In the last 5 years, Lake Victoria water level has seen a dramatic fall that has caused alarm to water resource managers. Since the lake basin contributes about 20% of the lakes water in form of discharge, with 80% coming from direct rainfall, this study undertook a satellite analysis of the entire lake basin in an attempt to establish the cause of the decline. Gravity Recovery And Climate Experiment (GRACE), Tropical Rainfall Measuring Mission (TRMM) and CHAllenging Minisatellite Payload (CHAMP) satellites were employed in the analysis. Using 45 months of data spanning a period of 4 years (2002)(2003)(2004)(2005)(2006), GRACE satellite data are used to analyse the variation of the geoid (equipotential surface approximating the mean sea level) triggered by variation in the stored waters within the lake basin. 776 J.L. Awange et al. TRMM Level 3 monthly data for the same period of time are used to compute mean rainfall for a spatial coverage of .25 • × .25 • (25 × 25 km) and the rainfall trend over the same period analyzed. To assess the effect of evaporation, 59 CHAMP satellite's occultation for the period 2001 to 2006 are analyzed for tropopause warming. GRACE results indicate an annual fall in the geoid by 1.574 mm/year during the study period 2002-2006. This fall clearly demonstrates the basin losing water over these period. TRMM results on the other hand indicate the rainfall over the basin (and directly over the lake) to have been stable during this period. The CHAMP satellite results indicate the tropopause temperature to have fallen in 2002 by about 3.9 K and increased by 2.2 K in 2003 and remained above the 189.5 K value of 2002. The tropopause heights have shown a steady increase from a height of 16.72 m in 2001 and has remained above this value reaching a maximum of 17.59 km in 2005, an increase in height by 0.87 m.Though the basin discharge contributes only 20%, its decline has contributed to the fall in the lake waters. Since rainfall over the period remained stable, and temperatures did not increase drastically to cause massive evaporation, the remaining major contributor is the discharge from the expanded Owen Falls dam.

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Current satellite missions dedicated to global mapping of the Earth's gravity field are providing accurate global models of the geopotential. Harmonic (Stokes) coefficients of the geopotential derived from satellite observations of its functionals, a potential gradient vector and/or gravity gradient tensor, correspond to the (time-averaged) gravitational potential that is generated by the geoid, topography and atmosphere. The manuscript deals with the effect of static topographical and atmospheric masses on spaceborne observations of the potential gradient vector and gravity gradient tensor that should be applied during their inversion into the geopotential at the geoid level. They would allow for derivation of a harmonic representation of the potential generated only by solid masses inside the geoid and ocean, i.e., harmonicity of the geopotential would apply to the entire space outside the geoid. The geopotential could be then synthesized without problems with diverging harmonic series, i.e., the fundamental condition for application of a truncated harmonic series everywhere outside the geoid would be met. Due to its large numerical values, compensation of topographical masses is outlined using a singlelayer potential. Although not entirely sufficient from a geophysical point of view, this model is often applied in geoid-related computations. Corresponding parameters are evaluated in manners that are consistent with spatial resolution and accuracy of current spaceborne data using spherical harmonic representation of topographical heights and corresponding mass distributions. Theoretical formulations are followed by numerical evaluations.

Several procedures for solving, in a closed form the GPS pseudo-ranging four-point problem P4P in matrix form already exist. We present here alternative algebraic procedures using Multipolynomial resultant and Groebner basis to solve the same problem. The advantage is that these algebraic algorithms have already been implemented in algebraic software such as "Mathematica" and "Maple." The procedures are straightforward and simple to apply. We illustrate here how the algebraic techniques of Multipolynomial resultant and Groebner basis explicitly solve the nonlinear GPS pseudo-ranging four-point equations once they have been converted into algebraic (polynomial) form and reduced to linear equations. In particular, the algebraic tools of Multipolynomial resultant and Groebner basis provide symbolic solutions to the GPS four-point pseudo-ranging problem. The various forward and backward substitution steps inherent in the classical closed form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Multipolynomial resultant and Groebner basis approaches eliminate several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of univariate polynomial equations (in this case quadratic equations for the range bias expressed algebraically using the given quantities) whose roots can be determined by existing programs (e.g., the roots command in MATLAB).

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