564 SEISMIC METHODS [Chap. 9 



or 



where tm is the mean time in the mean distance Xm . 



In this equation the depth z (normal to the bed) appears. It may be 

 obtained from eq. (9-77rf) with its correct value or be calculated from the 

 mean time under the assumption of horizontal bedding. For many dip 

 calculations, other approximations are satisfactory. One of these is to 

 draw a travel-time curve and to extend it toward the shot point. If the 

 shot-point travel time thus obtained is ^o = 2z/v, eq. (9-78a) becomes 



sin^ = ^.^-^. (9-786) 



to Ax vto 



In further approximation, let 2z = IR = vtm (Fig. 9-86). Then (9-78a) 

 becomes 



sin,p = v.--^. (9-78c) 



Ax Vtm 



Finally, the last term of this expression may be dropped so that 



sin (^ = — .V (approx.). (9-78d) 



Ax 



Since At/ Ax is equal to the reciprocal of the apparent velocity, eq. 

 (9-78c?) is identical with eq. (9-62), and the angle of incidence at the 

 surface, io (at R in Fig. 9-86), is assumed to be equal to the angle of dip. 

 For vertical incidence upon the bed (distances close to shot point) formula 

 (9-78d) is rigorous. 



The apparent up-dip and down-dip velocities of reflection impulses may 

 be obtained from a differentiation of the up-dip and down-dip travel times 

 given by eqs. (9-7 5a) and (9-756), so that the down-dip gradient (D.G.) 

 is given by 



^ = , f ^i"^ + ^^ ^ ^ D.G., (9-79a) 



dxd yViz^ + 4:ZXd sm cp -\- xl 



and the up-dip gradient (U.G.) is 



dtu — 2z sin (p -{- Xu 



dxu yV4:Z^ - 4zxu sin <p -|- xl 

 Hence, 



= U.G. (9-796) 



dtd _ 2z sin (p -{- xa , dtu _ —1z sin <p -\- Xu cq_7Q 'j 



dxd y^td dxu y^tu 



