In seismic Reverse-Time Migration (RTM), the random boundary condition (RBC) method can be used to reduce the amount of storage required to store the source or receiver wavefield. However, the RBC method generates low-frequency coherent noise in the wavefield, which is a remnant of the artificial reflections at the random boundaries. To solve this problem, we have proposed a frequency modulation method for the RBC-based RTM in which the dominant frequency of both the source and receiver wavefields is shifted upwards. This method not only mitigates the low-frequency coherent noise of the RBC method, but also improves the resolution of the RTM image.

Time–frequency analysis is widely used in the processing and interpretation of seismic data. The fractional-order Fourier transform can process nonstationary data by introducing a rotation angle into the time–frequency plane. To take advantage of the fractional-order Fourier transform and the$W$-transform, we introduce the fractional-order$W$-transform (FrWT) by introducing rotation and scaling parameters in the hope of producing time–frequency spectra with high-energy concentration and resolution. The rotation angle of the time–frequency plane introduces more characteristics into the time–fractional-frequency domain, making the algorithm more applicable to geophysical problems. We also propose a computational strategy to speed up the calculation. Numerical investigation of synthetic and real data shows that the proposed algorithm can achieve a time–frequency spectrum with higher resolution and energy concentration but at the cost of negligible additional computational effort.

In seismic full waveform inversion (FWI), the shot-encoding technique can reduce computational costs. It randomly encodes all individual shot gathers and combines them into a super shot gather. This allows the number of wavefield simulations to be reduced. To mitigate crosstalk noise caused by interference between different shots, shot-encoded FWI requires the shot encoding to be sufficiently random. However, this may destabilize the standard limited memory BFGS (L-BFGS) algorithm and lead to instability and nonconvergence in the interactive inversion of FWI. In this article, we propose the improved L-BFSG (iL-BFGS) method, which uses the formula of the outer product of the Hessian matrix to calculate the gradient difference for the recursive L-BFSG computation. We also redesigned the workflow for shot-encoded FWI to increase the randomness within the shot encoding. We have shown that the proposed method improves the image quality of shot-encoded FWI, especially for intermediate and deep formations, accelerates the convergence of iterative inversion, and improves the ability of FWI to suppress crosstalk effects.

In seismic inversion, the subsurface model can be parameterized by a truncated Fourier series, and the inversion problem is then the inversion of the Fourier coefficients. To improve the efficiency of the evaluation of the Fourier coefficients and the reconstruction of the model from the inverted coefficients, we propose to use the efficient implementation of the fast Fourier transform (FFT) to speed up these two calculations. By using the FFT pair, the computation time for 3D subsurface models with realistic size could be reduced by two to three orders of magnitude compared to conventional methods. When this model parameterization scheme is applied to seismic impedance inversion, we proposed two strategies to further improve the efficiency. One is to invert the Fourier coefficients from small-valued numbers to large-valued numbers, and the other is to divide the seismic data into subgroups and use part of them for the inversion of the Fourier coefficients. Both strategies are helpful for efficient inversion of the Fourier coefficients from the seismic data. Moreover, thanks to this model parameterization scheme, the Fourier coefficients are inverted in a multi-trace manner, and the impedance model reconstructed from the inverted Fourier coefficients has good spatial continuity. The scheme is able to generate stable and continuous impedance models from the inversion of seismic data with missing traces, with affordable computation times.

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 thel2norm 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.

For shot-encoded seismic waveform tomography, the restarted limited-memory BFGS (L-BFGS) algorithm is an effective technique to suppress the crosstalk effect among encoded seismic shots. It restarts the L-BFGS calculation at each iteration segment, consisted of a group of iterations, and recodes the individual shots randomly not only at the beginning but also at the inside of the iteration segment. Here, we simplified this scheme using an invariant shot-encoding within each iteration segment and recoding individual shots only at the beginning of the segment. This simplification did compromise the image quality at the early stage of inversion, as the crosstalk effect appeared on the inversion result of the low-frequency data. However, it eventually achieved both the computation efficiency and the good quality for a multiscale inversion procedure, which inverts the seismic data from low-frequency components to high-frequency components in sequence.

When seismic waves propagate through the earth, their high-frequency energy is absorbed by subsurface viscoelastic media. Seismic wavelet appears to be stretched out, as it is dominated by low-frequency components. In order to enhance seismic resolution, we propose here a wavelet compression method that utilizes the scale characteristic in the Fourier transform. The novelty of the scheme is a frequency-dependent scaling that extends the amplitude spectrum to both high- and low-frequency axes simultaneously. This is for the first time to make this frequency-dependent proposal, instead of a constant scaling scheme in the classic Fourier theory. It compresses seismic wavelet in the time domain and also simplifies the wavelet form effectively. This frequency-dependent scaling scheme leads to a transferring filter that is applicable to seismic field data. It results in an improvement in data resolution and in the ability of thin-layer identification, which will facilitate further seismic inversion and reservoir characterization.