How Various Sources of Uncertainty Affect Retrieval Uncertainty in the Optimal Estimation Framework Using a Non-precipitating Liquid Clouds Example

Optimal estimation (OE) is a commonly used inverse method in the geosciences. In a Bayesian context, a set of measurements (y) is related to the state vector to be retrieved (x) by the forward model F(x). Assuming Gaussian statistics, OE returns an optimal solution and its associated uncertainty by minimizing the cost function that consists of the state vector-a priori state difference weighted by the a priori uncertainty and the measurement-forward model difference weighted by the uncertainties of observation and forward model. OE algorithms are easy to implement and are finding increasing use within communities attempting to derive, for instance, cloud and precipitation microphysical properties from remote sensing data. However, even though OE algorithms are simple to implement, obtaining rigorous uncertainty estimates from them is a significant challenge. Our objective with this work is to illustrate the growth of retrieval uncertainty within the OE framework due to various sources using simple real world examples of non-precipitating liquid clouds.

Within the OE retrieval, several sources of uncertainties contribute to the overall retrieval uncertainty (S_x), including the measurement uncertainty (S_y), the uncertainties in a priori information (S_a) and uncertainties in the forward model due to imperfectly known parameters (S_b). In this study, two examples are given to demonstrate how uncertainties in S_y, S_a and S_b affect the ultimate retrieval uncertainty S_x. We apply OE technique to retrieve cloud liquid water content (LWC) and total number from measurements of radar reflectivity and extinction obtained in 2005 Marine Stratus/Stratocumulus Experiment (MASE). In the first example, the forward model is assumed perfect, which means all parameters are certain and S_b is zero. Then we perturb S_y and S_a separately and observe the response of S_x. We find the observation error S_y contributes significantly to the retrieval uncertainty under the assumption of “perfect model”. In our second example, we redo the first example, but assuming one of the gamma particle size distribution parameters, the shape parameter, is uncertain. We find that S_b is a dominant term that impacts the ultimate retrieval uncertainty and the effects of forward model uncertainties typically dominate the uncertainty in measurements.