3D forward modeling of upgoing and downgoing wavefields using Hilbert transform

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. F1-F8 ◽  
Author(s):  
Yikang Zheng ◽  
Yibo Wang ◽  
Xu Chang
Keyword(s):  
Fourier Transform ◽  
Hilbert Transform ◽  
Forward Modeling ◽  
Ocean Bottom ◽  
Reverse Time ◽  
Time Migration ◽  
Computation Efficiency ◽  
Wavefield Simulation ◽  
The Fourier Transform ◽  
The Hilbert Transform

The separation of upgoing and downgoing wavefields is an important technique in the processing of vertical seismic profiling data and ocean bottom cable data. It is also used in reverse time migration (RTM) based on the two-way wave equation to suppress low-frequency, high-amplitude noises and false images. Therefore, we model upgoing and downgoing wavefields directly in the wavefield propagation. There are several methods to obtain separated wavefields. The methods using the Fourier transform require storage of the wavefields, which is not practical due to the extremely high disk-space requirements. Methods using Poynting vectors have an ambiguity problem when crossing a peak or a trough of the wavefields. To improve the accuracy and stability of the modeled upgoing and downgoing wavefields in a complicated velocity model, we evaluate an efficient forward-modeling approach purely based on the Hilbert transform in 3D acoustic wavefield simulation. This method is implemented by the Hilbert transform along the time and depth axis, instead of the Fourier transform. We explicitly derive the formulas for upgoing and downgoing wavefield propagation and attach reproducible source codes. Applications to synthetic models indicate that this method can forward propagate upgoing and downgoing wavefields effectively and improve the imaging quality in migration. This method has various potential applications, e.g., 3D seismic imaging with high computation efficiency.

The properties of generalized offset linear canonical Hilbert transform and its applications

2017 ◽  
Vol 15 (04) ◽  
pp. 1750031 ◽  
Author(s):  
Shuiqing Xu ◽  
Li Feng ◽  
Yi Chai ◽  
Youqiang Hu ◽  
Lei Huang
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Hilbert Transform ◽  
Analytic Signal ◽  
Linear Canonical Transform ◽  
Single Sideband ◽  
Generalized Hilbert Transform ◽  
Offset Linear Canonical Transform ◽  
The Fourier Transform ◽  
The Hilbert Transform

The Hilbert transform is tightly associated with the Fourier transform. As the offset linear canonical transform (OLCT) has been shown to be useful and powerful in signal processing and optics, the concept of generalized Hilbert transform associated with the OLCT has been proposed in the literature. However, some basic results for the generalized Hilbert transform still remain unknown. Therefore, in this paper, theories and properties of the generalized Hilbert transform have been considered. First, we introduce some basic properties of the generalized Hilbert transform. Then, an important theorem for the generalized analytic signal is presented. Subsequently, the generalized Bedrosian theorem for the generalized Hilbert transform is formulated. In addition, a generalized secure single-sideband (SSB) modulation system is also presented. Finally, the simulations are carried out to verify the validity and correctness of the proposed results.

Analysis of direction-decomposed and vector-based elastic reverse time migration using the Hilbert transform

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. S599-S617
Author(s):  
Ting Hu ◽  
Hong Liu ◽  
Xuebao Guo ◽  
Yuxin Yuan ◽  
Zhiyang Wang
Keyword(s):  
Hilbert Transform ◽  
Computational Cost ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Spatial Direction ◽  
Time Migration ◽  
Imaging Artifacts ◽  
Dip Angle ◽  
The Hilbert Transform ◽  
Poynting Vectors

Straightforward implementations of elastic reverse time migration (ERTM) often produce imaging artifacts associated with incorrectly imaged mode conversions, crosstalk, and back-scattered energies. To address these issues, we introduced three approaches: (1) vector-based normalized crosscorrelation imaging conditions (VBNICs), (2) directional separation of wavefields to remove low-wavenumber noise, and (3) postimaging filtering of the dip-angle gathers to eliminate the artifacts caused by nonphysical wave modes. These approaches are combined to create an effective ERTM workflow that can produce high-quality images. Numerical examples demonstrate that, first, VBNICs can produce correct polarities for PP/PS images and can compute migrated dip-angle gathers efficiently by using P/S decomposed Poynting vectors. Second, they achieve improved signal-to-noise and higher resolution when performing up/down decomposition before applying VBNICs, and left/right decomposition enhances steep dips imaging at the computational cost of adding the Hilbert transform to a spatial direction. Third, dip filtering using slope-consistency analysis attenuates the remaining artifacts effectively. An application of the SEG advanced modeling program (SEAM) model demonstrates that our ERTM workflow reduces noise and improves imaging ability for complex geologic areas.

A space of slowly decreasing functions with pleasant Fourier transforms

1991 ◽  
Vol 119 (1-2) ◽  
pp. 73-86
Author(s):  
L. E. Fraenkel
Keyword(s):  
Banach Space ◽  
Fourier Transform ◽  
Bounded Variation ◽  
Hilbert Transform ◽  
Fourier Transforms ◽  
Weighting Function ◽  
The Fourier Transform ◽  
The Hilbert Transform ◽  
Decreasing Functions

SynopsisThe space in question is Aµ(R):=L1(R) + Bµ(R), where Bµ(R) is a Banach space that contains the “tails” (the dominant parts for large values of |x|) of certain slowly decreasing functions from R to R. Functions in Bµ(R) are of bounded variation, and the norm involves their variation and a weighting function. Theorems are proved only for Bµ(R), because those for L1(R) are known. The results concern the convolution of a function in Bµ(R) with one in L1(R), the Fourier transform acting on Bµ(R), and the signum rule for the Hilbert transform of functions in Bµ(R).

A high efficiency wavefield decomposition method based on Hilbert transform

Geophysics ◽  
2021 ◽  
pp. 1-68
Author(s):  
Xue Guo ◽  
Ying Shi ◽  
Weihong Wang ◽  
Xuan Ke ◽  
Hong Liu ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Hilbert Transform ◽  
Decomposition Method ◽  
High Efficiency ◽  
Computational Cost ◽  
Difference Approximation ◽  
Reverse Time ◽  
Time Migration ◽  
The Hilbert Transform ◽  
Wavefield Decomposition

Wavefield decomposition can be used to extract effective information in reverse time migration (RTM) and full waveform inversion (FWI). The wavefield decomposition methods based on the Hilbert transform (HTWD) and the Poynting vector (PVWD) are the most commonly used. The HTWD needs to save the wavefields at all time steps or introduce additional numerical simulation, which increases the computational cost. The PVWD cannot handle multi-wave arrivals, and its performance is poor in complex situations. We propose an efficient wavefield decomposition method based on the Hilbert transform (EHTWD). The EHTWD constructs two wavefields to replace the original wavefield and the wavefield after Hilbert transform. The first wavefield is obtained by using the dispersion relation to modify the frequency components. The other wavefield is obtained by time difference approximation. Therefore, there is a 90° phase change between the two wavefields. In EHTWD, we only need two wavefields at different moments, which avoids the additional numerical simulation. The EHTWD is also suitable for wavefield decomposition in arbitrary directions. Compared with HTWD, the computational complexity can be greatly reduced with the decrease of the number of imaging time slices. The numerical examples of wavefield decomposition demonstrate that the proposed method can realize wavefield decomposition in any direction. The examples of imaging decomposition and real data also show that the EHTWD suppresses the imaging noise effectively.

An exact adjoint operation pair in time extrapolation and its application in least-squares reverse-time migration

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. H27-H33 ◽  
Author(s):  
Jun Ji
Keyword(s):  
Least Squares ◽  
Adjoint Operator ◽  
Synthetic Data ◽  
Forward Modeling ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Product Test ◽  
Least Squares Migration ◽  
Time Extrapolation

To reduce the migration artifacts arising from incomplete data or inaccurate operators instead of migrating data with the adjoint of the forward-modeling operator, a least-squares migration often is considered. Least-squares migration requires a forward-modeling operator and its adjoint. In a derivation of the mathematically correct adjoint operator to a given forward-time-extrapolation modeling operator, the exact adjoint of the derived operator is obtained by formulating an explicit matrix equation for the forward operation and transposing it. The programs that implement the exact adjoint operator pair are verified by the dot-product test. The derived exact adjoint operator turns out to differ from the conventional reverse-time-migration (RTM) operator, an implementation of wavefield extrapolation backward in time. Examples with synthetic data show that migration using the exact adjoint operator gives similar results for a conventional RTM operator and that least-squares RTM is quite successful in reducing most migration artifacts. The least-squares solution using the exact adjoint pair produces a model that fits the data better than one using a conventional RTM operator pair.

Acoustic-elastic coupled equation for ocean bottom seismic data elastic reverse time migration

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. S333-S345 ◽  
Author(s):  
Pengfei Yu ◽  
Jianhua Geng ◽  
Xiaobo Li ◽  
Chenlong Wang
Keyword(s):  
Boundary Conditions ◽  
Particle Velocity ◽  
Ocean Bottom ◽  
Receiver Side ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
New Equation ◽  
Obs Data ◽  
Coupled Equation

Conventionally, multicomponent geophones used to record the elastic wavefields in the solid seabed are necessary for ocean bottom seismic (OBS) data elastic reverse time migration (RTM). Particle velocity components are usually injected directly as boundary conditions in the elastic-wave equation in the receiver-side wavefield extrapolation step, which causes artifacts in the resulting elastic images. We have deduced a first-order acoustic-elastic coupled equation (AECE) by substituting pressure fields into the elastic velocity-stress equation (EVSE). AECE has three advantages for OBS data over EVSE when performing elastic RTM. First, the new equation unifies wave propagation in acoustic and elastic media. Second, the new equation separates P-waves directly during wavefield propagation. Third, three approaches are identified when using the receiver-side multicomponent particle velocity records and pressure records in elastic RTM processing: (1) particle velocity components are set as boundary conditions in receiver-side vectorial extrapolation with the AECE, which is equal to the elastic RTM using the conventional EVSE; (2) the pressure component may also be used for receiver-side scalar extrapolation with the AECE, and with which we can accomplish PP and PS images using only the pressure records and suppress most of the artifacts in the PP image with vectorial extrapolation; and (3) ocean-bottom 4C data can be simultaneously used for elastic images with receiver-side tensorial extrapolation using the AECE. Thus, the AECE may be used for conventional elastic RTM, but it also offers the flexibility to obtain PP and PS images using only pressure records.

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