Tall buildings and flexible structures require a better characterization of long period ground motion spectra than the one provided by current seismic building codes. Motivated by that, a methodology is proposed and tested to improve recorded and synthetic ground motions which are consistent with the observed co-seismic displacement field obtained from interferometric synthetic aperture radar (InSAR) analysis of image data for the Tocopilla 2007 earthquake (Mw=7.7) in Northern Chile. A methodology is proposed to correct the observed motions such that, after double integration, they are coherent with the local value of the residual displacement. Synthetic records are generated by using a stochastic finite-fault model coupled with a long period pulse to capture the long period fling effect.It is observed that the proposed co-seismic correction yields records with more accurate long-period spectral components as compared with regular correction schemes such as acausal filtering. These signals provide an estimate for the velocity and displacement spectra, which are essential for tall-building design. Furthermore, hints are provided as to the shape of long-period spectra for seismic zones prone to large co-seismic displacements such as the Nazca-South American zone.