• machine learning
• hybrid machine learning
• discrete cosine transform
• DCT
• convolutional neural network
• CNN
• Monte Carlo approach
• petrophysical seismin inversion
• transfer learning

We implement a machine-learning inversion approach that uses a convolutional neural network (CNN) to solve the petrophysical seismic inversion. A discrete Cosine Transform (DCT) is used to compress both the input and output response of the network, and hence the network is trained to predict the nonlinear mapping between the DCT-transformed seismic data and the DCT-transformed petrophysical model. This transformation is used as an additional feature extraction technique that not only reduces the dimensionality of the input and output of the network but also guarantees the preservation of the assumed temporal and spatial continuity pattern in the estimated model. A theoretical rock-physics model (RPM) based on granular media models constitutes the link between the elastic and the petrophysical space, whereas the exact Zoeppritz equations map the elastic properties onto the seismic pre-stack domain. A direct sequential co-simulation with joint probability distribution generates the training and validation sets under the assumption of a stationary non-parametric prior and a Gaussian variogram model. We apply a Monte Carlo simulation strategy to propagate onto the final estimates both the uncertainties associated to the noise contamination in the data and the modeling error introduced by the network approximation. We discuss synthetic inversions to a realistic subsurface model that simulates a real gas-saturated reservoir hosted in a turbiditic sequence. We assess the robustness and stability of our trained CNN in case of erroneous assumptions about the noise statistics, errors in the calibrated RPM, and errors in the estimated source wavelet. The outcomes of the proposed approach are compared with those achieved by a more standard linearized inversion in which each seismic gather is inverted separately. Lastly, we demonstrate that transfer learning avoids retraining the network from scratch when the statistical properties of the training and test sets differ. Our experiments confirm that the implemented CNN inversion successfully solves the petrophysical seismic inversion and guarantees more stable and accurate predictions with respect to the standard inversion approach. In particular, once the network has been trained, the implemented inversion retrieves petrophysical properties and associated uncertainties in near real-time.