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Abstract. Four different basin functions are developed to estimate water storage variations within individual river basins from time variations in the Stokes coefficients now available from the GRACE mission. The four basin functions are evaluated using simulated data. Basin functions differ in how they minimize effects of three major error sources: measurement error; leakage of signal from one region to another; and errors in the atmospheric pressure field removed during GRACE data processing. Three of the basin functions are constant in time, while the fourth changes monthly using information about the signal (hydrologic and oceanic load variations). To test basin functions performance, Stokes coefficient variations from land and ocean models are synthesized, and error levels 50 and 100 times greater than pre-launch GRACE error estimate are used to corrupt them. Errors at 50 times pre-launch estimates approximately simulate current GRACE data. GRACE recovery of water storage variations is attempted for five different river basins (Amazon, Mississippi, Lena, Huang He and Oranje), representing a variety of sizes, locations, and signal variance. In the large basins (Amazon, Mississippi and Lena), water storage variations are recovered successfully at both error levels. As the error level increases from 50 to 100 times, basin functions change their shape, yielding less atmospheric pressure error and more leakage error. Amplitude spectra of measurement and atmospheric pressure errors have different shapes, but the best results are obtained when both are used in basin function design. When high-quality information about the signal is available, for example from climate and ocean models, changing the basin function each month can reduce leakage error and improve estimates of time variable water storage within basins. [ABSTRACT FROM AUTHOR]