Theoretical and efficient practical procedures for the generation of inflation factors for ES-MDA

The ensemble smoother with multiple data assimilation (ES-MDA) has proved to be a powerful assisted history-matching method. The main drawback of ES-MDA is that the inflation factors for damping the changes in model parameters have to be determined before starting the history-match. Although various authors have provided suggestions for determining the inflation factors adaptively as the history-match proceeds, these methods often result in a large number of data assimilation steps which can make ES-MDA too computationally inefficient for practical application to large-scale field problems. Here, we provide a theoretical procedure to determine exactly the minimum inflation factor at each data assimilation step that ensures the discrepancy principle is satisfied. Like previous adaptive ES-MDA methods, this method does not allow one to specify a priori the number of data assimilation steps to be done. Thus, using the exact theoretical procedure as a guide, we provide a practical efficient method for determining the inflation factors which allows one to specify a priori the number of data assimilation steps to be done with ES-MDA which still ensures that the initial inflation factor is chosen so that the discrepancy principle is approximately satisfied.