I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose levels lie "between" subsequent levels of cellular filtrations (and cones of comparison maps are shifted Moore spectra). An interesting particular case is the filtration that gives the "canonical" filtration on singular homology (note that this is not the Postnikov $t$-structure filtration, since the latter is "much further from the cellular one"); the levels of this filtration for finite spectra are finite as well (and "quotients" are the shifted Moore spectra corresponding to $H_*(E)$).
Did anybody study anything similar previously? Can you suggest me any possible applications for filtrations of this sort?
