Yu Geng

Seismic wave propagation and imaging using time‐space wavelets

Using a tensor product of a local exponential frame vector as the time-frequency atom (a drumbeat) and a local cosine basis function as the space-wavenumber atom (a beamlet), we construct a localized ( t ω − , x ξ − ) atom (a drumbeatbeamlet). The imaging condition in the local t ω − domain is introduced and the propagator matrix in the local ( t ω − , x ξ − ) domain is derived. The compression of seismic data using the new decomposition is tested and simple numerical tests on the propagator matrix are performed. The sparseness in both data and propagator decompositions is demonstrated through these examples. The method has potential application in efficient representation of seismic data and wave-theory based seismic imaging using compressed data.

Dreamlet transform applied to seismic data compression and its effects on migration

Local cosine/sine basis is a localized version of cosine/sine basis with a smooth window function. It has orthogonality and good time and frequency localization properties together with a fast algorithm. In this paper, we present a new method combining the main idea of local cosine/sine bases, multi-scale decomposition and the dispersion relation to form a multi-scale, multi-dimensional selfsimilar dreamlet transform. Meanwhile, a storage scheme based on the dreamlet decomposition is proposed by using zig-zag sequence. We apply this method to the SEG-EAGE salt model synthetic poststack data set for data compression. From the result, almost all the important features of the data set can be well kept, even in high compression ratio. Using the reconstructed data for migration, we can still obtain a high quality image of the structure. Through comparing with other decomposition and compression schemes, we believe that our sheme is more closely related to the physics of wavefield and has a better performance for seismic data compression and migration.

Gabor‐frame‐based Gaussian packet migration

We present a Gaussian packet migration method based on Gabor frame decomposition and asymptotic propagation of Gaussian packets. A Gaussian packet has both Gaussian-shaped time–frequency localization and space–direction localization. Its evolution can be obtained by ray tracing and dynamic ray tracing. In this paper, we first briefly review the concept of Gaussian packets. After discussing how initial parameters affect the shape of a Gaussian packet, we then propose two Gabor-frame-based Gaussian packet decomposition methods that can sparsely and accurately represent seismic data. One method is the dreamlet–Gaussian packet method. Dreamlets are physical wavelets defined on an observation plane and can represent seismic data efficiently in the local time–frequency space–wavenumber domain. After decomposition, dreamlet coefficients can be easily converted to the corresponding Gaussian packet coefficients. The other method is the Gabor-frame Gaussian beam method. In this method, a local slant stack, which is widely used in Gaussian beam migration, is combined with the Gabor frame decomposition to obtain uniform sampled horizontal slowness for each local frequency. Based on these decomposition methods, we derive a poststack depth migration method through the summation of the backpropagated Gaussian packets and the application of the imaging condition. To demonstrate the Gaussian packet evolution and migration/imaging in complex models, we show several numerical examples. We first use the evolution of a single Gaussian packet in media with different complexities to show the accuracy of Gaussian packet propagation. Then we test the point source responses in smoothed varying velocity models to show the accuracy of Gaussian packet summation. Finally, using poststack synthetic data sets of a four-layer model and the two-dimensional SEG/EAGE model, we demonstrate the validity and accuracy of the migration method. Compared with the more accurate but more time-consuming one-way wave-equation-based migration, such as beamlet migration, the Gaussian packet method proposed in this paper can correctly image the major structures of the complex model, especially in subsalt areas, with much higher efficiency. This shows the application potential of Gaussian packet migration in complicated areas.

Physical wavelet defined on an observation plane and the Dreamlet

Summary Wavefield or seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways. The timespace distributions must observe causality which is dictated by the wave equation. Wave solutions can only exist on the light cone in the 4D Fourier space. Physical wavelet is a localized wave solution by extending the light cone into complex causal tube. In this study we establish the link between the physical wavelet defined by Kaiser using AST (analytic signal transform) and the dreamlet (drumbeatbeamlet). We prove that dreamlet can be considered as a type of physical wavelet defined on an observation plane (earth surface or a plane at depth z during extrapolation). Causality (or dispersion relation) built into the wavelet (dreamlet) and propagator is a distinctive feature of physical wavelet which is advantageous for applications in wavefield decomposition, propagation and imaging. One example of dreamlet decomposition on seismic data is given.

Imaging in Compressed domain using Dreamlets

Summary The objective of this paper is to further investigate the theory and algorithm of wave propagation and imaging in the dreamlet domain for direct application of compressed seismic data. In our former work, dreamlet shows great application potential in efficient representation of seismic data. Based on the combination of dreamlet compression and wave propagation theory, we discuss one-way dreamlet wave propagation and imaging in terms of data processing in the compressed domain, that allows us to process only a few percentage of the decomposition coefficients of the whole data set to get the accurate imaging result. By using a 5-layer velocity model as numerical example, we show that the dreamlet coefficients of the seismic data is also decreasing with the depth of migration in the receiver side, meanwhile the imaging quality using the compressed data remains in a similar accuracy.

Efficient Gaussian Packets Representation And Seismic Imaging

In this paper we discuss an efficient way to apply Gaussian Packet method to data representation and seismic imaging. Similar to Gaussian Beam method, wavefield radiating from a seismic source as a set of Gaussian Packets can represent synthetic seismic data, and recorded wavefield at the surface can be expressed and downward continued by a set of Gaussian Packets at surface as well. The evolution of Gaussian Packet is determined by parameter of central frequency, local time, local space and ray emergent angle, and the shape of Gaussian Packet is determined by its initial value. Each Gaussian Packet is directly related to the ray emergent angle, and propagates along a central ray. Therefore, representation of seismic data using Gaussian Packets provides the local time slope and location information at certain central frequency, while summation of Gaussian Packets’ evolutions constructs the corresponding propagated wavefield. These properties also make seismic imaging using Gaussian Packets easily be understood and implemented. The method becomes efficient because 1) to represent seismic data, Gaussian Packets with given initial value can be used and inner product can be applied to obtain the useful information of seismic data; 2) with proper initial value, only Gaussian Packets with few central frequencies are needed for representing propagated seismic wavefield. Numerical examples on impulse responses and a 4layer zero-offset data are calculated to demonstrate the valid of the method.

Poincaré wavelet techniques in depth migration

A method based on space-time wavelets is developed for the migration problem in a smooth layered medium. The problem is to restore reflection boundaries inside the medium if signals emitted from the surface of the medium and reflected wavefield received on the same surface are known. Boundaries are determined as maxima of a function of sub-surface fields: a forward-propagated radiated field and a back-propagated received one. We represent the subsurface fields in terms of localized solutions running in the medium. Initial amplitudes of these localized solutions are calculated by means of the continuous space-time wavelet analysis for the boundary value (seismic) data. An example with seismograms calculated by the finite differences method is presented.

Evolution of Gaussian Packets In Inhomogeneous Media

Time-dependent Gaussian Packets are high-frequency asymptotic space-time particle-like solutions of the wave equation. A Gaussian Packet is a Gaussian shape timelocalized Gaussian Beam. We study the dependence of Packet shape to the initial values and discuss the evolution of Gaussian Packet in smooth media and strong inhomogeneous media. Through comparison between paraxial Gaussian Packets and the results from finite difference simulations in strongly inhomogeneous media, such as the SEG/EAGE salt model, we see the severe deviation of paraxial Gaussian Packet from the accurate solution at large propagation time and near the salt boundaries. The application to strongly heterogeneous media needs to be further studied.

Local Angle Domain Target Oriented Illumination Analysis And Imaging Using Beamlets

Summary Prestack depth migration has been widely used for seismic imaging recently. However, the imaging of sub-salt structure is still a hard issue. Here, we use local angle domain target oriented illumination analysis to study how the acquisition geometry affects the image quality, especially for some reflectors with certain dip angle in the sub-salt area. Numerical tests based on 2D SEG/EAGE salt model helps us to better understand the capability of prestack migration under the given acquisition system. This analysis can also provide us an easy and efficient way to do target-oriented migration for the given reflector.