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HOW TO IMPROVE FORECASTING WITH BRED
VECTORS BY USING THE GEOMETRIC NORM
Diego Pazó, Miguel A. Rodríguez, Juan M. López
Instituto de Física de Cantabria (IFCA), CSIC-UC, Santander (Spain)
https://sites.google.com/site/diegopazo/
2nd Workshop on Nonlinear Processes in Oceanic and Atmospheric Flows
OUTLINE
1) Introduction – Ensemble forecasting.
2) Definition of bred vectors.
3) Structure and growth of bred vectors, the role of the norm.
4) Enhancement of ensemble diversity with the geometric norm.
5) Improve of forecasting with bred vectors by using the geometric norm.
ENSEMBLE FORECASTING
Leith (1974): Ensemble mean outperforms deterministic forecasting.
truth
Ensemble mean
ENSEMBLE FORECASTING
ensemble
reality
SPATIO-TEMPORAL CHAOTIC SYSTEM: LORENZ '96 MODEL
d u x , t 
=−u  x−1, t[u  x−2, t −u  x1, t ]−u x , t F
dt
Lyapunov spectrum (F=8 ):
with
x=1,, L=128
BRED VECTOR
ERROR BREEDING – Toth & Kalnay (NCEP, 1993)
The 'breeding method': inexpensive procedure for generating ensembles.
Bred vectors (BVs) are finite perturbations periodically 'scaled down'. u'(t)
b(tm)
v
b(tm+1)
u(t)
T
t=0
 ut m =u ' t m −u t m  
b t m = 
m=m+1
 u t m 
∥ u t m ∥

u ' t m =ut m b t m 
 : THE LYAPUNOV VECTOR
The Lyapunov vector (LV) exhibits dynamical localization.
In log scale the LV belongs to the universality class of the Kardar-ParisiZhang (KPZ) equation:
∂t h = ξ + 2h + (h)2
Stretched-exponential localization of the Lyapunov vector: ∣g  x∣~e
−k ∣x −x 0∣
In log-scale the Lyapunov exponent λ becomes a speed:  L=∞− L~ L
−1
BRED VECTORS
The “bred vectors surface” is roughly piecewise copies of the LV surface
There is a nonlinear barrier that breaks the BV in several pieces !
EXPONENTIAL GROWTH RATE = FINITE-SIZE LYAPUNOV EXPONENT
The LE, λ, does not depend on the norm type.
The “bred exponent” (or FSLE), =


∥ u∥
1
ln
, depends on the norm type.
T

In the surface picture (i.e. log-scale) the most natural norm is the geometric norm.
L
h = 1 ∑ h  x , t =ln∥ u  x , t ∥0
L x=1
L
∥ u∥0 =∏ ∣ u  x ,t ∣1 / L
x=1
“Logarithmic bred vectors”
 u t m 
 u t m =u ' t m −u t m   b t m = 0
∥ ut m ∥0
m=m+1

u ' t m =u t m bt m 
SCALING LAW FOR THE FINITE-SIZE LYAPUNOV EXPONENT
 L=∞− ~
0
1
 ln B−ln 0 
2
 L=∞−   L~
0
 L=∞− L~ L
−1
f 
L
=
 ln B−ln 0 
L
2
ENSEMBLE of BVs
DEFINITION OF THE NORM
We have selected a spectrum of q­norms:
[
∥ u∥q =
L
1
q
∣ u x , t ∣
∑
L x=1
q=2 is the Euclidean norm.
q=∞ is the supremum norm: ∥ u∥∞ = supr {∣ u x , t∣}x=1,  , L
L
1/L
∥
u∥
=
∣
u

x
,t
∣
∏
q=0 is the geometric norm:
0
x=1
“Logarithmic bred vectors”
]
1 /q
ENSEMBLE DIVERSITY = ENSEMBLE DIMENSION
An ensemble of k bred vectors {bq(i)}i=1,...k.
We define the ensemble diversity in terms of the ensemble dimension.
2
∑  
k
D en t =
i =1
k
i
∑  i
i eigenvalues of the kk covariance matrix C ij t =
〈 b qi , b q j 〉
L∥bqi∥2 ∥b q j∥2
i=1
Den=3
Den2
Den1
q tunes the average ensemble dimension k=10 BVs
THE GEOMETRIC NORM (q=0) vs. THE OTHER q-NORMS
The ensemble dimension Den(t) exhibit smaller fluctuations for q=0
<Den> 3
The PDF of Den shows that Den=1 is statistically significant for q≠0
= 〈 [ Den t −〈 D en 〉]2 〉
1/ 2
THE GEOMETRIC NORM (q=0) vs. THE OTHER q-NORMS
Larger projection on the leading Lyapunov vector
i  t =∢  g t  , b qi  t  
LV
BV's

LV

BV's
THE GEOMETRIC NORM (q=0) vs. THE OTHER q-NORMS
〈
Larger growth rate (FSLE) = 1 ln
T
∥ u t mT ∥2
∥b q t m ∥2
〉
ensemble FORECASTING with BVs
ENSEMBLE FORECASTING: q=0 vs. q=2
We carry out a simple experiment of ensemble forecasting with bred vectors. We
compare the Euclidean (q=2) and the geometric (q=0) norms.
Assimilation of the true state of the system (independent of the forecasting)
The model is assumed to be perfect. Truth is represented by an independent run.
Observations are performed every 0.05 t.u. with Gaussian errors of variance 2=0.01.
The state of the system is assimilated via a standard 3D-Var method:
uca : Control analysis
ucf : Control forecast
y : Observations
B : forecast covariance matrix
R : error covariance matrix
u ca =u cf B BR−1  y−u cf 
ENSEMBLE FORECASTING: q=0 vs. q=2
We breed five perturbations: At the assimilation times they are rescaled down
to a size εq (and centered at the analysis) .
Forecast: the five perturbations (+ the negative of them) are integrated for a time tlead
The performance of the forecast is quantified by the root-mean-square error
between the truth and the ensemble mean.
CONCLUSIONS
Bred vectors DEPEND on the norm type.
An ensemble of logarithmic (q=0) bred vectors exhibits weaker fluctuations
while its members
1. are more strongly projected on the leading LV.
2. have growth rates that rapidly approach the LE.
The geometric norm yields better forecasts with bred vectors.
See also: https://sites.google.com/site/diegopazo/