Probabilistic seismic inversion based on physics-guided deep mixture density network

Deterministic inversion based on deep learning has been widely utilized in model parameters estimation. Constrained by logging data, seismic data, wavelet and modeling operator, deterministic inversion based on deep learning can establish nonlinear relationships between seismic data and model parameters. However, seismic data lacks low-frequency and contains noise, which increases the non-uniqueness of the solutions. The conventional inversion method based on deep learning can only establish the deterministic relationship between seismic data and parameters, and cannot quantify the uncertainty of inversion. In order to quickly quantify the uncertainty, a physics-guided deep mixture density network (PG-DMDN) is established by combining the mixture density network (MDN) with the deep neural network (DNN). Compared with Bayesian neural network (BNN) and network dropout, PG-DMDN has lower computing cost and shorter training time. A low-frequency model is introduced in the training process of the network to help the network learn the nonlinear relationship between narrowband seismic data and low-frequency impedance. In addition, the block constraints are added to the PG-DMDN framework to improve the horizontal continuity of the inversion results. To illustrate the benefits of proposed method, the PG-DMDN is compared with existing semi-supervised inversion method. Four synthetic data examples of Marmousi Ⅱ model are utilized to quantify the influence of forward modeling part, low-frequency model, noise and the pseudo-wells number on inversion results, and prove the feasibility and stability of the proposed method. In addition, the robustness and generality of the proposed method are verified by the field seismic data.