Kirchhoff Migration Research Articles

Geomechanical modelling of injection-induced seismicity: the case study of the Muara Laboh geothermal plant

SUMMARY In this work, we study the induced seismicity recorded during an injection operation at the Muara Laboh geothermal plant (Indonesia). The swarm, consisting of three bursts activating a normal fault zone, is characterized by rapid earthquake (km d−1) migration. We use a 2-D rate-and-state asperity model to better understand the physical mechanisms controlling the evolution of this induced swarm. The model suggests that the observed rapid seismic migration can be explained by the interaction among asperities through the expansion of slow post-seismic slip fronts. Also, it shows that the amount of seismicity generated by the fluid injection is strongly controlled by the background seismicity of the system, that is by the seismicity determined by the tectonic load charging the fault. This close correlation between natural and induced seismicity suggests that the injection in Muara Laboh principally stimulates critically stressed faults, which release the seismicity determined by their natural seismic cycle.

Enhancing the resolution of three‐dimensional migration images based on space‐variant point spread function deconvolution

AbstractImproving the resolution of seismic migration images plays an important role for geophysical interpreters to characterize underground reservoirs. However, the classical image domain least‐squares migration method based on the local‐stationary assumption cannot obtain a satisfactory high‐resolution seismic image due to the significant spatial variant characteristics of the point spread function. To mitigate this problem, we proposed a high‐resolution point spread function deconvolution method and applied it to two‐dimensional cases. Nevertheless, extending the two‐dimensional method to three‐dimensional problems directly would fail due to the intrinsic complexity in three‐dimensional cases. In this study, we resolve the differences encountered in the point spread function deconvolution method for two‐ and three‐dimensional cases and provide specific strategies for achieving high‐resolution imaging with low computational cost when extending the point spread function deconvolution method to three‐dimensional cases. The main schemes include (1) incorporating the analytical expression of the point spread function to guide the generation of three‐dimensional point spread function distributions, (2) extending the point spread function filter calculation method from a two‐dimensional square to a three‐dimensional rectangular prism and (3) interpolating to obtain more compact point spread functions for reducing migration artefacts. Results from the three‐dimensional synthetic Overthrust model and field data set demonstrate that our techniques could effectively enhance the spatial resolution of the migration images with reduced migration artefacts. With these specific strategies, the space‐variant point spread function deconvolution algorithm shows superior performance on three‐dimensional cases at a much lower computational cost compared with the classical least‐squares migration method and the local‐stationary deblurring method. Synthetic tests and real data applications confirm that the space‐variant point spread function deconvolution method has distinct advantages over both two‐ and three‐dimensional problems and can be widely adopted in practice.

Image-domain seismic inversion by deblurring with invertible recurrent inference machines

In complex geologic settings and in the presence of sparse acquisition systems, seismic migration images manifest as nonstationary blurred versions of the unknown subsurface model. Thus, image-domain deblurring is an important step to produce interpretable and high-resolution models of the subsurface. Most deblurring methods focus on inverting seismic images for their underlying reflectivity by iterative least-squares inversion of a local Hessian approximation; this is obtained by either direct modeling of the so-called point-spread functions (PSFs) or by a migration-demigration process. In this work, we adopt a novel deep-learning (DL) framework, based on invertible recurrent inference machines (i-RIMs), which allows approaching any inverse problem as a supervised learning task informed by the known modeling operator (convolution with PSFs in our case): our algorithm can directly invert migrated images for impedance perturbation models, assisted with the prior information of a smooth velocity model and the modeling operator. Because i-RIMs are constrained by the forward operator, they implicitly learn to shape/regularize output models in a training-data-driven fashion. As such, the resulting deblurred images indicate great robustness to noise in the data and spectral deficiencies (e.g., due to limited acquisition). The key role played by the i-RIM network design and the inclusion of the forward operator in the training process is supported by several synthetic examples. Finally, using field data, we find that i-RIM-based deblurring has great potential in yielding robust, high-quality relative impedance estimates from migrated seismic images. Our approach could be of importance toward future DL-based quantitative reservoir characterization and monitoring.

A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration

AbstractWe have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration.

Linearized waveform inversion for vertical transversely isotropic elastic media: Methodology and multi-parameter crosstalk analysis

Seismic migration and inversion are closely related techniques to portray subsurface images and identify hydrocarbon reservoirs. Seismic migration aims at obtaining structural images of subsurface geologic discontinuities. More specifically, seismic migration estimates the reflectivity function (stacked average reflectivity or pre-stack angle-dependent reflectivity) from seismic reflection data. On the other hand, seismic inversion quantitatively estimates the intrinsic rock properties of subsurface formulations. Such seismic inversion methods are applicable to detect hydrocarbon reservoirs that may exhibit lateral variations in the inverted parameters. Although there exist many differences, pre-stack seismic migration is similar with the first iteration of the general linearized seismic inversion.Usually, seismic migration and inversion techniques assume an acoustic or isotropic elastic medium. Unconventional reservoirs such as shale and tight sand formation have notable anisotropic property. We present a linearized waveform inversion (LWI) scheme for weakly anisotropic elastic media with vertical transversely isotropic (VTI) symmetry. It is based on two-way anisotropic elastic wave equation and simultaneously inverts for the localized perturbations (ΔVp0/Vp0,ΔVs0/Vs0,Δϵ,Δδ) from the long-wavelength reference model. Our proposed VTI-elastic LWI is an iterative method that requires a forward and an adjoint operator acting on vectors in each iteration. We derive the forward Born approximation operator by perturbation theory and adjoint operator via adjoint-state method. The inversion has improved the quality of the images and reduces the multi-parameter crosstalk comparing with the adjoint-based images. We have observed that the multi-parameter crosstalk problem is more prominent in the inversion images for Thomsen anisotropy parameters. Especially, the Thomsen parameter δ is the most difficult to resolve. We also analyze the multi-parameter crosstalk using scattering radiation patterns.The linearized waveform inversion for VTI-elastic media presented in this article provides quantitative information of the rock properties that has the potential to help identify hydrocarbon reservoirs.

Real-time detection of small objects in transverse electric polarization: Evaluations on synthetic and experimental datasets

<p>It is well-known that if one applies Kirchhoff migration (KM) to identify small objects when their values of magnetic permeabilities differ from those of the background (or transverse electric polarization), their location and outline shape cannot be satisfactorily retrieved because rings of large magnitudes centered at the location of objects appear in the imaging results. Fortunately, it is possible to recognize the existence and approximated location of objects in the 2D Fresnel dataset through the traditional KM, but no theoretical explanation for this phenomenon has been verified. Here we show that the imaging function of KM when tested on the Fresnel dataset can be expressed as squared zero-order and first-order Bessel functions and as an infinite series of Bessel functions of integer order greater than two. We also explain why the existence and approximate location of objects can be identified. This theoretical result is supported by numerical simulations on synthetic and experimental data.</p>

Seismic reverse time migration with random boundary and frequency modulation

Seismic reverse time migration with random boundary and frequency modulation

A Modified Kirchhoff Migration for Microwave Imaging in Superluminal Propagation Region

A Modified Kirchhoff Migration for Microwave Imaging in Superluminal Propagation Region

Fast Factorized Kirchhoff Migration Algorithm for Near-Field Radar Imaging With Sparse MIMO Arrays

Fast Factorized Kirchhoff Migration Algorithm for Near-Field Radar Imaging With Sparse MIMO Arrays

Real-time tracking of moving objects from scattering matrix in real-world microwave imaging

<abstract><p>The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with real-data are conducted.</p></abstract>

Hessian-Assisted Iterative Self-Training Learning for Seismic Migration

Hessian-Assisted Iterative Self-Training Learning for Seismic Migration

Seismic Migration Imaging of Mountain Tunnel Using Surface Observation System

Seismic Migration Imaging of Mountain Tunnel Using Surface Observation System

Simplified TTI pure qP-wave equation implemented in the space domain and applied for reverse time migration in TTI media

The assumption that the earth’s subsurface is a homogeneous and isotropic medium enables the imaging of important geologic structures but results in information loss, especially in more complex geologic media. Therefore, considering anisotropy is crucial for seismic imaging, particularly the most common anisotropy in geophysics: the transversely isotropic medium. However, this also means a considerable increase in the computational cost of reverse time migration (RTM). Thus, a new pseudoacoustic wave equation for pure qP-wave in tilted transversely isotropic (TTI) media, which can also be efficiently implemented using the finite-difference (FD) method with the unit vector method (UVM), is proposed, aiming to reduce the computational cost of the RTM. The proposed equation solved with fast Fourier transforms is exact and faster for seismic migration than the base equations found in the literature; however, a greater efficiency is achievable by using FD to compute the second derivatives. In contrast, when solved with the UVM, it is kinematically accurate and significantly faster, although dynamically inaccurate as it is an acoustic approximation. The efficiency and efficacy of this new equation are demonstrated by modeling and migrating synthetic TTI data found in the literature.

Modified pure-viscoacoustic wave propagation and compensated reverse-time migration in transversely isotropic media

Previous studies demonstrated that seismic attenuation and anisotropy can significantly affect the kinematic and dynamic characteristics of wavefields. If these effects are not incorporated into seismic migration, the resolution of the imaging results will be reduced. Considering the anisotropy of velocity and attenuation, we derive a new pure-viscoacoustic wave equation to simulate P wave propagation in transversely isotropic (TI) attenuating media by combining the complex dispersion relation and modified complex modulus. Compared to the conventional complex modulus, the modified modulus is derived from the optimized relationship between angular frequency and wavenumber, which can improve the modeling accuracy in strongly attenuating media. Wavefield comparisons illustrate that our pure-viscoacoustic wave equation can simulate stable P wavefields in complex geological structures without S-wave artifacts and generate similar P wave information to the pseudo-viscoacoustic wave equation. During the implementation, we introduce two low-rank decompositions to approximate the real and imaginary parts and then use the pseudo-spectral method to solve this new equation. Since the proposed equation can simulate decoupled amplitude attenuation and phase dispersion effects, it is used to perform Q-compensated reverse-time migration (Q-RTM). Numerical examples demonstrate the accuracy and robustness of the proposed method for pure-viscoacoustic wavefield simulations and migration imaging in transversely isotropic attenuating media.

Multi-image, reverse time, and Kirchhoff migrations with compact Green’s functions

Using compact representations of Green’s functions we derive a common framework for reverse time migration (RTM) and Kirchhoff migration. These compact Green’s functions (CGFs) are 3D volumes containing traveltimes and amplitudes for the N most representative events in the upcoming/downgoing decomposed 4D wavefields originating from a point source. Within this framework, we implement an RTM algorithm using a multivalued excitation time/amplitude imaging condition. This new approach produces four complementary imaging volumes (different combinations of source and receiver decomposed wavefields) and angle/azimuth gathers with computational effort less than 15% greater than that of plain (one image and no gathers) RTM algorithms. The advantages of separating the image volume into four complementary volumes are well established in the literature (low-frequency noise separation and turning wave imaging); however, its use has been limited by the computational cost. Despite using two source propagations to decompose the source wavefield, we reduce the computations to less than 20% of a single source propagation by performing finite-difference propagation with half the frequency limit used in the receiver wavefield propagation. The combination of CGF and an excitation time/amplitude imaging condition allows receiver wavefield decomposition with only one wavefield propagation. Our RTM algorithm constructs angle/azimuth gathers using a postmigration computation of the source and receiver wavefield’s propagation directions. To compute the propagation directions after migration, we use a new concept: the cumulative wavefield volumes, which are 3D, imaging-condition-guided compressions, of the 4D source and receiver wavefields. We also use CGF to implement a Kirchhoff migration algorithm that produces four complementary image volumes with RTM-like quality. Furthermore, we present synthetic and field data examples to clarify the new concepts and illustrate the results obtained using these methods.

Seismic reverse time migration based on biaxial wavefield decomposition

Seismic images generated by reverse time migration (RTM) are often contaminated by low-wavenumber artifacts. Decomposition of up- and downgoing waves during propagation can effectively reduce these artifacts. However, decomposition along the vertical axis is not sufficient to eliminate the migration artifacts. We introduce biaxial wavefield decomposition to decompose seismic wavefields in the lateral and vertical directions. After decomposition, there are 16 crosscorrelation terms between different components, and there are only four crosscorrelation terms that can be used separately to image flat layers and tilted interfaces. Within these four terms, the remaining artifacts can be easily identified and reduced by other methods, and the resolution can be further improved by adjusting the weights of the four terms. The decomposition method based on the analytic wavefield via the Hilbert transform has been shown to agree with the viscoacoustic wave equation with fractional spatial derivatives. Therefore, this biaxial wavefield decomposition method is suitable for decomposing the wavefields for the attenuation-compensated RTM. The decomposed and recombined migration results can better image a complex subsurface structure with high resolution.

Frequency-dependent Q simulation and viscoacoustic reverse time migration based on the fractional Zener model

Seismic attenuation is a basic physical property of the earth, which significantly affects the characteristics of seismic wavefields. Accurately simulating wave propagation in the earth is essential to image subsurface structures. Some prevailing methods (e.g., the standard linear solid and fractional Laplacian equation) to describe seismic wave propagation in attenuating media are mainly based on the constant- Q model (CQM), which is valid at room temperature and pressure. However, laboratory measurements suggest that the quality factor Q is a function of frequencies in some regions. To simulate the frequency-dependent Q effect, we derive a viscoacoustic wave equation from the stress-strain relationship of the fractional Zener model (FZM) with variable fractional orders. During the implementation, we separate the real and imaginary parts of the modulus and introduce a low-rank decomposition method to solve the FZM equation. Because the amplitude dissipation and phase dispersion are decoupled, we establish a compensated reverse time migration ( Q-RTM) algorithm to mitigate adverse effects caused by seismic attenuation and improve the quality of seismic migration in frequency-dependent attenuating media. A two-layer model and the BP gas chimney model are used to perform Q-RTM tests. A low-pass filter with a Tukey window function is applied to suppress numerical instability during the compensation. Numerical results demonstrate that our FZM Q-RTM approach can produce high-resolution images with corrected reflector positions and amplitudes. Because the CQM equation ignores the frequency dependence of Q, it may lead to overcompensation in Q-RTM.

A stable Q-compensated reverse-time migration method using a modified fractional viscoacoustic wave equation

Seismic wave propagation inside the earth is usually accompanied by amplitude dissipation and velocity dispersion. It is essential to accurately characterize these effects and make the corresponding compensations in seismic reverse-time migration (RTM) and the following interpretation. Due to the approximation of [Formula: see text] in the derivation, the conventional fractional viscoacoustic wave equation in strongly attenuative media is unsatisfactory. This paper develops a modified viscoacoustic wave equation by combining the relationship between angular frequency and complex wavenumber. The numerical simulation demonstrates its advantage of high accuracy in describing constant- Q effects, especially in strongly attenuative media with small Q values. A truncated Taylor-series expansion algorithm and a pseudospectral method are developed to solve this equation during wave propagation. To address the issue of numerical instability in Q-compensated RTM ( Q-RTM), a stabilized technology with regularization is further developed. Unlike the conventional filtering method implemented in the wavenumber domain, our approach only adds an explicit regularization term, which can be directly computed in the space domain. Furthermore, this regularization depends on velocities and quality factors, significantly improving its applicability in complicated structures. Noise-free data tests demonstrate that our Q-RTM method can effectively correct phase distortion, compensate for amplitude attenuation, and produce high-resolution migrated images without any cutoff-frequency artifacts caused by filtering. Noisy data experiments further verify the antinoise performance and robustness of our method.

Real-time identification of small anomalies from scattering matrix without background information

Several researches have confirmed the possibility of localizing small anomalies via Kirchhoff migration (KM); however, when the background information is unknown, small anomalies cannot be satisfactorily retrieved. This fact can be examined through the simulation results; however, related theoretical result to explain the reason of such phenomenon has not yet been investigated. In this contribution, we show that the imaging function of the KM can be expressed by an infinite series of the Bessel function of the first kind, material properties, and antenna arrangement, and applied alternative value of the background wavenumber. Based on the theoretical result, we explain why the exact location and shape of anomalies cannot be retrieved. The simulation results with synthetic data exhibited to support the theoretical result.

Reverse time migration surface offset gathers by attribute migration and constrained least-squares inversion

Abstract Surface offset gathers (SOGs) are crucial for velocity updating in seismic data migration and muting the stretched waveform in shallow parts of migrated seismic data. Kirchhoff-based migration methods can output SOGs because they take every seismic trace as the input. As a result, the imaging results can be rearranged according to the offset value of the input data to obtain SOGs. However, regarding more accurate wave equation methods (such as reverse time migration, RTM), the SOGs cannot be obtained directly with a single migration calculation. Attribute migration is an easy-to-implement method to output SOGs of wave equation migration methods. It calculates the offset value of each imaging point by dividing the migration result modulated by the offset attribute with the conventional migration result. However, the division error may cause the calculation result outside the given offset value. This paper proposes a constrained least-squares inversion method for stable calculation of the RTM offset range. The method ensures that the calculated results are within the given offset range. We tested the method against the direct division and least-squares methods without constraints using the Marmousi model and a real 3D dataset. We show that the SOGs obtained by the constrained least-squares inversion attribute migration method had the same characteristics as those derived using the division-based attribute migration method, but they also had higher vertical resolution, better energy distribution, and improved lateral continuity. In a further study, we expect to develop a stable and direct method to efficiently calculate SOGs for RTM and an iterative closed loop for RTM velocity updating.