
50
y = 1.1707x - 5.1282
R2 = 0.9335
Station Temperature
45
y = 1.2658x - 9.5544
R2 = 0.9104
40
1458
130
35
9181
30
Linear (1458)
Linear (9181)
25
Linear (130)
y = -0.0711x + 29.528
R2 = 0.0009
20
15
15
20
25
30
35
40
45
CD Temperature
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50
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:


   
 x X 
abs
 cutoff for Student ' s t distributi on
sx  

 

 n  



x is the last 10  year mean of the differences between station and CD temperature

X is the clim atolog ical (1971  2000) mean of the differences
s x is the last 10  year standard deviation of the differences
n  10 is the number of years
Student ' s cutoff ( for sample of 10 members) for Confidence level 95% is 2.306
NOAA NWS and OAR
Downscaling Temperature Forecasts
2 
 2 

 year  
 * TST , year  TCD , year   1 
 *  year 1 , where N  10 years
 N 1
 N  1
ai , adj  ai  ( year   mean for 19712000 )
12
Δ (°F)
10
8
6
4
2
1961
1971
1981
SLC-CD83
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1991
Trend
2001
Downscaling Temperature Forecasts
1.0
Spread of Station Forecast
Climatological Spread
ρ
0.9
(CD fcst/obs corr)
0.5
0.7
0.8
0.9
1
0.8
0.7
0.6
Confident
Prediction
0.5
0.5
0.6
0.7
0.8
0.9
ri – Station/CD Correlation
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1
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
0.04
Mean = 60.7
St.Dev.= 13.6
Median = 59.5
Mode = 52.0
Skewness = 0.225
Kurt = -0.526
0.03
0.02
Temperature is a
normally distributed
variable, therefore the
downscaling method
based on regression can
provide good estimates
0.01
0
0
10
20
30
40
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.
50
60
70
80
90 100 110
1.2
1
0.8
0.6
0.4
0.2
0
0 NWS
0.5 and
1 OAR
1.5
NOAA
Mean = 0.30
St. Dev.= 0.38
Median = 0.19
Mode = 0.01
Skewness = 3.11
Kurtosis = 14.67
2
2.5
3
3.5
4
4.5
Downscaling Precipitation Forecast
Distributions of seasonal precipitation totals are too skewed
Station
CD
Relative Frequency
0.3
0.2
0.1
0
1
2
3
4
5
Precipitation amount bins
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6
7
Downscaling Precipitation Forecast
15
10
5
0
-5
0.5
1.5
2.5
3.5
4.5
5.5
-10
6.5
7.5
8.5
9.5 10.5 11.5 12.5
Lead (month)
All:FMA,MAM,AMJ
ENSO:FMA,MAM,AMJ
Other:FMA,MAM,AMJ
20
Heidke Skill Score
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
Heidke Skill Score
20
15
10
5
0
-5
0.5
1.5
2.5
3.5
4.5
5.5
-10
6.5
7.5
8.5
9.5 10.5 11.5 12.5
Lead (month)
All:DJF,JFM,FMA
Other:DJF,JFM,FMA
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ENSO:DJF,JFM,FMA
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
Percentage of Non-Viable Stations for
DS using regression
Which distribution is an appropriate assumption for
precipitation?
Normal
Lognormal
Gamma
120%
100%
80%
60%
40%
20%
0%
FMA MAM AMJ MJJ
JJA
JAS ASO SON OND NDJ DJF JFM
Season
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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
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):
 e
  12  2
 e
2
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2   2
(e
2
 1)
Downscaling Precipitation Forecast
Parameters of the linear regression are quantiles of standard
normal distribution
Station
3
CD
Station Q transformed data
7
Precipitation
6
5
4
3
2
1
y = 0.9x + 0.007
R2 = 0.83
2
1
0
-4
-2
0
-1
-2
0
-3
-2
-1
0
1
2
3
Quantiles of Standard Normal
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-3
CD Q tranform ed data
2
4
Temporal Downscaling
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=
JFM
+
FMA
+
3
NOAA NWS and OAR
MAM
CRPS Skill Scores:
.051
.045
.027
.029
.041
.034
.026
.023
.020
.021
.024
.024
.040
.036
.026
.030
.094
.103
.074
.090
.055
.059
.055
.058
.013
.016
.027
.026
.044
.038
.050
.047
.065
.055
.042
.035
1-Month Lead, All initial times
Temperature
.002
.001
.011
.004
-.009
.002
-.006
-.008
Skill
.035
.030
High
.012
.015
.10
Moderate
.05
Low
.01
None
FD
.031
CD
.023 3-Mo
.028
.019 1-Mo
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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):
Ti  T  A * I i  phase    k z i  k   i
40
35
600
k 1
30
500
Insolation (Watts/m2)
Temperature (ºC)
n
_
45
β
25
20
15
400
α
300
10
200
5
0
-5
100
TMN
PHASE
-10
1
181
361
541
721
901
0
1081
days
Observed T
Fitted T
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Insolation
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 nonstationary 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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