Determining the electrical conductivity of the upper mantle using Sq Field

The atmospheric electric currents that varying in daily, seasonal, and latitude patterns above the Earth's surface act as a source that induces currents to flow in the conducting layers of the Earth. The magnitude, direction, and depth of penetration of the induced currents are determined by the characteristics of the source currents as well as the distribution of electrically conducting materials in the Earth. The solar quiet (Sq) magnetic field variation is a manifestation of an ionospheric current system. Heating at the dayside and cooling at the nightside of the atmosphere generates tidal winds which drive ionospheric plasma against the geomagnetic field inducing electric fields and currents in the dynamo region between 80-200 km in height. The current system remains relatively fixed to the Earth-sun line and produces regular daily variations which are directly seen in the magnetograms of geomagnetic “quiet” days, therefore the name Sq. The Sq field variations are dominated by 24-, 12-, 8-, and 6-hr spectral components and can penetrate in the conductive earth to depth between 100 to 600 km. For the situation in which field measurements are available about a spherical surface that separates the source from the induced currents (and a current doesn’t flow across this surface), Gauss (1838) devised a special solution of the differential electromagnetic field equations that is separable in the spherical coordinates r, θ , and φ . In Gauss’s solution, the field terms that represent radial dependence appear as two series. One with increasing powers of the sphere radius, r, and one with increasing powers of 1/r. As the value of r becomes larger (outward from the sphere) the first series produces an increased field strength, as if approaching external current sources. As the value of r decreases (toward the sphere center) the second series of 1/r terms indicate increased field strength, as if approaching internal current sources. Gauss had devised the way to separately represent the currents that were external and internal to his analysis At the Earth's surface the observed mixture of fields from the source and induced currents can be separated by spherical harmonic analyses and the relationship between the internal and external amplitudes and phases can be used to infer the Earth's conductivity profile at great depths. A spherical harmonic analysis (SHA) was applied to obtain a separation into internal and external field coefficients. The magnetic scalar potential, V, in colatitude θ and longitude φ described at the Earth's surface by
 
 
 
In which the cosine (A) and sine (B) coefficients of the expansion for the external (ex) and internal (in) parts are taken to be:
  
 
 
 
 
After separating the geomagnetic field into internal and external part by SHA we can use Schmucker’s (1970) method for profiling the Earth's substructure. In the method outlined by Schmucker formulas are developed that provide the depth (d) and conductivity () of apparent layers that would produce surface-field relationships similar to the observed components.
 
 
 
 
 
These profile values, need to be determined for each n, m set of SHA coefficients using the real z and imaginary p parts of a complex induction transfer function, , given as:
 
We calculate the Electrical conductivity properties of the upper by employing the 6, 8, 12, 24 hour spectral components of the quiet-day geomagnetic field variation. The Gauss coefficients obtained from an spherical harmonic analysis of the two components of the quiet daily variation field for the solar-quiet year 2009 were applied to Schmucker's model (Schmucker, 1970). The findings coincide with the results of previous solar quite years and demonstrate that electrical conductivity varies exponentially with depth between 150 and 530 Km.