I could refine and rewrite forever, but I have to stop and submit it someday.
Nevertheless, I believe the reference list is quite complete, and should provide broad exposure to all sides of the argument.
This mathematical tool was invented by the ubiquitous German mathematician, Carl Freidrich Gauss, circa 1835, for the purpose of evaluating the Earth's magnetic field.
This ingenious method uses an infinite sum of trigonometric functions to evaluate a field, on the surface of a sphere embedded in the field.
The physical model for describing the Earth's magnetic field carries the impressive looking name magnetohydrodynamics (MHD), but is also commonly called dynamo theory.
Simply stated, it has been shown that turbulent motions within an electrically conductive fluid will generate magnetic fields.
These models are actively tested and refined until they match the observed field to within the limits set by natural and unavoidable experimental uncertainties.
It is very important to keep in mind that the empirical model is built entirely from a statistical analysis of the data themselves.But, as you might imagine, modern computers can add a lot of numbers together very fast, so while this may have been a fundamental problem for Gauss, it is no longer all that bothersome.One can approximate the true shape of the field by extending the sum to an arbitrarily large number of terms, the only limitation being the practicalities involved.It is essentially independent from whatever physics might be involved in generating the field.One of Barnes' many mistakes is to insist that only the dipole component of the magnetic field is generated by currents within the Earth, and that all lesser components (called 'higher order components' in physics jargon) are generated by some other process, such as magnetic rocks or telluric currents (electric currents induced in the crust, for example, by lightning in thunderstorms, or induced as a reaction to currents in the ionosphere).In practice, the magnetic field is measured constantly at a number of official magnetic observatories all over the world, as well as at universities, or by other scientific teams and expeditions, and now several spacecraft measure the field well above the Earth, and out into deep space.