200 



EXPLORATION GEOPHYSICS 



21 log r dx. The potential due to the entire surface between x = Xi and 



X = X2 is 



V 



r-2 ( 



= 2 \ I log r dx = 2 1 



/ log r rf^ = 2 \ [/ log {V ^-{x- ^o)'}^0 dx 



The X and 3; components of the magnetic force due to the surface layer at 

 the point {xQya) are 



(g)- 



s 



91ogr 

 dxQ 



dx = 21 





{x — Xq) dx 



V+ {X- XqY 



m-'\ 



= I (log [yo^ + (X2 - xo)^} - log [yo^ + (xi - ^0)^] } 



yo dx 



(86) 



dyo I yo^+{x-xo)^ 



= 2/ ftan-i ^^^^^ - tan-i i£l-Il£ol"| 

 L yo yo J 



(87) 



The vertical and horizontal components due to the end of the thick layer 

 are given by the end effect equations for a thin layer. The total magnetic 



anomaly produced by a thick layer may 

 be calculated by adding the effects pro- 

 duced by its surfaces and its end. 



Special Cases 



1. Vertical Anomaly Near the End of a 

 Thick Horizontal Layer. — Assume that the 

 upper surface of the layer is buried at a 

 depth of 200 feet and that the layer is 1,000 

 feet thick. It will be convenient to consider 

 separately the effects of the upper surface, 

 the lower surface, and finally, the end of the 

 layer. For a horizontal layer, the vertical 



anomalies produced by the surfaces are equal to the yo components. Hence, the vertical 



component due to the upper surface of the layer is : 



dV 



Fig. 90. — Sketch showing quantities which 

 enter into the calculation of the magnetic 

 potential produced by a thick layer at an ex- 

 ternal point P. 



oyo L yo yo J 



If it is assumed that one end of the layer is vertical and the other end is very far 

 removed from the region in which anomalies are being measured, the first quantity 

 in the bracket approaches ir/2. The second quantity in the bracket may be simplified by 

 setting xi equal to zero. When these substitutions have been made, the last equation 

 becomes : 



|K = 2/J-5- + tan-^1 

 oyo L 2 :vo J 



where yo = 200 and Xi = 0. Similarly the vertical anomaly due to the lower surface is 



