Authors’ abstract: Data assimilation obtains improved estimates of the state of a physical system by dynamically combining imperfect model results with sparse and noisy observations of reality. Not all observations used in data assimilation are equally valuable. The ability to characterize the usefulness of different data points is important for analyzing the effectiveness of the dynamic data assimilation system, for data pruning, for dynamically configuring existing sensor systems, and for designing future sensor networks. This paper focuses on the four-dimensional variational (4D-Var) data assimilation framework. Metrics from information theory are used to quantify the contribution of observations to decreasing the uncertainty with which the system state is known. An important relationship between different information-theoretic metrics and the variational cost function/gradient is established under Gaussian, linear, and time-independent assumptions. This insight allows us to derive an ensemble-based computational procedure to estimate the information content of various observations in the context of 4D-Var. (A short paper summarizing the theoretical results, and showing numerical experiments with a linear system, was presented at the International conference on computational science ICCS 2012.) The approach is first illustrated on a small nonlinear test problem. Next, the methodology is applied to quantify the degrees of freedom for signal of satellite observations used in a global chemical data assimilation problem. The assimilation of a subset of data points characterized by the highest information content yields an analysis comparable in quality to the one obtained using the entire data set.