Li-Yun Fu

Rough Surface Scattering: Comparison of Various Approximation Theories for 2D SH Waves

A major impetus to study the topographic scattering effects is because most of our seismic observations are either at or very near to the earth’s surface. The sensitivity of regional phases to topographic roughness in the crust has been widely observed. Comparisons of several approximation solutions to rough surface scattering are conducted in this study for an analytical description of the close relation of topographic statistics and regional phase attenuation. These approximations include Kirchhoff approximation theory, Taylor expansion-based perturbation theory, two-scale model, Rytov phase approximation, and Born series method, with each valid for a range of roughness scales. Kirchhoff approximation ignores multiple scatterings between any two surface points. In general, it has been considered valid for the large-scale roughness components. Perturbation theory based on Taylor series expansion is valid for the small-scale roughness components where the surface heights deviate from a planar at z = 0 by less than a wavelength. Rytov phase approximation to large-scale topographic roughness is not subject to the stringent restrictions that apply to the Kirchhoff approximation. Tests with the Gaussian and semicircular convex topographies show that the Rytov approximation improves the Kirchhoff approximation in both amplitude and phase. For a two-scale topography that consists of two extreme roughness components (large and small scales), some assumptions are valid to combine the Kirchhoff and perturbation theories for rough surface scattering. The realistic methods for the multiscale surfaces come with the Born series approximation that accounts for multiple scattering between surface points. For instance, the second-order Born approximation might be sufficient to guarantee the accuracy for general rough surfaces without infinite gradients and extremely large surface heights. It must be stressed that the approximation solutions described in this article miss the conversion of energy between SH and P-SV waves that is one of the main features of the crustal wave guide in real situations. Extension to the elastic case must be conducted in the future.

Energy Partition and Attenuation of Regional Phases by Random Free Surface

The earths topography is generally rough at various scales. Numerical simulation techniques are used in this study to investigate the energy attenuation of regional phases across a randomly rough topography. We demonstrate a clear statistical correlation of the distance-dependent energy distribution with path topographic properties parameterized by the surface correlation length a and the surface root-mean-square height σ . Numerical experiments show that interference of randomly scattered waves by topography can cause regional wave amplitude undergoing strong variations. The topographic-scattering-driven energy distribution over a long distance is usually characteristic of an attenuation trend on the long distance scale, accompanied by amplitude fluctuations on the smaller distance scale. Total energy attenuation can be divided into large-scale and small-scale components that are correlated, respectively, in quite different manners, with along-path topographic statistics. On the one hand, the small-scale energy component is strongly related to the near-receiver topographic geometry. It has a striking similarity to the corresponding topographic curve. The spatial fluctuation of the small-scale component depends on a , whereas its amplitude amplification/deamplification is mainly related to σ , wavelength, and local incident angles. On the other hand, the large-scale component of energy curve, described by a scattering Q , demonstrates a scale-dependent relation with topographic statistics. A two-step analysis method is presented to evaluate the mantle leakage loss due to topographic scattering. The resultant topographic scattering Q is comparable with some observations with Q measured as a mean value in the crust. In summary, the study suggests the concept that topographic scattering might be a powerful mechanism to attenuate regional waves. Manuscript received 15 December 2000.

Two effective approaches to reduce data storage in reverse time migration

Prestack reverse time migration (RTM) requires extensive data storage since it computes wavefields in forward time and accesses wavefields in reverse order. We first review several successful schemes that have been proposed to reduce data storage, but require more computational redundancies. We propose two effective strategies to reduce data storage during RTM. The first strategy is based on the Nyquist sampling theorem, which involves no extra computational cost. The fact is that the time sampling intervals required by numerical algorithms or given by field records is generally several times smaller than that satisfied by the Nyquist sampling theorem. Therefore, we can correlate the source wavefields with the receiver wavefields at the Nyquist time step, which helps decrease storage of time history. The second strategy is based on a lossless compression algorithm, which is widely used in computer science and information theory. The compression approach reduces storage significantly at a little computational cost. Numerical examples show that the two proposed strategies are effective and efficient. Two approaches are proposed to reduce data storages in reverse time migration.The Nyquist scheme uses an imaging condition to reduce data storages.The Nyquist scheme introduces with no extra computational costs.The compression scheme employs compression algorithm to reduce data storages.The two approaches are proved to be effective via numerical examples.

Infinite boundary element absorbing boundary for wave propagation simulations

In the boundary element (BE) solution of wave propagation, infinite absorbing elements are introduced to minimize diffractions from truncated edges of models. This leads to a significant simplification and reduction of computational effort, especially for 3-D problems. The infinite BE absorbing boundary condition has a general form for both 2-D and 3-D problems and for both acoustic and elastic cases. Its implementation is facilitated by the introduction of the corresponding shape functions. Numerical experiments illustrate a nearly perfect absorption of unwanted diffractions. The approach overcomes some of the difficulties encountered in conventional absorbing techniques and takes less memory space and less computing time.

Numerical study of generalized Lipmann-Schwinger integral equation including surface topography

Rugged topographies and general heterogeneities in the near-surface sediments have a significant influence on exploration seismic data recorded at the free surface. To remove the near-surface effect on seismic data, a thorough understanding of wave propagation in the shallow subsurface region is needed. The physical effect of wave propagation in the near-surface region characterized by volume heterogeneities and rugged topographies can be described by a generalized Lipmann-Schwinger integral equation (GLSIE). By virtue of the GLSIE, in this paper I develop an efficient numerical method for the study of wave scattering in complex near-surface areas.

A Hybrid BE-GS Method for Modeling Regional Wave Propagation

Abstract — We present a hybrid boundary-element (BE) and generalized screen propagator (GSP) method for the 2-D SH problem to model the combined effects of arbitrarily irregular topography, large-scale crustal variation, and the associated small-scale heterogeneities on regional wave propagation. We develop a boundary connection technique to couple the wave fields calculated by the BE method with those of the GSP method. Its validity is tested by numerical experiments. For a long crustal waveguide, the relatively short sections with severe surface topography can be modeled by the time-consuming BE method to high frequencies, and the exterior field in the relatively weak heterogeneous media of large volume can be calculated by the GSP method. For the waveguide with severe topography, the BE method can be used section by section via the boundary connection technique to model the combined effects of rough topography and large-scale structural variation on Lg wave propagation at extended regional distances.¶Numerical comparisons with independent methods showed that the hybrid method is relatively accurate for Lg simulation. We apply the hybrid method to Lg wave propagation in two real crustal waveguides in the Tibet region; one with Lg blockage and another without blockage. We found that the most characteristic effect from the irregular topography is the strong scattering by the topographic structures. The scattering by local irregular topographies leads to anomalous near-receive effects and tends to remove energy from the guided waves, which causes decay of amplitude and waveform distortion. It can be expected that rough surface topography and random heterogeneities with scale length close to the dominant wavelength will be very efficient in attenuating regional waves. The dramatic lateral variation of the topography-Moho large-scale structure combined with the small-scale rough topography and random heterogeneities could be the cause of Lg anomalous attenuation and blockage observed in this region. More quantitative assessment of the topographic effects must be conducted in the future.

Forward problem of nonlinear Fredholm integral equation in reference medium via velocity-weighted wavefield function

A nonlinear Fredholm integral equation of the first kind for the perturbation potential can be derived by interpreting the acoustic velocity as a perturbation of a reference velocity. We present an accurate and efficient method to formulate and numerically solve this equation with no restriction on the size of the perturbation by introducing a velocity‐weighted wavefield function (i.e., the scalar product of the potential function with the perturbed velocity function inside a scatterer), which is an intermediate function associating the observed wavefield with the perturbed velocity function. We start with a singularity analysis of the Fredholm integral equation when the observation point coincides with the scattering point to establish a linear singular integral equation with respect to the velocity‐weighted wavefield function. Then we formulate the relation between the velocity‐weighted wavefield and scattered field in the f‐k domain. A numerical scheme is developed to solve the forward problem for the ...

Comparison of Different One-Way Propagators for Wave Forward Propagation in Heterogeneous Crustal Wave Guides

Crustal wave guides are usually heterogeneous on many scales, some of which can be investigated for regional phases by one-way approximation methods. The advantage of one-way propagation methods is the greater saving of computing time and memory, often up to several orders, than the full-waveform numerical methods. By slicing a half-space crustal wave guide into a number of slabs perpendicular to the propagation direction, Wu et al. (2000) formulate a wide-angle pseudoscreen method for the SH half-space problem, which leads to a generalized screen propagator (gsp) for simulating Lg propagation. In this article we introduce a broadband constant-coefficient propagator for the SH half-space problem, which accounts for wide angles in large-contrast media while allowing implementation using Fourier transforms alone. Particular attention is paid to the first-order separation-of-variables screen propagator (svsp1) that significantly improves the split-step Fourier (ssf) method for large lateral variations at the cost of one more Fourier transform in each slab. Advancing wave fields by svsp1 is actually a linear interpolation in the wavenumber domain between two split-step terms. We benchmark the ssf, gsp, and svsp1 synthetic seismograms against the full-waveform boundary-element synthetics for flat, belling, and necking crustal wave guides, which shows that the svsp1 method can model the Lg phase and the mantle wave (head wave) quite well in both travel time, energy, and waveform for most common mantle velocity perturbations. These numerical comparisons also demonstrate some limitations (especially in waveform) of the one-way propagation methods to model the Lg code attributed by forward scatterings as a result of lateral irregularities of the Moho.

P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation

A boundary element (be) method is developed to calculate the two-dimensional P-SV elastic response for crustal wave guides with irregular topographic features. To simulate long-range propagation of regional waves, a connection technique is proposed to avoid large matrix inversions that become formidable for long-range, high-frequency problems. By using this technique, a long crustal wave guide can be divided into relatively shorter sections, and the be method can be used section by section to model the effects of rough topography on wave propagation at extended regional distances. The validity of the technique is tested by comparison with a direct calculation. Numerical simulations with this scheme show that rough topography can scatter the P and Rayleigh waves and attenuate the energy propagating in the wave guide. This method can be used in computing the site effects on sites such as canyons, mountains, and valleys. The connection technique expands this method to deal with large earth models with irregular topography.

Joint inversion of seismic data for acoustic impedance

I propose a joint inversion scheme to integrate seismic data, well data, and geological knowledge for acoustic impedance estimation. I examine the problem of recovering acoustic impedance from band-limited seismic data. Optimal estimation of impedance can be achieved by combined applications of model-based and deconvolution-based methods. I incorporate the Robinson seismic convolutional model (RSCM) into the Caianiello neural network for network mapping. The Caianiello neural network provides an efficient approach to decompose the seismic wavelet and its inverse. The joint inversion consists of four steps: (1) multistage seismic inverse wavelets (MSIW) extraction at the wells, (2) the deconvolution with MSIW for initial impedance estimation, (3) multistage seismic wavelets (MSW) extraction at the wells, and (4) the model-based reconstruction of impedance with MSW for improving the initial impedance model. The Caianiello neural network offers two algorithms for the four-step process: neural wavelet estimation and input signal reconstruction. The frequency-domain implementation of the algorithms enables control of the inversion on different frequency scales and facilitates an understanding of reservoir behavior on different resolution scales. The test results show that, with well control, the joint inversion can significantly improve the spatial description of reservoirs in data sets involving complex continental deposits.