Unformatted text preview: 9/12/11 8. SPHERICAL PROJECTIONS (I) I Main Topics A What is a spherical projecFon? B Equal
angle (stereographic) projecFon of a line [two parts] C Equal
angle (stereographic) projecFon of a plane D IntersecFon of two arbitrary planes 9/12/11 GG303 1 8. SPHERICAL PROJECTIONS (I) Focal Mechanism, Sendai (Tohoku) Earthquake, Mw 9.0, March 11, 2011 hWp://www.virtualuppermantle.info/2011
Sendai
Japan.htm 9/12/11 GG303 2 1 9/12/11 8. SPHERICAL PROJECTIONS (I) II What is a spherical projecFon? A A 2
D projecFon describing the orientaFon of 3
D features. B A projecFon of the intersecFon of a line or plane with the surface of a (hemi)sphere, where the line or plane passes through the center of the (hemi)sphere. 9/12/11 GG303 3 8. SPHERICAL PROJECTIONS (I) II What is a spherical projecFon? (cont.) C Uses in geology and geophysics 1 Maps 2 RepresentaFon of the orientaFon of planar features (e.g., bedding, fractures, crystal faces) 3 RepresentaFon of the orientaFon of linear features (e.g., poles to planes, fold axes) 4 RepresentaFon of ﬁrst moFon data from earthquakes 9/12/11 GG303 4 2 9/12/11 8. SPHERICAL PROJECTIONS (I) III Equal
angle (stereographic) projecFon of a line A A line through the center of the projecFon sphere intersects the lower hemisphere at a point 9/12/11 GG303 5 8. SPHERICAL PROJECTIONS (I) III Equal
angle (stereographic) projecFon of a line (cont.) B The projecFon is where the line between the zenith and the point on the projecFon sphere intersects the horizontal projecFon plane. ProjecFon of line ProjecFon Plane PrimiFve Circle ProjecFon Sphere 9/12/11 GG303 6 3 9/12/11 8. SPHERICAL PROJECTIONS (I) III Equal
angle (stereographic) projecFon of a line (cont.) A A line is at the intersecFon of two planes: 1 a verFcal plane containing the line 2 an inclined plane with a dip matching the plunge of the line. B Plots as a point 9/12/11 GG303 7 8. SPHERICAL PROJECTIONS (I) III Equal
angle (stereographic) projecFon of a line (cont.) C Trend 1
Measured along a horizontal great circle 2
The projecFon (red dot) plots away from the origin in the direcFon of the trend of the line D Plunge (inclinaFon from horizontal) 1
Measured along a verFcal plane containing the line 2
As plunge increases, projecFon nears center of primi%ve circle PrimiFve circle 9/12/11 GG303 8 4 9/12/11 8. SPHERICAL PROJECTIONS (I) IV Equal
angle (stereographic) projecFon of a plane A Great circle: intersecFon of the surface of a sphere with a plane that passes through the center of the sphere (e.g., lines of longitude) B Small circle: intersecFon of the surface of a sphere with a plane that does not pass through the center of the sphere (e.g., lines of laFtude). A line rotated about an axis traces a cone that yields a small circle too. 9/12/11 GG303 9 8. SPHERICAL PROJECTIONS (I) IV Equal
angle (stereographic) projecFon of a plane C A plane plots as the cyclographic trace of a great circle (see Figs. A,B,D) D Strike: measured around the perimeter of the primiFve circle (Fig. D) E Dip: measured along a verFcal great circle perpendicular to the line of strike (Fig. D) 9/12/11 GG303 10 5 9/12/11 8. SPHERICAL PROJECTIONS (I) IV Equal
angle (stereographic) projecFon of a plane F Pole 1 Trend (Fig. D) a Strike of plane – 90° B Opposite azimuth of dip 2 Plunge (Figs. C and D) a 90°
dip of plane b Plots 90° away from “dip vector” (X) 9/12/11 GG303 11 8. SPHERICAL PROJECTIONS (I) III Spherical projecFon of a line (resumed.) C Rake (pitch) 1 Measured along the cyclographic trace of the plane containing the line 2 Measured from the direcFon of strike 3 Like measuring in the plane of a protractor 9/12/11 GG303 12 6 9/12/11 8. SPHERICAL PROJECTIONS (I) V IntersecFon of two arbitrary planes A Line of intersecFon projects as a point. B Represented by intersecFon of the cyclographic traces of the two planes. 9/12/11 GG303 13 8. SPHERICAL PROJECTIONS (I) V IntersecFon of two arbitrary planes (cont.) C Plane C, perpendicular to line of intersecFon c, is perpendicular to both intersecFng planes (A&B) D Pole to each intersecFng plane lies in the plane perpendicular to the line of intersecFon (i.e., a & b lie in plane C) E The line of intersecFon c is normal to the plane with poles a and b F Analogous to ﬁnding the cross product between poles a and b 9/12/11 GG303 14 7 9/12/11 8. SPHERICAL PROJECTIONS (I) V IntersecFon of two arbitrary planes (cont.) G Plane C, perpendicular to line of intersecFon c, is perpendicular to both intersecFng planes (A&B) H Pole to each intersecFng plane lies in the plane perpendicular to the line of intersecFon (i.e., a & b lie in plane C) I The line of intersecFon c is normal to the plane with poles a and b J Analogous to ﬁnding the cross product between poles a and b 9/12/11 GG303 Plane C b c a 15 8 ...
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