Gaussian Beam Summation Research Articles

Application of Gaussian Beam Summation Migration in Reflected In‐seam Wave Imaging

AbstractThe geological conditions for coal mining in China are complex, with various structural issues such as faults and collapsed columns seriously compromising the safety of coal mine production. In‐seam wave exploration is an effective technique for acquiring detailed information on geological structures in coal seam working faces. However, the existing reflected in‐seam wave imaging technique can no longer meet the exploration precision requirements, making it imperative to develop a new reflected in‐seam wave imaging technique. This study applies the Gaussian beam summation (GBS) migration method to imaging coal seams' reflected in‐seam wave data. Firstly, with regard to the characteristics of the reflected in‐seam wave data, methods such as wavefield removal and enveloped superposition are employed for the corresponding wavefield separation, wave train compression and other processing of reflected in‐seam waves. Thereafter, imaging is performed using the GBS migration technique. The feasibility and effectiveness of the proposed method for reflected in‐seam wave imaging are validated by conducting GBS migration tests on 3D coal‐seam fault models with different dip angles and throws. By applying the method to reflected in‐seam wave data for an actual coal seam working face, accurate imaging of a fault structure is obtained, thereby validating its practicality.

Computing the Radar Cross-Section of Dielectric Targets Using the Gaussian Beam Summation Method

Computing the Radar Cross-Section (RCS) of a given object is a topic of major importance for many applications, e.g., target detection and stealth technology. In this context, high-frequency asymptotic methods are widely used. In this article, we derive a Gaussian Beam Summation (GBS) method for both metallic and dielectric targets. The basic idea is to use the GBS method to compute the scattering far fields generated by the equivalent currents flowing on the surfaces. The validity of the proposed method is then investigated in the X-band. To accomplish this, the results obtained using this technique were compared to those obtained using other sufficiently accurate methods such as the ray tracing of FEKO and the ray asymptotic solution. As an example of the method’s accuracy, the GBS method was used to obtain the wave field in a homogeneous medium by fitting the results to a point source. In the same way, the method was used to compute the RCS of dielectric cuboids.

A new implementation of the beam migration by using a model-driven variable grid strategy

Beam migration offers higher imaging accuracy, better flexibility, stability and adaptability than conventional ray migration, while the variable grid method achieves lower storage and higher efficiency under the same accuracy by means of different grid generation of the velocity model. However, the efficiency and accuracy of Gaussian beam migration are strongly affected by the algorithm used for Gaussian beam summation whose key step is to find the underground point at which the local beam contributes relevantly to the reconstruction of the Green's function. Conventionally, one uses Hale's recursive search method that firstly searches on a global coarse grid and then interpolated to a local fine one. Practical experience shows that this method can greatly improve the efficiency of the beam scanning, but causes some wavy discontinuities and imaging artifacts as well. To solve this problem, we present a new method called “model-driven variable grid strategy”. In this strategy, a key step is to establish a relationship between the vertical gradient of the velocity model and the local grid constants through local velocity gradient with respect to the global average velocity. Numerical tests on synthetic and field data show that the new method has a minimal need on computer resources and can largely improve the accuracy and efficiency of the beam migration in regions with complex geological structures.

Gaussian Beam Summation Migration of Deep Reflection Seismic Data: Numerical Examples

The deep reflection seismic technique is essential to detect the basement of petroliferous basins and the fine structure of lithosphere. The traditional stacking or time migration is usually adopted for imaging deep reflection seismic data, but the accuracy is too limited to acquire precise imaging profiles. Development of an effective prestack depth migration method for deep reflection seismic data is the urgent requirement in the field of lithosphere structure detection. The Gaussian beam summation (GBS) migration requires no local slant stacking or phase approximation. It has higher imaging precision and better flexibility and applicability to the acquisition system, which can be well applied to imaging long-array and large-trace-interval deep reflection seismic data. In this letter, the GBS migration is extended into depth-domain imaging of deep reflection seismic data. Firstly, the basic principles of GBS migration are reviewed. Then, the validity of the method is verified by a migration test on a simple crustal-scale model. Finally, according to the crust-mantle structure of typical areas, a crustal model of western Sichuan and eastern Tibet and a classical collision model are established for migration experiments, and accurate deep-domain images are obtained. We mainly conducted numerical tests for the GBS migration of deep reflection seismic data in this letter. It was verified through imaging studies by the typical crust-mantle structure model that the proposed method was a prestack depth migration method suitable for the precise imaging of deep reflection seismic data, which provided a numerical basis for its subsequent imaging application in field data.

PS-Wave Angle-Domain Imaging With Gaussian Beam Summation in 2-D TTI Media

PS-Wave Angle-Domain Imaging With Gaussian Beam Summation in 2-D TTI Media

Least-squares Gaussian beam migration in viscoacoustic media

Because of amplitude decay and phase dispersion of seismic waves, conventional migrations are insufficient to produce satisfactory images using data observed in highly attenuative geologic environments. We have developed a least-squares Gaussian beam migration method for viscoacoustic data imaging, which can not only compensate for amplitude decay and phase dispersion caused by attenuation, but it can also improve image resolution and amplitude fidelity through linearized least-squares inversion. We represent the viscoacoustic Green’s function by a summation of Gaussian beams, in which an attenuation traveltime is incorporated to simulate or compensate for attenuation effects. Based on the beam representation of the Green’s function, we construct the viscoacoustic Born forward modeling and adjoint migration operators, which can be effectively evaluated by a time-domain approach based on a filter-bank technique. With the constructed operators, we formulate a least-squares migration scheme to iteratively solve for the optimal image. Numerical tests on synthetic and field data sets demonstrate that our method can effectively compensate for the attenuation effects and produce images with higher resolution and more balanced amplitudes than images from acoustic least-squares Gaussian beam migration.

Gaussian Beam Born Modeling for Single-Scattering Waves in Visco-Acoustic Media

A new Gaussian beam (GB) Born modeling method is presented in this letter to simulate the wavefields of single scattering events in visco-acoustic media. This approach allows the accurate computation of synthetic seismograms or waveforms for the visco-acoustic media. Our method relies on the GB summation method, in which the attenuation effects are taken into account by incorporating a frequency-dependent dissipation function into beam calculation. The key to efficiency is that the Born modeling kernel is evaluated in the time domain based on the filter-bank technique. The proposed method is tested with both 2-D and 3-D numerical examples and the results demonstrate its capability to simulate wavefields in complex structures, as well as its marked advantage in efficiency over finite-difference wave equation modeling.

Migration of Converted PS-Waves Directly From Irregular Surfaces by Using the Gaussian Beam Summation Method

Processing pressure shear (PS)-wave data is more challenging than PP-wave data because of the asymmetry of the source-to-receiver ray paths, particularly for irregular surfaces and sparse acquisition. Gaussian beam summation (GBS) migration is an effective method for imaging seismic data from irregular surfaces. In this letter, we introduce a converted PS-wave migration method for irregular surfaces using GBS, in which a scalar wavefield is used for wavefield propagation imaging. Cross correlation imaging is performed using forward-continued source wavefields calculated with the P-wave and reverse-continued wavefields calculated using the S-waves from the receivers. The corresponding Green's function is constructed as a superposition integral of the Gaussian beams emitted from the source and the receivers on the irregular surfaces, respectively. Numerical tests demonstrate that the method is a flexible and effective alternative for accurate imaging of converted PS-wave data from irregular surfaces.

2D anisotropic multicomponent Gaussian-beam migration under complex surface conditions

Elastic-wave migration in anisotropic media is a vital challenge, particularly for areas with irregular topography. Gaussian-beam migration (GBM) is an accurate and flexible depth migration technique, which is adaptable for imaging complex surface areas. It retains the dynamic features of the wavefield and overcomes the multivalued traveltimes and caustic problems of Kirchhoff migration. We have extended the GBM method to work for 2D anisotropic multicomponent migration under complex surface conditions. We use Gaussian beams to calculate the wavefield from irregular topography, and we use two schemes to derive the down-continued recorded wavefields. One is based on the local slant stack as in classic GBM, in which the PP- and PS-wave seismic records within the local region are directly decomposed into local plane-wave components from irregular topography. The other scheme does not perform the local slant stack. The Green’s function is calculated with a Gaussian beam summation emitted from the receiver point at the irregular surface. Using the crosscorrelation imaging condition and combining with the 2D anisotropic ray-tracing algorithm, we develop two 2D anisotropic multicomponent Gaussian-beam prestack depth migration (GB-PSDM) methods, i.e., using the slant stack and nonslant stack, for irregular topography. Numerical tests demonstrate that our anisotropic multicomponent GB-PSDM can accurately image subsurface structures under complex topography conditions.

Investigation of RCS of modified targets using experimental measurements and GB methods

In this study, the investigation of the electromagnetic scattering by modified targets is presented. A metallic notched plate and disc are considered. Two Gaussian beam (GB) techniques have been applied: GB launching (GBL) and GB summation (GBS). The theoretical formulations of GBL and GBS are established. The simulations of the radar cross-section (RCS) of the flat and notched targets (plate and disc) are presented. The application and evaluation of GBS and GBL methods are extended through the studied variations of the RCS in several scattering angles and different radar frequencies. The numerical RCS results are compared with those given by the physical optic model and the method of moments. Moreover, the numerical results are evaluated with measurements conducted in an anechoic chamber.

Gaussian-type beams in longitudinally inhomogeneous, lens-like media. Gradual transition from waveguide to antiwaveguide

We address propagation of paraxial Gaussian-type beam modes in longitudinally inhomogeneous axisymmetric lens-like media, with a specific case of a transition of waveguide into antiwaveguide. Higher-order modes are described in a unified manner using a so-called secondary parabolic equation. Analyzing a linear transition from waveguide to antiwaveguide where beams are described in terms of Airy functions, we conclude that the width of a localized Gaussian-type beam in inhomogeneous media can dramatically grow. This fact should be taken into account in numerical procedures employing summation of Gaussian beams.

Experimental Results and Numerical Simulation of the Target RCS using Gaussian Beam Summation Method

This paper presents a numerical and experimental study of Radar Cross Section (RCS) of radar targets using Gaussian Beam Summation (GBS) method. The purpose GBS method has several advantages over ray method, mainly on the caustic problem. To evaluate the performance of the chosen method, we started the analysis of the RCS using Gaussian Beam Summation (GBS) and Gaussian Beam Launching (GBL), the asymptotic models Physical Optic (PO), Geometrical Theory of Diffraction (GTD) and the rigorous Method of Moment (MoM). Then, we showed the experimental validation of the numerical results using experimental measurements which have been executed in the anechoic chamber of Lab-STICC at ENSTA Bretagne. The numerical and experimental results of the RCS are studied and given as a function of various parameters: polarization type, target size, Gaussian beams number and Gaussian beams width.

Time-domain least-squares migration using the Gaussian beam summation method

Time-domain least-squares migration using the Gaussian beam summation method

A fast algorithm for depth migration by the Gaussian beam summation method

A fast algorithm for depth migration by the Gaussian beam summation method

Born modeling for heterogeneous media using the Gaussian beam summation based Green's function

Born approximation is a commonly used approximation in the simulation of seismic wave propagation. Calculation of the Green's function in Born approximation integral is essential for Born modeling. We derive a new Born formula based on the Gaussian beam representations of Green's functions. This procedure can be used to mitigate the problems like the caustic, shadow region, and multivalued traveltime caused by multipathing that traditional geometric ray theory cannot deal with. However, due to the characteristic of complex traveltime in the Gaussian beam, we present a new isochronous stack method for Gaussian beam based Born modeling. Additionally, two basic issues, background velocity and integral region selection, are discussed. Numerical results demonstrate the accuracy and efficiency of the Gaussian beam based Born theory and implementation.

Reverse time migration with elastodynamic Gaussian beams

Elastic migration has been widely paid attention by employing the vector processing of multicomponent seismic data. Ray based elastic Kirchhoff migration has such properties as high flexibility and high efficiency. However, it has failed to solve many problems caused by multipath. On the other hand, elastic reverse-time migration (RTM) based on the two-way wave equation is known to be capable of dealing with these problems, but it is extremely expensive when applied in 3D cases and velocity model building. Based on the elastic Kirchhoff-Helmholtz integral, we calculate decoupled backward-continued wavefields by introducing elastic Green functions for P- and S-waves, which is expressed by a summation of elastodynamic Gaussian beams. The PP and polarity-corrected PS images are obtained by calculating the correlation between downward and decoupled backward-continued vector wavefields, where polarity correction is performed by analyzing the relation between the polarization direction of converted PS waves and incident angle on the interface. To a large extent, our method combines the high efficiency of ray-based migration with the high accuracy of wave-equation based reverse-time migration. Application of this method to multicomponent synthetic datasets from the fault model and Marmousi2 model demonstrates the validity, flexibility and accuracy of the new method.

Q-compensated migration by Gaussian beam summation method

Migration by the Gaussian beam summation method (MGBSM) combines the high efficiency and flexibility of Gaussian beam migration with the high accuracy of wave equation reverse-time migration, which can overcome the problems of caustics, handle all arrivals, yield good images of steep flanks, and is readily extendible to target-oriented implementation. The viscoacoustic prestack migration is of practical significance because it considers the viscosity of subsurface media. In this paper, we propose a MGBSM to perform seismic data compensation for frequency dependent absorption and dispersion. First, we derive the expressions for the attenuation equation and analyse the precision of the viscoacoustic Green function based on Gaussian beams. Then, the principle and procedure of compensation is presented. Finally we propose Q-compensated MGBSM. Numerical examples indicate that Q-compensated MGBSM has a higher imaging resolution than acoustic MGBSM when the viscosity of the subsurface is considered.

The Gaussian Beam Summation and the Gaussian Launching Methods in Scattering Problem

This paper is mainly devoted to application of the Gaussian beam summation technique in electromagnetic simulations problem. Gaussian beams are asymptotic solutions of the Helmholtz equation within the paraxial approximation. Since they are insensitive to ray transition region, several techniques based on Gaussian beam are used to evaluate high frequency EM wave equation, which overcome partially or fully the difficulties of singular regions (caustics, zero field in shadow zones). This paper concentrates on the explicit formulation of the electromagnetic field scattered from radar target. In this approach, when the incident field illuminates the target, the scattering is accounted in a complex weighing function. The wave field at a receiver is evaluated as superposition of Gaussian beams concentrated close to rays emerging from the target, passing through the neighbor of the receiver.

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

Symmetric hyperbolic systems include many physically relevant systems of PDEs such as Maxwell's equations, the elastic wave equations, and the acoustic equations [L. Ryzhik, G. Papanicolaou, and J. Keller, Wave Motion, 24 (1996), pp. 327--370]. In the current paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with polarized waves, in which the dispersion matrix of the hyperbolic system has eigenvalues with constant multiplicity greater than one over the domain of computation. The new results in this paper include new Gaussian beam equations in the presence of multiple eigenvalues, error estimates for Gaussian beam summation in the symmetric hyperbolic system case, and a new multidirectional Eulerian summation formula which maintains accuracy after the formation of caustics.

Kirchhoff modeling for attenuative anisotropic media using Gaussian beams

Seismic wave propagation in attenuative media can be efficiently modeled with ray-based methods. We present a methodology to generate reflection data from attenuative anisotropic media using the Kirchhoff scattering integral and summation of Gaussian beams. The Green’s functions are computed in the reference elastic model by Gaussian-beam summation, and the influence of attenuation is incorporated as a perturbation along the central ray. The reflected P-wave is obtained by substituting the approximate Green’s functions into the Kirchhoff scattering integral. Numerical examples for a transversely isotropic medium above a horizontal reflector and for a structurally complex acoustic model with a salt body confirm the accuracy of the method.