THERMAL METHODS 979 



measurements made intermittently throughout the year, or daily, at the 

 same time of day. 



The annual variation can be observed to a depth of between 75 and 

 100 feet, depending on the thermal properties of the rocks. The periodic 

 variations in temperature may show local fluctuations due to changes in 

 meteorological conditions and to differences in topography and overburden. 



Below 100 feet, the temperature depends, in general, only on the flow 

 of heat from the center of the earth. 



Diurnal variations (when they have not been excluded from measure- 

 ments by a sufficient depth of bore hole) are corrected for in a manner 

 very similar to that used in correcting for the diurnal magnetic variations, 

 (see Chapter III), or by calculation, as described in the following para- 

 graph. 



Calculation for Periodic Temperature Variations 



The theory to be applied in this case rests in Equation 5, the general equation 

 excluding internal sources or sinks. 



-^v==r = ^ (5) 



ca ot ^ 



The one-dimensional form of Equation 5 may be used without introducing any 

 appreciable error. This impHes the assumption that the surface of the earth is a plane 

 in the region of investigation.* The surface is thus uniformly radiated, leaving the 

 temperature a function of the depth s only. 



k ?fT _3T 

 ca -dz" dt ^^^^ 



Since the radiation of the sun is roughly periodic, the change in heat flow through 

 points close to the surface of the earth is likewise periodic in both diurnal and annual 

 variations. At the surface, then, the following periodic boundary condition must be 

 imposed. 



T=. To sin wt at 2 = (12) 



Equation 5a is linear and homogeneous, and a particular solution has the form, 



T = Ae'"*^' (13) 



which, by substitution into Equation 5a, is a solution if, and only if, 



a z=. /3 



ca 



With proper substitutions,! and noting that Equation 13 is a particular solution, 

 a total of four particular solutions may be written as follows : 



T = Be-''sm(yt-R) (14) 



T = B'e'' sin (yt + R) (15) 



T = Cc-"" cos (yt - R) (16) 



T = Ce'' cos (yt + R) (17) 



* In areas of rugged topography and complicated geology, precise mathematical treatment is 

 almost impossible, and the interpreter must then depend upon his knowledge of probable subsurface 

 geology and prior experience. 



t The discussion follows that given by Ingersoll and Zobel, "Mathematical Theory of Heat 

 Conduction," Ginn and Co. 



