Earthquake Analysis by 3-D Affine Deformations

Changes in coordinates at stations affected by earthquakes have been monitored successfully, for years, with precision using satellite navigation.  Results of interest have then been produced in the past by processing the outcome, e.g., investigating the history of triangles formed thereby.  The first application to earthquakes of entirely different criteria (affine deformation states) has produced results with encouraging prospects for prediction, both in time (more than two weeks prior to the 2011 Tohoku quake) and spatially (departures from the affine model at the station nearest epicenter).

The fifteen independent states of a standard 3-D 4×4 affine transformation can be categorized in five sets of three, each set having x-, y-, and z-components for translation, rotation, perspective, scaling, and shear.  Immediately the three degrees-of-freedom associated with perspective are irrelevant for purposes here.  In addition both translation and rotation, clearly having no effect on shape, can be analyzed separately — and the same is likewise true of uniform scaling.  It is thus widely known that there are five “shape states” involved in 3-D affine deformation, three for shear and two for nonuniform scaling.  One way to describe shape states is to note their effects in 2-D, where there is only one for nonuniform scaling (which deforms a square into a rectangle) and one for shear (which deforms a rectangle into a parallelogram).  Therefore it is noted here that added insight into earthquake investigation can be obtained by analyzing affine features – with specific attention given to their individual traits (degrees-of-freedom).

For investigating earthquakes from affine degrees-of-freedom, methodology of another very different field — anatomy — is highly relevant but ironically lacking a crucial feature.  As currently practiced, physiological studies of affine deformations concentrate heavily on two-dimensional representations.  While full affine representation is very old, its inversion  — i.e., optimal estimation of shape states from a given overdetermined coordinate set — has previously been limited to 2-D.

Immediately then, extension was required for adaptation.  The fundamentals, however, still remain applicable. Instead of designated landmark sets coming from a group of patients, here they are associated with a series of days (e.g., from several days before to some days after a quake).  Each landmark set is then subjected to a series of procedures (centroiding, normalization, rotation) for fitting landmark sets from one day to another.

The Procrustes representation from procedural steps just described provides sequences of centroid shifts in each direction, rotations about each axis, and amounts of uniform scaling needed for each separate day.  In addition to those seven time histories, there are five more that offer potential for greater insight (again, shape states — three shear and two nonuniform scaling).  All were obtained for landmark coordinate sets reported before and after the 2011 Tohoku quake.  From sample recorded coordinates provided along with shape state values, readers of the manuscript  are enabled to verify results.