In this paper, we present an analytical study on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from a pulsating vertical throughflow. We applied two methods to analyze this problem: the frozen profile and the averaged equations. We found that the two approaches lead to essentially the same results and discussed physical implications of this finding, in particular, with respect to fundamental differences between the throughflow modulation and the temperature modulation problems.