Transcript Slide 1

Automated Forward Modelling –
Overview and Prospects
Martin Connors
Athabasca University
Presented at University of Alberta, April 18, 2007
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.
A single magnetogram tells little in fact and
can be misleading
Even several magnetograms need
interpretation by a ‘geomagician’
Inversion tells us more by giving simple
parameters extracted from several ground stations
Good ground spatial coverage, with reliable
inversion methods, is essential. Magnetic inversion
is best done along magnetic meridians,
Ability to match input data is best near the middle of the
chain (although often not in Z due to electrojet structure)
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 1970’s by Rostoker and Kisabeth
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)
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
2 a is the parameter vector
N y
i - F( a , xi ) 
(x,y) the N data points
2
 = 
 and weighting is applied
i

i=1 
Arriving at a minimum in χ2 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.
L-M in more complex χ2 space
Initially follows gradients
(more or less)
Then uses local
curvature
Diagram: A. Zisserman,
Oxford U.
Test on simulated magnetic data;
note ordinate is logarithmic
For meridian data, AFM adjusts
current and borders: in this test
case all parameters were varied
Physical Parameter of current in
test: starts to improve at Iteration 6
Variation of Geometric parameters:
note big changes around Iteration 6
Numerical Recipes’ version of L-M:
note importance of scale parameter λ
Return to test case, things happen
when scale parameter λ correct
Scale parameter
should be small
when –nearsolution, here it
was started too
small so the
algorithm
adjusted it to be
appropriate, then
as the solution
was approached,
reduced it again
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
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, postonset shows Churchill
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
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.
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.
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.
Frey’s FUV width is wider than AFM
gives.
AE and AFM match on average and
can be cross-calibrated. Weimer’s
ate-bt+c
parametrization is very good on
average.
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
October 3-4 2000 Event in CANOPUS Churchill
Meridian X Component (quiet time in middle is day)
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
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
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.
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
(Hallowe’en storm 2003).
This is one of few cases found
with good ground-based
optical data.
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
Low-Altitude Satellites view near-Earth FACs
Satellite overpasses can provide information about
field-aligned currents in the auroral zone.
Comparison of optical borders and inversion
results for growth phase (Feb 22 1997)
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.
Spacecraft and the Auroral Oval
Evening sector eastward electrojet
B eastward
B equatorward
• Spacecraft
traversing the
auroral oval
respond primarily
to the solenoidal
current system
comprised of fieldaligned currents at
its poleward and
equatorward
borders.
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]:
P 
B y
1.256E
Here ΣP=3.5 mho
Other
comparisons are
possible, for
example of
inverted
electrojet(grey
bar) with
electrons
detected by FAST
(black) or with
optical emissions
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 ΣH=4 mho, comparable to ΣP
“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
Regional
use can
contain
more
information
than
meridian
use
More current
systems can
be added to
see the
morning and
evening
sector
electrojets
as well as
the SCW
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
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)
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 King’s UC astronomer Brian Martin) in December 2004
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
An important feature of Canada is the usual presence
of two GOES footpoints (red arrows) and these can
facilitate ground-near-Earth comparisons
GOES 7 – CANOPUS Conjugacy in 1990 – Profile on chain
allowed detailed study of Pc 5 (presumably FLR)
[Ziesolleck et al., JGR, 1996, 1997]
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