When the molecules and atoms of the atmosphere receive enough external energy, one or more electrons are dissociated from the molecules or atoms. This process is called ionization. The solar ultraviolet (EUV) radiation and particle precipitation are the two primary energy sources in the ionization. Also cosmic radiation contributes to this ionization. This layer of atmosphere is called ionosphere. The ionosphere is that part of the atmosphere in which the number of free electrons is so high that, it significantly affects the propagation of radio waves. Ionospheric refraction is one of the main error sources on GPS signals. This effect is proportional to the total electron content (TEC). TEC is a projection of electron density along signal path extending from the satellite to the receiver on the ground. The unit of TEC is TECU and 1 TECU equals 10^{16} electrons/m^{2}. Production of free electrons in the ionosphere depends on many factors, such as solar, geomagnetic, gravitational and seismic activity period.
There are many methods to obtain electron density or TEC profiles and predictions. In early time, direct measurements such as ionosonde was a kind of effective instrument to achieve this purpose. Later, some empirical and mathematical models were developed. For example, IRI (international reference ionosphere) model, PIM (the parameterized ionospheric model) are empirical models. Mathematical models divided to two categories: single-layer (2-D) and multi-layer (3-D & 4-D). The existing 2-D methods of modeling the electron density can be classified to non-grid based and grid based techniques. The former modeling techniques are based on the least squares estimation of a functional model for certain types of observables derived from the GPS carrier phase and code measurements. Polynomials and spherical harmonics are some of the base functions that are commonly in use. In grid based modeling, the spherical shell of free electrons is developed into a grid of rectangular elements. Special reconstruction algorithms are then used for estimating the electron density within the every element of the shell. Neglecting the vertical gradient of the electron density is the main deficiency of the two dimensional modeling techniques.
To study the physical properties of the ionosphere, computerized tomography (CT) demonstrated an efficient and effective manner. Due to the sparse distribution of GPS stations and viewing angle limitations, ionospheric electron density (IED) reconstruction is an ill-posed inverse problem. Usually, iterative or non- iterative algorithm used for electron density reconstruction. Non- iterative algorithms are known regularization methods. Using these methods to solve the ill posed problems will produce bias in final results. In this paper, we used hybrid regularization algorithm for solving ionosphere tomography. This method is a combination of two regularizations methods: Tikhonov regularization and total variation (TV). Tikhonov regularization is a classical method for solving ill-posed inverse problem and total variation effectively resists noise in results. To apply the method for constructing a 3D-image of the electron density, GPS measurements of the Iranian permanent GPS network (at 3-day in 2007) have been used. The modeling region is between 24^{0} to 40^{0} N and 44^{0} to 64^{0} W. The result of hybrid regularization method has been compared to that of the zero order Tikhonov regularization method and NeQuick model outputs. The minimum relative error for hybrid method is 1.55% and the maximum relative error is 19.52%. Also, maximum and minimum absolute error is computed 1.32×10^{11} (ele/m^{3}) and 6.67×10^{11} (ele/m^{3}), respectively. Experiments demonstrate the effectiveness, and illustrate the validity and reliability of the proposed method. |