hypothetical locations A-I (cf. Fig. 4) under the additional assumption that these equidistant locations are located on the surface of a sphere. Whereas the most flexible splines of m = 2 and m = 3 (red and green lines) directly or approximately intersect with the observed data points (green circles), this is not necessarily the case for splines of intermediate (m = 4, blue) or reduced (m = 5, orange) flexibility. At the same time, less flexible splines provide increasingly smoother estimates, and all spline interpolations provide estimates for intermediate locations and those beyond the borders of the array. Consequently, the (negative) second spatial derivatives corresponding to these spherical spline interpolations (Fig. 7, bottom) represent continuous surface Laplacian estimates, which provide appropriate gradient transitions across locations, being sharper and enhanced for more flexible splines but smoother and more gradual for less flexible splines. The shapes of these spline-based surface Laplacian estimates (with the exception of m = 5) are similar to that of the discrete second spatial derivative (Fig. 4), and all spline-based surface Laplacian estimates are also reference-free (i.e., invariant to the addition of a constant to the data series).
