The representation of dynamic ocean topography (&zgr;) through spherical harmonic (SH) and orthonormal (ON) expansions was studied using TOPEX altimeter data, three potential coefficient models used to define geoid undulations, and three estimates of &zgr; from oceanographic data and global circulation models (GCMs). The ON expansions are desirable when one wishes to study the spectral characteristics of a function in a defined domain such as the ocean. The potential coefficient models tested were JGM-2, JGM-3, and GRIM4--C4b. Each model was augmented with the OSU91A potential coefficients from degree 71 to 360. The &zgr; models were those of Levitus <1982> and values implied by the POCM--4B (Semtner/Chervin) model and a Los Alamos National Laboratory Model POP(96) (Malone, Smith, Dukowicz). The latter two models were defined over a 2-year time period. Values of &zgr; were computed from 2 years of TOPEX data using the three potential coefficient models. The ON expansions of &zgr; from the TOPEX data were then compared to the estimates from the oceanographic data. The differences, to ON degree 14, with the POCM--4B model and the TOPEX results were ¿14.0 cm (JGM-2), ¿12.4 cm (JGM-3), and ¿14.4 cm (GRIM4--C4b). A comparison with the other &zgr; estimates using TOPEX/JGM-3 gives differences of ¿14.3 cm (Levitus) and ¿13.3 cm . The comparisons were made only to degree 14 because (1) the correlation between the &zgr; coefficients from TOPEX data and POCM--4B fell off beyond degree 14 and (2) the geoid undulation accuracy, in the ocean region, was equal to the &zgr; signal near degree 14. These results suggest &zgr; estimates made above degree 14 may be contaminated by geoid undulation errors. Also suggested from the comparisons was that the TOPEX/JGM-3 estimates of &zgr; were more reliable than those from oceanographic data to degree 8 (2500-km resolution). The &zgr; estimates from the POCM--4B and POP(96) models, 2-year averages, agreed well north of 40 ¿S. Below this the differences could reach 40 cm in the Antarctic Circumpolar Current (60 ¿S, 215¿). The differences between the TOPEX/JGM-3 and POCM--4B &zgr; estimates exceeded 20 cm in a number of places . The largest differences (-62 cm) occurred in the Banda Sea. The &zgr; representations were used to calculate upper ocean geostrophic velocities in the east/west and north/south directions. Excluding a 10¿ band on either side of the equator, the difference (TOPEX versus POCM--4B) was ¿2.5 cm/s with the magnitude of the total velocity being 4.8 cm/s. The difference was consistent with the error estimates of the velocities implied by the errors in the JGM-3 coefficients to degree 14. The &zgr; estimates were also determined from four recent mean sea surface grids and the results compared to the POCM--4B model through the ON representation. The MSS grids used were the OSUMSS95, the UTCSRMSS95, the GFZ/D-PAF MSS95A, and the CNES/GRGS MSS95. The best agreement, to degree 14, was found with the OSUMSS95 (¿11.1 cm) and the CSRMSS95 (¿11.5 cm). The comparisons were poorer (¿15 cm) when a mean sea surface was used where no mean inverted barometer correction had been applied to the gridded data. Although substantial progress has been made in the past 10 years in the determination of the Earth's gravitational potential, the accuracy limitations of geoid undulation determination still hinder the comparison and assimilation of altimeter data and oceanographic data. The need for a dedicated gravity satellite mission, to yield improved geoid undulation determinations, is clearly seen. ¿ American Geophysical Union 1996