A Semi-Ordered Fast Iterative Method (SOFI) for monotone front propagation in simulations of geological folding

This paper present a novel algorithm for monotone front propagation of anisotropic nature. In several examples the new algorithm is shown to be fast and able to solve a general class of front propagation problems. The algorithm is inspired by Huygens’ principle in that the front is described using a list of nodes that are used as source points to evolve the front. Nodes affected by the source points are either directly used as source points or temporarily paused, depending on their solution value and the average solution value of all source points. This feature makes the algorithm semi-ordered. Still, nodes may be used as source points several times, making the algorithm iterative of nature. Together, these features create the Semi-Ordered Fast Iterative (SOFI) method. Unlike other iterative algorithms the performance does not depend strongly on the domain geometry or variations in front velocity. Instead, the performance seems closer to that of the more stable Ordered Upwind Methods. We compare the computational time between the SOFI and Fast Marching method for an increasing grid on two isotropic examples. The computational time of the SOFI method is shorter than that of the Fast Marching method, especially on large grids. Ordered Upwind Methods have a computational scaling of O(N logN), where N is the total number of unknown nodes. The logN factor stems from the sorting needed for the front propagation. The SOFI method needs no sorting, and our numerical experiments indicate that it is of order O(N). On isotropic examples the SOFI method solutions are identical to those from the Fast Marching method assuming the same stencil is used in both methods. On problems with anisotropy the solutions are identical to those from the Fast Sweeping method when the same stencils are used. The SOFI method has many similarities with two recently introduced iterative methods, the Fast Iterative method and the Two Queue method. The Fast Iterative method lacks the semi-ordering, and its performance is therefore very problem dependent. The Two Queue method also pauses nodes to get a partially ordered method, but is only applicable to isotropic problem formulations. Stencils of different forms can be used with minor modifications of the algorithm. We present examples where the stencil uses only edge connected nodes, and also when diagonal nodes are included in the stencil. The SOFI method can use any consistent local wave approximation (stencil), and may therefore keep the constant velocity assumption to a small area unlike the Ordered Upwind Methods which often assume that the velocity profile is constant in a larger neighbourhood. In geoscience, the simulation of an expanding front is used for the modelling of structural folds. Modelling of geological folding is a key component of the shared earth model the Compound Earth Simulator, developed by the oil and gas company Statoil. Non-parallel folds are modelled using anisotropic front propagation, where the velocity of the front depends on the direction the front is moving. Three different classes of folds are illustrated in our example section, all created using the SOFI method.