Background

Many techniques have been developed to infer the instantaneous state of the electrostatic potential in the high-latitude ionosphere. One common method utilizes electric field measurements from drift meters on low alititude satellites such as OGO 6, DE 2, DMSP to construct synoptic maps of $\Phi$ [Heppner, 1977,Heppner and Maynard, 1987,Weimer, 1995]. A drawback to studies of this type is the limited spatial extent of the measurements. Multiple satellite passes are required to produce synoptic or global patterns which, as a result, become statistical or averaged in nature. To estimate $\Phi$ over the entire high-latitude ionosphere based on a single satellite pass requires either numerous assumptions about the convection at large distances from the satellite track or the liberal use of statistical data.

A procedure called the assimilative mapping of ionospheric electrodynamics (AMIE) technique was developed to overcome such limitations by incorporating a variety of different types of observations. Direct measurements of the convection velocity from radars or satellites are combined with indirect measurements of magnetic variations from magnetometers or satellites to fabricate global maps of electrostatic potential [Richmond and Kamide, 1988]. While this technique is widely used for the purpose of constructing maps of $\Phi$ over the entire high-latitude region, uncertainties in the specification of ionospheric conductances, necessary in the inversion of magnetograms, can greatly affect the solution. The reliance on magnetometers has been due to the availability of these data over large areas. Recently, it has become possible to base the global solution of $\Phi$ more on direct measurements of convection velocity.

Ruohoniemi and Baker [1998] presented a technique comparable to AMIE that is tailored to direct measurements of convection from HF radars. In their approach, line of sight (LOS) Doppler velocities from the Super Dual Auroral Radar Network (SuperDARN) are fitted to an expansion of $\Phi$ in terms of spherical harmonic functions. The LOS Doppler velocity measurements of the ionospheric convection velocity provided by SuperDARN are augmented by additional velocity vectors from a statistical model to constrain the solution in regions where no SuperDARN data are available. The statistical model currently used is the Applied Physics Laboratory (APL) convection model, which was derived from nearly six years of HF radar observations [Ruohoniemi and Greenwald, 1996]. The necessity of using statistical data was discussed by Ruohoniemi and Baker [1998], and they point out that any model could be used.

The need for statistical model data can be understood from the following considerations. A best-fit global solution for $\Phi$ could indeed be determined from a set of localized radar velocity measurements. The solution would be optimal in the sense that the differences between the measured velocities and those implied by the fitting are minimized in a least-squares sense. The physical expression of the solution is a set of coefficients for the terms of the spherical harmonic expansion of $\Phi$. Over the area of measurements the values of the coefficients are constrained in such a way as to reproduce the observations. Outside of this area no constraints exist and straightforward application of the set of coefficients will lead, in general, to wildly unrealistic results for $\Phi$. If a plausible global solution is required the fitting must be suitably constrained over the outlying areas.

In the fitting algorithm of Ruohoniemi and Baker [1998], a pattern from the statistical model is sampled for velocity values that bound the values of the coefficients in the spherical harmonic expansion of $\Phi$. In this way, the solution for $\Phi$ beyond the area of radar observations is effectively constrained to realistic values. To increase realism, the selection of model data is keyed to the prevailing IMF conditions at the magnetopause. The fitting with model data is of course somewhat less optimal in terms of reproducing the direct measurements of convection velocity. The mapping of $\Phi$ and the determination of $\Phi_{\sf PC}$ will be more sensitive to the statistical model contribution when coverage of the measurements is not sufficient to fix the total potential variation. For example, when the coverage spans the dusk sector, $\Phi$ will be in reasonable agreement with the observations in the dusk convection cell while $\Phi$ will be determined mostly by the model pattern in the dawn cell. The solution of $\Phi$ will thus be undesirably dependent on the choice of statistical model. This situation preserves the uncertainty characteristic of earlier studies of $\Phi$ and $\Phi_{\sf PC}$.