A neural network approach to estimating a posteriori distributions of Bayesian retrieval problems

A neural-network-based method, quantile regression neural networks (QRNNs), is proposed as a novel approach to estimating the a posteriori distribution of Bayesian remote sensing retrievals. The advantage of QRNNs over conventional neural network retrievals is that they learn to predict not only a single retrieval value but also the associated, case-specific uncertainties. In this study, the retrieval performance of QRNNs is characterized and compared to that of other state-of-the-art retrieval methods. A synthetic retrieval scenario is presented and used as a validation case for the application of QRNNs to Bayesian retrieval problems. The QRNN retrieval performance is evaluated against Markov chain Monte Carlo simulation and another Bayesian method based on Monte Carlo integration over a retrieval database. The scenario is also used to investigate how different hyperparameter configurations and training set sizes affect the retrieval performance. In the second part of the study, QRNNs are applied to the retrieval of cloud top pressure from observations by the Moderate Resolution Imaging Spectroradiometer (MODIS). It is shown that QRNNs are not only capable of achieving similar accuracy to standard neural network retrievals but also provide statistically consistent uncertainty estimates for non-Gaussian retrieval errors. The results presented in this work show that QRNNs are able to combine the flexibility and computational efficiency of the machine learning approach with the theoretically sound handling of uncertainties of the Bayesian framework. Together with this article, a Python implementation of QRNNs is released through a public repository to make the method available to the scientific community.