Seismic inversion allows the prediction of subsurface properties from seismic reflection data and is a key step in reservoir modeling and characterization. With the generalization of machine learning in geophysics, deep learning methods have been proposed as efficient seismic inversion methods. However, most of these methods lack a probabilistic approach to deal with the uncertainties inherent in the seismic inversion problem and/or rely on complete and representative training data, which often is partially or scarcely available. We have explored the ability of deep convolutional neural networks to extract meaningful and complex representations from spatially structured data, combined with geostatistical simulation, to efficiently invert poststack seismic data directly for high-resolution models of acoustic impedance. Our model incorporates physics constraints and sparse direct measurements while leveraging the use of imprecise but widely distributed indirect measurements as represented by the seismic data. The models generated with geostatistical simulation provide additional information with higher spatial resolution than the original seismic data and allow assessing uncertainty in the model predictions by generating multiple realizations of the subsurface properties. Our method can (1) provide an uncertainty assessment of the predictions, (2) model the complex and nonlinear relationship between data and model, (3) extend the seismic bandwidth at low and high ends of the frequency parameters spectrum, and (4) lessen the need for large, annotated training data. Our method is applied to a 1D synthetic example and a real 3D application example from a Brazilian reservoir. The predicted impedance models are compared with those obtained from a full iterative geostatistical seismic inversion. Our method allows retrieving similar models but has the advantage of generating alternative solutions in greater numbers, providing a larger exploration of the model parameter space in less time than the iterative geostatistical seismic inversion.