Weighted Envelope Correlation-Based Waveform Inversion Using Automatic Differentiation

Full-waveform inversion (FWI) is a popularly used high-resolution seismic inversion method. It relies on the measure of the misfit between observed data and predicted data. Due to the sinusoidal nature of seismic waves, a direct comparison of observed data and predicted data using the l ₂ norm may cause cycle skipping. A variety of objective functions for FWI have been proposed to resolve this issue over the years. Based on the gradient optimization method, an explicit expression of the model gradient of the defined objective function is needed to be derived and calculated. This complicated step can be circumvented by using an automatic gradient calculation technique, called automatic differentiation (AD). AD allows calculation the gradients of the model parameters, as well as those of the inputs using the chain rule. Taking advantage of the deep-learning framework, FWI with different objective functions can be automatically optimized using AD. To improve the accuracy and applicability of FWI on real data, we propose a new objective function that we refer to as the weighted envelope-correlation inversion (WECI), which combines two correlation-based waveform inversions. The weights imposed on these two terms in this new objective function can be dynamically adjusted by the sigmoid function during the optimization process. We show the versatility and effectiveness of AD-based waveform inversions using different objective functions through numerical tests. We also demonstrate the superiority of the proposed WECI method on synthetic data and real data.