DeepNet: an ultrafast neural learning code for seismic imaging

A feed-forward multilayer neural net is trained to learn the correspondence between seismic data and well logs. The introduction of a virtual input layer, connected to the nominal input layer through a special nonlinear transfer function, enables ultrafast (single iteration), near-optimal training of the net using numerical algebraic techniques. A unique computer code, named DeepNet, has been developed, that has achieved, in actual field demonstrations, results unattainable to date with industry standard tools. virtual input layer allows for a specific type of preprocessing of the input vectors. The associated computations between the actual and virtual input layers differ from the usual inter-layer neural network computations. 1. Background and purpose The ability to accurately predict the location of remaining oil in the neighborhood of existing production wells is of vital economic importance to the petroleum industry. For practical purposes, one typically targets volumes of fluid 10 meters thick and 200 meters in lateral extent at a distance of 200 meters from each well, requiring a resolution accuracy of 5% in terms of the distance from the observation well. Available oilfield information incorporates many datasets with different scales, uncertainties, sample volumes, and relevance. Well logs (e.g., porosity, gamma ray response, and resistivity) provide the most accurate possible sensor-based characterization of the geological formations encountered along the path of a well [l]. On the other hand, lowresolution seismic dam are generally used to conduct large-scale field assessments [2]. The specific focus of the research we report in this paper was to develop a methodology that would enable fast and accurate prediction of well pseudo logs from seismic data across an entire oil field. 2. Network Architecture We consider a generalized multilayer feed-forward neural network. In general, the nodes (neurons) are organized in layers, namely: (i) input, (ii) (one or several) hidden, and (iii) output. Ehch layer 1 contains N, nodes. In addition to these traditional layers, we introduce a virtual input layer between the input layer and the (first) hidden layer. The Figure 1: Neural net with virtual input layer To simplify the description, and without loss of generality, we discuss a network architecture having No = I input nodes, Nr = V virtual input nodes, one hidden layer with NZ = H nodes, and N3 = 0 output nodes. The goal in the learning process is to determine the synaptic interconnection matrices (W, , WHO}. This will be achieved by minimizing (some norm of) the difference between the output values calculated by the net and the target outputs. In our approach, we decouple the nonlinearities of the transfer functions at each layer from the linear interlayer pattern propagation. This paradigm follows the seminal observations of Biegler-K&rig and Barman [3], and Tam and Chow [4]. The essence of their approach was to minimize, solving a sequence of least squares problems, the learning error on each layer separately, rather than globally for the entire network. Since for most practical applications the number of training examples exceeds the dimensionality of the training patterns, approximate results are typically obtained. Our proposed algorithm, on the other hand, implements a sequence of alternating direction singular value decompositions (SVD), on a network architecture having a monotonically decreasing number of nodes in successive layers. We modify the traditional neural network architecture by introducing a virtual input layer connected to the input nodes through a nonlinear transfer function. The virtual layer acts as a preprocessor of the input vectors, and replaces a highly over-determined linear system with a uniquely solvable one. 3. Network Training The learning algorithm begins at the output layer. A full iteration consists of a backward pattern propagation to the virtual layer, followed by a forward sweep back to the output layer. Our notation is as follows. A bar indicates a quantity calculated by the network (as opposed to a “target” or initialized quantity); a superscript f denotes a forward calculation. Each node implements a nonlinear transfer function Q : %+(O,l), typically a sigmoid. Since Q, is bijective, the inverse p -’ is well defined. The K seismic signatures used for training are stored as rows of matrices; the number of columns of each matrix equals the number of nodes of the corresponding processing layer. For convenience, the matrix dimensions are explicitly indicated as subscripts. The following two equations relate the presynaptic inputs T to the postsynaptic outputs S at the output layer.