ELECTRICAL PROBING WITH AN APPARATUS IN WATER 177 



The obtained expression is basic, being the most general integral expression 

 of potential for an arbitrary position of the current source and the point 

 of observation ^vithin the limits of the upper layer. 



When working with the method of electrical sonde, the apparatus is 

 always placed horizontally, consequently in formula (13) ^ve can put z — 0: 



oo 



^o--^ + j(l I [e-^'""" + 2^1 + 5i (e2'«- + e-2"^''o)] /„ (mrd) m. (14) 







Let us suppose that the thicknesses of the individual layers Jiq, h^, h^, ..., 

 ^^_j have a general measurement, which we can represent by h such that 



n-l 



S' 



i =0 



K =PqK hQ+h^ = PiK hQ+h^ + h^ =p^h, ... , 2 /',• =/>„-! ^'' 



where Po,Pi,P2i •••■>Pn-i ^^e whole numbers. 

 The function 



Pnig) 



^1 



QnigY 



where g = e~^"^^ is a rational fraction which can be split up into a series in 

 increasing powers of g by dividing the polynomial in the numerator by the 

 polynomial in the denominator. In a general form we have 



^1 = 2 ?n 



n = l " 



-2mnh 



where q^ = the coefficients of emission. 



Substituting the expression for the coefficient 5^ in formula (14) we obtain 



UQ=-^+-^q j e-^""'" lQ{mr) dm+ qj^ qn e'^""'^ Io{mr) dm + 



2r 



+ y ^ V 5n /e-2mnft ^Q^mzo + e-^'"^") ^mr) dm, (15) 



n = l J 



Using the Weber-Lip shitz formula, we obtain an expression mth which 

 it is possible to calculate directly the potential at points on one level with 



Applied geophysics 12 



