Automated Forward Modelling – Overview and Prospects

Martin ConnorsAthabasca 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

a is the parameter vector

(x,y) the N data points

and weighting is applied

2

i=1

Ni i

i=

y - F( , x )

a2

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 –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

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, post-onset 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

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.

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 observationBlue 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)

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.

FAST

Spacecraft and the Auroral Oval

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

B eastward

B equatorward

Upw

ard FA

C

Dow

nwar

d F

AC

Evening sector eastward electrojet

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]: E

ByP 256.1

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