Only with satellites it is possible to cover the entire Earth densely with gravityfield related measurements of uniform quality within a short period of time. However,due to the altitude of the satellite orbits, the signals of individual local massesare strongly damped. Based on the approach of Petrovskaya and Vershkov we determinethe gravity gradient tensor directly from the spherical harmonic coefficientsof the recent EIGEN-GL04C combined model of the GRACE satellite mission. Satellitegradiometry can be used as a complementary tool to gravity and geoid informationin interpreting the general geophysical and geodynamical features of theEarth. Due to the high altitude of the satellite, the effects of the topography and theinternal masses of the Earth are strongly damped. However, the gradiometer data,which are nothing else than the second order spatial derivatives of the gravity potential,efficiently counteract signal attenuation at the low and medium frequencies.In this article we review the procedure for estimating the gravity gradientcomponents directly from spherical harmonics coefficients. Then we apply thismethod as a case study for the interpretation of possible geophysical or geodynamicalpatterns in Iran. We found strong correlations between the cross-components ofthe gravity gradient tensor and the components of the deflection of vertical, and weshow that this result agrees with theory. Also, strong correlations of the gravityanomaly, geoid model and a digital elevation model were found with the diagonalelements of the gradient tensor.