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Mapping time-varying gravity via satellite-to-satellite tracking systems holds great potential as a new way to monitor the Earth's global climate system. Measurement noises and systematic deficiencies in sampling, both in time and space, cause global geoid or surface mass solutions to have a structured spherical harmonic error spectrum, with strong degree and order dependences and cross-correlations. To extract average values of geoid or surface mass variations around global gridpoints on Earth's surface and over various geographic regions, both the shape of the averaging kernel and the resulting average uncertainties must be considered quantitatively and statistically. We investigate two methods of the Backus and Gilbert continuous geophysical inverse formalism for optimal averages around points on Earth's surface. The first averaging kernel optimally approximates the Dirac-δ function. With an equivalent measure of deviation from the Dirac-δ function, the optimal average has greater (up to 2.6 times) accuracy than does the most widely used isotropic Gaussian filter for GRACE analysis. The second method was crafted to decrease the kernel weight as the distance from the point of interest increases. A new method is presented to use a modified Gaussian averaging kernel that reduces average uncertainties with minimum loss of resolution. The modified method has some advantages over using the kernel that optimally approximates the Dirac-δ function. Both methods are computationally efficient and are applied to simulated and real GRACE data to compute improved averages around fine-resolution global gridpoints and used with non-diagonal covariance matrices to intelligently reduce effects of correlated errors. The optimal probabilistic method of least squares with a priori information is discussed in the spherical harmonic domain. The property of optimality will be preserved when the estimates are mapped to the geographic domain for spatial averages. A regionally-bounded Gaussian a priori function is derived in the spherical harmonic domain to better represent different change regimes separated by major geographic boundaries. We also introduce an algorithm to derive the optimal regional average incorporating a constraint such that the average weight over the region is unity. Applications of such more realistic a priori information (and/or constraint) can produce improved average estimates using satellite gravity data. [ABSTRACT FROM AUTHOR]