 0 -20 -15 -10 -5  0  5  10  15  20 k (cm$^{-1}$) 1.6 x 10 -4 1.4 x 10 -4 1.2 x 10 -4 1.0 x 10 -4 8.0 x 10 -5 6.0 x 10 -5 4.0 x 10 -5 2.0 x 10 -5 -2.0 x 10 -5 -4.0 x 10 -5 -6.0 x 10 -5 T e xt b 1,s ( R,k ) k r=1 r=3 r=5 r=7 r=9 r=11  0.5  0.4 0.3  0.2  0.1 -0.1 -0.2 -0.3 -4 0 -2 0 2 4 kernel !"# $ % & ’ (  0  0.0001  0.0002  0.0003  0.0004  0.0005  0.0006 -20 -15 -10 -5  0  5  10  15  20 RMS Value of C [6] 1,c (z) using 12 Seeds z (cm) -10 -7.5 -5 -2.5 2.5 5 7.5 $_{10}$k 0.25 0.5 0.75 1 1.25 1.5 1.75 2 !"#$% &’ (" )" !*+,*-" FIG. 18: The kernels k $^{m}$βr$_{m}$( k ) / Se $^{′}$$_{r}$( U, q ) for the case m = 1 and r = 1 , 3 , 5 , 7 , 9 , 11, as a function of k , with q and k related by (40c). FIG. 15: The quantity ˜ b$_{1}$$_{,s}$ ( R, k ) computed from uniform random noise (61) using a circular cylinder with R = 2 cm. FIG. 19: The function C [6] 1 ,c as computed from a mesh contain- ing random noise (61) at each mesh point. Damping of this noise illustrates the effect of smoothing. (Solid line) Result obtained using a circular cylinder with R = 2 cm. (Dashed line) Result obtained using an elliptical cylinder with semimi- nor axis of 2 cm and a semimajor axis of 4 cm. FIG. 16: The kernel k m − $^{1}$/I ′ $_{m}$( kR ) as a function of k for the case m = 1 and R = 2 cm. FIG. 17: The product k m − 1 ˜ b$_{m,s}$ ( R, k ) /I ′ m appearing in (22), as computed from uniform random noise (61) with m = 1 and R = 2 cm.  0 -20 -15 -10 -5  0  5  10  15  20 Product x -4 x -4 x -4 x -4 x -5 x -5 k (cm$^{-1}$) 2.5 10 2.0 10  1.5 10 1.0 10 5.0 10 -5.0 10 3-dimensional finite element modeling code OPERA-3d for the field components B$_{x}$ , B$_{y}$ , and B$_{z}$ on a mesh of spacing 0 . 4 × 0 . 2 × 0 . 2 cm in a volume 10 . 4 × 5 . 2 × 480 cm, extending beyond the fringe-field region. The field components are provided to a precision of 0.1 G rela- tive to a peak field of 16.7 kG. An elliptic cylinder with semimajor axis 4 . 4 cm and semiminor axis 2 . 4 cm was placed in the domain of the data, and the field on the elliptic cylinder boundary was constructed using bicubic interpolation. See Figs. 3 and 5. The interior field was computed using the on-axis gra- dients through terms of degree 6 in x, y over the do- main of the original data. This solution for the interior field was then compared to the original data at each grid point. Fig. 21 displays the vertical field B$_{y}$ off-axis at 14