Download Presentazione di PowerPoint

Targetting and Assimilation: a dynamically consistent approach
Anna
(*)ISAC-CNR
(*)
Trevisan , Alberto
Bologna, [email protected];
(°)
(°)
Carrassi
and Francesco
(§)
Uboldi
Dept. of Physics University of Ferrara, [email protected];
( §)no
affiliation, [email protected]
Results
Summary
A new assimilation method and results of its application in an adaptive observations experiment are presented.
Comparison between two different experiments:
Targetting and assimiliation are considered as two faces of the same problem and are addressed with a dynamically
consistent approach.
 Experiment type I: all observations, fixed and adaptive, are assimilated by means of 3DVAR
Assimilation increment is confined within the unstable subspace where the most important components of the error
are expected to grow.
 Experiment type II: the fixed observations are assimilated by means of 3DVAR; the adaptive observations by means of the proposed method
The adaptive observation in both experiments is placed at the location where the current bred vector attains its maximum amplitude
The analysis cycle is considered as an observationally forced dynamical system whose stability is studied.
Perfect Observations
Unstable vectors of the Data Assimilation System are estimated by a modified Breeding technique.
Noisy Observations
Mathematical Formulation
The analysis increment is confined to the N-dimensional unstable subspace spanned by a set of N vectors en:
x  x  Ea
a
(1)
f
where xa is the analysis, xf is the forecast (background) state, E is the I x N matrix (I being the total number of
degrees of freedom) whose columns are the en vectors and the vector of coefficients a, the analysis increment in the
subspace, is the control variable . This is equivalent to estimate the forecast error covariance as:
P f  E  Γ  ET
(2)
where  represents the forecast error covariance matrix in the N-dimensional subspace spanned by the columns of E.
The solution is :
x  x  E Γ HE 
a
f
T
HE Γ HE
T
R
 y
1
o
 Hx f

(3)
here H is the (Jacobian of the) observation operator, R is the observation error covariance matrix, and yo is Mdimensional observation vector. It is intended here that N  M, so that for each vector en there is at least one
observation.
FIG (1): Normalized RMS analysis error as a function of time. The error is expressed in potential
enstrophy norm and it is normalized by natural variability. Upper panel: Experiment I-P (3DVAR),
adapt. obs. is a sounding; Middle panel: Experiment II-P, adapt. obs. is one temperature value; Lower
panel: Experiment II-P-2, adpt. obs. is temperature and meridional velocity at same grid point.
FIG (2): Same as fig (1) but using observations affected by error. Upper panel: Experiment I-N;
Middle panel: Experiment II-N Lower Panel: Experiment II-N-5, adapt. obs. are 5 temperature
values at grid points surrounding the targeted location.
For the theoretical discussion and detailed mathematical formulation see: [4] and [5].
Stability Analysis
Outline of usage
 Estimate the unstable subspace obtained by Breeding on the Data Assimilation System;
 Targetted observations are placed at locations where bred vectors en have maximum component;
 A wide (2500km) Gaussian modulating function is used to isolate the regional structure of the bred vector;
 A small number N of Bred Vectors are used to construct the matrix E;
 Use E to estimate Pf and confine the analysis increment in the unstable subspace: xa = xf + Ea;
Experiment with a Quasi-geostrophic Model
 64-longitudinal x 32-latitudinal x 5 levels periodic channel model, (Rotunno and Bao, 1996);
 Land (longitudinal grid points 1-20) completely covered by observations;
FIG (3): Background error (color) at two randomly chosen instants and analysis increment (black
 1 adaptive observation over the Ocean (longitudinal grid points 21-64);
contour). Upper panels: exp. I-P (3DVAR); Lower panel: exp. II-P and Gaussian masking function
(red contour). The area shown is a portion of the whole domain; note also that the interval between
isolines of the analysis increment is different in the two panels.
 Perfect Model hypothesis;
 Standard Analysis scheme: 3D-Var assimilation over land, (Morss et al., 1999);
 A single unstable vector e is used at each analysis time;
 Breeding on the Data Assimilation System (BDAS) to estimate unstable vectors e;
 Observations are assimilated in control solution and perturbed trajectories;
 Breeding time: 10 days;
 Bred vector discarded after use and a new perturbation introduced at each analysis time;
 Perfect (labeled P) and noisy (labeled N) observations experiments;
 Simulation Length: 2 years;
FIG (4): Leading Lyapunov exponent of the assimilation system as a function of time
(perfect observations). Values are averaged over all previous instants. The growth rate is
expressed in units of days-1. Dotted line: Experiment type I-P (3DVar); Continuous Line:
Experiment type II-P.
Conclusion
The main conclusion of this work is that the benefit of adaptive observations is
greatly enhanced if their assimilation is based on the same dynamical principles as
targetting. Results show that a few carefully selected and properly assimilated
observations are sufficient to control the instabilities of the Data Assimilation
System and obtain a drastic reduction of the analysis error. Further results (not
shown, see [5]) obtained in the context of a primitive equation ocean model and
regular observational network also proves the ability of the proposed assimilation
method to reduce the error and efficiently exploit the estimated unstable subspace.
The application of the method to a regular observational network and its extention
to the four dimensional case are presently under study. The success obtained so far
makes us very hopeful that, even in an operational environment with real adaptive
observations, this method can yield significant improvement of the assimilation
performance.
References
 [1]: detailed results and theoretical discussion: Carrassi A., A. Trevisan and F. Uboldi, 2004. Deterministic Data Assimilation and Targeting by Breeding
on the Data Assimilation System. In review for J. Atmos. Sci..
 [2]: on the 3DVar scheme: Morss R., 1999. Adaptive Observations: Idealized sampling strategy for improving numerical weather prediction. PhD thesis,
Massachussets of Technology, Cambridge, MA, 225 pp.
 [3]: on the QG model: Rotunno R., and J. W. Bao, 1996. A case study of cyclogenesis using a model hierarchy. Mon. Wea. Rev., 124, 1051-1066
 [4]: a previous study with a small nonlinear model: Trevisan A., and F. Uboldi, 2004. Assimilation of standard and targeted observation within the
unstable subspace of the observation-analysis-forecast cycle system. J. Atmos. Sci., 61, 103-113
 [5]: update of applications in progress and oceanic application: Uboldi F., A. Trevisan and A. Carrassi, 2005. Developing a Dynamically Based
Assimilation Method for Targeted and Standard Observations. Nonlinear Process in Geophysics, 12, 149-156
Acknowledgments
The authors thank Rebecca Morss and Matteo Corazza for providing the code of QG-Model and 3D-Var and
for getting us started with its use. We also thank Eugenia Kalnay, Zoltan Toth and Mozheng Wei for their
useful comments.