  Quantifying forecast uncertainty is a key aspect of state-of-the-art
numerical weather prediction and data assimilation systems. Ensemble-based data
assimilation systems incorporate state-dependent uncertainty quantification
based on multiple model integrations. However, this approach is demanding in
terms of computations and development. In this work a machine learning method
is presented based on convolutional neural networks that estimates the
state-dependent forecast uncertainty represented by the forecast error
covariance matrix using a single dynamical model integration. This is achieved
by the use of a loss function that takes into account the fact that the
forecast errors are heterodastic. The performance of this approach is examined
within a hybrid data assimilation method that combines a Kalman-like analysis
update and the machine learning based estimation of a state-dependent forecast
error covariance matrix. Observing system simulation experiments are conducted
using the Lorenz'96 model as a proof-of-concept. The promising results show
that the machine learning method is able to predict precise values of the
forecast covariance matrix in relatively high-dimensional states. Moreover, the
hybrid data assimilation method shows similar performance to the ensemble
Kalman filter outperforming it when the ensembles are relatively small.
