A new technique, the method of spherical cap harmonic analysis, has provided the means to construct an analytical model of the vertical field above 40¿N from Magsat data. With this technique, proper consideration can be taken of altitude variaton and of the source-free constraint that the vertical field be the vertical derivative of a potential function. Maps can be produced at any altitude. Here they are given at the lower and upper limits of the Magsat satellite, 6700 and 6900 km radial distance or approximately 335 and 535 km geodetic altitude. A novel decimation procedure, applied to NASA ''Investigator B'' data screened for low disturbance levels, results in an approximately uniform distribution in area, about 1 point per (150 km)2. Each of the 5087 data points represents an average over 5 s of time or 37 km of satellite track. The harmonic model, of maximum index 22, includes only those basis functions that are necessary for fitting a potential which need not be differentiable in colatitude and that are therefore completely orthogonal over the spherical cap. Although there are 276 coefficients in the theoretical model, only 190 were found to be statistically significant. The resulting scatter, or standard error of estimate, was 3.6 nT.