UNCERTAINTIES IN FIELD-LINE TRACING IN THE MAGNETOSPHERE .2. THE COMPLETE INTERNAL GEOMAGNETIC-FIELD

The discussion in the preceding paper is restricted to the uncertainti es in magnetic-field-line tracing in the magnetosphere resulting from published standard errors in the spherical harmonic coefficients that define the axisymmetric part of the internal geomagnetic field (i.e. g (n)(0) +/- delta g(n)(0)). Numerical estimates of these uncertainties based on an analytic equation for axisymmetric field lines are in exce llent agreement with independent computational estimates based on step wise numerical integration along magnetic field lines. This comparison confirms the accuracy of the computer program used in the present pap er to estimate the uncertainties in magnetic-field-line tracing that a rise from published standard errors in the full set of spherical harmo nic coefficients, which define the complete (non-axisymmetric) interna l geomagnetic field (i.e. g(n)(m) +/- delta g(n)(m)) and h(n)(m) +/- d elta h(n)(m)). An algorithm is formulated that greatly reduces the com puting time required to estimate these uncertainties in magnetic-field -line tracing. The validity of this algorithm is checked numerically f or both the axisymmetric part of the internal geomagnetic field in the general case (1 less than or equal to n less than or equal to 10) and the complete internal geomagnetic field in a restrictive case (0 less than or equal to m less than or equal to n, 1 less than or equal to n less than or equal to 3). On this basis it is assumed that the algori thm can be used with confidence in those cases for which the computing time would otherwise be prohibitively long. For the complete internal geomagnetic field, the maximum characteristic uncertainty in the geoc entric distance of a held line that crosses the geomagnetic equator at a nominal dipolar distance of 2 R(E) is typically 100 km. The corresp onding characteristic uncertainty for a field line that crosses the ge omagnetic equator at a nominal dipolar distance of 6 R(E) is typically 500 km. Histograms and scatter plots showing the characteristic uncer tainties associated with magnetic-field-line tracing in the magnetosph ere are presented for a range of illustrative examples. Finally, estim ates are given for the maximum uncertainties in the locations of the c onjugate points of selected geophysical observatories. Numerical estim ates of the uncertainties in magnetic-field-line tracing in the magnet osphere, including the associated uncertainties in the locations of th e conjugate points of geophysical observatories, should be regarded as ''first approximations'' in the sense that these estimates are only a s accurate as the published standard errors in the full set of spheric al harmonic coefficients. As in the preceding paper, however, all comp utational techniques developed in this paper can be used to derive mor e realistic estimates of the uncertainties in magnetic-field-line trac ing in the magnetosphere, following further progress in the determinat ion of more accurate standard errors in the spherical harmonic coeffic ients.