260 GRAVITATIONAL METHODS [Chap. 7 



masses 45° below the horizon. The gradient anomahes of two- and three- 

 dimensional masses are therefore quite similar in respect to vertical angle 

 and differ only in regard to distance power. In the curvatures, an exten- 

 sive strike dimension produces sensitivity in the vertical direction in addi- 

 tion to that in horizontal direction. The depth to center, D, and radius 

 may be determined, as in eq. (7-926), from the relations"^ 



D = xn= I.IZxg = 0.578a;c 1 



*"> '-'max. ^max. 



. \ (7-936) 



R = 0.564VCn.ax.-I> = 0.700\/G„ax.-A J 



where Xc„ is the abscissa of the point of zero curvature anomaly (see 



Fig. 7-99). 



Other two-dimensional features are illustrated in Fig. 7-98, such as 



anticlines, synchnes, incHned dikes, vertical steps, slabs, and vertical 



dikes. Their anomalies may be calculated readily from those of a slope 



(Fig. 7-986). The calculation involves essentially the integration of 



formula (7-916). For the gradient this may be written, with the notation 



r^ P 2x dx 

 of Fig. 7-986, Ux'z = ^kb / zdz \ — ~— , where x ^ xa on the sloping 



edge. Using Tq = a;o + z^ ^ = a — v'o , J'o dipQ = ds sin a, and dz = ds 



sin a, we have U^'z = 2kd I d(po sin a sin ^o/sin \p or Ux'z = A;5 2 sin^ a 



fi L 



loge - — sin 2a (^2 — <pi) \. Thus, with the dip angle i = ir — a, and 



similar analysis for the curvature. 



Ux'z = 2k8 sin i sin i loge - -|- cos i(<p2 — <pi) 

 — Ui,' = 2k8 sin i sin t (^ — v^i) — cos i loge — . 



(7-93c) 



For direct interpetation it is useful to know that the center between 

 the extremes in the curvatures is situated above the center of the sub- 

 surface slope. With this point as origin, the abscissas of the maximum 

 gradient and curvature values are given by 



a^G„,ax. = 2(2) .J, ^) ^^^'^'^ ^'' ^c,„ax. = W^Dd + (D - df cotan^ i. 



For complete depth and dip determinations, diagrams have been con- 

 structed by Jung."^ 



""Jung, Zeit. Geophys., 3(6), 267-280 (1927). 

 "T Ibid. 



