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ABSTRACT We present an approach for generating stochastic scenario rupture models and semistochastic broadband seismic waveforms that include validated P waves, an important feature for application to early warning systems testing. There are few observations of large magnitude earthquakes available for development and refinement of early warning procedures; thus, simulated data are a valuable supplement. We demonstrate the advantage of using the Karhunen–Loève expansion method for generating stochastic scenario rupture models, as it allows the user to build in desired spatial qualities, such as a slip inversion, as a mean background slip model. For waveform computation, we employ a deterministic approach at low frequencies (<1  Hz) and a semistochastic approach at high frequencies (>1  Hz). Our approach follows Graves and Pitarka (2010) and extends to model P waves. We present the first validation of semistochastic broadband P waves, comparing our waveforms against observations of the 2014 Mw 8.1 Iquique, Chile, earthquake in the time domain and across frequencies of interest. We then consider the P waves in greater detail, using a set of synthetic waveforms generated for scenario ruptures in the Cascadia subduction zone. We confirm that the time-dependent synthetic P-wave amplitude growth is consistent with previous analyses and demonstrate how the data could be used to simulate earthquake early warning procedures.