Transcript Downscaling - SAMA-IPSL

Downscaling:
Time-space dependent
Downscaling of Wind Stress Data
For Ocean modellings
by
Jun-Ichi Yano
Hiromi Kobayashi
Pascale Bouruet-Aubertot
Downscaling:
?
Principal Approach
Oceanographic
Application
Downscaling:
?
Downscaling:
Spatial scale: DX
?
Dx (< DX )
original
output/
observational
scale
(e.g., atmosphere)
= 50 – 200 km
Alternatively:
Temporal scale: DT
?
required
information/
model-input
scale
(e.g., oceans)
= 5 – 10 km
Dt (< DT )
Principal Approach
Power Law in Tubulence
(Scaling Law)
+
Wavelets
Downscaling
(Spatial-Temporal Heterogeneity)
Oceanographic
Application
Oceanographic
Context:
Wind Stress
(Alexandra Bozec)
Data:
DX=60km
Zonal Component
Data:
DX=125km
Model:
DX=80km
Response to
the 3 types
of Wind Stress
Inputs
in 4 regions
(rows):
Depth of Mixed
Layer
(Alexandra Bozec)
Data:
Data:
Model:
DX=60km DX=120km DX=80km
Downscaling Principle: Fully-Nonlinear
Turbulent System:Power-law Spectra :
log P(k)
Question:
a = a(t)?:
Modification
of variability
with time&
space:
Heterogeneity
-a
log k
Extrapolation
?
1/DX
1/Dx
Methods:Observational Data Analysis:
Data Set: Buoy Time Series (off coast Nice):
Wind Speed:Mar 1999-Dec 2001, Dt=1h
x
Methods:Observational Data Analysis:
Data Set: Buoy Time Series (off coast Nice):
Wind Speed:Mar 1999-Dec 2001, Dt=1h
Methodology: Taylor’s frozen hypothesis:
k=w/u: k
w
Seek a power law: P(w,t) ~w-a(t) as a function
of time (Morlet wavelet spectra)
How to estimate: a=a(t) ?
Wavelet Transform of the Wind Speed Time
series:
Power Law
•Time sereis (segment)
U j et S j (Ti ) pour j = 6000 : 6500
•Wavelet Spectrum
(instantaneous)
Spe ctre à t = 999h
Strategy for the Downscaling
?
log P(T)
Dt
DT
log T
Strategy for the Downscaling
a=?
log P(T)
Tc=?
Dt
log T
Strategy for the Downscaling
•Estimation of wavelet coefficients:
<U, Yu,s> = |<U, Yu,s>| eij(u,s)
i) |<U, Yu,s>| ~ sa(u)/2 :present work
ii) j(u,s) = ? :future work
NB: stochastic : probability
NB: u
t
•Reconstruction of a time series
Strategy for the Downscaling
a=?
log P(T)
Tc=?
Dt
log T
Determination of the Power Exponent:
Seek a spectrum of the form : P(T) ~Ta
for a limited period
Spectrum at t=15787h
Mean over 2n+1 spectra, n=24
Determination of the Power Exponent:
Seek a spectrum of the form : P(T) ~Ta
for Ti < Tc:
a=?
Tc = ?
Spectre en loi de puissance
On cherche un spectre de la forme : P(T) ~Ta
pour Ti < Tc:
a=?
Tc = ?
Probability Distribution of the Exponent:
a (t) : exponent of the spectrum
a=1
Probability
Density :
p(a)
a=5/3
a
Total energy
How to Estimate the Exponent a from the
Other Conditions?: Joint-Probabilities
?
a
Conclusions
Buoy Data: Surface Wind-Speed time series
with Dt = 1 h:
Power-Law Spectra in Wavelet Space:
Two Regimes:
Most likely exponent (14% chance): a = 5/3
2nd Regime with a = 1
Preliminary anlyses for the joint-probabilites
Future Work: Statistics for the Phase: A
1st-order Markov Model?
http://www.ipsl.jussieu.fr/CLIMSTAT/CARGESE/TA
LKS/JUNICHI/downscale.ppt
ftp://ftp.lmd.jussieu.fr/pub/yano/cargese/review.
ftp://ftp.lmd.jussieu.fr/pub/yano/cargese/hiromi.ps
ftp://ftp.lmd.jussieu.fr/pub/yano/cargese/poster_bozec.
ppt
Final Remark: Two Schools in Downscaling:
Climatological
Planetary-Scale
104 km
Synotpic
Scale
Meso-Convective
Scale
103 km
10 - 102 km
I. DX*
Dx
« Regionalization » (CL12, HS9, …)
Linear-Wave Dynamics
II. DX*
Dx
Nonlinear-Turbulent
How to Estimate the Exponent a from the
Other Conditions?: Joint-Probabilities
Tc
a
Total energy
How to Estimate the Exponent a from the
Other Conditions?: Joint-Probabilities
Tc
Outline
Review:
what is downscaling?
why necessary?: oceanographic context
scale dependence: types I & II
approaches for the type I: linear approaches
limitations of linear approaches
approaches for the type II: nonlinear
An explorative study for the downscaling of
the wind stress over the Mediterranean Sea
Downscaling Type I
determining factor
« slaved »
Climatological
State
Synotpic
Scale
DX* =104 km
Dx =103 km
e.g., ENSO, NAO
e.g., European Rainfall
Methodologies: Linear
•Classification methods (Pattern Recognition)
•Linear statistical methods: CCA, SVD
« Weakly » Nonlinear Approaches
•Artificial Neural Netwrok (ANN)
•Kriging but with one-to-one correspondence
(i.e., almost linear)
Linear statistical methods: CCA, SVD
Singular Vector Decomposition,
~
~
SVD: <j(x)c(x’)>t = Sl ll jl(x) cl(x’)
~2
~2
where <jl >x= 1, <cl >x= 1, & l1> l2> l3>….>0
Canonical Correlation Analysis,
CCA:
<j(x)j(x’)>t-1/2<c(x)c(x’)>t–1/2<j(x)c(x’)>t
~
~
= Sl ll jl(x) cl(x’)
Limitations of the Downscaling Type I
(Linear Statistical Approach)
Physically-related two variables are not
necessarily identified by this method (e.g.,
y & z = D2y , Newman & Sardeshmakh 1995 JC)
Length of data required to establish a
statistically-robust result could be huge &
hard to estimate (van den Dool 1994 Tellus)
(Remeady DX* -> +oo, Zorita&von Stroch 1999)
Synoptic-scale is not perfectly « slaved » to
a climatological state
« stochasticity »
Transfer from the Type-I Regime to Type II
Dx
increase of Nonlinearity
~ Ro = U/WL
Linear-Wave
Dynamics
Teleconnections
(Hoskins&Karoly 1981 JAS)
Semi-Determinisitc
Nonlinear
Dynamics
Locally-defined
Stochasitc
Previous Attempts for the Downscaling
Type II (Synoptic
Meso-Convective)
Topographically-induced rains:
local rainfall ~ vH.grad (h) (e.g., Sinclair 1994)
Weather generators: stochastic generation
of local time series with a given climate
state (Wilks, Wilby, ……)
Scaling-law based approaches: fractal,
wavelets, etc. (e.g., Deidda 2000 WRS, Kuligowski
&Barros 2001 JAM, Croa&Wood 2002 JH)
Generalization to a heterogeneous case
A Physical Basis for the Downscaling of the FullyNonlinear System (Type II)
multiscale
mode
log P(k)
interactions
-a
log k
Linear Method
Does not work
Bouée «Côte d’Azur»
de Météo-France :
vitesse du vent, à
dt=1h, mars 1999 décembre 2001
Correction des
erreurs de
chronologie de la
série de mesures
Série temporelle Ui=1,M
avec M=24514,
dont L=19522
termes qui seront
analysés
m/s
Data Set (Time series)
Morlet Wavelet (continuous wavelet)
Ondelette - mère : Y  L2 ( R )


-
Y (t )dt = 0 et
Ex : Ondelette de Morlet Y (t ) =
Y =1
1
(  )
2
1
4
- t 2 i t
exp( 2 ) e
2
Famille d ' ondelettes :
1
t -u
Y(
) et Yu , s = 1
s
s
Transformée en ondelettes d ' une fonction f  L2 ( R ) : Wavelet

1 * t -u
W ( f )(u , s ) = f ,Yu , s =  f (t )
Y (
) dt
-
s
s
u , s  R  0; Yu , s (t ) =
transform:
Théorème de reconstruction
Soit Y  L2 ( R) vérifiant la condition d ' admissibil ité :
CY = 

0
ˆ (w )
Y
w
2
dw
<

Alors  f  L2 ( R)
1  
1
t -u
ds
f (t ) =
W ( f )(u , s )
Y(
) du 2


0

CY
s
s
s
NB: overcompleteness
Transformation en ondelettes de la série U
Power Law
Transformée en ondelettes de U :
W (U )(t j , Ti ) avec j = 1 : M
Spectre de U à la date t j :
 i S j (Ti ) = W (U )(t j , Ti )
2
Time-mean
Spectre à t = 999h
Fourier
Methods:Observational Data Analysis:
Data Set: Buoy Time Series (off coast Nice):
Wind Speed:Mar 1999-Dec 2001, Dt=1h
Methodology: Taylor’s frozen hypothesis:
k=w/u: k
w
Seek a power law: P(w,t) ~w-a(t) as a function
of time (wavelet spectra)
How to estimate: a=a(t) ?