A new method is proposed to derive from data magnetospheric magnetic field configurations without any a priori assumptions on the geometry of electric currents. The approach utilizes large sets of archived satellite data and uses an advanced technique to represent the field as a sum of toroidal and poloidal parts, whose generating potentials Ψ1 and Ψ2 are expanded into series of radial basis functions (RBFs) with their nodes regularly distributed over the 3-D modeling domain. The method was tested by reconstructing the inner and high-latitude field within geocentric distances up to 12 RE on the basis of magnetometer data of Geotail, Polar, Cluster, Time History of Events and Macroscale Interactions during Substorms, and Van Allen space probes, taken during 1995–2015. Four characteristic states of the magnetosphere before and during a disturbance have been modeled: a quiet prestorm period, storm deepening phase with progressively decreasing SYM-H index, the storm maximum around the negative peak of SYM-H, and the recovery phase. Fitting the RBF model to data faithfully resolved contributions to the total magnetic field from all principal sources, including the westward and eastward ring current, the tail current, diamagnetic currents associated with the polar cusps, and the large-scale effect of the field-aligned currents. For two main phase conditions, the model field exhibited a strong dawn-dusk asymmetry of the low-latitude magnetic depression, extending to low altitudes and partly spreading sunward from the terminator plane in the dusk sector. The RBF model was found to resolve even finer details, such as the bifurcation of the innermost tail current. The method can be further developed into a powerful tool for data-based studies of the magnetospheric currents.