Determination of the electrical conductivity and temperature of the upper mantle using geomagnetic quiet-daily variations

The electrical conductivity of the earth is its physical parameter, which can be studied in global scales. The investigations of the earth's electrical conductivity are based on geomagnetic field variations. Time variations of the geomagnetic field consist of long-term and short-term (transient) variations. External sources induce electric currents into the earth. The induced electric currents give rise, in turn, to an internal component of the magnetic variations observed at the surface. Transient magnetic fields (external sources) pass through electrically conducting earth with amplitude reduction and phase rotation. We are concerned here with the inductive skin-effect of natural geomagnetic variations which they undergo within the earth's interior. These variations are very slowly oscillating and can be regarded as quasi-stationary on a global scale. There is a measure of the penetration of an alternating magnetic field of frequency (or period ) into a conductor of conductivity . The penetration is often expressed by the skin depth S, given by (in terms of kilometers):
(1)
Geomagnetic induction studies involve frequencies from a few cycles per minute to fractions of a cycle per day. So the penetration depth at which the main inductive attenuation of geomagnetic variation occurs depends on the period of the variation and electrical conductivity of the earth.
The inductive response of the earth's interior to the spectrum of geomagnetic variations and thereby the internal electrical conductivity distribution can be studied by two complementary methods: we can observe (i) the vertical magnetic component Z (magnetic method) or (ii) the tangential electrical component E (magnetotelluric method) of a transient surface field, setting either one of them in relation to the horizontal magnetic component H. The magnetic method is applied in this study. The available geomagnetic data consists of records of three components X, Y, Z (horizontal
north, horizontal east and vertical components) or Z, D, H (vertical, declination and horizontal components) of the earth's magnetic field as functions of time recorded at a number of points distributed over the surface of the earth. By separating magnetic variations into parts of internal (i) and external (e) origin, we can determine the electromagnetic response of the earth to a particular input (e). The ratio of the parts of the magnetic field of internal and external origin is a measure of the response and is dependent on both the external current system and the distribution of electrical conductivity within the earth.
The geomagnetic quiet-daily variations are the most persistent of all the geomagnetic variations that occur in days when the magnetic activity level is very low. In these days the magnetograms from any (except a high latitude) observatory show smooth pattern with no, or only very small, rapid fluctuations. index is the measure for selecting the quiet day. The geomagnetic bays are the events that occur in geomagnetic quiet days and their periods are between 30 minutes to 3 hours.
Once the geomagnetic variations at each observatory have been analyzed in terms of frequency, the spatial behavior can be expressed by expanding each frequency component in a series of spherical harmonics over the surface of the earth. The geomagnetic field outside the earth can be expressed as the gradient of a scalar potential as (Banks 1969):
(2)
The potential can be represented as a series of spherical harmonics; in this particular case the harmonics are purely zonal:
(3)
The coefficients and , corresponding to the internal and external parts of the field, respectively. The horizontal and vertical components of the geomagnetic field at the earth's surface are derived from as: