V. SMOOTHING In this section, we investigate the smoothing of numer- ical noise that results from the use of the surface fitting techniques described in Section III. Consider computing the on-axis gradients C [ n ] m,α from a grid of numerical field values B using a circular or elliptical cylinder, as de- scribed in the previous sections. Observe that (22) and (55) are linear in each of the values B$_{x}$ and B$_{y}$ at the grid points. The inclusion of additive numerical noise Δ B at each grid point therefore results in on-axis gradients of the form C [ n ] m,α + Δ C [ n ] $_{m,α}$, where the contribution Δ C [ n ] m,α is determined by the values Δ B according to the proce- dures of Section III. Note that only the field values Δ B$_{x}$ and Δ B$_{y}$ at grid points near the surface of the circular cylinder affect the functions Δ C [ n ] $_{m,α}$. To examine the effect of additive noise, we gener- ate a random noise field Δ B whose components are proportional, at the 1% level, to the strength of the monopole-doublet on-axis vertical field. Consider the grid [ − 4 . 4 , 4 . 4] × [ − 2 . 4 , 2 . 4] × [ − 300 , 300] cm used in Sec- tion IV for fitting the field of the monopole doublet, with mesh points indexed by j = 1 , · · · N . Let B$_{y}$ (0 , 0 , z ) de- note the value of the monopole-doublet on-axis vertical field at longitudinal location z , as determined from (59). At each point ( x$_{j}$, y$_{j}$, z$_{j}$ ) we set according to (61). Figs. 19-20 illustrate the on-axis gra- dients C [6] 1 ,c and C [0] 7 ,c as computed using these field values. The solid line in Fig. 19 illustrates the rms value of the on-axis gradient C [6] 1 $_{,c}$, as computed using a circular cylin- der of radius R = 2 cm according to (22). The dashed line in Fig. 19 illustrates the rms value of the on-axis gradient C [6] 1 $_{,c}$, as computed using an elliptical cylinder of semiminor axis 2 cm and semimajor axis 4 cm according to (55). In Fig. 20, similar results are shown for the on-axis gradient C [0] 7 $_{,c}$. The attentive reader might wonder why we have dis- played the C [ n ] m,c for noise while the the monopole doublet field is governed by the gradients C [ n ] $_{m,s}$. The reason is that the C [ n ] m,s are produced by fields that are predominantly in the vertical ( y ) direction, and for such fields there is rel- atively less difference between circular and elliptical sur- face fitting because the semi-minor axis of the elliptical cylinder is the same as the radius of the circular cylinder. Note also that in both cases only the component of the field normal to the surface is used. Horizontal fields drive primarily the C [ n ] $_{m,c}$. However, in the horizontal direction, the semi-major axis of the elliptical cylinder is substan- tially larger than the radius of the circular cylinder. We therefore expect the advantage of using elliptical cylin- ders compared to circular cylinders will be most apparent in the C [ n ] $_{m,c}$. Indeed, this is what Figs. 19 and 20 show. The effects of errors in the surface data are suppressed more when the bounding surface is farther away from the field observation point. As a result of this suppression, it is advantageous to use a fitting surface that is as far away as possible. Suppose we compare the C [ n ] m,c due to noise and shown in Figs. 19 and 20 with the corresponding C [ n ] m,s in Figs. 12 and 13. This is reasonable because a horizontal monopole doublet would produce C [ n ] m,c analogous to the C [ n ] m,s shown in Figs. 12 and 13. Remarkably, we find that, as a result of smoothing, the effect of 1% noise in the field data produces, on average, only on the order of 0.01% changes in the on-axis gradients. Here the δ$_{x}$ ( j ) and δ$_{y}$ ( j ) are uniformly distributed ran- dom variables taking values in the interval [ − 1 , 1], and ϵ = 0 . 01. After interpolating these values onto the sur- face of a circular cylinder with R = 2 cm and z ∈ [ − 300 , 300] cm, we use the procedure described in Section IIIA to compute the on-axis gradients (22). In Fig. 15, we have displayed the computed quan- tity ˜ b$_{m,s}$ ( R, k ) appearing in (17) for the case m = 1. It is a function of the spatial frequency k , having ran- dom variations of approximately uniform variance over the interval [ − 20 , 20] cm − $^{1}$. Fig. 16 displays the kernel k m − $^{1}$/I ′ $_{m}$( kR ) multiplying ˜ b$_{m,s}$ ( R, k ) in (22) for the case m = 1 and R = 2 cm. Note the rapid decay of this func- tion for large | k | . Finally, Fig. 17 displays the product of these functions, illustrating the dramatic suppression of high- k contributions to the Fourier integral appearing in (22). A similar phenomenon occurs when fitting is performed using an elliptical cylinder. In this case, a sequence of kernels contributes to (55) for each fixed m . In Fig. 18 we have displayed the kernels contributing to the case m = 1, with x$_{max}$ = 4 cm and y$_{max}$ = 2 cm. Kernels take their maxima at k = 0, and these maxima decrease monotonically with increasing index r . All kernels de- crease rapidly with increasing | k | [3, 4]. To study the effect of noise on the on-axis gradients, twelve distinct random fields were generated on a mesh A less stringent test of the accuracy of surface meth- ods is that the magnetic field, as computed from surface data using field values on a 3-dimensional mesh, should reproduce the magnetic field at the interior mesh points. (This is also a test of the quality of the magnetic data on the mesh.) We computed such an interior fit, and the associated transfer map, for the modified CESR-c design of the Cornell wiggler, which has been adopted as the design prototype for use in International Linear Collider studies [9, 10]. Cornell provided data obtained from the VI. APPLICATIONS A. ILC Damping Ring Wiggler Fields Δ B$_{x}$ ( x$_{j}$, y$_{j}$, z$_{j}$ ) = ϵB$_{y}$ (0 , 0 , z$_{j}$ ) δ$_{x}$ ( j ) , (61a) Δ B$_{y}$ ( x$_{j}$, y$_{j}$, z$_{j}$ ) = ϵB$_{y}$ (0 , 0 , z$_{j}$ ) δ$_{y}$ ( j ) . (61b) 13