  Let E be a ring spectrum for which the E-Adams spectral sequence converges.
We define a variant of Mahowald's root invariant called the `filtered root
invariant' which takes values in the E_1 term of the E-Adams spectral sequence.
The main theorems of this paper concern when these filtered root invariants
detect the actual root invariant, and explain a relationship between filtered
root invariants and differentials and compositions in the E-Adams spectral
sequence. These theorems are compared to some known computations of root
invariants at the prime 2. We use the filtered root invariants to compute some
low dimensional root invariants of v_1-periodic elements at the prime 3. We
also compute the root invariants of some infinite v_1-periodic families of
elements at the prime 3.
