An inversion algorithm is commonly used to estimate the elastic properties, such as P-wave velocity ([ ]), S-wave velocity ([ ]), and density ([ ]) of the earth’s subsurface. Generally, the seismic inversion problem is solved using one of the traditional optimization algorithms. These algorithms start with a given model and update the model at each iteration, following a physics-based rule. The algorithm is applied at each common depth point (CDP) independently to estimate the elastic parameters. Here, we have developed a technique using the convolutional neural network (CNN) to solve the same problem. We perform two critical steps to take advantage of the generalization capability of CNN and the physics to generate synthetic data for a meaningful representation of the subsurface. First, rather than using CNN as in a classification type of problem, which is the standard approach, we modified the CNN to solve a regression problem to estimate the elastic properties. Second, again unlike the conventional CNN, which is trained by supervised learning with predetermined label (elastic parameter) values, we use the physics of our forward problem to train the weights. There are two parts of the network: The first is the convolution network, which takes the input as seismic data to predict the elastic parameters, which is the desired intermediate result. In the second part of the network, we use wave-propagation physics and we use the output of the CNN to generate the predicted seismic data for comparison with the actual data and calculation of the error. This error between the true and predicted seismograms is then used to calculate gradients, and update the weights in the CNN. After the network is trained, only the first part of the network can be used to estimate elastic properties at remaining CDPs directly. We determine the application of physics-guided CNN on prestack and poststack inversion problems. To explain how the algorithm works, we examine it using a conventional CNN workflow without any physics guidance. We first implement the algorithm on a synthetic data set for prestack and poststack data and then apply it to a real data set from the Cana field. In all the training examples, we use a maximum of 20% of data. Our approach offers a distinct advantage over a conventional machine-learning approach in that we circumvent the need for labeled data sets for training.

The physics that describes the seismic response of an interval of saturated porous rocks with known petrophysical properties is relatively well understood and includes rock physics, petrophysics, and wave propagation models. The main goal of seismic reservoir characterization is to predict the rock and fluid properties given a set of seismic measurements by combining geophysical models and mathematical methods. This modeling challenge is generally formulated as an inverse problem. The most common geophysical inverse problem is the seismic (or elastic) inversion, i.e., the estimation of elastic properties, such as seismic velocities or impedances, from seismic amplitudes and traveltimes. The estimation of petrophysical properties, such as porosity, lithology, and fluid saturations, also can be formulated as an inverse problem and is generally referred to as rock-physics (or petrophysical) inversion. Several deterministic and probabilistic methods can be applied to solve seismic inversion problems. Deterministic algorithms predict a single solution, which is a “best” estimate or the most likely value of the model variables of interest. In probabilistic algorithms, on the other hand, the solution is the probability distribution of the model variables of interest, which can be expressed as a conditional probability density function or a set of model realizations conditioned on the data. The probabilistic approach provides a quantification of the uncertainty of the solution in addition to the most likely model. Our goal is to define the terminology, present an overview of probabilistic seismic and rock-physics inversion methods for the estimation of petrophysical properties, demonstrate the fundamental concepts with illustrative examples, and discuss the recent research developments.

Exploration seismology uses reflected and refracted seismic waves, emitted from a controlled (active) source into the ground, and recorded by an array of seismic sensors (receivers) to image the subsurface geologic structures. These seismic images are the main resources for energy and resource exploration and scientific investigation of the crust and upper mantle. We survey recent advances in applications of machine-learning methods, more specifically deep neural networks (DNNs), in exploration seismology. We provide a technically oriented review of DNN applications for seismic data acquisition; data preprocessing tasks such as interpolation/extrapolation, denoising, first-break picking, velocity picking, and seismic migration; data processing tasks such as geologic and structural interpretations; and data modeling tasks such as the inference of subsurface structures and lithologic and petrophysical properties. DNNs have entered almost every sector of exploration seismology. They have outperformed many traditional algorithms for the automation of seismic data acquisition, data preprocessing, data processing, interpretations, and data modeling tasks. However, despite the impressive performances of DNN-based approaches, the out-of-distribution generalization and interpretability of these models remain challenging. To overcome these challenges, incorporating domain knowledge into the DNNs is a promising path and a focus of current deep-learning research in seismology.

We have built convolutional neural networks (CNNs) to obtain petrophysical properties in the depth domain from prestack seismic data in the time domain. We compare two workflows — end-to-end and cascaded CNNs. An end-to-end CNN, referred to as PetroNet, directly predicts petrophysical properties from prestack seismic data. Cascaded CNNs consist of two CNN architectures. The first network, referred to as ElasticNet, predicts elastic properties from prestack seismic data followed by a second network, referred to as ElasticPetroNet, that predicts petrophysical properties from elastic properties. Cascaded CNNs with more than twice the number of trainable parameters as compared to end-to-end CNN demonstrate similar prediction performance for a synthetic data set. The average correlation coefficient for test data between the true and predicted clay volume (approximately 0.7) is higher than the average correlation coefficient between the true and predicted porosity (approximately 0.6) for both networks. The cascaded workflow depends on the availability of elastic properties and is three times more computationally expensive than the end-to-end workflow for training. Coherence plots between the true and predicted values for both cases show that maximum coherence occurs for values of the inverse wavenumber greater than 15 m, which is approximately equal to 1/4 the source wavelength or λ/4. The network predictions have some coherence with the true values even at a resolution of 10 m, which is half of the variogram range used in simulating the spatial correlation of the petrophysical properties. The Monte Carlo dropout technique is used for approximate quantification of the uncertainty of the network predictions. An application of the end-to-end network for prediction of petrophysical properties is made with the Stybarrow field located in offshore Western Australia. The network makes good predictions of petrophysical properties at the well locations. The network is particularly successful in identifying the reservoir facies of interest with high porosity and low clay volume.