Development of Inference Methods for Rotational Motions on Ground Surface and Application to Microtremors Acquired with a Small-Size Dense Array

Abstract

In this study, we develop two methods for the inference of rotation vector on ground surface, two rocking rotations and a single torsional rotation. The first, termed nth-order elastic method, is based on the elasticity of the ground surface, and the rotation vector is constructed from the first derivative of ground motions which are approximated by nth-order Taylor expansion. Meanwhile, the second, termed rigid method, is based on the rigidity of ground surface and the rotation vectors are directly obtained by the least square method which minimize the sum of the squared difference motions recorded and predicted by the equation of the rigid motions. Furthermore, the second is divided into two methods. The one, multi-site rigid method, uses the differential motions at multi-sites, while the other, single-site rigid method, uses the differential motions at a single site. Also, we show that the 1st-order elastic method is the same as the multi-site rigid method. Appling the nth-order elastic method to microtremor recordings acquired with a small-size dense array, we successfully infer the rotation vector and find the rigid zone with a radius of 5m for the torsional motions. We furthermore obtain the two findings. The first is that the rotation vector can be inferred with a simpler array of a small size and a fewer observation sites. The second is that the root mean squared (RMS) amplitudes of the torsional motions inferred by the single site rigid method are approximately the same as the RMS amplitudes inferred by the 1st-order elastic method in a zone close to the reference site.