Targetting and Assimilation: a dynamically consistent approach

1. Targetting and Assimilation: a dynamically consistent approach Anna Trevisan(*), Alberto Carrassi(°) and Francesco Uboldi(§) (*)ISAC-CNR Bologna, A.Trevisan@isac.cnr.it; (°) Dept. of Physics University of Ferrara, carrassi@fe.infn.it; ( §)no affiliation, uboldi@magritte.it Results Summary • Comparison between two different experiments: • Experiment type I: all observations, fixed and adaptive, are assimilated by means of 3DVAR • Experiment type II: the fixed observations are assimilated by means of 3DVAR; the adaptive observations by means of the proposed method • A new assimilation method and results of its application in an adaptive observations experiment are presented. • Targetting and assimiliation are considered as two faces of the same problem and are addressed with a dynamically consistent approach. • Assimilation increment is confined within the unstable subspace where the most important components of the error are expected to grow. • The analysis cycle is considered as an observationally forced dynamical system whose stability is studied. • Unstable vectors of the Data Assimilation System are estimated by a modified Breeding technique. • Mathematical Formulation • The analysis increment is confined to the N-dimensional unstable subspace spanned by a set of N vectors en: • (1) • 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 envectors 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: • (2) • where  represents the forecast error covariance matrix in the N-dimensional subspace spanned by the columns of E.The solution is : • (3) • here H is the (Jacobian of the) observation operator, R is the observation error covariance matrix, and yo is M-dimensional observation vector. It is intended here that N  M, so that for each vector en there is at least one observation. • For the theoretical discussion and detailed mathematical formulation see: [4] and [5]. • 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 Pfand 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; • 1 adaptive observation over the Ocean (longitudinal grid points 21-64); • Perfect Model hypothesis; • Standard Analysis scheme: 3D-Var assimilation over land, (Morss et al., 1999); • A single unstable vector eis 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; The adaptive observation in both experiments is placed at the location where the current bred vector attains its maximum amplitude Perfect Observations Noisy Observations 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. 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. Stability Analysis 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. FIG (3): Background error (color) at two randomly chosen instants and analysis increment (black 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. 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.