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We propose a full-wave equation based method for seismic illumination analysis. The full-wave finite-difference method is used to calculate the waves from the source and receiver to the subsurface target. A time domain local slowness analysis technique is used to decompose the wavefield into angle domain beams. Based on the angle domain information, we formulate the illumination matrix from which different illumination measurements can be derived. Unlike the one-way wave equation based method, the current approach does not have the angle limitation and the reflection/transmission energy can be accurately calculated. The new illumination method can handle turning waves which are crucial for imaging steep structures. Thus, this method is particularly useful for providing illumination analysis in reverse-time migration. A group of numerical examples are calculated to demonstrate the time-domain wavefield decomposition and the creation of different illumination measurements. The illumination in the 2D BP model is investigated. Our special attention is focused on the illumination of vertical and overhung structures.

We propose a full-wave equation based illumination analysis method with target oriented capability. The fullwave finite-difference propagator is used to extrapolate the source and receiver side wavefields to the subsurface target area. A time-domain local-slowness analysis method is used to decompose the wavefields into local angle domain. The local illumination matrix can be constructed and different types of illumination measurements can be derived. This illumination analysis method does not have angle limitations. Thus this approach can handle structures with their dipping angles beyond 90 degrees and is particularly useful to provide illumination analysis for reverse-time migration.

The De Wolf approximation has been introduced to overcome the limitation of the Born and Rytov approximations in long range forward propagation and backscattering calculations. The De Wolf approximation is a multiple-forescattering-single-backscattering (MFSB) approximation, which can be implemented by using an iterative marching algorithm with a single backscattering calculation for each marching step (a thin-slab). Therefore, it is also called a one-return approximation. The marching algorithm not only updates the incident field step-by-step, in the forward direction, but also the Green's function when propagating the backscattered waves to the receivers. This distinguishes it from the first order approximation of the asymptotic multiple scattering series, such as the generalized Bremmer series, where the Green's function is approximated by an asymptotic solution. The De Wolf approximation neglects the reverberations (internal multiples) inside thin-slabs, but can model all the forward scattering phenomena, such as focusing/defocusing, diffraction, refraction, interference, as well as the primary reflections. In this chapter, renormalized MFSB (multiple-forescattering–single-backscattering) equations and the dual-domain expressions for scalar, acoustic and elastic waves are derived by using a unified approach. Two versions of the one-return method (using MFSB approximation) are given: one is the wide-angle, dual-domain formulation (thin-slab approximation) (compared to the screen approximation, no small-angle approximation is made in the derivation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be clearly seen that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, are mainly controlled by impedance perturbations. Later in this chapter the validity of the thin-slab and screen methods, and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface, derived from numerical simulations by the wide-angle method, are shown to match the theoretical curves well up to critical angles. The methods are applied to the fast calculation of synthetic seismograms. The results are compared with finite difference (FD) calculations for the elastic French model. For weak heterogeneities ( ± 15 % perturbation), good agreement between the two methods verifies the validity of the one-return approach. However, the one-return approach is about 2–3 orders of magnitude faster than the elastic FD algorithm. The other example of application is the modeling of amplitude variation with angle (AVA) responses for a complex reservoir with heterogeneous overburdens. In addition to its fast computation speed, the one return method (thin-slab and complex-screen propagators) has some special advantages when applied to the thin-bed and random layer responses.

In the crustal waveguide environment, the major part of wave energy is carried by forward propagating waves, including forward scattered waves. Therefore, neglecting backscattered waves in numerical modeling will not modify the main features of regional waves in most cases. By neglecting backscattering in the theory, the wave modeling becomes a forward marching problem in which the next step of propagation depends only on the present values of the wavefield in a transverse cross-section and the heterogeneities between the present cross-section and the next one (wavefield extrapolation interval). The saving of computation time and computer memory is enormous. A half-space screen propagator (generalized screen propagator) has been developed to accommodate the free-surface boundary condition for modeling SH wave propagation in complex crustal waveguides. The SH screen propagator has also been extended to handle irregular surface topography using conformal or non-conformal topographic transforms. The screen propagator for modeling regional SH waves has been calibrated extensively against some full-wave methods, such as the wavenumber integration, finite-difference and boundary element methods, for different crustal models. Excellent agreement with these full-wave methods demonstrated the validity and accuracy of the new one-way propagator method. For medium size problems, the screen-propagator method is 2–3 orders of magnitude faster than finite-difference methods. It has been used for the simulation of Lg propagation in crustal models with random heterogeneities and the related energy partition, attenuation and blockage. It is found that the leakage attenuation of Lg waves caused by large-angle forward scattering by random heterogeneities, which scatters the guided waves out of the trapped modes and leaking into the mantle, may contribute significantly to Lg attenuation and blockage in some regions. In the case of P-SV elastic screen propagators, plane wave reflection calculations have been incorporated into the elastic screen method to handle the free surface. Body waves, including the reflected and converted waves, can be calculated by real wavenumber integration; while surface waves (Rayleigh waves) can be obtained with imaginary wavenumber integration. Numerical tests proved the validity of the theory and methods.

We define and formulate the resolution of an imaging system based on the inverse theory and local angle domain decomposition of Green’s functions. The resolution defined in this way includes both the effects of acquisition system and imaging (migration) process. The theory and method are based on wave theory and no asymptotic approximation is made in the calculation. It represents and quantifies the actual resolution we observed in the migrated images. The resolution taking into account only the influence of the acquisition system (frequency band and spatial aperture) and neglecting the factors such as the errors in backpropagation (migration) can be considered as the resolution limit of the system, the best resolution an acquisition system can get. Theoretical analysis and numerical examples are given to show the importance of propagator accuracy in the evaluation of resolution in complex media.

Summary A depth migration method combining the full-wave reverse-time calculation and local one-way propagation is proposed. The full-wave finite-difference propagator is used to extrapolate the source and receiver waves in the model. The wavefields are extracted from selected locations and then a local one-way propagator is used to extrapolate the wavefield towards different directions. The one-way propagator is also served as a one-way filter, separating waves propagating along different directions. Finally, the image is calculated from directional waves generated by the one-way propagator and the artifacts from backscattered waves of both source and receiver sides are effectively eliminated. This method also dramatically reduces the size of output files which is important when dealing with large 3D models.

In this project, we develop new theories and methods for multi-domain one-way wave-equation based propagators, and apply these techniques to seismic modeling, seismic imaging, seismic illumination and model parameter estimation in 3D complex environments. The major progress of this project includes: (1) The development of the dual-domain wave propagators. We continue to improve the one-way wave-equation based propagators. Our target is making propagators capable of handling more realistic velocity models. A wide-angle propagator for transversely isotropic media with vertically symmetric axis (VTI) has been developed for P-wave modeling and imaging. The resulting propagator is accurate for large velocity perturbations and wide propagation angles. The thin-slab propagator for one-way elastic-wave propagation is further improved. With the introduction of complex velocities, the quality factors Qp and Qs have been incorporated into the thin-slab propagator. The resulting viscoelastic thin-slab propagator can handle elastic-wave propagation in models with intrinsic attenuations. We apply this method to complex models for AVO modeling, random media characterization and frequency-dependent reflectivity simulation. (2) Exploring the Information in the Local Angle Domain. Traditionally, the local angle information can only be extracted using the ray-based method. We develop a wave-equation based technique to process the local angle domain information. The approach can avoid the singularity problem usually linked to the high-frequency asymptotic method. We successfully apply this technique to seismic illumination and the resulting method provides a practical tool for three-dimensional full-volume illumination analysis in complex structures. The directional illumination also provides information for angle-domain imaging corrections. (3) Elastic-Wave Imaging. We develop a multicomponent elastic migration method. The application of the multicomponent one-way elastic propagator and the wide-angle correction preserve more dynamic information carried by the elastic waves. The vector imaging condition solves the polarization problem of converted wave imaging. Both P-P and P-S images can be calculated. We also use converted waves to improve the image of steep sub-salt structures. The synthetic data for the SEG/EAGE salt model are migrated with a generalized screen algorithm and for the converted PSS-wave path. All the sub-salt faults are properly imaged.

The transmission coefficient and WKBJ approximation can be adopted for constructing the true amplitude propagators. For vertically inhomogeneous media, the WKBJ approximation or transmission coefficients can be applied on the plane waves at each depth to get the true amplitudes. While for the media with lateral variation, the correction coefficients vary with horizontal locations, and cannot be applied on the global plane waves. However, the beamlet propagators decompose the wavefield into localized beamlets, and the laterally varying WKBJ or transmission correction coefficients can be conducted on a localized basis at each depth. In this paper, the theory for the localized WKBJ and transmission corrections are proposed and implemented with the local cosine basis (LCB) beamlet propagator. Numerical examples of the impulse responses are calculated to demonstrate the feasibility of the local WKBJ and local transmission corrections.

The resolution of seismic migration imaging is often given as some function of the data aperture, the size of the Fresnel zone at the image location, the frequency of the data used in the migration and the illumination of the target region. Traditional resolution analysis is based on calculation of a Point Spreading Function (PSF) in the wavenumber domain from the scattering wavenumbers at the image point. This spectral representation can be converted into a space-domain PSF by Fourier Transform. However, other issues influence image resolution or distort migrated images. For example, the velocity model is generally not reliably known and variations of the model from the true structure can have a significant impact on image focus, thus reducing resolution. Another factor that limits resolution is the type and accuracy of the propagator used in imaging. To compare the resolution of images obtained when using ray-based Kirchhoff migration with that obtained from wave equation migration, we developed a 2D heterogeneous model that can be numerically simulated. Poststack migrations of exploding reflector data obtained for the model show that the PSF are influenced by the migration operator and are not necessarily the same as expected from the range of scattering wavenumbers estimated using a simple analysis approach.

Summary For beamlet migration methods, the wave fields normally are statically and regularly windowed, and the wave field in e ach window is propagated with beamlet propagators based on local cosine basis (LCB) or Gabor-Daubechies frame (GDF), followed by local perturbations. These methods can provide pretty good imaging results when compared to traditional methods. However, with the LCB beamlet method, some artifacts are present at steeply inclined interface of l arge velocity contracts. The artifacts can be greatly reduc ed by dynamically shifting windows in the LCB beamlet method. The numerical results on 2D and 3D models will show the improvement in the imaging quality.

Summary Based on beamlet decomposition of wave field and Green’s function, we formulated an amplitude correction method in angle domain. The formulation relates the local image matr ix (LIM), which bears the footprints of the acquisition apertur e and propagation path effects, and the local scattering mat rix (LSM), which is directly related to the medium propert y. From the formulation, two types of amplitude correction ar e proposed: one is the correction for common reflection-angle image for AVA analysis. The other is the correction fo r total strength image. From the imaging results of the four-laye r model and the SEG/EAGE model, we see significant improvement in amplitude fidelity and image quality.

Summary In this paper, we compare one-way wave propagator and full-wave propagator such as finite difference (FD) method i n the evaluation of illumination and acquisition dip response (ADR) maps. For the case of 2D SEG/EAGE salt model, the illumination and ADR maps by the one-way method and the FD method have similar general distributions in space and angle domain. However, some difference exists in absolut e values, especially for the subsalt area. This is due to the strong effects of transmission loss caused by salt boundary scat tering, reflection and defocusing. The difference in wide-angle amplitude behavior is also noticeable. Therefore, in most cases illumination analysis calculated by one-way wave propagators can provide reliable evaluation.

Employing Gabor-Daubechies frame bases, beamlet domain decomposition and propagation provide localizations in both space and direction of the wave fields. First directional illumination is discussed and then the acquisition-aperture efficacy matrix and dip-response function are derived. As numerical examples, we calculate the directional illumination maps and acquisition efficacy maps for a lens model and the SEG-EAGE salt model. We investigate the influences of acquisition geometry and overlaying structures on the prestack migration image quality using the acquisition dip-response.

A wave-equation based method is developed for seismic illumination analysis. The approach uses the one-way screen propagator to provide fast and accurate wavefield extrapolation. A local plane wave analysis is used to determine energy propagation directions. The method can avoid the singularity problem usually linked to the high frequency asymptotic method. It provides a practical tool for three-dimensional full-volume illumination analysis in complicated structures. To demonstrate the potential application of this method, numerical examples for the 3D SEG/EAGE salt model are calculated.

The recent development of wave equation based migration methods provided accurate propagators for seismic wave extrapolation. These propagators brought the possibility that many analysis and inversion can be made at the depth using migrated wavefield. In this study, we propose an approach for extracting angle domain information from migrated wavefield. The method is based on localized plane wave analysis and can be used for almost any migration method. Then, using the concept of the local image matrix, useful information (angle domain image gathers, reflector dips, etc.) can be further extracted from this angle related information. Numerical examples are conducted to demonstrate the applications of this method.

Multicomponent data increased the information of seismic data. Recent developments of ocean bottom seismic (OBS) technology make more multicomponent data available. From these data, both P-P and P-S sections can be obtained. Since these waves carry different properties of the target reflectors, combining them together can provide us with petrophysical parameters that cannot be obtained with P-wave data alone. However, extracting the new information requires more physically rigorous techniques. Many attempts have been made to carry out elastic wave migrations. Chang and McMechan (1987, 1994) conducted 2D and 3D elastic reverse-time migrations using a full wave finite-difference method. Zhe and Greenhalgh (1997) used potentials, instead of displacements, to propagate P and S waves. Hokstad (2000) proposed a multicomponent Kirchhoff migration method and a specially designed imaging condition for multicomponent elastic waves. On the other hand, others chose to treat elastic waves as scalar waves with different speeds and used scalar wave propagator to do the migration. Using scalar wave propagators to simulate elastic wave propagation, although straightforward, sometimes over simplifies the real physical processes. On the other hand, the phase screen method for scalar wave migration has been used for many years (Stoffa et al., 1990). Several modifications of this method have been made to improve its accuracy under wide-angle and large velocity contrasts (e.g., Ristow and Ruhl, 1994; Jin, et al., 1998; Xie and Wu, 1998; Huang, et al., 1999; Huang and Fehler, 2000; Xie et al., 2000). Compared with other methods, this one-way wave equation based method provided an efficient, high quality propagator for scalar wave migration. The scalar wave screen propagator has also been extended into elastic wave case (Wu, 1994, 1996; Xie and Wu, 1996, 2000; Wild and Hudson ,1998). Wu and Xie (1994) did some primary work to test the elastic screen method as a backpropagator for multi-component elastic migration. Based on the above-mentioned progresses, it is natural to extend the scalar wave screen migration method to the multicomponent elastic wave case. In the following sections, we propose an elastic screen propagator and vector imaging conditions for prestack depth migration and imaging of multicomponent seismic data.

The long-term goal is to develop a new method and the code to simulate 3D acoustic/elastic wave propagation and interaction with the ocean water and ocean bottom environment. The method will be applied to numerical simulations and imaging to study the wave/sea-bottom interaction, energy partitioning, scattering mechanism and other problems that are crucial for many ocean bottom-surveying techniques. Our understanding on shallow water acoustic wave propagation and its interaction with sediments can be improved. Objectives are to improve and develop the ECS (Elastic Complex-Screen), a new one-way elastic wave propagation method, and apply it to the ocean bottom environment to study the elasto-acoustic wave propagation in complex laterally heterogeneous media, including rough interface and random volume heterogeneity.

Summary One-way wave equation method is very efficient. It can be used as forward and backward propagators in seismic modeling and migration. However, unlike the full wave method, the one-way method is usually derived for a source free situation. The source is added to the calculation as a boundary condition. The free surface effect at both the source side and receiver side are often not considered. Based on the plane wave decomposition, this study gives a free surface and source condition for elastic one-way wave equation method. This boundary condition can be applied at either the source side or the receiver side. With this boundary condition, many of the free surface effects, e.g., reflected P and S-waves, secondary surface waves, etc., can be generated. This boundary condition can be easily applied to reflection, VSP and cross hole modelings and seismic migration. It is especially useful for one-way methods based on the wavenumber domain Fourier transforms.

A new method of calculating synthetic seismograms of primary reflections or backscattering for surface reflection surveying is introduced using the De Wolf MFSB approximation and phase-screen one-way propagations. Two versions of algorithm are presented, both of which use the dual domain technique for computational efficiency. One is the direct use of MFSB approximation, in which the step length of calculating the backscattered waves uses the grid spacing, but the step length of forward propagation uses the screen interval which is much greater than the grid spacing. The other version is the screen approximation for both the forward and backward scattered waves and therefore both step lengths can adopt the screen interval, resulting in great computational efficiency. The screen-approximation involves small-angle approximation and may have some effects on large-angle backscattered waves. Numerical examples using both versions of algorithm show good results. Good agreements with finite difference calculation on the second example indicate that even though the screen approximation involves the small-angle approximation, satisfactory results can be obtained for many practical applications with a great computational efficiency.

In this paper. we first give a brief summary of the formulation for calculating reflections from a 3D elastic structure based on a complex-screen method. The incident wave and reflected wave are propagated by using a complexscreen propagator. The reflections are calculated based on local Born approximation. When using a small angle approximation, the backscattering can also be formulated into a screen reflection which has a high computation efficiency and needs relatively small computer memory. As expected from the scattering theory. the forward transmitted waves, like P to P or to are controlled by the Pand S-wave velocity perturbations, while the reflections are controlled by the impedance perturbations. The converted waves are mainly controlled by the density and shear module perturbations. Numerical examples are given to show the accuracy of the method. For two special cases, the synthetic seismograms obtained with this method are compared with that calculated by elastic finite-difference method. The results show general consistency.

Due the huge amount of computation and internal memory required, wave backpropagation becomes the bottleneck of prestack migration or other 3D imaging/inversion procedures. We propose to use the multi-screen backpropagator for 3D prestack migration in laterally inhomogeneous background (depth migration). Multi-screen (phase-screen for scalar waves, elastic complex-screen for elastic waves) backpropagator shuttles between space-domain and wavenumber-domain using FFT and therefore avoids the time-demanding matrix multiplication. The time saving is tremendous for large-size elastic wave problems. Because it needs to store the medium parameters only one grid-plane for each step, the enormous computer memory saving makes it capable of handling large 3D problem prohibitive to other methods. The method of elastic complex screen (ECS) is a one-way propagation algorithm by neglecting the backscattered waves. However, all the forward multiple-scattering effect, such as the focusing/defocusing, diffraction, interference, wave conversion between P and S, interface waves, guided waves, etc., can be correctly handled. In this paper first the Love integral and Love migration integral are introduced. The formulation of elastic complex-screen as elastic wave one-way propagator is summarized. Numerical tests and comparisons with other full-wave methods (elastic wave finite difference and eigenfunction expansion method) are presented to show the validity of the propagator. Finally, two numerical examples of single-shot prestack migration using the ECS backpropagator, one for homogeneous background and the other for inhomogeneous background, are shown to demonstrate the feasibility of the proposed scheme.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The theory of spatial distribution of seismic energy density in one dimensional (1D) random media derived in part 1 (Wu, 1993) is tested by numerical experiments using a full wave propagation matrix method. The geometry of numerical experiment mimics the configuration of zero-offset VSP (Vertical Seismic Profiling) along a borehole. A procedure of octave-band frequency averaging is applied to the measured data to reduce fluctuation of spatial energy distribution, so that stable estimations of medium parameters can be achieved without resorting to ensemble averaging. Results from Monte-Carlo numerical experiments for both infinite random media and finite random slabs with or without bottom reflections show good agreement for dark-to-gray (weak to intermediate scattering compared with absorption) media. When scattering is very strong (when backscattering-absorption ratio S{sub b} > 3), results from single realization fluctuate substantially. However, most the practical situations of sedimentary rocks in the crust fall into the validity region of the energy transfer theory.

Due to the limitation of sparse seismic observations on the earth’s surface, the deterministic picture of small‐scale heterogeneities in the earth is difficult to obtain in general. Statistical characterization is a viable alternative to study the distribution and properties of the small‐scale heterogeneities. The pioneering work of Aki in 1973 modeled the heterogeneities with a single‐layer uniform random medium having only three parameters: scale, perturbation strength, and layer thickness. This simple model is consistent with the observations which are limited to only the transversal coherence functions of seismic wave (phase and amplitude) across an array and it has no depth resolution. The following work in 1970s and early 1980s were based on similar models. Flatte introduced the key idea of angular coherence functions using different earthquakes at large epicentral distances in the late 1980s (Flatte and Wu, 1988), which opened a new area of earth heterogeneity characterization by extending the single, uniform layer model to a depth‐dependence model. Later, the angular coherence function was extended to the joint transverse‐angular coherence function, which significantly increases the depth resolution of the heterogeneity spectra. In this talk, we summarize the research results along this direction conducted by a group of scientists in the earth science at University of California, Santa Cruz, collaborated with Flatte in different stages.

Using Lg-wave Q tomography, we construct a broadband crustal attenuation model for the Middle East. The Q Lg images reveal a relationship between attenuation and geological structures. Strong attenuation is found in the continental collision orogenic belt that extends from the Turkish and Iranian plateau to the Pamir plateau. We investigate the frequency dependence of Q Lg in different geologic formations. The results illustrate that Q Lg values generally increase with increasing frequency but exhibit complex relationships both with frequency and between regions. An average Q Lg value between 0.2 and 2.0 Hz, Q Lg (0.2–2.0 Hz), may be a critical index for crustal attenuation and is used to infer the regional geology. Low-Q Lg anomalies are present in the eastern Turkish plateau and correlate well with low Pn-velocities and Cenozoic volcanic activity, thus indicating possible partial melting within the crust in this region. Very strong attenuation is also observed in central Iran, the Afghanistan block, and the southern Caspian Sea. This in line with the previously observed high crustal temperature, high-conductivity layers, and thick marine sediments in these areas, suggests the high Lg attenuation is caused by abnormally high tectonic and thermal activities.