Numerous modelling tasks require the solution of ill-posed inverse problems where we seek to find a distribution of earth models that match observed data such as reflected acoustic waveforms or produced hydrocarbon volumes. We present a deep learning framework to create stochastic samples of posterior property distributions for ill-posed inverse problems using a gradient-based approach. The spatial distribution of petrophysical properties is created by a deep generative model and controlled by a set of latent variables. A generative adversarial network (GAN) is used to represent a prior distribution of geological models based on a training set of object-based models. Then we minimize the mismatch between observed ground-truth data and numerical forward-models of the generator output by first computing gradients of the objective function with respect to grid-block properties and then using neural network backpropagation to obtain gradients with respect to the latent variables. Synthetic test cases of acoustic waveform inversion and reservoir history matching are presented. In seismic inversion, we use a Metropolis adjusted Langevin algorithm (MALA) to obtain posterior samples. For both synthetic cases, we show that deep generative models such as GANs can be combined in an end-to-end framework to obtain stochastic solutions to geophysical inverse problems.


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