Errors on derived results

1
Precise current density determination for the
study of the magnetopause current layer with
multiple spacecraft
J. De Keyser, F. Darrouzet, E. Gamby Belgian
Institute for Space Aeronomy, Brussels,
Belgium johan.dekeyser_at_aeronomie.be
Errors on derived results The result of LSGC,
applied to a scalar field f, is ?f and the
associated error covariance matrix C(d?fi, d?fj).
This matrix is usually not diagonal Because of
the distribution of points and the coupled way in
which the gradient components in space and time
are computed, the errors on the gradient
components are not statistically independent.
When applying LSGC to vector fields, as in the
curlometer, there is in addition a coupling
between the errors on the components of the
gradients of Bx, By, and Bz, in particular
because of the div B 0 condition. It is
important to take these non-vanishing
cross-correlations into account while
establishing the error estimates on the current
components µ02 djx2 ( d?yBz - d?zBy
)2 (d?yBz)2 (d?zBy)2 - 2
d?yBzd?zBy µ02 djy2 ( d?zBx -
d?xBz )2 (d?zBx)2 (d?xBz)2 - 2
d?zBxd?xBz µ02 djz2 ( d?xBy -
d?yBx )2 (d?xBy)2 (d?yBx)2 - 2
d?xByd?yBx where the colored terms are
cross-correlations between errors on different
components. Of course, there are also
cross-correlations between the errors on the
current components µ02 djx djy (
d?yBz - d?zBy ) ( d?zBx - d?xBz )
d?yBz d?zBx d?zBy d?xBz -
d?yBz d?xBz - d?zBy d?zBx µ02 djx djz
( d?yBz - d?zBy ) ( d?xBy - d?yBx )
d?yBz d?xBy d?zBy d?yBx
- d?yBz d?yBx - d?zBy d?xBy µ02 djy
djz ( d?zBx - d?xBz ) ( d?xBy - d?yBx
) d?zBxd?xBy d?xBzd?yBx
- d?zBxd?xBy - d?xBzd?xBy For the
divergence, which one may a priori impose to
vanish, one has (d(?B))2 ( d?xBx
d?yBy d?zBz ) 2 (d?xBx)2
(d?yBy)2 (d?zBz)2 2
d?xBxd?yBy d?xBxd?zBz d?yByd?zBz
Implementation We are currently preparing a
free and portable stand-alone implementation of
LSGC, in the form of a Matlab library. This
library consists of computing routines (apply to
scalar and vector fields) producing ?f and
C(d?fi, d?fj) lsgc_avds, lsgc_avd, lsgc_avs,
lsgc_ads, lsgc_av, lsgc_ad, lsgc_as, lsgc
computes the gradients in a set of points, with
automatic determination of velocity (v),
homogeneity directions (d) and/or scales (s),
when the other parameters are given lsgc_inst_ads,
lsgc_inst_ad, lsgc_inst_as, lsgc_inst
computes instantaneous gradients (uses spatial
homogeneity scales for estimating total error and
weighting the system) lsgc_cgc classical
gradient computation (no weighting) for more
than 4 simultaneous data, a least-squares
solution is returned and of routines for
retrieval of the results lsgc_f returns the
field error margins lsgc_gradx, lsgc_gradt
returns the spatial/temporal gradient error
margins lsgc_curl, lsgc_div returns the curl
or divergence of a vector field error
margins lsgc_vframe, lsgc_lc, lsgc_sign
returns the frame velocity, homogeneity scales,
curvature signs with which the end result was
obtained lsgc_neq, lsgc_svd, number of
equations used, singular values or the problem
Conclusion A robust curlometer can be obtained
with least-squares gradient computation, with the
divergence-free constraint built into the method.
It provides error estimates on the current
density vector This is essential to assess the
significance of the current densities,
particularly for narrow structures as the
magnetopause, or for weak currents. References D
arrouzet, F., J. De Keyser, P. M. E. Décréau, J.
F. Lemaire, and M. W. Dunlop. Spatial gradients
in the plasmasphere from Cluster. Geophys. Res.
Lett. 33, L08105, 2006. De Keyser, J., F.
Darrouzet, M. W. Dunlop, and P. M. E. Décréau.
Least-squares gradient calculation from
multi-point observations of scalar and vector
fields Methodology and applications with Cluster
in the plasmasphere. Ann. Geophys. 25, 971987,
2007. De Keyser, J., Least-squares
multi-spacecraft gradient calculation with
automatic error estimation, Ann. Geophys., 26,
32953316, 2008.

• Determining the homogeneity parameters
• The detection of magnetospheric currents is often
  quite difficult, since currents typically occur
  in sheets, like the magnetopause, in which case
  one has to deal with small-scale structures that
  are rarely sampled very well, or as distributed
  current systems, like the ring current, in which
  case the current density is rather low. It is
  therefore important to be able to reliably
  estimate the error bars on the obtained gradients
  and current density vectors, in order to be able
  to assess the physical significance of the
  gradient and current density vectors that one
  obtains.
• The error bars depend on the measurement errors,
  but also on the spacecraft configuration relative
  to the scale sizes of the magnetic field
  structures, i.e., on the homogeneity parameters.
  In practice, you must be able to compute the
  homogeneity parameters automatically.
• Determination of scales
• Assume that the directions uj are given. We then
  try to determine the length scales lj
  automatically by looking at the residuals of the
  overdetermined system and
• requiring the ratio A ?² / (N-M) ? 1
• or by analyzing the pattern of residuals to
  improve the lengths with a heuristic so that Aj
  ljnew/lj ? 1
• In both cases, we use BFGS multi-dimensional
  optimization to minimize the target function
• F A A-1 or F Sj ( Aj Aj-1 )
• This target function usually has several minima
  we have devised an algorithm to find an initial
  guess close to the global minimum. Various
  strategies can help to reduce BFGS computer time
  consumption, which is especially welcome in the
  multi-dimensional case.
• There is often not enough information in the
  sampled data set to really determine all lj. We
  use simple heuristics to deal with this.
• Determination of homogeneity directions

• Abstract
• In this contribution we discuss the abilities of
  a least-squares curlometer (an application of
  least-squares gradient computation techniques,
  LSGC) to provide precise current density vectors,
  together with error estimates, something that is
  of considerable importance for the study of the
  magnetopause. The success of such a curlometer
  depends on the reliable determination of the
  motion of the current layer, as that helps to
  ensure that the optimal set of data is used for
  the computation of the currents. Another aspect
  is the use of constraints to enforce a zero
  magnetic field divergence. Additional geometric
  constraints may be imposed. The technique can be
  applied without any a priori limitation on the
  number of spacecraft or their configuration.
• Classical 4-point gradient computation
• Classical gradient computation (CGC) needs 4
  simultaneous measurements to find the spatial
  gradient.
• - Homogeneity all sampling positions must be
  within the gradient region
• Large relative errors are inevitable with
  numerical differentiation, so you
• need accurate and well-calibrated data
• Ill-conditioning of the problem if sampling
  positions are nearly co-planar
• Least-squares gradient computation
• We compute the gradient of a field f in x0 from
  the measurements fi at xi (red dots) by a number
  of spacecraft. In the neighborhood of x0, f is
  approximated by
• where f0 and are the value and the
  space-time gradient at x0, with ?x x - x0. This
  neighborhood is the homogeneity domain, described
  by orthogonal unit vectors U uj and
  homogeneity scales lj. The error dfi on each fi
  comes from the measurement error dfi,meas and the
  approximation error (quadratic upper bound for a
  linear approximation)
• dfi,approx dfi,meas ?xi²