Correcting the Negative Values of the Retrieved Ionospheric Electron Density Profiles Using the NNLS Algorithm

F. Ghafoori

Abstract:	The Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) is a joint project between the United States and Taiwan for weather forecasting, climate monitoring and research on space weather. The COSMIC six micro satellites were launched successfully on April 15, 2006 with the aim of tracking the radio signals from the Global Positioning System (GPS) as they pass through Earth's atmosphere. During the radio occultation event a micro satellite (herein referred to as Low Earth Orbit (LEO) satellite) rises or sets relative to one of the 24 GPS satellites at the Earth's horizon. Thus the atmospheric layers are sounded at different altitudes by radio waves propagating from the GPS satellite to the LEO satellite. From the observed phase path increases, which are due to refraction of the radio waves by the ionosphere and the neutral atmosphere, the atmospheric refractivity, ionospheric electron density and other parameters including temperature and pressure in the neutral atmosphere are calculated with high accuracy. In the current work ionospheric electron density profiles are derived based on radio occultation technique using (1) the Abel integral inversion and (2) the standard Least Square (LS) estimation technique. The comparison between the results of these two approaches shows mean error values of less that %1. It often happens that the retrieved electron density profiles contain some negative values mostly at lower altitudes (<100 km). Since electron density is a positive parameter, it is believed that the negative values are probably due to the poor geometry, erroneous, biased and noisy data or poor mathematical approaches. For solving the problem of negative values in the retrieved electron density profiles the Non-Negative Least Squares (NNLS) estimation algorithm by Lawson and Hanson (1974) is employed in order to obtain the final results in the least squares mode and at the same time to apply some constraints on the solutions to be positive or stay close to zero as much as possible. Since the invention of NNLS in 1974, it has extensively been used in different fields including applied mathematics, physics, chemistry, medical science, economics, etc. However, The NNLS estimation is quite a new topic in the field of GPS applications and ionospheric studies and is introduced for the first time in this paper for deriving the ionospheric electron density profiles. An introduction to NNLS and the solution algorithm is given in detail in one of the following sections. The final results show lower RMSE in NNLS estimation technique compared with the LS estimation and the Abel inversion algorithm.
Published in:	Proceedings of the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2009) September 22 - 25, 2009 Savannah International Convention Center Savannah, GA
Pages:	2518 - 2529
Cite this article:	Ghafoori, F., "Correcting the Negative Values of the Retrieved Ionospheric Electron Density Profiles Using the NNLS Algorithm," Proceedings of the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009, pp. 2518-2529.
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