Data assimilation is the application of Bayes' theorem to condition the states of a dynamical systems model on observations. Any real-world application of Bayes' theorem is approximate, and therefore, we cannot expect that data assimilation will preserve all of the information available from models and observations. We outline a framework for measuring information in models, observations, and evaluation data in a way that allows us to quantify information loss during (necessarily imperfect) data assimilation. This facilitates quantitative analysis of trade-offs between improving (usually expensive) remote sensing observing systems versus improving data assimilation design and implementation. We demonstrate this methodology on a previously published application of the ensemble Kalman filter used to assimilate remote sensing soil moisture retrievals from Advanced Microwave Scattering Radiometer for Earth (AMSR-E) into the Noah land surface model.