There are two principal challenges with geological observations: the time problem and the space problem. Geological observations are obtained in the present, but are used to infer the history of the Earth over millions of years. When developing models of the Earth's evolution on those time scales, we cannot validate the model with direct observations of its past state: the time problem. Further complicating the problem is our inability to dissect the Earth: we can only hope to probe it in a few places or using fields observable at the surface such as the magnetic or gravity field. To solve this problem, we are investigating ways of best using the available data to constrain model parameters, choose a particular model over another or quantify the uncertainty of our model based on our assumptions.
As mentioned, models of the diffusion type can explain the depositional history of a sedimentary basin. Although such models are currently in use by sedimentologists and geomorphologists to simulate the process of sedimentary deposition, the impact of the models has been quite limited. This fact is mainly due to the difficulty for geologists to estimate values of the diffusion coefficients entering the models. The uncertainty in these coefficients leads to an unknown overall uncertainty in the simulations. We try to attack estimation of diffusion coefficients from two different angles: (i) by estimating parameters fra observed present-day data, using techniques adopted from the Inverse Problems project, and (ii) by representing the uncertain coefficients as stochastic quantities and then computing the overall uncertainty in simulations.
In the latter stochastic approach, one assigns probability distributions for the uncertain parameters. Using the probabilistic collocation method (PCM) one can compute the expectation and variance of the response of the model from a series of deterministic simulations. Since the deterministic simulations just involve the original non-stochastic simulator (in this case the commercial code Dionisos), PCM is a non-intrusive method that can be used for a wide range of problems. Toward the end of 2009, PCM was also applied to electromechanical simulations in the heart. The application of PCM for depositional modeling showed promise and involved real field data from the Ebro basin in Spain.
Stochastic simulation remains an important area of the Computational Geoscience group's research, and in 2009 we established a close collaboration with Texas A&M University and the Stochastic Mechanics Group at the Zachry Department of Civil Engineering.
