Continuous Data Assimilation with Blurred-in-Time Measurements of the Surface Quasi-Geostrophic Equation

Abstract

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

Keywords

Data assimilation / Nudging / Time-Averaged observables / Surface quasi-geostrophic equation

References

[1] Albanez D A F, Nussenzveig Lopes H J, Titim E S. Continuous data assimilation for the three-dimensional Navier-Stokes-α model. Asymptotic Anal., 2016, 97(1-2): 139-164

[2] Azouani A, Olson E, Titi E S. Continuous data assimilation using general interpolant observables. J. Nonlinear Sci., 2014, 24(2): 277-304

[3] Azouani A, Titi E S. Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction diffusion paradigm. Evol. Equ. Control Theory, 2014, 3(4): 579-594

[4] Bergemann K, Reich S. An ensemble Kalman-Bucy filter for continuous data assimilation. Meteorl. Z., 2012, 21: 213-219

[5] Bessaih H, Olson E, Titi E S. Continuous assimilation of data with stochastic noise. Nonlinearity, 2015, 28: 729-753

[6] Blocher J. Chaotic Attractors and Synchronization Using Time-Averaged Partial Observations of the Phase Space, 2016

[7] Blocher J, Martinez V R, Olson E. Data assimilation using noisy time-averaged measurements. Physica D, 2018, 376: 49-59

[8] Bloemker D, Law K J H, Stuart A M, Zygalakis K. Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation. Nonlinearity, 2013, 26: 2193-2219

[9] Caffarelli L, Vasseur A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Annals of Math., 2010, 171(3): 1903-1930

[10] Carrillo J A, Ferreira C F. The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations. Nonlinearity, 2008, 21: 1001-1018

[11] Charney J, Halem M, Jastrow R. Use of incomplete historical data to infer the present state of the atmosphere. J. Atmos. Sci., 1969, 26: 1160-1163

[12] Cheskidov A, Dai M. The existence of a global attractor for the forced critical surface quasi-geostrophic equation in L². J. Math. Fluid Mech., 2018, 20(1): 213-225

[13] Constantin P, Coti-Zelati M, Vicol V. Uniformly attracting limit sets for the critically dissipative SQG equation. Nonlinearity, 2016, 29: 298-318

[14] Constantin P, Glatt-Holtz N, Vicol V. Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations. Comm. Math. Phys., 2014, 330(2): 819-857

[15] Constantin P, Majda A, Tabak E. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 1994, 7: 1495-1533

[16] Constantin P, Tarfulea A, Vicol V. Long time dynamics of forced critical SQG. Commun. Math. Phys., 2014, 335(1): 93-141

[17] Constantin P, Vicol V. Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Fund. Anal., 2012, 22(5): 1289-1321

[18] Constantin P, Wu J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math Anal., 1999, 30(5): 937-948

[19] Coti-Zelati M. Long time behavior of subcritical SQG in scale-invariant Sobolev spaces. J. Nonlinear Sci., 2018, 28(1): 305-335

[20] Coti-Zelati M, Vicol V. On the global regularity for the supercritical SQG equation. Indiana Univ. Math. J., 2016, 65: 535-552

[21] Dong H. Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness. Discrete Gontin. Dyn. Syst. Series A, 2010, 26(4): 1197-1211

[22] Farhat A, Jolly M S, Titi E S. Continuous data assimilation for the 2D Benard convection through velocity measurements alone. Phys. D, 2015, 303: 59-66

[23] J. Math. Fluid Mech., 2016, 18 1

[24] Farhat A, Lunasin E, Titi E S. Data assimilation algorithm for 3D Benard convection in porous media employing only temperature measurements. J. Math. Anal. Appl., 2016, 438(1): 492-506

[25] Farhat A, Lunasin E, Titi E S. On the Charney conjecture employing temperature measurements alone: the paradigm of 3D planetary geostrophic model. Math. Glim. Weather Forecast, 2016, 2: 61-74

[26] Foias C, Mondaini C, Titi E S. A discrete data assimiliation scheme for the solutions of the 2D Navier-Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst., 2016, 15(4): 2109-2142

[27] Ibdah H A, Mondaini C F, Titi E S. Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm, 2018

[28] Jolly M S, Martinez V R, Titi E S. A data assimilation algorithm for the subcritical surface quasi-geostrophic equation. Adv. Nonlinear Stud., 2017, 17(1): 167-192

[29] Ju N. The maximum principle and the global attractor for the dissipative 2D quasi-g eostrophic equations. Comm. Math. Phys., 2005, 255: 161-181

[30] Kalnay E. Atmospheric Modeling, Data Assimilation, and Predictability, 2003, New York: Cambridge University Press

[31] Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equation. Comm. Pure. Appl. Math., 1988, 41(7): 891-907

[32] Kenig C E, Ponce G, Vega L. Well-posedness of the initial value problem for the Korteweg-de-Vries equation. J. Am. Math. Soc., 1991, 4(2): 323-347

[33] Khouider B, Titi E S. An inviscid regularization for the surface quasi-geostrophic equation. Commun. Pure. Appl. Math., 2008, 61: 1331-1346

[34] Kiselev A, Nazarov F. Variation on a theme of Caffarelli and Vasseur. J. Math. Sci., 2010, 166(1): 31-39

[35] Kiselev A, Nazarov F, Volberg A. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math., 2007, 167: 445-453

[36] Markowich P A, Titi E S, Trabelsi S. Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model. Nonlinearity, 2016, 24(4): 1292-1328

[37] Mondaini C F, Titi E S. Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm. SIAM J. Numer. Anal., 2018, 56(1): 78-110

[38] Pedlosky J. Geophysical Fluid Dynamics, 1987, New York: Springer-Verlag

[39] Resnick S G. Dynamical problems in non-linear advective partial differential equations, 1995 1-86