Full waveform inversion method using envelope objective function without low frequency data

•An envelope-based objective function FWI method is proposed.•The gradient is computed efficiently with the adjoint state method.•Our method can decrease the nonlinearity of the traditional FWI.•The adjoint source is rich in low frequency and a smooth gradient can be obtained.•Our method can provide an accurate initial model for the traditional FWI.

Full waveform inversion (FWI) has been a successful tool to build high resolution velocity models, but it is affected by a local minima problem. The conventional multi-scale strategy to tackle this severe problem may not work for real seismic data without long offsets and low frequency data. We use an envelope-based objective function FWI method to provide the long wavelength components of the velocity model for the traditional FWI. The gradient can be computed efficiently with the adjoint state method without any additional computational cost. Simple models are used to prove that the envelope-based objective function is more convex than the traditional misfit function, thus the cycle-skipping problem can be mitigated. Due to the envelope demodulation effect, the adjoint source of the envelope-based FWI contains abundant low frequency information, therefore the gradient tends to sense the low wavenumber model update. A Marmousi synthetic data example illustrates that the envelope-based FWI method can provide an adequately accurate initial model for the traditional FWI approach even when the initial model is far from the true model and low-frequency data are missing.