A simple method is developed for predicting ground motions for future large earthquakes for specific sites by summing and filtering recordings of adjacent small earthquakes. This method is tested by simulating strong-motion records for the Loma Prieta earthquake (M 7.0) using aftershocks (M 3.7 to 4.0) recorded at the same sites. I use an asperity rupture model where the rms stress drop averaged over the fault plane is constant with moment. The observed spectra indicate that stress drop remains constant from the M 3 aftershocks up to the M 7 mainshock, about six orders of magnitude in seismic moment. Each simulation sums the seismogram of one aftershock with time delays appropriate for propagating rupture and incorporates directivity and site response. The simulation scales the spectrum in accordance with a constant stress drop, ω−2 source model. In this procedure, the high-frequency energy of the aftershock sum above the corner frequency of the aftershock is not reduced when it is convolved with the mainshock slip velocity function, unlike most previous methods of summation. For most cases, the spectra (0.6 to 20 Hz), peak accelerations, and durations of the simulated mainshock records are in good agreement with the observed strong-motion records, even though only one aftershock waveform was used in each simulation. This agreement indicates that the response of these soil sites is essentially linear for accelerations up to about 0.3 g. The summed aftershock records display the same site-dependent values of fmax as the mainshock records, implying that fmax is a site effect rather than a property of the mainshock rupture process.