# On the dual advantage of placing observations through forward sensitivity analysis

By: Shady E Ahmed, Omer San, Sivaramakrishnan Lakshmivarahan, John M Lewis

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the ... more

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the dual
advantage of placing the observations where the square of the sensitivity of
the model solution with respect to the unknown control variables, called
forward sensitivities, attains its maximum. Therefore, we can force the
observability Gramian to be of full rank, which in turn guarantees efficient
recovery of the optimal values of the control variables, which is the first of
the two advantages of this strategy. We further show that the proposed strategy
of placing observations has another inherent optimality: the square of the
sensitivity of the optimal estimates of the control with respect to the
observations (used to obtain these estimates) attains its minimum value, a
second advantage that is a direct consequence of the above strategy for placing
observations. Our analytical framework and numerical experiments on linear and
nonlinear systems confirm the effectiveness of our proposed strategy.
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