Automated Forward Modelling

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Automated Forward Modelling Overview and
Prospects

• Martin Connors
• Athabasca University
• Presented at University of Alberta, April 18, 2007

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Ground Magnetic Measurements

• Ground measurements allow determination of
  ionospheric currents through inversion of
  magnetic data
• Can be combined with imaging to determine
  morphology and characteristics of particle
  precipitation.

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A single magnetogram tells little in fact and can
be misleading
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Even several magnetograms need interpretation by
a geomagician
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Inversion tells us more by giving simple
parameters extracted from several ground stations
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Good ground spatial coverage, with reliable
inversion methods, is essential. Magnetic
inversion is best done along magnetic meridians,
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Ability to match input data is best near the
middle of the chain (although often not in Z due
to electrojet structure)
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In practice, how can one do inversions?

• A forward model based on characteristics of
  field-aligned and ionospheric (and induced)
  currents can be made
• The parameters of such a model can be adjusted so
  that ground magnetic fields from the model match
  those observed
• Pioneering work in this was done at UofA in the
  1970s by Rostoker and Kisabeth

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Automated Forward Modelling

• Making the model to be matched to the data is
  referred to as Forward Modelling
• Doing the fit by hand takes a lot of time and
  perhaps does not allow the parameter space to be
  fully explored
• There is motivation to take the Forward approach
  and automate it
• A powerful and generally used inversion technique
  is the Levenberg-Marquardt approach (i.e.
  recently incorporated into MATLAB)

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How does L-M work?

• L-M combines gradient descent plus a Newton
  solver in a space where an objective function is
  defined dependent on parameters
• In fitting data a suitable objective function is

a is the parameter vector (x,y) the N data
points and weighting is applied
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Arriving at a minimum in ?2 space
Far from minimum, following the gradient
works BUT Near the minimum, the gradient is zero
and a quadratic form solution must be used L-M
COMBINES these approaches Diagram A. Zisserman,
Oxford U.
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L-M in more complex ?2 space
Initially follows gradients (more or less) Then
uses local curvature Diagram A. Zisserman,
Oxford U.
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Test on simulated magnetic data note ordinate is
logarithmic
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For meridian data, AFM adjusts current and
borders in this test case all parameters were
varied
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Physical Parameter of current in test starts to
improve at Iteration 6
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Variation of Geometric parameters note big
changes around Iteration 6
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Numerical Recipes version of L-M note
importance of scale parameter ?
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Return to test case, things happen when scale
parameter ? correct
Scale parameter should be small when near-
solution, here it was started too small so the
algorithm adjusted it to be appropriate, then as
the solution was approached, reduced it again
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Use of L-M in studying substorms and sawtooth
events. The meridian form has been quite useful.
Work done with R. L. McPherron and J. Ponto
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ALert

• The AL (or AE) index can be misleading
• Here the AFM results are extremely clear for a
  substorm with strong growth phase
• AL or even the inverted current mislead as to
  onset time
• AL pre-onset shows Alaska conditions, post-onset
  shows Churchill

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Statistical Properties of Substorms

• A large-scale inversion project was undertaken
  for 1997 Churchill meridian data
• Baselining the data is essential yet was
  challenging
• Approximately 65 onsets were found to be very
  robustly inverted, comparable to the number of
  events in some other statistical studies
• We have studied internal relations of parameters
  and not yet relation to external parameters such
  as solar wind in any great detail

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Our results indicate a westward electrojet at
time of onset of about 0.1 MA and also show the
latitude at onset to increase with lesser
current. The former is a quantitative measure but
likely an overestimate the latter is a
well-known result made quantitative.

• Frey et al. (2004) found a distribution of Image
  FUV onsets skewed toward the evening sector. Our
  onsets straddle midnight. FUV onsets are due to
  bright evening sector auroras the currents are
  in fact roughly symmetric around midnight.

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Post-Onset AVERAGE Behaviour

• The current increases rapidly (20 min) to about
  0.45 MA (an overestimate), black dots AFM, open
  dots AE in MA, curve Weimer (1993)
• The electrojet poleward border rises rapidly by
  about 5º (black dots AFM, open dots Frey et al.,
  2004). The equatorward border does not move.
  Freys FUV width is wider than AFM gives.
• AE and AFM match on average and can be
  cross-calibrated. Weimers
• ate-btc
• parametrization is very good on average.

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Sawtooth Events

• Several sawtooth events were selected from a list
  supplied by Joe Borovsky from LANL injections.
  Our final selection was then based on inspection
  of CANOPUS magnetometer data
• It is hard to determine to what degree this
  sample may be biased toward large events
• Inspection of the CANOPUS stackplots already
  makes clear the large latitudinal extent of
  sawtooth events. AE will be biased downward in
  such cases.
• Large currents make sawtooth events good for AFM

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October 3-4 2000 Event in CANOPUS Churchill
Meridian X Component (quiet time in middle is day)
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Oct 3-4 2000 Oct 3 is not discussed much here
but note good Image WIC data. ACE shows extended
periods of -BZ. Quiet time at CANOPUS likely due
to BZ when on dayside
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Inversion Results for Oct 4 2000

• Early UT hours are quiet, during BZ
• Growth-like signature ca. 4 UT accompanies slow
  southward turning
• Onsets are like substorms but width of electrojet
  very large
• Currents of up to 2.5 MA are several times those
  of average substorms
• Current density in electrojets may not be
  extreme AL proxy

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Inversion Verification

• Black is X (north), dots for model, solid for
  observation
• Blue is Z (downward)
• Z is small at N edge likely due to current wedge
  width being chosen too small.

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Other Inverted Cases

• Several other cases were inverted with generally
  similar results very wide electrojets and very
  large currents. This case of Nov. 8 2004 did not
  invert well due to lack of stations far enough
  south. Nevertheless there are indications of
  currents of 7 MA, comparable to the largest ever
  seen to cross a meridian (Halloween storm 2003).
• This is one of few cases found with good
  ground-based optical data.

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Low-altitude Satellite-ground Comparison

• The AFM meridian chain inversions give
  latitudinal boundaries between which current
  flowed
• These parameters can be compared with results of
  low-altitude satellite overflights such as those
  for e-POP/CASSIOPE
• This can give Hall/Pedersen conductivity ratios

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Low-Altitude Satellites view near-Earth FACs
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Satellite overpasses can provide information
about field-aligned currents in the auroral zone.
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Comparison of optical borders and inversion
results for growth phase (Feb 22 1997)
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Comparison of optical borders and inversion
results for growth phase (Feb 22 1997)
FAST
Two electrojet model results are shown superposed
on 557.7 nm optical meridian scan data from
Gillam. The growth phase arc is poleward of the
evening sector eastward electrojet.
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Spacecraft and the Auroral Oval
Evening sector eastward electrojet

• Spacecraft traversing the auroral oval respond
  primarily to the solenoidal current system
  comprised of field-aligned currents at its
  poleward and equatorward borders.

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Conductivity

• If electric and magnetic field measurements are
  possible, and with some small assumptions, it is
  possible to determine the integrated Pedersen
  conductivity of the auroral oval. Smiddy et al
  1980

Here SP3.5 mho
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Other comparisons are possible, for example of
inverted electrojet(grey bar) with electrons
detected by FAST (black) or with optical emissions
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Combined Satellite E field and current from
inversion

• This is the same orbit (1997) whose data were
  previously shown, with average E of 25 mV/m
• The inversion (actually a single electrojet was
  more appropriate) gives a width of 440 km and
  total current of 0.04 MA
• This leads to SH4 mho, comparable to SP

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3d or global AFM

• This does not work as well because the parameters
  are not as well constrained as they can be
  locally
• There are more parameters
• Data can be sparse so they cannot be well
  constrained

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Regional use can contain more information than
meridian use
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More current systems can be added to see the
morning and evening sector electrojets as well as
the SCW
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Prospectus

• AFM is still likely best used on meridians where
  it is well constrained
• Regional use can study regions comparable to the
  SCW
• Full global use is not usually well constrained
  due to lack of stations
• In all envisaged uses, more stations are needed

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Also need more meridians to do good
satellite-ground correlation PLUS we can do FAC
studies if we can invert several meridians and
have space data (THEMIS)
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New technology UCLA magnetometers are ideal with
real-time data access and 2 Hz cadence.
EDMO mag installed by Martin Connors (Tom
Sawyer-like technique applied to Kings UC
astronomer Brian Martin) in December 2004
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Revived EDA Magnetometers

• An unfunded consortium including Don Wallis,
  George Sofko, Dieter Andre, Martin Connors,
  Gordon Rostoker, and others, has attempted to
  revive older EDA magnetometers
• These are used with 200 A/D cards and old
  computers
• Results are generally good but progress is slow
• Saskatoon SuperDARN site (mid-October) seen at
  right

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An important feature of Canada is the usual
presence of two GOES footpoints (red arrows) and
these can facilitate ground-near-Earth comparisons
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GOES 7 CANOPUS Conjugacy in 1990 Profile on
chain allowed detailed study of Pc 5 (presumably
FLR) Ziesolleck et al., JGR, 1996, 1997
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What is needed to extract the most data from
ground and satellite magnetics?

• Inversion techniques such as AFM allowing phase
  of substorm to be identified
• More stations allowing multiple meridians to
  characterize currents
• More stations allowing 3-d modelling
• Correlative studies from several satellites