Multivariate predictive analysis is a widely used tool in the petroleum industry in situations in which the deterministic nature of the relationship between a variable that requires prediction and a variable that is used for the purposes of such prediction is unknown or very complex. For example, to perform a sweet-spot analysis, it is necessary to predict potential oil and gas production rates on a map, using various geologic and geophysical attribute maps (porosity, density, seismic attributes, gravity, magnetic, etc.) and the initial oil and gas production rates of several control or training wells located in the area of interest. We have developed a new technology that allows for building a stable nonlinear predictive operator by using the combination of a neural network, a genetic algorithm, and a controlled gradient method. The main idea behind the proposed technology is to combine stochastic and deterministic approaches during the construction of the predictive operator at the training stage. The proposed technology avoids many disadvantages of the genetic algorithm and gradients methods, such as a high level of dependency on the initial values; the phenomenon of over-fitting (overtraining), which results in creation of an operator with unstable predictability; and a low speed of decreasing error during iteration, and, as a result, a low level of prediction quality. However, the above-mentioned combination uses the advantages of both methods and allows for finding a solution significantly closer to a global minimum for the objective function, compared to simple gradient methods, such as back propagation. The combination of these methods together with Tikhonov regularization allows for building stable predictions in spatial or/and time coordinates.

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The paper describes the method for solving the inverse problems of gravimetry based on the func tional representations, which follow from the variational principles in the uniform metrics with respect to density models of the geological medium. The functional representations are obtained for both the linear problem (which study local density distributions) and nonlinear problem (which study a system of structural models). The explicit formulas for calculating density models are derived for the particular cases based on the introduced functional representations of the obtained solutions. In the general case, the converging iterative processes providing the solution for both the density distribution models and structural models are con structed. The relationship is established between the functional representations implementing the variational principle in the uniform metric and linear integral representations corresponding to the optimization in the quadratic norm, on one hand, and the other known density models, on the other hand.

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