We calculate near-source broadband (0–10 Hz) seismograms by combining low-frequency three-dimensional (3D) finite-difference seismograms (0–0.5 Hz) computed in a 3D velocity model using site-specific scattering Green’s functions for random, isotropic scattering media. The scattering Green’s functions are convolved with a slip-rate function to form local scattering operators (scatterograms), which constitute the high-frequency scattered wave field. The low-frequency and high-frequency scatterograms are then combined in the frequency domain to generate broadband waveforms. Our broadband method extends the Mai et al. (2010) approach by incorporating dynamically consistent source-time functions and accounting for finite-fault effects in the computation of the high-frequency waveforms. We used the proposed method to generate broadband ground motions at 44 sites located 5–100 km from the fault, for M_w 7.7 earthquake scenarios (TeraShake) on the southern San Andreas fault, which include north-to-south, south-to-north, and bilateral rupture propagation from kinematic and spontaneous dynamic rupture models. The broadband ground motions computed with the new method are validated by comparing peak ground acceleration, peak ground velocity, and spectral acceleration with recently proposed ground-motion prediction equations (GMPEs). Our simulated ground motions are consistent with the median ground motions predicted by the GMPEs. In addition, we examine overturning probabilities for 18 precariously balanced rock sites (PBR). Our broadband synthetics for the M_w 7.7 TeraShake scenarios show no preferred rupture direction on the southern San Andreas fault but are inconsistent with the existence of PBRs at several of the sites analyzed.