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Appendix: Modifications of APL Global Convection Mapping Technique

Two modifications have been made to the mapping technique of Ruohoniemi and Baker [1998]. The fitted convection patterns now produced by the APL group, including those presented in this paper, incorporate these modifications unless otherwise specified. Here we describe the changes and the reasons for adopting them.

The first concerns the specification of the equatorward boundary of the convection zone. In Ruohoniemi and Baker [1998] this boundary, referred to here as $\Lambda^{\sf circ}$, was taken to be a circle of constant invariant latitude. We use the value of $\Lambda^{\sf circ}$ at midnight MLT, $\Lambda^{\sf circ}_0 \equiv \Lambda^{\sf circ} (0\ {\rm MLT})$, to identify this circular boundary. The actual value of $\Lambda^{\sf circ}_0$ was either set to a constant (usually 60$^\circ$) or varied from scan to scan to accommodate the varying size of the convection zone. The equatorward boundary was specified as a circle in the derivation of the statistical convection model of Ruohoniemi and Greenwald [1996] and this has been the accepted convention in studies of this kind. However, we have found that the SuperDARN data invariably indicate that the convection boundary is located at higher latitudes on the dayside than on the nightside. The definition of the boundary used in the fitting should reflect this character. As a practical matter, we have found that using a circular boundary that accommodates the flow on the nightside allows the flow on the dayside to extend to unrealistically low latitudes.

The solution is to introduce an MLT dependence in the specification of the boundary latitude. We have explored several options for the boundary, including a global shifting of the coordinate flows so that the convection pattern is centered on a point 4$^\circ$ equatorward of the magnetic pole towards the nightside. In the end, we adopted a solution based on the work of Heppner and Maynard [1987]. These authors studied the dependence of the latitude of the boundary on MLT and geomagnetic activity level using electric field measurements from the DE satellite. Their Figure 10 indicates that this boundary, referred to here as $\Lambda^{\sf HMB}$, has a similar shape through all activity levels, but expands and contracts. Figure A1 shows several examples of $\Lambda^{\sf HMB}$ boundaries for varying sizes of the convection zone. The boundary is almost circular on the nightside but rises steeply in latitude after crossing the dawn and dusk meridians, reaching a highest latitude near the 11 MLT meridian.We characterize the size of the boundary by referring to the latitude of its intercept with the midnight MLT meridian, $\Lambda^{\sf HMB}_0 \equiv
\Lambda^{\sf HMB}(0\ {\rm MLT})$. For a given scan, a value of $\Lambda^{\sf HMB}_0$ is determined such that all the significant convection observations are contained within the boundary. Some freedom remains in choosing the limit to what defines significant flow, but 100 m/s is a typical value.

Figure A1: Examples of lower latitude convection boundaries used in the improved fitting algorithm of Ruohoniemi and Baker [1998]. Four boundaries, $\Lambda^{\sf HMB}$, based on the work of Heppner and Maynard [1987] are shown for midnight MLT latitude intersections of $\Lambda^{\sf HMB}_0 =
58^\circ, 62^\circ, 66^\circ$, and $70^\circ$. The thick curve indicates the $\Lambda^{\sf HMB}$ boundary used in the fittings presented in this paper. The dashed line shows the circular boundary $\Lambda^{\sf circ}$, defined by $\Lambda^{\sf circ}_0 = 62^\circ$, used in previous studies [ Ruohoniemi and Greenwald, 1998] are shown.

The spherical harmonic fitting continues to be performed over the entire region poleward of $\Lambda^{\sf circ}_0$, where numerically $\Lambda^{\sf circ}_0 = \Lambda^{\sf HMB}_0$. Grid cells in the crescent-shaped region on the dayside located between the latitude of $\Lambda^{\sf circ}$ and $\Lambda^{\sf HMB}$ (see Figure A1) are padded with zero velocity vectors. Our implementation of $\Lambda^{\sf HMB}$ allows some small, but non-zero, potential contours to exist equatorward of the HMB boundary (see for example, Figure 3). The resulting patterns more closely reproduce the character of convection at lower latitudes.

A consequence of using the lower latitude convection boundary defined by $\Lambda^{\sf HMB}$ is a minor difference in the statistical model data used in the fitting. A careful inspection of Figure 4b reveals subtle differences from the statistical model of Ruohoniemi and Greenwald [1996]. The patterns derived by of Ruohoniemi and Greenwald [1996] were fitted using a circular lower latitude convection boundary ($\Lambda_0$) corresponding to 60$^\circ\Lambda$. When the fitting technique of Ruohoniemi and Baker [1998]. is used, $\Lambda_0$ is determined by the extent of the LOS observations and the statistical model is scaled accordingly, thus modifying the pattern somewhat.

The second improvement over the original fitting technique described by Ruohoniemi and Baker [1998] is a weighting scheme which reduces the tendency for the statistical model data to dominate the global solution of $\Phi$ at higher order fittings. By the order of the fitting we refer back to the expansion of $\Phi$ used by Ruohoniemi and Baker [1998] in terms of spherical harmonic functions of order L and degree M. The values of L and M determine the resulting spatial filtering of the velocity data.

The original technique of Ruohoniemi and Baker [1998] weighted the statistical model data relative to the LOS velocity data without regard to the order of the fitting. The weight assigned to a velocity value from the model was set to the geometric mean of the radar velocity measurements. Fittings at higher order required progressively more statistical model vectors ($\propto L^2$) in order to constrain the behavior of the greater number of terms in the spherical harmonic expansion over the regions of no radar observations. A fixed weighting scheme, therefore, causes the solution to be more influenced by the statistical data at higher orders. The consequence was an undesirable compromise in selecting the order of the fitting at a level high enough to adequately reproduce finer-scaled features in areas where data were present, but low enough so the statistical model didn't dominate the solution.

An improvement has been made whereby the weight of the statistical data is adjusted according to the order of the fitting. The result is that the degree to which the model vectors affect the solution is less dependent on the order of the fit. In essence, the weight defined above as the geometric mean of the radar velocity measurements is reduced by the factor 42/L2. The effect is to roughly equalize the contribution of the statistical model to the fitting solution for $L \ge 4$ (fittings of order L < 4 are not considered useful). The cost associated with this progressive de-weighting might be the emergence of erratic behavior in the patterns over the areas of no radar observations. However, we have found the results to be quite satisfactory with these settings, at least up to fittings of order L = 10. Higher order fittings can now be used to reproduce smaller-scale features described by the LOS measurements and the statistical model data simply guide the solution realistically in regions where no radar data are available.

Figure A2 shows residual potentials for varying orders of fit (L = 4,6,8,10 from top to bottom) for both the fixed and varied (left and right column) weighting schemes. Inspection of the residuals in the left column of Figure A2 shows the problem of using a high order fit with the fixed weighting scheme of Ruohoniemi and Baker [1998]. Increasingly large (>12 kV) residual contours are evident as the order is increased. In this example, the increasing fitting order is inadvertently causing the global solution to converge on the solution defined by the statistical model.

Figure A2: Residual electrostatic potential contours for order L = 4, 6, 8, and 10 fittings (top to bottom) using fixed and adjusted weights (left and right columns) for the statistical velocity data. The two APL models used to obtain the residuals correspond to IMF orientation Bz-/By- and Bz+ with IMF magnitude 6$\le$BT$\le$12 nT for both fittings.

The right column of Figure A2 reveals a slight trend toward larger residuals with increasing fitting order, implying that the statistical data is influencing the solution to a greater extent in the higher order fittings. However, in regions where LOS velocity data are present (e.g., 15-18 MLT and 75-85 $^\circ\Lambda$), the maximum deviation in the residual is $\le\sim$3 kV. Such minor change in the residual contours from order 4 to 10 demonstrates that the improved weighting scheme allows the fitting technique to be virtually independent of order in regions were velocity data are present. Larger residual deviations (up to $\sim$10 kV) are seen in the postmidnight-dawn region where no velocity measurements exist. Such a trend is expected at higher orders. The additional statistical data used in the higher order fittings causes the differences between model patterns to become more evident in locations where no data exists.

Contrasting the right column in Figure A2 with the left column shows the marked improvement the adjusted weighting scheme has on reducing the impact of the statistical model at higher orders. Smaller-scale features present in the velocity data can now be reproduced in the global patterns without significant influence from the statistical model data.

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Next: Acknowledgments Up: Electrostatic potential patterns in Previous: Summary

Simon Shepherd 2000-07-13