Blog Layout

Earthquake Analysis by 3-D Affine Deformations

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.

 

By James Farrell 09 May, 2023
A look back in time by James L Farrell, PHD - 2023
11 Apr, 2020
Apologies for little posting lately. Much activity included some with deadlines; this will focus primarily on the few years leading up to Covid.
11 Apr, 2020
GNSS Aided Navigation & Tracking
By James Farrell 30 Aug, 2018
Apologies for little posting lately. Much activity included some with deadlines; this will be limited to the past twelve months. In 2017 my involvement in the annual GNSS+ Conference again included teaching the satnav/inertial integration tutorial sessions with OhioU Prof. Frank vanGraas. Part I and Part II are likewise being offered for Sept 2018. Also...Read More
28 Jun, 2018
Once again I am privileged to work with Ohio University Prof. Frank vanGraas, in presenting tutorial sessions at the Institute of Navigation’s GNSS-19 conference. In 2019, as in several consecutive previous years, two sessions will cover integrated navigation with Kalman filtering.  Descriptions of the part 1 session and part 2 session are now available online. By way of...Read More
30 Apr, 2018
The Institute of Navigation’s GNSS+ 2018 Conference provides me the privilege of collaborating with two of the industry’s pillars of expertise. Ohio University Professor Frank van Graas and I are offering fundamental and advanced tutorials.  Then on the last day of the conference I’m coauthored with William Woodward, Chairman of SAE Int’l Aerospace Avionics Systems Division and hardware lead...Read More
24 Apr, 2018
A new SAE standard for GPS receivers is a natural complement to a newly receptive posture toward innovation unmistakably expressed at high levels in FAA and Mitre (ICNS 2018).  Techniques introduced over decades by this author (many on this site) can finally become operational. 1980s euphoria over GPS success was understandable but decision-makers, lulled into complacency, defined requirements in adherence...Read More
22 Mar, 2018
At April’s ICNS meeting (Integrated Communications Navigation and Surveillance) as coauthor with Bill Woodward (Chairman, SAE International Aerospace Avionics Systems Division), I’ll present “NEW INTERFACE REQUIREMENTS: IMPLICATIONS for FUTURE“.  By “future” we indicate the initiation of a task to conclude with a SAE standard that will necessitate appearance of separate satellite measurements to be included...Read More
16 Jul, 2016
A recent video describes a pair of long-awaited developments that promise dramatic benefits in achievable navigation and tracking performance.  Marked improvements will occur, not only in accuracy and availability; over four decades this topic has arisen in connection with myriad operations, many documented in material cited from other blogs here. 
12 Feb, 2016
For reasons, consider a line from a song in Gilbert-&-Sullivan’s Gondoliers: “When everybody is somebody, then nobody is anybody” — (too many cooks) For consequences, consider this question: Should an intolerable reality remain indefinitely intolerable? While much of the advocacy expressed in my publications and website have focused on tracking and navigation, this tract concentrates...Read More
More Posts
Share by: