T - Weather-Chaos Group

HOW TO IMPROVE FORECASTING WITH BRED
VECTORS BY USING THE GEOMETRIC NORM
Diego Pazó
Instituto de Física de Cantabria (IFCA), CSIC-UC, Santander (Spain)

Santander

OUTLINE
1) Lorenz '96 model.
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) Forecasting with bred vectors: Euclidean vs. geometric norm.

SPATIO-TEMPORAL CHAOTIC SYSTEM: LORENZ '96 MODEL
d ux , t
=−u x−1,t [u x−2,t −u x1,t ]−u x ,t F
dt

LORENZ '96 MODEL: 'LINEAR TANGENT EQUATIONS'
d ux , t
=−[u x−2, t−u x1, t] ux−1,t −ux−1,t[u x−2,t −u x1,t ]−u x ,t
dt
∥ut ∥≃∥ut=0∥ exp n t
Lyapunov spectrum (F=8 ):

THE LYAPUNOV VECTOR
The Lyapunov vector g(x,t) exhibits dynamical localization.
Log-normal
stats.
h(x,t)=ln|g(x,t)|
h(x,t)=ln|g(x,t)|
Normal
stats.
In log scale the LV belongs to the universality class of the Kardar-Parisi-
Zhang (KPZ) equation:
∂ t
h = ξ + 2 h + (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 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(t m
)
b(t m+1
)
t=0
T
u(t)
ut m
=u' t m
−ut m
bt m
= u t m
∥ u t m
∥ u' t m =ut m bt m
m=m+1

DEFINITION OF THE NORM
We have selected a spectrum of q­norms: ∥ u∥ q =[
q] 1 ∑ L
1/q
∣ ux , t ∣
L x=1
q=2 is the Euclidean norm.
q=∞ is the supremum norm:
q=0 is the geometric norm:
∥ u∥ ∞ =sup {∣u x ,t∣} x=1,, L
L
∥ u∥ 0 =∏∣u x ,t ∣ 1 / L
x=1
L
Note that h= 1 ∑ h x , t =ln∥ u x ,t∥
L 0
x=1
“Logarithmic bred vectors”
ut m
=u' t m
−ut m
b t m
= 0
ut m
∥ ut m
∥ 0
u' t m
=ut m
bt m
m=m+1

BRED VECTORS
The “bred vectors surface” is roughly a piecewise copy of the LV surface
There is a nonlinear barrier that breaks the BV in several pieces !

ENSEMBLE of BVs

ENSEMBLE DIVERSITY → ENSEMBLE DIMENSION
For an ensemble of k bred vectors {b q (i) } i=1,...k
we define the ensemble diversity
in terms of the ensemble dimension.
k
D en t= ∑ i=1
i
eigenvalues of the kk covariance matrix
k
∑
i=1
i
2
i
C ij
t =
〈b i
q
, b j q
〉
L∥b i
q ∥ 2 ∥b j q ∥ 2
D en
=3 D en
2 D en
1

The amplitude q controls the average ensemble dimension
k=10 BVs

THE GEOMETRIC NORM (q=0) vs. THE OTHER q-NORMS
The ensemble dimension D en
(t) exhibit smaller fluctuations for q=0
3

The PDF of D en
shows that D en
=1 is statistically significant for q≠0

=〈 [ D en
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 =∢ gt , b q i t
LV
BV's
LV
BV's

Larger localization. (Squared) surface width:
V t =[h x ,t −h ] 2

THE GEOMETRIC NORM (q=0) vs. THE OTHER q-NORMS
Larger growth rate (FSLE) = 1 T 〈 ln ∥ ut mT ∥ 2
∥b q t m ∥ 2
〉

ensemble FORECASTING with BVs

ENSEMBLE FORECASTING
Leith (1974): Ensemble mean outperforms deterministic forecasting.
truth
Ensemble mean

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:
u ca : Control analysis
u cf : Control forecast
y : Observations
u ca =u cf BBR −1 y−u cf
B : forecast covariance matrix
R : error covariance matrix

ENSEMBLE FORECASTING: q=0 vs. q=2
Breeding: we 'breed' five perturbations: At the assimilation times they are
rescaled down to a size ε q
(and centered at the analysis) .
Forecast: the five perturbations (+ the negative of them) are integrated for a time t lead
The performance of the forecast is quantified by the root-mean-square error
between the truth and the ensemble mean.

ENSEMBLE FORECASTING: q=0 vs. q=2
The Rank histogram

ENSEMBLE FORECASTING: q=0 vs. q=2
The continuous ranked probability score

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 larger growth rates.
The geometric norm improves forecasts with bred vectors.
Thanks to my collaborators: Miguel A. Rodríguez, Juan M. López,
Sarah Hallerberg, Ivan G. Szendro, Sixto Herrera, Jesús Fernández.