Mapping paleostress trajectories by means of the clustering of reduced stress tensors determined from homogeneous and heterogeneous data sets

It is easy to draw stress trajectories to investigate the present stress field by interpolating stress orientations determined at control points. However, challenges arise when we deal with the trajectories of paleostresses, because neighboring control points may have the stress orientations of different tectonic phases. We must choose coeval stresses to draw the trajectories. Recent stress inversion techniques can separate stresses from heterogeneous data from fault, dilational fractures, etc. Natural data sets from those structures are often heterogeneous, and age data are usually not enough to classify the stresses by age. As a result, an unsupervised classification problem of the inversion results must be solved to draw the trajectories. Here, we propose a simple and heuristic procedure for this problem. We assume smooth trajectories during each of the phases. The smoothness makes density-based clustering adoptable to solve the problem. The heterogeneity of data sets allows the additional partition of the clusters. As a worked exercise for this technique, the trajectories of minimum horizontal stress orientations were drawn based on the paleostresses determined from the attitudes of felsic dikes and quartz veins formed in mid Cretaceous orogeny in the North Kitakami Terrain, northern Japan. The orogen-parallel and orogen-perpendicular extensional stress fields delineated by the present technique were probably the manifestations, respectively, of the gravitational collapse of the orogen and of regional extensional tectonics in the Far East.