Nonlinear process control of wave-equation inversion and its application in the detection of gas

The wave equation describes how seismic waves propagate in the subsurface. Inversion methods based on the wave equation naturally take into account the complex behavior of propagating waves and can be used to make accurate estimates of model parameters. However, computational cost and poor convergence have not been overcome, and thus restrict the broad application of this technique. Preconditioned conjugate gradient inversion using back-propagation techniques is a simple, robust implementation of wave-equation inversion in which the step length for correcting the model for each iteration is a crucial factor affecting convergence and hence computational cost. The step length can be calculated by an adaptive controller based on the theory of model reference nonlinear control that ensures that the error energy of the complex system vanishes rapidly. Although the computational cost for each iteration remains the same, the inversion is robust and converges more rapidly than other methods. We tested our method on synthetic data generated from a three-layer fractured model. The inversion converges to the true model after five iterations, and different initial models give similar inversion results. The application to 2D real data from a gas field in western China illustrates that even two iterations yield unambiguous interpretable inversion sections.