downscaling climate variables downscaling:

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

NOAA NWS and OARNOAA NWS and OAR

Outline

• Introduction: Local Climate Variables• Downscaling Seasonal Temperature

Forecasts • Downscaling Seasonal Precipitation

Forecast• Temporal Downscaling• Summary


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


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


Introduction (cont.)

Downscaling requirements:

• Model Simplicity

• Validity of Distribution

• Existence of potential predictability


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.)


Introduction: Source of Predictability

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( = 2 / r )

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~ 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


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ENSOeffect

synoptic broadening

red noisebackground


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log

Time Averages

Periods

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( = 2 / r )

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ENSOeffect

synoptic broadening

red noisebackground


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?



• Why downscale?



When there is a climate signal, CPC has a reason to change the odds from climatological distribution


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


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


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

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All:DJF,JFM,FMA ENSO:DJF,JFM,FMA

Other:DJF,JFM,FMA

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All:FMA,MAM,AMJ ENSO:FMA,MAM,AMJ

Other:FMA,MAM,AMJ


He

idke

Ski

ll S

core

He

idke

Ski

ll S

core

Lead (month)

Lead (month)


Predictability of the Downscaling Source – Map



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


Downscaling Temperature ForecastsHow 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 isthedeterministic forecast (from1971 - 2000),

CD the1971 - 2000 std,and

thecorrelation skill for theTCD forecast

NOTE : Low skill pushes the forecast towards theclimatology


yclimatologthetowardondistributithepushesncorrelatioLowNOTE

CDandstationbetweentcoefficienncorrelatiotheisr

anddeviationdardtansstationtheiswhere

rTraTbaT

regressionbyestimatedarestdandmeanondistributiStation

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iiiCDiCD

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:

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:.3

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stdandmeanhasondistributiforecaststationtheandCombining

iiiCD

CD

ii

CDiii

/:

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2^^

^^



y = 1.2658x - 9.5544R2 = 0.9104

y = -0.0711x + 29.528R2 = 0.0009

y = 1.1707x - 5.1282R2 = 0.9335

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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)



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('

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10

)20001971(

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'

islevelConfidenceformembersofsampleforcutoffsStudent

yearsofnumbertheisn

sdifferencetheofdeviationdardtansyearlasttheiss

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etemperaturCDandstationbetweensdifferencetheofmeanyearlasttheisx

ondistributitsStudentforcutoff

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s

Xxabs

x

x



2

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1961 1971 1981 1991 2001

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)(

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1

2

20001971,

1,,

formeanyeariadji

yearyearCDyearSTyear

aa

yearsNwhereN

TTN


Δ (

°F)


ri – Station/CD Correlation

ρ (CD fcst/obs corr)

Sp

read

of

Sta

tion

Fore

cast

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Climatological Spread

Confident Prediction



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


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


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.


Downscaling Precipitation Forecast


Distributions of seasonal precipitation totals are too skewed

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Precipitation amount bins

Rel

ativ

e F

req

uen

cy

Station CD


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

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All:FMA,MAM,AMJ ENSO:FMA,MAM,AMJ

Other:FMA,MAM,AMJ

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All:DJF,JFM,FMA ENSO:DJF,JFM,FMA

Other:DJF,JFM,FMA


He

idke

Ski

ll S

core

He

idke

Ski

ll S

core

Lead (month)

Lead (month)


Predictability of CPC Precipitation Forecasts



• 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




Which distribution is an appropriate assumption for precipitation?

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FMA MAM AMJ MJJ JJA JAS ASO SON OND NDJ DJF JFM

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Normal Lognormal Gamma


• 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




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


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Quantiles of Standard Normal

Pre

cip

itat

ion

Station CD


Parameters of the linear regression are quantiles of standard normal distribution

y = 0.9x + 0.007R2 = 0.83

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CD Q tranformed dataS

tati

on

Q t

ran

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rmed

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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


+ +

.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

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1-Month Lead, All initial times


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


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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Observed T Fitted T Insolation

PHASETMN

Temperature (ºC)

Insolation (Watts/m2)

α

β

days


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.


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)


downscaling climate variables downscaling:

Documents

sardeshmukhnoaa nws

climate signal

scale existsnoaa nws

temperature equation

rachael craignoaa nws

month temperature

qo qnoaa nws

downscaling source