A new technique for recovering magnetic field maps that describe two-dimensional, coherent field structures observed in space is documented, benchmarked, and then applied to four magnetopause crossings by the spacecraft AMPTE/IRM (Active Magnetospheric Particle Tracer Explorers/Ion Release Module) in which the basic observed signatures were those associated with a tangential discontinuity. The calculations required for the recovery consist of the numerical solution of the Grad-Shafranov equation, using as initial values magnetic field and plasma data collected by a single spacecraft along a straight-line trajectory, produced when structures are convected past it. The integration proceeds in small spatial steps in both directions away from the trajectory. The integration domain, which is rectangular, is limited in the transverse direction by the appearance of numerically generated singularities. Nevertheless, the method offers a substantial field of view of the region surrounding the trajectory, within which the accuracy is a few percent. For the magnetopause events examined, it is found that the simple tangential-discontinuity structure is modified by embedded strings of magnetic islands, separated by X-type nulls in the transverse field. These configurations are interpreted as being the result of the tearing mode after it has reached its saturated state. Two or more islands contained within larger islands are observed, indicating that, during the active phase of the tearing mode, the reconnection rate was not the same at all X points. The possibility exists that one dominant X point produces a pair of narrow channels of open flux, connecting the magnetosphere to the magnetosheath. Even without such open flux, the presence of the islands should allow flow of plasma along magnetic field lines, from the outermost (magnetosheath) to the innermost (magnetosphere) parts of the magnetopause current layer, thus facilitating the overall plasma transport across the layer.