The Turning Bands Method for simulation of random fields using line generation by a spectral method

The turning bands method (TBM) for the simulation of multidimensional random fields is presented. These fields commonly occur in the Monte Carlo simulation of hydrologic processes, particularly groundwater flow and mass transport. The general TBM equations for two‐ and three‐dimensional fields are derived with particular emphasis on the more complicated two‐dimensional case. For stationary two‐dimensional fields the unidimensional line process is generated by a simple spectral method, a technique which can be generally applied to any two‐dimensional covariance function and which is easily extended to anisotropic and areal averaged processes. Theoretically and by example the TBM is shown to be ergodic even for a finite number of lines, and it is demonstrated that it rapidly converges to the true statistics of the field. Guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments. The TBM is compared to other methods in terms of cost and accuracy, demonstrating that the TBM is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion, such as the nearest neighbor approach. The unidimensional spectral technique presented here permits, for the first time, the inexpensive and accurate TBM simulation of any proper two‐dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes.