Data Assimilation as a Problem in Optimal Tracking: Application of Pontryagin's Minimum Principle to Atmospheric Science

• Published in: Journal of the atmospheric sciences Vol. 70; no. 4; pp. 1257 - 1277
• Main Authors: LAKSHMIVARAHAN, S, LEWIS, J. M, PHAN, D
• Format: Journal Article
• Language: English
• Published: Boston, MA American Meteorological Society 01.04.2013
• Subjects: Algorithms; Artificial satellites; Atmospheric sciences; Atmospherics; Burgers equation; Control systems; Control theory; Data assimilation; Data collection; Dynamical systems; Earth, ocean, space; Energy; Error correction; Error correction & detection; Exact sciences and technology; External geophysics; Feedback; Feedback control; Geophysics; Hamiltonian functions; Mechanics; Meteorology; Modelling; Optimization; Physics of the high neutral atmosphere; Principles; Space science; Stochastic models; Tracking; Tracking control; Trajectories; Weather forecasting; United States > US; feedback; Deterministic model; Model output; energy transfer; Error correction; Forcing; Dynamical system; Burgers equation; Transfer equation; trajectory; Data assimilation
• ISSN: 0022-4928; 1520-0469
• DOI: 10.1175/JAS-D-12-0217.1

A data assimilation strategy based on feedback control has been developed for the geophysical sciences-a strategy that uses model output to control the behavior of the dynamical system. Whereas optimal tracking through feedback control had its early history in application to vehicle trajectories in space science, the methodology has been adapted to geophysical dynamics by forcing the trajectory of a deterministic model to follow observations in accord with observation accuracy. Fundamentally, this offline (where it is assumed that the observations in a given assimilation window are all given) approach is based on Pontryagin's minimum principle (PMP) where a least squares fit of idealized path to dynamic law follows from Hamiltonian mechanics. This utilitarian process optimally determines a forcing function that depends on the state (the feedback component) and the observations. It follows that this optimal forcing accounts for the model error. From this model error, a correction to the one-step transition matrix is constructed. The above theory and technique is illustrated using the linear Burgers' equation that transfers energy from the large scale to the small scale.