We have addressed the geophysical problem of obtaining an elastic model of the subsurface from recorded normal-incidence seismic data using convolutional neural networks (CNNs). We train the network on synthetic full-waveform seismograms generated using Kennett’s reflectivity method on earth models that were created under rock-physics modeling constraints. We use an approximate Bayesian computation method to estimate the posterior distribution corresponding to the CNN prediction and to quantify the uncertainty related to the predictions. In addition, we test the robustness of the network in predicting impedances of previously unobserved earth models when the input to the network consisted of seismograms generated using: (1) earth models with different spatial correlations (i.e. variograms), (2) earth models with different facies proportions, (3) earth models with different underlying rock-physics relations, and (4) source-wavelet phase and frequency different than in the training data. Results indicate that the predictions of the trained network are susceptible to facies proportions, the rock-physics model, and source-wavelet parameters used in the training data set. Finally, we apply CNN inversion on the Volve field data set from offshore Norway. P-wave impedance inverted for the Volve data set using CNN showed a strong correlation (82%) with the log at a well.