Contents

## Main issues Main issues (continued) Atmospheric data Observational network Ocean data past

Eliason, Greg, News Director has reference to this Academic Journal, PHwiki organized this Journal Data Assimilation in Meteorology in addition to Oceanography Michael Ghil Ecole Normale Supérieure, Paris, in addition to University of Cali as long as nia, Los Angeles Joint work with Dmitri Kondrashov, UCLA, in addition to many others: please see http://www.atmos.ucla.edu/tcd/ Outline Data in meteorology in addition to oceanography – in situ & remotely sensed Basic ideas, data types, & issues how to combine data with models transfer of in as long as mation – between variables & regions stability of the fcst.assimilation cycle filters & smoothers Parameter estimation – model parameters – noise parameters at & below grid scale Subgrid-scale parameterizations – deterministic (classic) – stochastic dynamics & physics Novel areas of application – space physics – shock waves in solids – macroeconomics Concluding remarks Main issues The solid earth stays put to be observed, the atmosphere, the oceans, & many other things, do not. Two types of in as long as mation: – direct observations, in addition to – indirect dynamics (from past observations); both have errors. Combine the two in (an) optimal way(s) Advanced data assimilation methods provide such ways: – sequential estimation the Kalman filter(s), in addition to – control theory the adjoint method(s) The two types of methods are essentially equivalent as long as simple linear systems (the duality principle)

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Main issues (continued) Their per as long as mance differs as long as large nonlinear systems in: – accuracy, in addition to – computational efficiency Study optimal combination(s), as well as improvements over currently operational methods (OI, 4-D Var, PSAS). Outline Data in meteorology in addition to oceanography – in situ & remotely sensed Basic ideas, data types, & issues how to combine data with models filters & smoothers – stability of the fcst.-assimilation cycle Parameter estimation – model parameters – noise parameters at & below grid scale Subgrid-scale parameterizations – deterministic (classic) – stochastic dynamics & physics Novel areas of application – space physics – shock waves in solids – macroeconomics Concluding remarks Atmospheric data Total no. of observations = 0(105) scalars per 12h24h 0(102 ) observations/[(significant d-o-f) x (significant t)] Bengtsson, Ghil & Källén (eds.): Dynamic Meteorology, Data Assimilation Methods (1981) Drifting buoys: Ps 267 Cloud-drift: V 2×2259 Aircraft: V 2×1100 Ship & l in addition to surface: Ps, Ts , Vs 4×3446 Polar orbiting satellites: T 5×2048 Balloons : V 2x581x10 Radiosondes : T, V – 3x749x10

Observational network Quality control preliminary & as part of the assimilation cycle Ocean data past Total no. of oceanographic observations/met. obsns = O(104) as long as the past; & = O(101) as long as the future : Syd Levitus (1982). Ocean data present & future Altimetry sea level; scatterometry surface winds & sea state; acoustic tomography temperature & density; etc. Courtesy of Tony Lee, JPL

Space physics data Space plat as long as ms in Earths magnetosphere Basic ideas of data assimilation in addition to sequential estimation – I Simple illustration Want to estimate u – temperature of this room, based on the readings u1 in addition to u2 of the two thermometers. Estimate û = 1u1 + 2u2 Interpretation will be: u1 = uf – first guess (of numerical as long as ecast model) u2 = uo – observation (R/S, satellite, etc.) û = ua – objective analysis Basic ideas of data assimilation in addition to sequential estimation – II If u1 in addition to u2 are unbiased, in addition to û should be unbiased, then 1 + 2 = 1, so one can write û = u1 + 2(u2 – u1): updating (sequential) If u1 in addition to u2 are uncorrelated, in addition to have A1 =12, A2 =22: known st in addition to ard deviations, Then the minimum variance estimator() is û = u1 + A2 /( A2 – A1) (u2 – u1) in addition to its accuracy is Â = ( A1 + A2) max {A1, A2} BLUE = Best Linear Unbiased Estimator

Kalman Filter – I Kalman Filter – II Kalman Filter – III

Kalman Filter – IV Basic concepts: barotropic model Shallow-water equations in 1-D, linearized about (U,0,), fU = y U = 20 ms1, f = 104s1, = gH, H 3 km. PDE system discretized by finite differences, periodic B. C. Hk: observations at synoptic times, over l in addition to only. Ghil et al. (1981), Cohn & Dee (Ph.D. theses, 1982 & 1983), etc. Trade-off between variables Wind info. is better (here) than height info. Some variables are observed, others are not. Observing System Simulation Experiments (OSSE) Identical twins vs. real observations Height obsd Wind obsd

Conventional network good observations R Q P R; (ii) poor observations R >> Q P Q/(1 2); P = QR/[Q + (1 2)R] (iii) always (provided 2 < 1) P min {R, Q/(1 2)}. (a) Q = 0 P = 0 (b) Q 0 (i), (ii) in addition to (iii): Relative weight of observational vs. model errors Advection of in as long as mation b) {first guess} - {FGGE analysis} Halem, Kalnay, Baker & Atlas (BAMS, 1982) 300 {first guess} - {FGGE analysis} {6h fcst} - {conventional (NoSat)} 300 Upper panel (NoSat): Errors advected off the ocean Lower panel (Sat): Errors drastically reduced, as info. now comes in, off the ocean Evolution of DA I Transition from early to mature phase of DA in NWP: no Kalman filter Ghil et al., 1981() no adjoint Lewis & Derber (Tellus, 1985); Le Dimet & Talagr in addition to (Tellus, 1986) () Bengtsson, Ghil & Källén (Eds., 1981), Dynamic Meteorology: Data Assimilation Methods. M. Ghil & P. M.-Rizzoli (Adv. Geophys., 1991). Evolution of DA II Cautionary note: Pantheistic view of DA: variational ~ KF; 3- & 4-D Var ~ 3- & 4-D PSAS. Fashionable to claim its all the same but its not: God is in everything, but the devil is in the details. M. Ghil & P. M.-Rizzoli (1991, Adv. Geophys.) Outline Data in meteorology in addition to oceanography - in situ & remotely sensed Basic ideas, data types, & issues how to combine data with models stability of the fcst.assimilation cycle filters & smoothers Parameter estimation - model parameters - noise parameters at & below grid scale Subgrid-scale parameterizations - deterministic (classic) - stochastic dynamics & physics Novel areas of application - space physics - shock waves in solids - macroeconomics Concluding remarks Error components in as long as ecastanalysis cycle The relative contributions to error growth of analysis error intrinsic error growth modeling error (stochastic)

Assimilation of observations: Stability considerations as long as ecast state; model integration from a previous analysis Corresponding perturbative (tangent linear) equation If observations are available in addition to we assimilate them: Evolutive equation of the system, subject to as long as cing by the assimilated data Corresponding perturbative (tangent linear) equation, if the same observations are assimilated in the perturbed trajectories as in the control solution The matrix (I KH) is expected, in general, to have a stabilizing effect; the free-system instabilities, which dominate the as long as ecast step error growth, can be reduced during the analysis step. Joint work with A. Carrassi, A. Trevisan & F. Uboldi Free-System Dynamics (sequential-discrete as long as mulation): St in addition to ard breeding Observationally Forced System Dynamics (sequential-discrete as long as mulation): BDAS Stabilization of the as long as ecastassimilation system I Assimilation experiment with a low-order chaotic model Periodic 40-variable Lorenz (1996) model; Assimilation algorithms: replacement (Trevisan in addition to Uboldi, 2004), replacement + one adaptive obsn located by multiple replication (Lorenz, 1996), replacement + one adaptive obsn located by BDAS in addition to assimilated by AUS (Trevisan & Uboldi, JAS, 2004). Trevisan & Uboldi (JAS, 2004) Stabilization of the as long as ecastassimilation system II Observational as long as cing Unstable subspace reduction Free System Leading exponent: max 0.31 days1; Doubling time 2.2 days; Number of positive exponents: N+ = 24; Kaplan-Yorke dimension 65.02. 3-DVarBDAS Leading exponent: max 6×103 days1; AUSBDAS Leading exponent: max 0.52×103 days1 Assimilation experiment with an intermediate atmospheric circulation model 64-longitudinal x 32-latitudinal x 5 levels periodic channel QG-model (Rotunno & Bao, 1996) Perfect-model assumption Assimilation algorithms: 3-DVar (Morss, 2001); AUS (Uboldi et al., 2005; Carrassi et al., 2006)

Outline Data in meteorology in addition to oceanography – in situ & remotely sensed Basic ideas, data types, & issues how to combine data with models stability of the fcst.assimilation cycle filters & smoothers Parameter estimation – model parameters – noise parameters at & below grid scale Subgrid-scale parameterizations – deterministic (classic) – stochastic dynamics & physics Novel areas of application – space physics – shock waves in solids – macroeconomics Concluding remarks The main products of estimation() Filtering (F) video loops Smoothing (S) full-length feature movies Prediction (P) NWP, ENSO Distribute all of this over the Web to scientists, in addition to the person in the street (or on the in as long as mation superhighway). In a general way: Have fun!!! () N. Wiener (1949, MIT Press) Kalman smoother For a fixed interval, weak constrained 4-D Var is equivalent to the sequential (Kalman) smoother. Cohn, Sivakumaran & Todling (MWR, 1994)

The DA Maturity Index of a Field (Satellite) images (weather) as long as ecasts, climate movies The theoretician: Science is truth, dont bother me with the facts! The observer/experimentalist: Dont ruin my beautiful data with your lousy model!! Pre-DA: few data, poor models Early DA: Better data, so-so models. Stick it (the obsns) in direct insertion, nudging. Advanced DA: Plenty of data, fine models. EKF, 4-D Var (2nd duality). Post-industrial DA: Conclusion No observing system without data assimilation in addition to no assimilation without dynamicsa Quote of the day: You cannot step into the same riverb twicec (Heracleitus, Trans. Basil. Phil. Soc. Miletus, cca. 500 B.C.) aof state in addition to errors bMe in addition to ros c You cannot do so even once (subsequent development of flux theory by Plato, cca. 400 B.C.) = Everything flows General references Bengtsson, L., M. Ghil in addition to E. Källén (Eds.), 1981. Dynamic Meteorology: Data Assimilation Methods, Springer-Verlag, 330 pp. Daley, R., 1991. Atmospheric Data Analysis. Cambridge Univ. Press, Cambridge, U.K., 460 pp. Ghil, M., in addition to P. Malanotte-Rizzoli, 1991. Data assimilation in meteorology in addition to oceanography. Adv. Geophys., 33, 141266. Bennett, A. F., 1992. Inverse Methods in Physical Oceanography. Cambridge Univ. Press, 346 pp. Malanotte-Rizzoli, P. (Ed.), 1996. Modern Approaches to Data Assimilation in Ocean Modeling. Elsevier, Amsterdam, 455 pp. Wunsch, C., 1996. The Ocean Circulation Inverse Problem. Cambridge Univ. Press, 442 pp. Ghil, M., K. Ide, A. F. Bennett, P. Courtier, M. Kimoto, in addition to N. Sato (Eds.), 1997. Data Assimilation in Meteorology in addition to Oceanography: Theory in addition to Practice, Meteorological Society of Japan in addition to Universal Academy Press, Tokyo, 496 pp.

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