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We have developed an equivalent-source method for performing reduction to the pole and related transforms from magnetic data measured on unevenly spaced stations at different elevations. The equivalent source is composed of points located vertically beneath the measurement stations, and their magnetic properties are chosen in such a way that the reduced-to-the-pole magnetic field generated by them is represented by an inverse-distance Newtonian potential. This function, which attenuates slowly with distance, provides better coverage for discrete data points. The magnetization intensity is determined iteratively until the observed field is fitted within a certain tolerance related to the level of noise; thus, advantages in computer time are gained over the resolution of large systems of equations. In the case of induced magnetization, the iteration converges well for verticalor horizontal inclinations, and results are stable if noise is taken into account properly. However, for a range of intermediate inclinations near 35°, a factor tending to zero makes it necessary to perform the reduction through a two-stage procedure, using an auxiliary magnetization direction, without significantly affecting the speed and stability of the method. The performance of the procedure was tested on a synthetic example based on a field generated on randomly scattered stations by a random set of magnetic dipoles, contaminated with noise, which is reduced to the pole for three different magnetization directions. Results provide a good approximation to the theoretical reduced-to-the-pole field using a one- or a two-stage reduction, showing minor noise artifacts when the direction is nearly horizontal. In a geophysical example with real data, the reduction to the pole was used to correct the estimated magnetization direction that originates an isolated anomaly over Sierra de San Luis, Argentina.

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S U M M A R YDirect Global Positioning System measurement of geoid undulations on accurately levelled stations, usually tens of kilometres apart, can be interpolated by taking advantage of denser surveys of free-air gravity anomalies covering the same area. Using either a spherical or a planar earth model, a two-layer equivalent source is constructed, with the deepest masses located under the geoid stations and the shallower ones under the gravity stations, in such a way that the effect of the masses fits simultaneously, with different precisions, the anomalous potential related to the geoid and its vertical gradient or gravity anomaly. This poses a linear Bayesian problem, whose associated system of equations can be solved directly or by iterative procedures.The ability of the described method to predict the geoid elevation over the gravity stations is assessed in a synthetic example; and in the application to a real case, a gravity-enhanced geoid is mapped for an area of Buenos Aires province, Argentina, where local features are put in evidence.