Katsushi Sato

Short note: A spherical code and stress tensor inversion

A spherical code is a set of many points arranged at more or less uniform intervals on a hypersphere. Here, we present a spherical code comprising 60,000 points in five-dimensional space and the list of the uniformly distributed reduced stress tensors corresponding to the points. This code benefits tectonic studies. By the way of example, it is demonstrated how the code improves the resolution of stress tensor inversion.

Analysis of clustering in three-dimensional grain fabric

Sedimentological analysis of grain fabric has paid scant attention to grain shape. However, the information of grain orientation is inseparable from that of shape in three-dimensional fabric analysis. Not only should the dominant major-axis orientations be recognized, but so should the dominant combinations of shapes and orientations of grains. The aim of this paper is to demonstrate that such combinations can be identified by density-based cluster analysis in a five-dimensional parameter space, where a point represents a specific combination of the shape and orientation of a grain approximated by a triaxial ellipsoid. We tested the present method using an artificial data set. The data were successfully classified into correct groups. Next, we applied it to a data set obtained by X-ray computed microtomography from some 5000 sand grains deposited in an experimental flume. We show that triaxial grains, which have principal axes with distinctive radii, have major axes in the paleocurrent orientation and roller-shaped grains in the transverse orientation. Both grain types have vertical minor axes. Those orientations are not preconditioned for statistical analysis but are recognized as significant; this suggests the potential of the method in three-dimensional fabric analysis for applications in sedimentology. The method can be applied to the statistical processing of ellipsoidal objects including, for example, deformed grains, stress ellipsoids, and magnetic susceptibility ellipsoids.

Bias correction for the orientation distribution of slump fold axes: Application to the Cretaceous Izumi basin

Linear structures perpendicular to an outcrop surface are easily discovered, but those parallel to the surface are not, giving rise to a biased orientation distribution of the structures. Here, we propose a bias correction method: Statistical inversion was conducted to unbias the distribution of the axes of mesoscale slump folds in the Cretaceous Izumi Group, Japan using the orientation distribution of outcrop surfaces. The observed axes showed a cluster in the SE quadrant. Their unbiased distribution had a girdle pattern with a maximum concentration orientation in the same quadrant, but the unbiased one had a lower peak density than the observed one, and was more girdle-like than the observed one. The maximum concentration axis of the unbiased distribution was roughly perpendicular to the paleocurrents observed in the same area. Therefore, the popular view that the axes of slump folds are perpendicular to paleoslope applies to the folds in the area in a statistical sense. The hypothesis about the vergences of slump folds and paleoslope hold only about a half of the observed slump folds.

Fast multiple inversion for stress analysis from fault-slip data

The multiple inverse method is widely used to invert multiple stress tensors from fault-slip data caused by polyphase tectonics. A practical problem of the method is the time-consuming computation related to its iterative procedure. This paper describes a way of accelerating the computation by replacing an exhaustive grid search for the optimal stress tensor by direct calculation using an analytical solution. Furthermore, a technique to reduce noise in the result was developed based on the estimation of instabilities of solutions.

Tectonic evolution of northwestern Imbrium of the Moon that lasted in the Copernican Period

The formation ages of tectonic structures and their spatial distributions were studied in the northwestern Imbrium and Sinus Iridum regions using images obtained by Terrain Camera and Multiband Imager on board the SELENE spacecraft and the images obtained by Narrow Angle Camera on board LRO. The formation ages of mare ridges are constrained by the depositional ages of mare basalts, which are either deformed or dammed by the ridges. For this purpose, we defined stratigraphic units and determined their depositional ages by crater counting. The degradation levels of craters dislocated by tectonic structures were also used to determine the youngest limits of the ages of the tectonic activities. As a result, it was found that the contractions to form mare ridges lasted long after the deposition of the majority of the mare basalts. There are mare ridges that were tectonically active even in the Copernican Period. Those young structures are inconsistent with the mascon tectonics hypothesis, which attributes tectonic deformations to the subsidence of voluminous basaltic fills. The global cooling or the cooling of the Procellarum KREEP Terrane region seems to be responsible for them. In addition, we found a graben that was active after the Eratosthenian Period. It suggests that the global or regional cooling has a stress level low enough to allow the local extensional tectonics.Graphical AbstractThe formation ages of mare ridges are constrained by the depositional ages of mare basalts, which are either deformed (orange unit) or dammed (blue unit) by the ridge. The degradation levels of craters dislocated by tectonic structures were also used to determine the youngest limits of the ages of the tectonic activities.

Physical Meaning of Stress Difference for Fault-Slip Analysis

Recent stress tensor inversion methods for fault-slip analysis are used to distinguish between multiple stress states to elucidate spatiotemporal change of the earth’s crustal tectonics. An estimator named the stress difference has been a practicable tool to measure the difference between stress solutions of inversion analysis. This measure corresponds to the expected difference in shear stress direction on a randomly oriented fault plane, which is, however, an approximation including several degrees of deviation. This study investigated the formula of stress difference and found the exact physical meaning, specifically the expected difference in shear stress vector which carries information on magnitude as well as direction. The present discovery is based on the analytical proportionality between the second invariant of stress tensor and the root mean square magnitude of shear stress for all orientation of fault planes. The meaningless difference in non-dimensional shear stress magnitude was found to be incorporated into the value of stress difference. This fact is not convenient for fault-slip analysis dealing only with orientations.