Placement of Observations for Variational Data Assimilation: Application to Burgers’ Equation and Seiche Phenomenon

Abstract

Observation placement in variational data assimilation determines cost function structure in the space of control. The presence of flatness in the cost function’s gradient presents problems in the iterative passage to the cost function’s minimum. Determination of observation placement that avoids these flat zones generally permits expeditious passage to the cost function minimum. A contribution to this volume (Lakshmivarahan S, Lewis JM, Maryada SKR (2020b) Observability Gramian and its role in the placement of observations in dynamical data assimilation. In: Data assimilation for atmospheric, oceanic, and hydrologic applications. Springer Pub. Co., New York) has theoretically determined methodology that identifies observation placement that avoids these flat zones. The placement relies on the norm of a semi-definite positive Gramian matrix G—a matrix derived from forecast sensitivity to control. Two dynamical systems are tested with this methodology: (1) Burgers’ Equation, and (2) Seiche phenomenon, the normal mode oscillations in lakes. Analytic solutions to both constraints have been found. For each dynamical system, two sets of observation placement are considered: one where observations locations correspond to places where the norm of G is large, and one where the norm is small. Results indicate that observation placement where the norm of G was large led to well-defined structure of the cost function at the operating point in control space, a structure where the cost-function gradient was bound away from zero, whereas choices for observation locations where the norm of G was small-magnitude led to troublesome cost function structure, a structure where small-magnitude gradient presented difficulty in advancing toward the cost-function minimum.