For a partially hyperbolic attractor with a center bundle splitting in a dominated way into one-dimensional subbundles we show that for Lebesgue almost every point there is an empirical measure from x with a SRB component. Moreover if the center exponents are non zero, then x lies in the basin of an ergodic hyperbolic SRB measure and there are only  finitely many such measures. This gives another proof of the existence of SRB measures in this context, which was established firstly in [11] by using random perturbations. Moreover this generalizes results of [15, 18] which deal with a single one-dimensional center subbundle.