Chap. 10] 



ELECTRICAL METHODS 



717 



If conductivity ratios are reversed (good conductor below), the apparent 

 resistivity ratio lies between values of and 1 and approaches 0. 



3. Three horizontal layers. The theoretical treatment of this case pro- 

 ceeds along the same lines as that for the two-layer case. The mathe- 

 matical relations become more complex, however, because images of 

 sources and sinks are produced by reflection on two boundary surfaces. 

 The calculation has been carried out by Hummel** for the general case 

 and for a special condition where the thickness of the two top layers is 

 the same {h = hi) and the bottom layer is infinite in extent and of infinite 

 resistivity. Results obtained for the latter case indicate that (for large 

 electrode separations) the apparent resistances follow Kirchhoff's law for 

 two resistances connected in parallel, so that the average resistivity of 

 two infinite layers, of resistivity pi 

 and p2 and respective thickness h\ 

 and Az , is given by 



^1 + ^2 

 Pav. 



Pi PS 



(10-39) 



Fig. 10-53. 



Three-layer case (after 

 Hummel). 



This relation may be extended to 

 cover the case of more than two 

 layers of any thickness and makes 

 possible a graphic approximation in 

 the interpretation of resistivity 

 curves. Fig. 10-53 illustrates the 

 apparent resistivity curve for three 

 layers with resistivities p, pi , and 

 Pi and of thickness h, hi , and <» , 



with h = hi . The shape of the curve for distances large in com- 

 parison with the depths of the upper layers is almost independent of 

 the properties of these upper layers. For large spacings it is asymptotic 

 to the line pjp = 2, since the resistivity of the lower infinite medium is 

 twice that of the surface layer. Curves h and c are the "approximation" 

 curves obtained by combining the upper two layers and considering the 

 resultant layer with the bottom layer as a two-layer problem (curve 6). 

 Similarly, curve c follows by combination of the middle and bottom layer. 



In Fig. 10-54 the middle layer has a high and the bottom layer a low 

 resistivity. This curve is conunonly obtained in water table deter- 

 minations. 



The theoretical curves prove that even for large differences in con- 

 ductivity there is no abrupt change in apparent resistivity measured at the 

 surface. If irregular curves and "breaks" are obtained in the field, they 



»8 Ihid., p. 409-414. 



