## Low-rank viscoacoustic wave extrapolation

July 18, 2019 Documentation No comments

A new paper is added to the collection of reproducible documents: Viscoacoustic modeling and imaging using low-rank approximation

A constant-$Q$ wave equation involving fractional Laplacians was recently introduced for viscoacoustic modeling and imaging. This fractional wave equation has a convenient mixed-domain space-wavenumber formulation, which involves the fractional-Laplacian operators with a spatially varying power. We propose to apply low-rank approximation to the mixed-domain symbol, which enables a space-variable attenuation specified by the variable fractional power of the Laplacians. Using the proposed approximation scheme, we formulate the framework of the $Q$-compensated reverse-time migration ($Q$-RTM) for attenuation compensation. Numerical examples using synthetic data demonstrate the improved accuracy of using low-rank wave extrapolation with a constant-$Q$ fractional-Laplacian wave equation for seismic modeling and migration in attenuating media. Low-rank $Q$-RTM applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional RTM from acoustic data.

## Missing log data interpolation and semiautomatic seismic well ties

May 7, 2019 Documentation No comments

Relating well log data, measured in depth, to seismic data, measured in time, typically requires estimating well log impedance and a time-to-depth relationship using available sonic and density logs. When sonic and density logs are not available, it is challenging to incorporate wells into integrated reservoir studies as the wells cannot be tied to seismic. We propose a workflow to estimate missing well log information, automatically tie wells to seismic data and generate a global well-log property volume using data matching techniques. We first use local similarity scan to align all logs to constant geologic time and interpolate missing well log information. Local similarity is then used to tie available wells with seismic data. Finally, log data from each well is interpolated along local seismic structures to generate global log property volumes. We use blind well tests to verify the accuracy of well-log interpolation and seismic-well ties. Applying the proposed workflow to a 3D seismic dataset with 26 wells achieves consistent and verifiably accurate results.

## Seismic data interpolation using seislet transform

May 7, 2019 Documentation No comments

Data interpolation is an important step for seismic data analysis because many processing tasks, such as multiple attenuation and migration, are based on regularly sampled seismic data. Failed interpolations may introduce artifacts and eventually lead to inaccurate final processing results. In this paper, we generalize seismic data interpolation as a basis pursuit problem and propose an iteration framework for recovering missing data. The method is based on nonlinear iteration and sparse transform. A modified Bregman iteration is used for solving the constrained minimization problem based on compressed sensing. The new iterative strategy guarantees fast convergence by using a fixed threshold value. We also propose a generalized velocity-dependent (VD) formulation of the seislet transform as an effective sparse transform, in which the nonhyperbolic normal moveout equation serves as a bridge between local slope patterns and moveout parameters in the common-midpoint domain. It can also be reduced to the traditional VD-seislet if special heterogeneity parameter is selected. The generalized VD-seislet transform predicts prestack reflection data in offset coordinates, which provides a high compression of reflection events. The method was applied to synthetic and field data examples and the results show that the generalized VD-seislet transform can reconstruct missing data with the help of the modified Bregman iteration even for nonhyperbolic reflections under complex conditions, such as VTI media or aliasing.

## Predictive painting across faults

May 7, 2019 Documentation No comments

A new paper is added to the collection of reproducible documents: Predictive painting across faults

Predictive painting can effectively spread information in 3D volumes following the local structures (dips) of seismic events. However, it has troubles spreading information across faults with significant displacement. To address this problem, we propose to incorporate fault slip information into predictive painting to correctly spread information across faults. The fault slip is obtained by using a local similarity scan to measure local shifts of the different sides of a fault. We propose three different methods to utilize the fault slip information: 1) area partition method, which uses fault slip to correct the painting result after predictive painting in each divided area; 2) fault-zone replacement method, which replaces fault zones with smooth transitions calculated with the fault slip information to avoid sharp jumps; and 3) unfaulting method, where we use the fault slip information to unfault the volume, perform predictive painting in the unfaulted domain, and then map the painting result back to the original space. The proposed methods are tested in application of predictive painting to horizon picking. Numerical examples demonstrate that predictive painting after incorporating fault slip information can correctly spread information across faults, which makes the proposed three approaches of utilizing fault slip information effective and applicable.

## Streaming PF for random noise attenuation

May 6, 2019 Documentation No comments

In seismic exploration there are many sources of random noise, for example, scattering from a complex surface. Prediction filters (PFs) have been widely used for random noise attenuation, but these typically assume that the seismic signal is stationary. Seismic signals are fundamentally nonstationary. Stationary PFs fail in the presence of nonstationary events, even if the data are cut into overlapping windows (“patching”). We propose an adaptive PF method based on streaming and orthogonalization for random noise attenuation in the $t$-$x$ domain. Instead of using patching or regularization, the streaming orthogonal prediction filter (SOPF) takes full advantage of the streaming method, which generates the signal value as each new noisy data value arrives. The streaming signal-and-noise orthogonalization further improves the signal recovery ability of the SOPF. The streaming characteristic makes the proposed method faster than iterative approaches. In comparison with $f$-$x$ deconvolution and $f$-$x$ regularized nonstationary autoregression (RNA), we tested the feasibility of the proposed method in attenuating random noise on two synthetic datasets. Field data examples confirmed that the $t$-$x$ SOPF had a reasonable denoising ability in practice.

## Least-squares diffraction imaging

May 6, 2019 Documentation No comments

A new paper is added to the collection of reproducible documents: Least-squares path-summation diffraction imaging using sparsity constraints

Diffraction imaging aims to emphasize small-scale subsurface heterogeneities such as faults, pinch-outs, fracture swarms, channels, etc. and can help seismic reservoir characterization. The key step in diffraction imaging workflows is based on the separation procedure suppressing higher-energy reflections and emphasizing diffractions, after which diffractions can be imaged independently. Separation results often contain crosstalk between reflections and diffractions and are prone to noise. We propose an inversion scheme to reduce the crosstalk and denoise diffractions. The scheme decomposes an input full wavefield into three components: reflections, diffractions and noise. We construct the inverted forward modeling operator as the chain of three operators: Kirchhoff modeling, plane wave destruction and path-summation integral filter. Both reflections and diffractions have the same modeling operator. Separation of the components is done by shaping regularization. We impose sparsity constraints to extract diffractions, enforce smoothing along dominant local event slopes to restore reflections and suppress the crosstalk between the components by local signal-and-noise orthogonalization. Synthetic and field data examples confirm the effectivness of the proposed method.

## $Q$-compensated Least Squares RTM

May 3, 2019 Documentation No comments

Attenuation of seismic waves needs to be taken into account in order to improve the accuracy of seismic imaging. In viscoacoustic media, reverse time migration (RTM) can be performed with $Q$-compensation, which is also known as $Q$-RTM. Least-squares RTM (LSRTM) has also been shown to be able to compensate for attenuation through linearized inversion. However, seismic attenuation may significantly slow down the convergence rate of the least-squares iterative inversion process without proper preconditioning. We show that incorporating attenuation compensation into LSRTM can improve the speed of convergence in attenuating media, obtaining high-quality images within the first few iterations. Based on the lowrank one-step seismic modeling operator in viscoacoustic media, we derive its adjoint operator using non-stationary filtering theory. The proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation. Recognizing that, in viscoacoustic media, the wave equation Hessian may become ill-conditioned, we propose to precondition LSRTM with $Q$-compensated RTM. Numerical examples show that the resulting $Q$-LSRTM method has a significantly faster convergence rate than LSRTM, and thus is preferable for practical applications.

## EMD-Seislet

February 12, 2019 Documentation No comments

A new paper is added to the collection of reproducible documents: EMD-seislet transform

The seislet transform uses a prediction operator which is connected to the local slope or frequency of seismic events. In this paper, we propose combining the 1D non-stationary seislet transform with empirical mode decomposition (EMD) in the $f-x$ domain. We use the EMD to decompose data into smoothly variable frequency components for the following 1D seislet transform. The resultant representation shows remarkable sparsity. We introduce the detailed algorithm and use a field example to demonstrate the application of the new seislet transform for sparsity-promoting seismic data processing.

## Marmousi-2 modeling and processing

January 30, 2019 Documentation 2 comments

A new paper is added to the collection of reproducible documents: 2D modeling and basic processing with Madagascar

This reproducible document was created during the 2017 Working Workshop in Houston.

This document demonstrates finite difference modeling and simple processing using the Madagascar software package. The paper uses this simple processing sequence to teach the basics of Madagascar.
The SEGY file of the Marmousi2 model is downloaded from the Internet and used to create a few synthetic shots. Geometry is computed and loaded in the trace headers. Simple processing including amplitude correction, CMP sorting, velocity conversions, NMO correction, stack, Kirchhoff migration, and least squares Kirchhoff migration are applied. Displays of each processing stage are created. The full processing sequence takes less that 6 minutes on a 2013 MacBook Pro computer.

## Elastic low-rank wave propagation

November 19, 2018 Documentation No comments

Conventional solutions of elastic wave equations rely on inaccurate finite-difference approximations of the time derivative, which result in strict dispersion and stability conditions and limitations. In this work, we derive a general solution to the elastic anisotropic wave equation, in the form of a Fourier Integral Operator (FIO). The proposed method is a generalization of the previously developed recursive integral time extrapolation operators from acoustic to elastic media, and can accurately propagate waves in time using the form of the analytical solution in homogeneous media. The formulation is closely connected to elastic wave mode decomposition, and can be applied to the most general anisotropic medium. The numerical calculation of the FIO makes use of a low-rank approximation to enable highly accurate and stable wave extrapolations. We present numerical examples including wave propagation in 3D heterogeneous orthorhombic and triclinic models.