**Authors**

**Abstract**

A fluctuating electric current flowing in the Earth’s atmosphere causes corresponding electric currents to flow in the conducting Earth below the source current. The depth of penetration of the induced currents is determined by the characteristics of the source currents as well as the distribution of electrically conducting materials in the Earth. At the Earth’s surface observatories measure the composite of external (source) and internal (induced) field components from the currents. The quiet daily field variations, called Sq for 'solar quiet-time', provide a natural signal source with frequencies appropriate to upper mantle conductivity studies. This paper is concerned with the quiet-day field variations, their separation into external and internal contributions, and the use of this separation to profile the electrical conductivity of the Earth's upper mantle.

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,? and ? 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 increases (outward from the sphere) the first series produces 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 sphere. For these external and internal series, there are individual spherical harmonic analysis (SHA) polynomial (Legendre) terms, each having two indices, degree m and order n. The full field is then represented as paired (external and internal Legendre terms) elemental parts, each satisfying the physical laws.

Gauss applied the SHA method to the global field observations and verified that most of the Earth’s main field originated from internal sources. By the turn of the century, the method was also used to show that the daily, quiet-time geomagnetic field variations came mostly from sources of current external to the Earth (Schuster, 1889 and 1908); this finding led to the discovery of the ionosphere. We will analyze the quiet fields with this SHA to separate the Sq ionospheric source currents from the induced currents within the Earth.

On days undisturbed by solar-terrestrial field and particle activity the geomagnetic records from a surface observatory display a smooth variation of field during the daylight hours. These variations are dominated by 24-, 12-, 8-, and 6-hr spectral components that change slowly from day to day through the seasons. Some authors prefer to select quiet days by a limiting value of the day's Ap (e.g., Ap = 10). Others take a fixed number (e.g., five) of the quietest days for a given month, whatever the values may be. We have preferred to select those days for which no Kp index exceeded a limiting level (e.g., 2+) because active records can cause unrealistic conductivity computations.

The 19 North American observatories are selected in this study. There are 60 days in 1997 in which the global geomagnetic disturbance index, Kp, have all 8 daily values less than 2+. These days are taken as preliminary "quiet day" recordings. All observatories have 60-min sample records. The original recordings of field are in Universal Time (UT) as orthogonal north, east, and into-the-earth components of field as X, Y, and Z. The data for each component are Fourier analyzed for each quiet day.

In the method outlined by Schmucker (1970) for profiling the Earth's substructure, formulas are developed that provide the depth (d) and conductivity ( ) of apparent layers (substitute conductors) that would produce surface-field relationships similar to the observed components. These profile values,

(1)

and

(2)

need to be determined for each n, m set of SHA coefficients using the real and imaginary parts of a complex induction transfer function, , given as:

(3)

Schmucker (1970) showed that the transfer function is obtained from the ratio of the field components, for a given n, m, as:

(4)

(5)

where X, Y and Z are the northward, eastward, and into the Earth field components in gammas, R is the Earth's radius in kilometers, ? is the colatitude at the field measurement location, and is the Schmidt normalized associated Legendre polynomial. To use the above conductivity-depth formulation, it is now necessary to find the individual n, m SHA terms of the three field components at the field-measurement location. With the fitted potential function (V) given as:

(6)

in which the cosine (A) and sine (B) coefficients of the expansion for the external (ex) and internal (in) parts are taken to be:

(7)

Campbell and Anderssen (1983) showed that Equations (1) and (2) may be determined directly from the SHA coefficients with the expressions

(8)

(9)

Electrical conductivity properties of the upper mantle for a North American sector of the Earth have been determined using the 24-, 12-, 8-, and 6-hr spectral components of the quiet-day geomagnetic variations. Spherical harmonic coefficients obtained from an analysis of the three components of the quiet daily variation (Sq) field for the solar-quiet year of 1997 were applied to a modeling procedure that was modified from Schmucher's (1970) publication. From a depth of about 100 to 650 km, the conductivity, , may be represented by with d is the depth in kilometers. Small perturbations of conductivity indicating some layering at 100 to 400 and 400 to 650 km correspond to the similar behavior of the Earth's density in these regions. From temperature-depth models we infer that the multiphase bulk properties of the expected silicates in these regions behave approximately as with T as the temperature in Kelvin.

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,? and ? 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 increases (outward from the sphere) the first series produces 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 sphere. For these external and internal series, there are individual spherical harmonic analysis (SHA) polynomial (Legendre) terms, each having two indices, degree m and order n. The full field is then represented as paired (external and internal Legendre terms) elemental parts, each satisfying the physical laws.

Gauss applied the SHA method to the global field observations and verified that most of the Earth’s main field originated from internal sources. By the turn of the century, the method was also used to show that the daily, quiet-time geomagnetic field variations came mostly from sources of current external to the Earth (Schuster, 1889 and 1908); this finding led to the discovery of the ionosphere. We will analyze the quiet fields with this SHA to separate the Sq ionospheric source currents from the induced currents within the Earth.

On days undisturbed by solar-terrestrial field and particle activity the geomagnetic records from a surface observatory display a smooth variation of field during the daylight hours. These variations are dominated by 24-, 12-, 8-, and 6-hr spectral components that change slowly from day to day through the seasons. Some authors prefer to select quiet days by a limiting value of the day's Ap (e.g., Ap = 10). Others take a fixed number (e.g., five) of the quietest days for a given month, whatever the values may be. We have preferred to select those days for which no Kp index exceeded a limiting level (e.g., 2+) because active records can cause unrealistic conductivity computations.

The 19 North American observatories are selected in this study. There are 60 days in 1997 in which the global geomagnetic disturbance index, Kp, have all 8 daily values less than 2+. These days are taken as preliminary "quiet day" recordings. All observatories have 60-min sample records. The original recordings of field are in Universal Time (UT) as orthogonal north, east, and into-the-earth components of field as X, Y, and Z. The data for each component are Fourier analyzed for each quiet day.

In the method outlined by Schmucker (1970) for profiling the Earth's substructure, formulas are developed that provide the depth (d) and conductivity ( ) of apparent layers (substitute conductors) that would produce surface-field relationships similar to the observed components. These profile values,

(1)

and

(2)

need to be determined for each n, m set of SHA coefficients using the real and imaginary parts of a complex induction transfer function, , given as:

(3)

Schmucker (1970) showed that the transfer function is obtained from the ratio of the field components, for a given n, m, as:

(4)

(5)

where X, Y and Z are the northward, eastward, and into the Earth field components in gammas, R is the Earth's radius in kilometers, ? is the colatitude at the field measurement location, and is the Schmidt normalized associated Legendre polynomial. To use the above conductivity-depth formulation, it is now necessary to find the individual n, m SHA terms of the three field components at the field-measurement location. With the fitted potential function (V) given as:

(6)

in which the cosine (A) and sine (B) coefficients of the expansion for the external (ex) and internal (in) parts are taken to be:

(7)

Campbell and Anderssen (1983) showed that Equations (1) and (2) may be determined directly from the SHA coefficients with the expressions

(8)

(9)

Electrical conductivity properties of the upper mantle for a North American sector of the Earth have been determined using the 24-, 12-, 8-, and 6-hr spectral components of the quiet-day geomagnetic variations. Spherical harmonic coefficients obtained from an analysis of the three components of the quiet daily variation (Sq) field for the solar-quiet year of 1997 were applied to a modeling procedure that was modified from Schmucher's (1970) publication. From a depth of about 100 to 650 km, the conductivity, , may be represented by with d is the depth in kilometers. Small perturbations of conductivity indicating some layering at 100 to 400 and 400 to 650 km correspond to the similar behavior of the Earth's density in these regions. From temperature-depth models we infer that the multiphase bulk properties of the expected silicates in these regions behave approximately as with T as the temperature in Kelvin.

**Keywords**