In a previous CREWES talk, which was also published in The Leading Edge (Russell et al., 2002), we applied the multi-layer perceptron (MLP) neural network to the solution of a straightforward AVO classification problem. In that paper, we pointed out that the basis of the MLP is the sigmoidal, or S, function that is used in the training. The S function is one of the most useful functions in mathematics and physics, and has many applications besides its use in neural networks. The S function comes in various guises, such as the logistic function, the tanh function, the Cole-Cole equation, the population function, etc. In this article I will first describe the function itself and its origin and derivation. I will then discuss the various applications of this function, many of which I have talked about in several previous CREWES talks, notably the talk on solving an AVO problem using a neural network. These applications range from population growth through paramagnetism and heavy oil modeling to neural networks, making the S function one of the most ubiquitous functions used in geophysics.
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