lower loop 



e 



upper loop 



FIG. 3. Plots of F,(-y) from a numerical integration of Eq. (113). Re -Fih") is proportional to the loop contributions towards <X^ >. 

 Re F, (0) = i^ corresponds to Eq. (96) for a single upward (S positive) or downward (S negative) loop. The short-dashed curve 

 gives the apex approximation [Eq. (101)1 to F\.' 



A=n(Ar-r)/a),„{R , 



where {r, 2) is the position of the apex and (R is the ray 

 curvature at the apex. The function /i (A) cuts off rapid- 

 ly with increasing A, and therefore we can write, ap- 

 proximately, 



Fi(fl) = /•,(«) 



4 w' 1 r 



/i(A)rf.r = 



2(R«2 



(101) 



times the number of upward loops. 



For a single complete upward (downward) loop with 

 range R'{R') we denote Fi(R') simply by f }. Then for 

 a complete double loop, with range R'', we have 



F{-=Fi(Rn = Fi + Fl . 



If the double loop has an apex near the surface, then 

 Fl = Fi and Fi = 0; thus 



The results of a numerical evaluation of Fl and FJ 

 as a function of ray angle "5 are shown in Fig. 3, as is 

 the apex approximation ^1, which is seen to be an ex- 

 cellent approximation for 9t5° . The largest value of 

 Fl occurs at a ray angle 6 near 2°, and a corresponding 



apex depth of 750 m; deeper rays are reduced by the 

 smaller value of n', rays of shallower apex are reduced 

 by a smaller radius of curvature (R. 



We shall also need the variance of i = dX/dt; this is 

 found by inserting w^ in Eq. (66) under the integral sign. 

 The result is 



(\x\) = oj,,no({5C/C)l){j-')q''BRF2, 



where 



(102) 



Btt-V]" dx sec' 9,,' log {^\ ma) (103) 



and 



fzi^) 



(log^-i 



log 



1 



(i+a7_^+i 



■2(1+a2)'^2'°S (1+A2)>/2 



w 



log 



(104) 



For the axial ray (where/2 = l) and near surface rays 

 we have, respectively, 



F2 = 8i7-'(n/«„)'(fl/fl)logw/co.„ = 0.132i?/S , (105) 



Fi = arHI T,)"' (B\(Si\f 'Hry' {h/no)'iog{h/ioj . (loe) 



225 



