It is assumed that magnetic dipoles are useful as a first approximation to the electrical currents in the core that produce the earth's main magnetic field. For simplicity the model is restricted to a central dipole and several additional radial dipoles at equal distances from the center of the earth. A least‐squares method is used to adjust the amplitude, latitude, and longitude of each dipole for a best fit to the observed field components on the earth's surface. In the first of four studies the observed field was the field of the United States 1945 world charts. Originally 11 dipoles, 10 of them at the core‐mantle interface at 0.54 earth radii, were used. Progressively better fits were obtained as the dipoles were placed deeper, and two of the dipoles were eliminated at greater depths. The 29‐parameter, 9‐dipole model, with the radial dipoles at 0.28 earth radii, produced nearly as good a fit to the 1945 field as Vestine's 48 spherical harmonic coefficients. Models were also fitted to the United States 1955 world chart field, to the British Admiralty 1955 world chart field, and to the field synthesized from the Finch‐Leaton spherical harmonic coefficients for 1955. The last model produced the best fit. In all cases the radial dipoles are surprisingly deep and the central dipole is considerably stronger than the centered dipole given by the first three spherical harmonic coefficients. The great depth of the radial dipoles is qualitatively explained by a shielding effect from currents in the mantle and core. The spherical harmonic coefficients from the analyses of Vestine and of Finch and Leaton are compared with the spherical harmonic coefficients computed from the dipole parameters.

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A mathematical model of the geomagnetic field for 1965.0 is presented. The field is given by internal and external spherical harmonic coefficients to degree and order 12. This model was derived from approximately 425,000 measurements made since 1900. An important step in the derivation was the analytic smoothing of data, which produced the regularly spaced grid values used as input to a two‐stage spherical harmonic analysis. Consequently it has been possible to construct the 1965 world magnetic charts (published by the United States Naval Oceanographic Office) from the model, instead of the reverse. On the whole, the smoothed surface field defined by the model (and by the charts exactly drawn from the model) is believed to be at least as faithful to nature as that defined by any previously published model or set of charts. Plans for future analyses include the use of a modified truncation of the series expression for the potential and an allowance for the earth's oblateness.

We have investigated the spherical harmonic analysis (SHA) of scalar intensity F. The method used was the analysis of simulated data, derived by adding pseudorandom noise to surface values (with equiangular spacing) synthesized from existing spherical harmonic coefficients (SHC). An advantage of using simulated data is that this makes possible the computation of vector as well as scalar residuals. In general, similar results were obtained regardless of whether the number of input SHC used in the synthesis of simulated data was less than, equal to, or greater than the number of output SHC. The main result was that the largest vertical intensity residuals occurred in the equatorial region, the foci following the magnetic dip equator rather than the geographic equator. On account of the equiangular spacing, sine colatitude (sinθ ) weighting was usually used in the SHA. Analyses with other weights at θ = 85° and 95° resulted in no improvement. Substitution of vector for scalar data at these colatitudes gave a much better result. This offers some hope that an improved vector model might be obtained from a real data set composed chiefly of F measurements if it incorporated additional vector measurements in selected areas. Complete vector measurements would of course be still better.

A mathematical description of the geomagnetic field for 1970 has been derived by using 360,000 land, sea, and airborne measurements accumulated since 1939. The model is expressed by 168 spherical harmonic coefficients gnm, hnm to m = n = 12. It was used as the basis for the 1970 edition of the U.S. World Charts of Magnetic Variation.