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
[*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
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
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
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
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 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 . 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 . 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 .
In this way,
the solution for
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
and the determination of
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,
will be in
reasonable agreement with the observations in the dusk convection cell while
will be determined mostly by the model pattern in the dawn cell. The
solution of
will thus be undesirably dependent on the choice of
statistical model. This situation preserves the uncertainty characteristic of
earlier studies of
and .