334 



EXPLORATION GEOPHYSICS 



where G is the gravitational constant. {G equals 6.68 X 10^^ in the centi- 

 meter-gram-second system of units.) The total gradient in the x direction 

 due to the terrain is 



^^ = 3G 



'dx'dz 



m 



r^COS g {li — z) 



da dr dz 



z^ — 2hz 

 Because ^ ^ ^^ small, only a small error is introduced by writing the 



expression for the gradient in the form 





''i f- 



r^ cos a (JlZnr ~ y2Znr'^ ) 



(r2 + A2)5/2 



dadr 



where Znr is the value of z for a circle of radius r. Also, if the differences 

 in height are small, the term y2Znr^ is small relative to hznr, and the ex- 



pression for the gradient becomes 



'dx'ds 



3(7(7 



i 



-Vl I Zn 



COS a da dr 



or 





— OTrCnGcr 



i 



(r2 + /l2)5/2 



(r2 + /j2)5/2 



dr 



(95) 



In similar manner, it can be shown that <:„ is the only Fourier coefficient 



that must be known to determine 

 o27y 

 — and that d„ and €„ are the 



dyd2 



only Fourier coefficients that 

 must be known to determine the 

 curvature quantity. 



In practical field work, it is 

 impossible to specify ^ as a con- 

 tinuous function of a. Instead, 

 Schweydar utilized 8 azimuths 

 (0°, 45°, 90°, etc.). (For very 

 rugged terrain, 16 or more azi- 

 muths may be used.) If the level- 

 ing is done in eight azimuths (0°, 

 45°, etc.) and if the differences 

 in height on any circle are desig- 

 " """Jt.ft '". """ nated as zi, z^, Zz, etc. (Figure 



SCALE ~mrftrj ' ^> ^> \ O 



187), substitution in the Fourier 



Fig. 187. — Layout for terrain leveling. (After . /r>o\ • 1 j 



Schweydar. Zeitschrift fur Geophysik.) equation (9Z) yields : 



