480 EXPLORATION GEOPHYSICS 



2 3' Sg simple considerations indicate 



^:t^^^\^^-^^;^;^;^::^^^ \ ^^^ ^^P^ °^ solution to be ex- 



^5|^^^^^^^^^^^^^^P 4 pected : Suppose that the dis- 



^^^^^^^^^^^^^^^^— i tance between ^^i and S'. is 



:zzr z=r I=3^.,,.zr^_-=-^^'^ -~- -^— very small compared with ^ ; 



^^ z=z ii^l"'!!^::^ _zr All" -=i" only a small amount of current 



rz Tzi_ — —^^ — -^=~ =^ ^=^ -_ will flow into medium p2 and 



zr HZ zir ziz: zzz z=r z^Hr^_j:^£~ the measurement of the resis- 



^ ji— _ — L_ "Tn: Jizi- ^^_~^^r tivity by one of the methods de- 



FiG. 288.— Current lines between two electrodes in a SCribed in CaSC III will yield a 



layered medium. 



value of p nearly equal to pi. 

 If, on the other hand, the electrode spacing is very large compared with 

 the thickness A, the effect of the now relatively thin upper layer will gen- 

 erally be small, and the value of p obtained by one of the methods described 

 in Case III will be nearly equal to p2- The curve obtained on plotting p 

 against SiS^ resembles that shown in Figure 289. The region of transi- 

 tion AB occurs at values of SiS^ comparable in magnitude with A. 



DEPTH= K-S,S2 



Fig. 289. — Schematic resistivity curve for two-layer structure. 



Thus, in addition to enabling one to recognize a simple resistivity dis- 

 tribution, such considerations indicate the order of magnitude of the depth 

 of the buried medium. It would be very difficult, however, to determine 

 the exact depth from such physical considerations. It is much safer to refer 

 to the exact mathematical solution of the two-layer problem. 



Before considering the two-layer case, it will be of some interest to 

 illustrate the use of the exact method by solving two simple problems by 

 the method of images. f 



t See T. A. Stratton, "Electromagnetic Theory," McGraw-Hill, 1941, p. 193. 

 W. R. Smythe, "Static and Dynamic Electricity," McGraw-Hill, 1939, pp. 69, 87, 283. 

 J. H. Jeans, "Mathematical Theory of Electricity and Magnetism," Cambridge Univ. Press, 

 1923, p. 200. 



