Chap. 7] 



GRAVITATIONAL METHODS 



247 



from north by east), the two sets of transformation equations for gradients 

 and curvatures are: 



Ugt = Ux'z cos a — Uy'z sin a 



Uyt = Ux'z sin a + Uy't cos a 



Vu = f/A' cos 2a + 2Ux'y' sin 2a 



2Uxy = — Ul' sin 2a + 2V ^-v- cos 2a 



and 



(7-906) 



Ux'z = Uxt cos a + t/yz sin a 

 Uy't = — Uxz sin a + {/j,« cos a 

 U^' = f/A cos 2a — 2(7j;, sin 2a 

 2Ux'y' = f/A sin 2a -{- 2Uxy cos 2a. 



Graphical transformation follows a similar procedure as shown previ- 

 ously in Fig, 7-93, that is, single azimuth projection of components for 

 gradients, double azimuth projection for curvatures. 



4. Conversion to curves. Conversion of torsion balance data to curves 

 frequently gives considerable interpretational advantages, particularly 

 where these curves are used with geologic sections through "two-dimen- 

 sional" geologic bodies. Therefore, field traverses and interpretation pro- 

 files are laid out at right angles to the strike when possible. Because of 

 regional or other effects, the directions of gradient and curvature vectors 

 may not coincide with the direction of the profile. In this case they should 

 be projected on the profile, which may be done analytically or graphically. 

 If in Fig. 7-94, ^ is the azimuth of a gradient vector with reference to 

 the profile direction x', and ^ is the azimuth of a curvature vector, the 

 projected values are 



dg dg 



--, = — cos ^ 

 dx' ds 



and 



-Ua = Rc OS 2^. 



(7-90c) 



In the corresponding graphical construction the gradient vector is pro- 

 jected upon the profile direction as shown. For the projection of the 

 curvature, the construction of the auxiliary vector at twice the angle with 

 the profile is again necessary. 



For two-dimensional geologic features of virtually infinite extent in the 

 y direction, the (corrected) gradient vectors make the angles of 0° or 180° 

 with the profile direction. The corresponding angles of the curvatures 

 are 0° or 90°. If the subsurface feature is infinite in the y' direction, 

 dg/dy', 2Ux'v' , and d^U/dy\ = and -Ua' = d^U/dxl . A gradient vector 

 in the x' direction is then plotted as positive ordinate and a vector pointing 



