Zedong Wu

The natural combination of full and image‐based waveform inversion

Integrating migration velocity analysis and full waveform inversion can help reduce the high non-linearity of the classic full waveform inversion objective function. The combination of inverting for the long and short wavelength components of the velocity model using a dual objective function that is sensitive to both components is still very expensive and have produced mixed results. We develop an approach that includes both components integrated to complement each other. We specifically utilize the image to generate reflections in our synthetic data only when the velocity model is not capable of producing such reflections. As a result, we get the migration velocity analysis working when we need it, and we mitigate its influence when the velocity model produces accurate reflections (possibly first for the low frequencies). This is achieved using a novel objective function that includes both objectives. Applications to a layered model and the Marmousi model demonstrate the main features of the approach.

Migration velocity analysis using pre-stack wave fields

ABSTRACT Using both image and data domains to perform velocity inversion can help us resolve the long and short wavelength components of the velocity model, usually in that order. This translates to integrating migration velocity analysis into full waveform inversion. The migration velocity analysis part of the inversion often requires computing extended images, which is expensive when using conventional methods. As a result, we use pre‐stack wavefield (the double‐square‐root formulation) extrapolation, which includes the extended information (subsurface offsets) naturally, to make the process far more efficient and stable. The combination of the forward and adjoint pre‐stack wavefields provides us with update options that can be easily conditioned to improve convergence. We specifically use a modified differential semblance operator to split the extended image into a residual part for classic differential semblance operator updates and the image (Born) modelling part, which provides reflections for higher resolution information. In our implementation, we invert for the velocity and the image simultaneously through a dual objective function. Applications to synthetic examples demonstrate the features of the approach.

Source–receiver two-way wave extrapolation for prestack exploding-reflector modelling and migration

Most modern seismic imaging methods separate input data into parts (shot gathers). We develop a formulation that is able to incorporate all available data at once while numerically propagating the recorded multidimensional wavefield forward or backward in time. This approach has the potential for generating accurate images free of artiefacts associated with conventional approaches. We derive novel high-order partial differential equations in the source–receiver time domain. The fourth-order nature of the extrapolation in time leads to four solutions, two of which correspond to the incoming and outgoing P-waves and reduce to the zero-offset exploding-reflector solutions when the source coincides with the receiver. A challenge for implementing two-way time extrapolation is an essential singularity for horizontally travelling waves. This singularity can be avoided by limiting the range of wavenumbers treated in a spectral-based extrapolation. Using spectral methods based on the low-rank approximation of the propagation symbol, we extrapolate only the desired solutions in an accurate and efficient manner with reduced dispersion artiefacts. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach.

An Improved Full Waveform Inversion Procedure Based on Scattering Angle Enrichment

The gradient of standard full waveform inversion (FWI) attempts to map the residuals in the data to perturbations in the model. Such perturbations include smooth background updates from the transmission components and high wavenumber updates from the reflection components. A new optimization approach is used to update the background and perturbed velocity, simultaneously. We also use a more efficient scattering angle filter to enhance the large scattering angle for the background velocity update and enhance the small scattering angle for the perturbed velocity update. This efficient implementation of the filter is fast and requires less memory. Thus, the new FWI procedure updates mainly along the wave-path for both diving and reflected waves in the initial stages. At the same time, it updates the perturbation with mainly reflections (filtering the diving waves). An application to Marmousi model shows that this method converges starting with a linearly increasing velocity, and with data free of frequencies below 4 Hz.

A highly accurate finite-difference method with minimum dispersion error for solving the Helmholtz equation

Abstract Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is highly accurate and efficient.

Migration Velocity Analysis Using Prestack Wavefields

Using both the image and data domains to perform velocity inversion can help us resolve the long and short wavelength components of the velocity model, usually in that order. This translates to integrating migration velocity analysis (MVA) into full waveform inversion (FWI). The MVA part of the inversion often requires computing extended images, which is pricy using conventional methods. As a result, we use prestack wavefield (the double-square-root (DSR)) extrapolation, which includes the extended information (subsurface offsets) naturally, to make the process far more efficient and stable. The combination of the forward and adjoint prestack wavefields provides us with update options that can be easily conditioned to improve convergence. We, specifically, use a modified differential semblance operator to divide the extended image to the residual part for classic differential semblance operator (DSO) updates and the image modeling part, which provides reflections for higher resolution information. In our implementation, we invert for the velocity and the image simultaneously through a dual objective function. Application to the Marmousi model demonstrates the features of the approach.

Simultaneous Inversion of the Background Velocity and the Perturbation in Full Waveform Inversion

The gradient of standard full waveform inversion attempts to map the data residuals to perturbations in the model. Such perturbations may include smooth background updates from the transmission components or high wavenumber updates from the reflection components. However, if we fix the reflection components using imaging, the gradient of what is referred to as reflection FWI (RFWI) admits mainly transmission background-type updates to the velocity model. The drawback of existing RFWI methods is that they lack an optimal image capable of producing reflections within the convex region of the optimization. Since the influence of velocity on the data is given mainly by its propagator(background) and perturbed (reflectivity) components, we optimize both components simultaneously using a modified objective function. Specifically, we invert for the velocity and image simultaneously with an objective function that fits the summation of the modeled data from the source and the image to the observed data. Since the objective function is quadratic with respect to the image, the inversion for the image is fast which meant to absorb mainly the amplitude residual. An application to Marmousi model shows that this method converges starting with a linearly increasing velocity, and with data free of frequencies below 4 Hz.