Description of basis functions#
Our approach of reconstructing the GDF from SPAN-Ai measurements involves two steps:
Estimating the gyro-centroid \(\boldsymbol{v}_{\mathrm{gyro}}\) through which the magnetic field \(\boldsymbol{B}\) is anchored.
Inferring the gyrotropic distribution which best matches the FAC grids projected in the \((v_{\parallel},\, v_{\perp})\) plane.
The first step uses a Bayesian framework, which requires using a faster Slepian functions generation algorithm. This is why we use a 1D Slepian on-a-polar-cap (in angular space \(\theta\)) along with 1D cubic B-splines in radial space \(r = \sqrt{v_{\parallel}^2 + v_{\perp}^2}\). The overall setup for the polar cap fitting method is shown in Fig. 3.
Fig. 3 Fitting architecture when using 1D Slepians on a polar cap. Panel (A) shows the distribution of grid points in black after rotating to FAC followed by boosting the frame to induce a maximum angle of \(\Theta\) to the grid points about the origin. The set of B-splines corresponding to the SPAN-Ai grids projected into FAC is shown in panel (B). The local support in radius for the red B-spline highlighted is shown as the shaded red arc in panel (A). All grid points which encompassed by this B-spline is marked with a larger size. Panel (C) shows the Slepian functions optimally concentrated within angle \(\Theta\) about the gyroaxis. The distribution of the grid points within the red arc of panel (A) is overplotted on each of the Slepian functions \(S_{\alpha}(\theta)\). The vertical red dashed line marks \(\Theta\) used for the polar cap shown in panel (A).#
Radial knots for cubic B-splines#
The domain is discretized as a function of velocity \(v\) and polar angle \(\theta\). We use cubic B-splines with local supports at knots, which are computed for each timestamp, to render localized support on different velocity shells. The knots are placed logarithmically with a spacing of \(\Delta \log_{10}(v)=\) 0.0348. We arrive at this number after investigating the average log spacing between SPAN-i velocity shells as
The number of knots are calculated based on this logarithmic spacing and the farthest extents of the grids which have a non-zero count.
where, \(N_{\rm{knots}}\) is rounded down to the closest integer and \((v_{\rm{min}}, v_{\rm{max}})\) are the minimum and maximum velocity magnitudes of grids with non-zero counts. Finally, the knot locations in \(v\) are computed from binning \(\log_{10}(v)\) into \(N_{\rm{knots}}\) and finding the bin centers. The above prescription renders the first and the last points at the edge of the domain, resulting in zero B-spline support. In order to ensure that the very first and last points are supported by B-splines, we add two additional knots at \(\log_{10}(v_{\rm{min}}) - \Delta \log_{10}(v)\) and \(\log_{10}(v_{\rm{max}}) + \Delta \log_{10}(v)\). Finally, these knots are raised to the power of 10 and are used to generate the B-splines. These knots are computed for each timestamp as the grids change. The resultant B-splines for our chosen timestamp is shown in panel (B) of Fig. 3. We have highlighted one of the B-spline peaking around \(v \sim 470\) km/s in red. The arc of \((v_{\parallel}, v_{\perp})\) space is also shaded in red in panel (A). The cloud of grids which are spanned by this B-spline (using 0.1 in panel (B) as a threshold) are marked in large black dots in panel (A).
Polar-cap Slepians from localization coefficients#
In panel (C) of Fig. 3, we show the three Slepian functions which are optimally confined inside a \(\Theta = 60^{\circ}\) polar cap with a maximum wavenumber \(L_{\rm{max}} = 12\). These functions capture the variation of the distribution across the grid points as a function of polar angle for each B-spline velocity shell. The \(\Theta = 60^{\circ}\) is demarcated by a vertical red dashed line. The angular location of the grids spanned within the red arc in panel (A) are over-plotted on each Slepian function.