Evaluating nonlinear maximum likelihood optimal estimation uncertainty in cloud and aerosol remote sensing

Uncertainty estimates are important when retrieving properties of clouds and aerosols from satellites measurements. These measurements must be interpreted using a form of inverse theory, such as optimal estimation. In atmospheric remote sensing these inverse methods often assume that the forward model is linear in the region of uncertainty. This assumption is not necessarily valid. This paper presents an exact confidence procedure in contrast to the linear approximation using a maximum likelihood estimator. Two simple examples of retrieving the effective radius and optical depth of a volcanic ash cloud and water cloud show a discrepancy between the linear approximation and the exact procedure. The exact procedure is especially useful for inference where the entire parameter space has been forward modelled prior to or during the retrieval, such as using look up tables. When the inference method calculates the likelihood over the whole parameter space, it is less computationally expensive than a linear approximation.