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Producing reliable acoustic subsurface velocity models still remains the main bottleneck of the oil and gas industry's traditional imaging sequence. In complex geological settings, the output of conventional ray-based or wave-equation-based tomographic methods may not be accurate enough for full waveform inversion (FWI) to converge to a geologically satisfactory Earth model. We create a new method referred to as full waveform inversion by model extension (FWIME) in which a wave-equation migration velocity analysis (WEMVA) technique is efficiently paired with a modified version of FWI. We show that our method is more powerful than applying WEMVA and FWI sequentially, and that it is able to converge to accurate solutions without the use of a good initial guess or low-frequency energy. We demonstrate FWIME's potential on five realistic and challenging numerical examples that simulate complex geological scenarios often encountered in hydrocarbon exploration. We guide the reader step by step throughout the optimization process. We show that our method can simultaneously invert all wave types with the same simple mechanism and without the need for a user-intensive hyper-parameter tuning process. In an online repository, we provide a fully-reproducible open-source software solution implemented with general-purpose graphics processing units (GPU) and with a user-friendly Python interface.