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Sandy Antunes

Introduction: Modeling the Earth

The underlying structure of the Earth is, as noted in depth by the Naval Oceanographic Office, "a composite of several magnetic fields generated by a variety of sources". Electromagnetic theory provides a mathematical way to calculate a magnetic field strength given a specific configuration. In theory, one could derive the field strength at any position simply by having an accurate representation of the mechanisms generating the magnetic field. For example, given a dipole of known magnetization, the field can be explicitly derived.

For geomagnetic studies, however, the problem is reversed. Measurements are taken of field strengths, and from this a general model is created that describes the field in toto, and provides some information on the underlying generation system. Such a data-driven (rather than equation-based) model makes good use of basic magnetic vector potential calculations, in particular the dipole and multipole expansions.

In many applications, the Earth's magnetic field can be approximated as a dipole. It shrouds the earth and traps charged particles (electrons and protons) that stream from the sun. As the field lines converge at the poles, the magnetic field dips into the atmosphere and creates aurora and other effects. Far from the Earth, the approximation is valid and the broad interaction of the solar wind and the Earth is easily modeled. Standing on the Earth's surface, this dipole approximation is sufficient and a standard compass will handily point to magnetic north and navigate you home.

Go up a short distance into the atmosphere, however, and the basic dipole model insufficiently describes the real behavior. Most notable is the South Atlantic Anomaly (SAA), where the magnetic field lines dip surprisingly close to the surface, and the rampant particle streams traverse the lower atmosphere. Small dips, where the geomagnetic 'rigidity' is low, also allow particle influx to the lower atmosphere.

These charged particles can disrupt electronics and therefore, for low-earth orbit satellites, the specific field configuration is a highly significant matter of study. Modeling local behavior is crucial, and therefore parameterization of the field based on sampled data is an active field of study. In short, the broad dipole field is meaningless when compared with the local anomalistic effects that are actually measured.

A common model of the Earth's magnetic field is IGRF 95, released in 1996 by the American Geophysical Union. This data set gives multipole terms and moments for a sampling of longitude/latitude/altitude points. This model is created using space-based and land sampling, and in effect works the dipole problem backwards. Instead of using the field model to determine individual field values, it uses field values to create a numerical model of the Earth's magnetic field.

From this, then, one can determine the expected field value for any longitude/latitude/altitude point through interpolation of some form. Linear interpolation of a vector field in spherical geometry is not applicable, but spline models have been successful applied. A more accurate method is to use the assumptions of the underlying physics and examine a variety of dipole and multipole expansions used to accurately map out the local Earth earth's magnetic field.

Such a study has useful physical implications. Accurate modeling allows for prevention of electronics damage to powered hardware for low-earth orbit astronomical satellites such as the Rossi X-Ray Timing Explorer, ASCA and the upcoming Astro-E. We use models derived from earlier satellite work with Ginga and XTE to illustrate the possible approaches.

Vector Magnetic Potential for a Sphere

Consider a uniformly magnetized sphere, with radius a, magnetization M₀ along the z axis (ref: Fields and Waves in Communications Electronics, Ramo et al, 1994). It has a surface magnetic charge density = ₀M₀ cos . Solving the vector equivalent of the Poisson equation ²_m = - _m/₀ in spherical coordinates yields terms for r < a and r > a and ultimately provides two terms for the vector magnetic potential, H = -_m:

H = -(^M₀/₃) [ r cos - _v sin ] = -z ^M₀/₃ for r < a

H = +(^M₀a3/_{3 r³}) [ 2r cos + _v sin ] for r > a

These produce curves very similar to that of a magnetic dipole. Using these, it is fairly straightforward to calculate the field strength and direction for any given point. However, this assumes a homogeneous magnetization and a static field. The Earth's field has more of a dynamo effect, plus induced fields from the solar wind, as well as spatial inhomogeneities due to the Earth's construction. Therefore, we must use a more gradiated model.

The multipole expansion is useful for a more complex field distribution. When a field deviates from a pure dipole, adding additional coefficients provides a more accurate representation of the full field structure. The general multipole expansion for a vector potential is given by:

A(r) = (^₀/₄) _l=0 (¹/_r^l+1) _V' J(r')r'^lP_l(cos ') d'

Our monopole term is of course zero, so the leading term is the 2nd, or dipole, term. Our dipole term is as given earlier. The further n terms add with an 1/rⁿ, thus as we get further from our given center, the additional terms have less of a contribution. Giving just the expansion for the first three terms, with all integrals being over the volume V', we get:

A(r) = (/₄) [ ¹/_r J(r') d' + ¹/_r² J(r') (r r') d' + ¹/_r³ (¹/₂)J(r') [3 (r r')² - r'²] d' + ... ]

Armed with multipole expansion terms, we can now set about modeling local regions of the geomagnetic field using our data sets to provide the appropriate multipole coefficients for that given region.

Our Specific Case: The Earth's Magnetic Field

Our spherical case provides that, given a magnetization over a uniform Earth, we can calculate the field. The Earth's magnetic field has a dipole tilt of 10.70 deg N, 71.44 deg W from the rotation axis, and an approximate dipole moment of 7.788 x 10²² A m², with a dipole offset of (-401.2, 285.5, 194.6) km in a right-hand system. (Ref: Fraser-Smith, R. A., "Centered and Eccentric Geomagnetic Dipoles and Their Poles," Rev. Geophys., 25, 1, 1987).

The International Geomagnetic Reference Field (IGRF) 90 model specifies the Earth's magnetic dipole terms and moments. (Ref: Langel, R. A., "International Geomagnetic Reference Field, 1991 Revision," J. Geomag. Geoelectr., 43, 1007, 1991). The latest IGRF model is IGRF 95 (released in 1996). It presents data as follows. For a given longitude band and a given altitude, it presents a table of values (after an initial line that includes the number of multipole terms and the reference date). Note the fields are Degree, Order, coefficient G, and, if not zero, the next coefficient H, for example:

Degree	Order	G	H
1	0	G(1)	-
1	1	G(2)	G(3)

Appendix A contains one full sample table for a given positions from IGRF 90.

The full model includes the coefficients of the first 10 spherical harmonics of the Earth's magnetic field at the reference time (in this case, January 1, 1991), as well as the first derivatives of these coefficients. This is sufficient to provide a 10th-order multipole, which is typically used to generate the McIlwain L value. The McIlwain L is defined as the magnetic field strength using a 10th-order multipole centered on a tilted Earth that provides magnetic L values by following field lines from any given position to the magnetic equator.

A simpler version-- our basic dipole-- can be used when calculation speed is essential, and only calculates the fit to the IGRF data using a dipole field calculation for the L value. The output in any case produces the following sample table of values:

Accuracies of models

Some software to do the calculation is available from NOAA, for doing dipole calculations of geomagnetic coordinates, as well as spherical harmonics and Fourier components of harmonics. For this paper, we used several in-house routines provided by the ASCA and XTE missions.

Assuming that the IGRF data defines the true field at the points given, doing a full 10th-order fit actually replicates the calculations used to create the IGRF data set. Which is to say, the data set was assembled using unevenly sampled data, and then applying a 10th-order fit to create the evenly sampled IGRF data.

Therefore, there is in theory no deviation from the IGRF data using a 10th-order multipole interpolation, assuming you are using the same reference time as the data set. To extrapolate to a given reference time, the coefficients (given as change per year) can be applied, with a stated median difference of 0.2% and a maximum difference of no more than 1.7% over any given 5 year span.

That said, there is a computational cost for such accuracy. The time trials by Michael Stark et al (personal communication, 1999) took 3.14ms per calculation. In comparison, for a low-earth orbit satellite (600km, inclined 25 degrees), the simple dipole model has a median error of 3% and a maximum error of 20%, and took only 63us, or 1/50th the time. "MacL" generates a very fast McIlwain "L" value for a given satellite position. It uses a dipole offset for the field, with an IGRF 90 (1990) model of the dipole and moments. This is used as a fast approximation for the XTE real-time software. The code states a maximum deviation of 20%, with a median difference of 3%, for typical low earth orbit satellites; no comparison of deviation as a function of time since 1990 is available.

Below we show the field shape for our test case (600km, 25 degrees) using the explicit 10th order multipole (Figure 1), and the "cheaper" dipole calculation (Figure 2). The error between these two are shown in Figure 3.

Figure 1 IGRF model (10th order multipole) for 600km, 25 degree across full longitude band and 60 degree latitude range.

Figure 2 Simple dipole interpolations of IGRF data for 600km, 25 degree across full longitude band and 60 degree latitude range.

Figure 3 Percentile dipole error in L calculation

In comparison, we have data from a spline interpolation of the geomagnetic contours. In essence, this ignores any concept of underlying structure (no multipoles) and instead does a purely mathematical surface/curve fit to the data. This calculation (using ASCA software) uses rigidity values (GeV/c) instead of geomagnetic field values, and has an input table of 15 latitude and 27 longitude points for the given altitude of interest. When we compare the spline data fit to the more physics-based multipole expansion, we find there is a clear deviation (extending to 5% or more) of the spline fit, as shown in Figure 4.

Figure 4 Correlation of Spline and Multipole calculations along all longitudes as a function of latitude

Conclusion

The convergence of core electromagnetic theory, intelligent data gathering, and mathematics of interpolation highlights the necessity of a good theoretical understanding of the phenomenon before effective data gathering can take place. It also sets limits on the accuracy possible as well as suggesting limits on calculation difficulty required to achieve a specific error level in one's answers.

The most direct conclusion is that a basic understanding of the underlying vector magnetic field yields better results when interpolating measured geomagnetic data. Purely mathematical methods (spline fits, simple dipoles) have much less accuracy than a proper multipole expansion of the local field. Simple measurements of geomagnetic field strength (as the simple rigidity measurements showed) leads to inaccurate results when applying an interpolation.

The IGRF geomagnetic data sets in particular specifically measure field quantities in terms of an assumed underlying multipole structure. Therefore, when doing analysis with those data sets, understanding of the implicit structure lets you construct more accurate results. At the same time, knowledge of the basic multipole underpinnings also means that you can determine, for a given problem set, how many expansion terms are required.

This has great significant for multi-million dollar high energy satellite telescopes, at the very least. When the outcome of an inaccurate calculation or fallacious assumption can cause hardware damage, the realm of theory becomes something one can not neglect.

References and Supplemental Information

1. Electromagnetic Fields, Wangsness, R., 1979, John Wiley & Sons Inc
2. FELDG and SHELLG model for calculating the McIlwain L parameter, used as the basis of our in-house routines
3. Fields and Waves in Communications Electronics, Ramo, S., Whinnery, J. R., and Van Duzer, T., 1994, John Wiley & Sons Inc
4. Fraser-Smith, R. A., "Centered and Eccentric Geomagnetic Dipoles and Their Poles," Rev. Geophys., 25, 1, 1987
5. Geomagnetic software
6. HEASARC, home for many X-Ray astronomical observatories
7. IGRF 95, released in 1996 by the American Geophysical Union.
8. Langel, R. A., "International Geomagnetic Reference Field, 1991 Revision," J. Geomag. Geoelectr., 43, 1007, 1991
9. Naval Oceanographic Office discussion of the source of the Earth's magnetic field.
10. Space-based and land sampling of the geomagnetic field