Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme.

One of reverse-time migration’s main limitations is that an unscaled adjoint operator is prone to produce images with low resolution, inaccurate amplitudes, and even artifacts. Least-squares reverse-time migration (LSRTM) was subsequently introduced to mitigate this inadequacy via the use of an approximation to an inverse operator. LSRTM suffers from its own limitations, most importantly from poor condition, which often manifests itself as image artifacts. One approach to ameliorate this issue is to constrain the optimization problem by introducing a penalty term to the cost function. Penalizing estimated parameters for sparsity is one such constraint that has been shown to be effective. A drawback of this technique is that it introduces a trade-off between data fitting and image sparsity. Furthermore, if using the Cauchy constraint, an additional trade-off is introduced due to the requirement to estimate a hyper-parameter. We introduce an alternative approach that mitigates these trade-offs by combining a multiplicative cost function with an effective means for determining the Cauchy hyper-parameter. We also introduce a new formulation of the multiplicative cost function that avoids over-penalization by the constraint via the introduction of a relaxation term. Finally, we seek to improve the computational efficiency by introducing a new approach for computing the step length. As such, our method introduces three novel aspects to constrained LSRTM: (i) a relaxed multiplicative cost function, (ii) semi-automatic estimation of the Cauchy hyper-parameter, and (iii) efficient computation of the step length. We discuss the theory and implementation, followed by application to three synthetic datasets and a real ultrasonic dataset. Given the presence of large salt bodies, elasticity, and noise, along with the directivity of piezoelectric ultrasonic transducers, these datasets provide a challenging test of the approach outlined. Results demonstrate that our method is robust in handling the challenges imposed by these scenarios.

Adaptive waveform inversion (AWI) is one of a new breed of full-waveform inversion (FWI) algorithms that seek to mitigate the effects of cycle skipping (Warner & Guasch, 2016). The phenomenon of cycle skipping is inherent to the classical formulation of FWI, owing to the manner in which it tries to minimize the difference between oscillatory signals. AWI avoids this by instead seeking to drive the ratio of the Fourier transform of the same signals to unity. One of the strategies most widely employed by FWI practitioners when trying to overcome cycle skipping, is to introduce progressively the more nonlinear components of the data, referred to as multiscale inversion. Since AWI is insensitive to cycle skipping, we assess here whether this multiscale approach still provides an appropriate strategy for AWI.

Finite difference is the most widely used method for seismic wavefield modeling. However, most finitedifference implementations discretize the Earth model over a fixed grid interval. This can lead to irregular model geometries being represented by 'staircase' discretization, and potentially causes mispositioning of interfaces within the media. This misrepresentation is a major disadvantage to finite difference methods, especially if there exist strong and sharp contrasts in the physical properties along an interface. The discretization of undulated seabed bathymetry is a common example of such misrepresentation of the physical properties in finite-difference grids, as the seabed is often a particularly sharp interface owing to the rapid and considerable change in material properties between fluid seawater and solid rock. There are two issues typically involved with seabed modeling using finite difference methods: firstly, the travel times of reflections from the seabed are inaccurate as a consequence of its spatial mispositioning; secondly, artificial diffractions are generated by the staircase representation of dipping seabed bathymetry. In this paper, we propose a new method that provides a solution to these two issues by positioning sharp interfaces at fractional grid locations. To achieve this, the velocity The Unconventional Natural Gas Institute, China University of Petroleum (Beijing), Beijing 102249, China model is first sampled in a model grid that allows the center of the seabed to be positioned at grid points, before being interpolated vertically onto a regular modeling grid using the windowed sinc function. This procedure allows undulated seabed bathymetry to be represented with improved accuracy during modeling. Numerical tests demonstrate that this method generates reflections with accurate travel times and effectively suppresses artificial diffractions.

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