We can measure declination, inclination and intensity at different places around the globe, but not everywhere all the time. Yet it is often handy to be able to predict what these components are. For example, it is extremely useful to know what the deviation is between true North and declination in order to find our way with maps and compasses. In principle, magnetic field vectors can be derived from the magnetic potential ψ_{m} as we showed in Chapter 1 . For an axial dipolar field, there is but one scalar coefficient (the magnetic moment m of a dipole source). For the geomagnetic field, there are many more coefficients, including not just an axial dipole aligned with the spin axis, but two orthogonal equatorial dipoles and a whole host of more complicated sources such as quadrupoles, octupoles and so on. A list of coefficients associated with these sources allows us to calculate the magnetic field vector anywhere outside of the source region. In this section, we outline how this might be done.
As we learned in Chapter 1 , the magnetic field at the Earth’s surface can be calculated from the gradient of a scalar potential field (H = ∇ψ_{m}), and this scalar potential field satisfies Laplace’s Equation:
 (2.5) 
For the geomagnetic field (ignoring external sources of the magnetic field which are in any case small and transient), the potential equation can be written as:
 (2.6) 
where a is the radius of the Earth (6.371 × 10^{6} m). In addition to the radial distance r and the angle away from the pole θ, there is ϕ, the angle around the equator from some reference, say, the Greenwich meridian. Here, θ is the colatitude and ϕ is the longitude. The g_{l}^{m}s and h_{l}^{m}s are the gauss coefficients (degree l and order m) for hypothetical sources at radii less than a calculated for a particular year. These are normally given in units of nT. The P_{l}^{m}s are wiggly functions called partially normalized Schmidt polynomials of the argument cosθ. These are closely related to the associated Legendre polynomials. [When m = 0 the Schmidt and Legendre polynomials are identical.] The first few of P_{l}^{m}s are:
To get an idea of how the gauss coefficients in the potential relate to the associated magnetic fields, we show three examples in Figure 2.3. We plot the inclinations of the vector fields that would be produced by the terms with g_{1}^{0},g_{2}^{0} and g_{3}^{0} respectively. These are the axial (m = 0) dipole (l = 1), quadrupole (l = 2) and octupole (l = 3) terms. The associated potentials for each harmonic are shown in the insets.
In general, terms for which the difference between the subscript (l) and the superscript (m) is odd (e.g., the axial dipole g_{1}^{0} and octupole g_{3}^{0}) produce magnetic fields that are antisymmetric about the equator, while those for which the difference is even (e.g., the axial quadrupole g_{2}^{0}) have symmetric fields. In Figure 2.3a we show the inclinations produced by a purely dipolar field of the same sign as the present day field. The inclinations are all positive (down) in the northern hemisphere and negative (up) in the southern hemisphere. In contrast, inclinations produced by a purely quadrupolar field (Figure 2.3b) are down at the poles and up at the equator. The map of inclinations produced by a purely axial octupolar field (Figure 2.3c) are again asymmetric about the equator with vertical directions of opposite signs at the poles separated by bands with the opposite sign at midlatitudes.
As noted before, there is not one, but three dipole terms in Equation 2.6, the axial term (g_{1}^{0}) and two equatorial terms (g_{1}^{1} and h_{1}^{1}). Therefore, the total dipole contribution is the vector sum of these three or . The total quadrupole contribution (l = 2) combines five coefficients and the total octupole (l = 3) contribution combines seven coefficients.
So how do we get this marvelous list of gauss coefficients? If you want to know the details, please refer Langel (1987). We will just give a brief introduction here. Recalling Chapter 1, once the scalar potential ψ_{m} is known, the components of the magnetic field can be calculated from it. We solved this for the radial and tangential field components (H_{r} and H_{θ}) in Chapter 1. We will now change coordinate and unit systems and introduce a third dimension (because the field is not perfectly dipolar). The north, east, and vertically down components are related to the potential ψ_{m} by:
 (2.7) 
where r, θ, ϕ are radius, colatitude (degrees away from the North pole) and longitude, respectively. Here, B_{V } is positive down, B_{E} is positive east, and B_{N} is positive to the north, the opposite of H_{r} and H_{θ} as defined in Chapter 1. Note that Equation 2.7 is in units of induction, not Am^{1} if the units for the gauss coefficients are in nT, as is the current practice.
Going backwards, the gauss coefficients are determined by fitting Equations 2.7 and 2.6 to observations of the magnetic field made by magnetic observatories or satellite for a particular time. The International (or Definitive) Geomagnetic Reference Field or I(D)GRF, for a given time interval is an agreed upon set of values for a number of gauss coefficients and their time derivatives. IGRF (or DGRF) models and programs for calculating various components of the magnetic field are available on the internet from the National Geophysical Data Center; the address is http://www.ngdc.noaa.gov. there is also a program igrf.py included in the PmagPy package (see Appendix F.1).
In practice, the gauss coefficients for a particular reference field are estimated by leastsquares fitting of observations of the geomagnetic field. You need a minimum of 48 observations to estimate the coefficients to l = 6. Nowadays, we have satellites which give us thousands of measurements and the list of generation 10 of the IGRF for 2005 goes to l = 13.

In order to get a feel for the importance of the various gauss coefficients, take a look at Table 2.1, which has the Schmidt quasinormalized gauss coefficients for the first six degrees from the IGRF for 2005. The power at each degree is the average squared field per spherical harmonic degree over the Earth’s surface and is calculated by R_{l} = ∑ _{m}(l + 1)[(g_{l}^{m})^{2} + (h_{l}^{m})^{2}] (Lowes, 1974). The socalled Lowes spectrum is shown in Figure 2.4. It is clear that the lowest order terms (degree one) totally dominate, constituting some 90% of the field. This is why the geomagnetic field is often assumed to be equivalent to a magnetic field created by a simple dipole at the center of the Earth.

