An Iterative Inversion Of Seismic Data Constrained By Wells Using Artificial Neural Networks
Authors: Liu Z.L. , Liu J.Q. , Castagna J.P. , Pan C.H. 
Published: October 2008

Abstract
In this paper we present a data-driven joint inversion of seismic and well data. Rather than using a theoretical operator, we learn the forward operator using a BP neural network trained by the data set with well data as input and seismic recordings near the well as desired output. Then, we start the inversion by inputing the initial model of each trace into the trained BP network and propagating the output errors backward through the network iteratively to modify the input stratum model. The inversion is verified and its robustness analyzed using synthetic data. A field case involving a migrated seismic section and two wells was inverted to obtain the desired pseudo-log sonic and density profiles. The research results show that the technique can perform multiparameter inversion constrained with multiple wells, and deal with complex nonlinear problems.

Introduction
The strategy of seismic inversion is commonly in looking for an optimal geophysical model which minimizes the residuals between the model responses and observed data. The responses of the model are conventionally calculated by a theoretical forward operator, such as a convolution or wave equation operator1,2. Artificial neural networks(ANN) have recently attracted a great deal of attention for their ability to "learn," or "estimate" the mapping operator from training data which describe the mapping relationship. Among neural networks for function approximation, the feed forward network trained by error back propagation (BP)3 algorithm is one of the most effective and popular neural networks. It has been shown that the BP neural network with one hidden layer is a universal function approximator 4. In the learning of the BP (error back-propagation) neural network, the output error is propagated backward through the network to modify the link weights of the network, and the mapping relationship between the input and desired output vectors is encoded in the weights with a distributed form. Linen and Kinderman5 have shown that the same mechanism of the weight learning can also be applied to update the input of the network to iteratively invert the input parameter model of the network. In their approach, the output errors of the network are ascribed to the input errors of the network, rather than the weight errors. The iterative inversion of a BP network is, thus, to search the input signals in the input space to minimize the output errors, while the learning of it is to find a set of weights in the weight space to minimize the output errors Hoskins and Hwang6 apply the iterative inversion technique to their adaptive controllers. They learn the forward model of the plant, and perform the iterative inversion to generate control commands. They conclude that the proposed control approach allows the controllers to respond on line to changes in plant dynamics. Here, we apply the technique to the joint inversion of well-log and seismic data. We first learn the forward operator using a BP network trained by the data set consisting of well data as input and seismic recordings located at or near the well as desired output. Then upon the trained BP network, we iteratively invert seismic recordings into the stratum parameters trace by trace. As the approximation of the forward operator is adaptively driven by well-log and seismic data and there are no assumptions about the operator and the medium, the inversion can be nonlinear with a high degree of complexity.