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Lattice structures of the earth magnetic field
 
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8 - Fourier analysis of the earth magnetic field

8.1 - Preparation to the Fourier analysis

Starting point of the fourier analysis is the field of the total intensity as it was already introduced in the chapter 2– Illustration 2.1 and is shown in Illustration 8.1 (IGRF 1980) once again. With these graphics a table was generated for the values of the complete intensity.

conditions:
Longitude: -180° ≤ λ ≤ +180°
Latitude: -90° ≤ φ ≤ +90°

 
 Worldmap of the magnetic total intensity   A table of the values can be found here Table of earth field

The whole field is disassembled in single cuts.

Every cut proceeds along a circle of latitude. Besides, zero point is the beginning value of -13,5 degrees of west ascertained in the chapter 4.

From +90 north to -90 south the cuts became invested in the distance of 7.5 degrees - this then produces 24 cuts.
Illustration 8.1 - World map of the magnetic total intensity    
 
One submits every cut to a linear (numerical) fourier analysis, with the variable λ and the step width 7.5 degrees, i.e. with 48 points per cut. A total of 1106 points arise thus for the analysis of the earth field.
(see also numerical harmonious analysis in " mathematics for engineers from Brauch, Dreyer, Haacke - page 490)

As a basis of the whole analysis serves a numerical (linear) harmonious procedure as it is described in the book "Mathematik für Ingenieure" by Brauch/Dreyer/Haacke and also is known as an algorithm by Goertzel (and Reinsch).

The linear numerical procedure functions with an angle from 0 to 360 degrees. For this first numerical harmonious analysis of the cuts along the parallels, so with the degree of longitude λ, the condition on the value area is fulfilled. Merely the parametre λ must be adapted. It is worth:
 
 limits for lambda
 
This procedure delivers for a given numerical order, so here to the values of a cut, the suitable Fourier coefficients. With these coefficients the cuts can be shown as a sum of trigonometric functions.

 

 

8.1.1 - Two-dimensional Fourier analysis

Then 24 cuts by which the first Fourier analysis causes, can be shown in general so:
 
 Harmonical analysis for the first cut until  Harmonical analysis for the m cut
 
If one writes out these equations, the following picture arises:
 
 Numerical harmonic analysis

 

By the first numerical harmonious analysis occurs an equation system with m+1 formulas with in each case n+1 limbs owns. Then an other Fourier analysis, with the variables can be carried out about the columns of the coefficient matrix (with the Am und Bm) and the variable φ (Phi). This second numerical harmonious analysis delivers the final coefficients with the Aji und Bji.
 
 limits for phi  
 
If one separates the zonal, sectoral part from the tesseral portion, than arise for every point on the earth with the parametres λ, φ this formula:
 
 End formula 1
 
Attention: in the book a misprint has appeared in this equation!!!
   
The first Term for Y (zonal, sectoral part) can be still complemented so:
   
 zonal, sectoral part  Expanding the zonal, sectoral part
 
Then the qualitative final equation can be shown for a point on the earth's surface so:
 
 End formula for the magnetic field
 
There is to be noted here that λ, φ do not show directly the geographic coordinates, but are parametres, which represent these coordinates.
If one multiplies the clips, so merely Terme of sines and Cosines funktions appear, which are tied together multiplikativ with each other. Socalled spherical harmonics or grids.
 
The magnetic field of the earth can be described by a sum of grids

 

 

8.2 - The quantitative result

A quantitative evaluation of the given table of the earth magnetic field by means of the described numerical Fourier analysis, generated for the earth-magnetic flux density B, the following result:
all values in μT (mükrotesla)
 

 formula for the earth magnetic flux density B

 
As to be seen the variable itself disassembled the magnetic field in a static one, a sectoral (BS), a zonal (BZ) and a tesseral (BT) portion (see chapter 5).
There are only terms on in the equation like in the formula of Gauß (see
chapter 2.7). Then it is worth:
 
B = 47,2183 + BZ + BS + BT    µT (mükrotesla)

 

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