Ru-Shan Wu

Diffraction tomography and multisource holography applied to seismic imaging

Seismic tomography is emerging as an imaging method for determining subsurface structure. When the view‐angle coverage is limited and the scale of the medium inhomogeneities is comparable with the wavelength, as is often true in geophysical applications, the performance of ordinary ray tomography becomes poor. Other tomographic methods are needed to improve the imaging process. Here we study diffraction tomography and multisource holography and evaluate their performances for surface reflection profiling (SRP), vertical seismic profiling (VSP), and cross‐hole measurements. Theoretical formulations are derived for two‐dimensional geometry in terms of line sources along a source line and line receivers along a receiver line. The theory for diffraction tomography is based on the Born or Rytov approximation. The performances of diffraction tomography and multisource holography are evaluated by examining the information coverage in the spatial frequency domain and by numerical examples. Multisource holography,...

Scattering characteristics of elastic waves by an elastic heterogeneity

Elastic wave scattering by a general elastic heterogeneity having slightly different density and elastic constants from the surrounding medium is formulated using the equivalent source method and Born approximation. In the low‐frequency range (Rayleigh scattering) the scattered field by an arbitrary heterogeneity having an arbitrary variation of density and elastic constants can be equated to a radiation field from a point source composed of a unidirectional force proportional to the density contrast between the heterogeneity and the medium, and a force moment tensor proportional to the contrasts of elastic constant. It is also shown that the scattered field can be decomposed into an “impedance‐type” field, which has a main lobe in the backscattering direction and no scattering in the exact forward direction, and a “velocity type” scattered field, which has a main lobe in the forward scattering direction and no scattering in the exact backward direction. For Mie scattering we show that the scattered far f...

Wide-angle elastic wave one-way propagation in heterogeneous media and an elastic wave complex-screen method

In this paper a system of equations for wide-angle one-way elastic wave propagation in arbitrarily heterogeneous media is formulated in both the space and wavenumber domains using elastic Rayleigh integrals and local elastic Born scattering theory. The wavenumber domain formulation leads to compact solutions to one-way propagation and scattering problems. It is shown that wide-angle scattering in heterogeneous elastic media cannot be formulated as passage through regular phase-screens, since the interaction between the incident wavefield and the heterogeneities is not local in both the space domain and the wavenumber domain. Our more generally valid formulation is called the “thin-slab” formulation. After applying the small-angle approximation, the thin-slab effect degenerates to that of an elastic complex-screen (or “generalized phase-screen”). Compared with scalar phase-screen, the elastic complex-screen has the following features. (1) For P-P scattering and S-S in-plane scattering, the elastic complex-screen acts as two separate scalar phase-screens for P and S waves respectively. The phase distortions are determined by the P and S wave velocity perturbations respectively. (2) For P-S and S-P conversions, the screen is no longer a pure phase-screen and becomes complex (with both phase and amplitude terms); both conversions are determined by the shear wave velocity perturbation and the shear modulus perturbation. For Poisson solids the S wave velocity perturbation plays a major role. In the special case of α0 = 2β0, S wave velocity perturbation becomes the only factor for both conversions. (3) For the cross-coupling between in-plane S waves and off-plane S waves, only the shear modulus perturbation δμ has influence in the thin-slab formulation. For the complex-screen method the cross-coupling term is neglected because it is a higher order small quantity for small-angle scattering. Relative to prior derivations of vector phase-screen method, our method can correctly treat the conversion between P and S waves and the cross-coupling between differently polarized S waves. A comparison with solutions from three-dimensional finite difference and exact solutions using eigenfunction expansion is made for two special cases. One is for a solid sphere with only P velocity perturbation; the other is with only S velocity perturbation. The Elastic complex-screen method generally agrees well with the three-dimensional finite difference method and the exact solutions. In the limiting case of scalar waves, the derivation in this paper leads to a more generally valid new method, namely, a scalar thin-slab method. When making the small-angle approximation to the interaction term while keeping the propagation term unchanged, the thin-slab method approaches the currently available scalar wide-angle phase-screen method.

Multiple scattering and energy transfer of seismic waves—Separation of scattering effect from intrinsic attenuation II. Application of the theory to Hindu Kush region

In order to separate the scattering effect from intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In paper I (Wu, 1985), radiative transfer theory is applied to seismic wave propagation and the energy density distribution (or the average intensity) in space for a point source is formulated in the frequency domain. It is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves. In this paper, the data from digital recordings in the Hindu Kush region are used as an example of application of the theory. We also discuss two approximate solutions of coda envelope in the time domain: the single scattering approximation and the diffusion approximation and discuss the relation with the frequency domain solution. We point out that in only two cases can the apparent attenuation be expressed as an exponential decay form. One is thedark medium case, i.e., whenB0≪0.5, whereB0 =ηs/(ηs +ηa) is the seismic albedo,ηs is the scattering coefficient,ηa is the absorption coefficient. In this case the absorption is dominant, the apparent attenuationb can be approximated by the coherent wave attenuationb =ηs +ηa. The other case is thediffuse scattering regime, i.e., whenB0≫0.5 (bright medium) andR≫Ls,t ≪ τs, whereR andt are the propagation distance and lapse time,Ls and τs are the scattering lengths (mean free path) and scattering time (mean free time), respectively. However, in this case the envelope decays with a rate close to the intrinsic attenuation, while the intensity decreases with distance with a coefficientb ≈d0(ηs +ηa) ≈dsηs, whered0 andds are the diffusion multipliers (0<d0,ds<1).For the Hindu Kush region, by comparing the theory with data from two digital stations of 53 events distributed up to depths of 350 km, we find that the scattering is not the dominant factor for the measured apparent attenuation ofS waves in the frequency range 2–20 Hz. From the observation on high frequency (f>20 Hz) seismograms, we suggest the existence of a stron-scattering surface layer with fine scale heterogeneities in the crust, at least for this region.

Wave-equation-based seismic illumination analysis

We present a wave-equation-based method for seismic illumination analysis. A one-way wave-equation-based, generalized screen propagator is used to extrapolate the wavefields from sources and receivers to the subsurface target. A local plane-wave analysis is used at the target to calculate localized,directionalenergyfluxesforbothsourceandreceiver wavefields. We construct an illumination matrix using these energy fluxes to quantify the target illumination conditions. The target geometry information is used to manipulate the illumination matrix and generate different types of illumination measures. The wave-equation-based approach can properly handle forward multiple-scattering phenomena, including focusing/defocusing, diffraction, and interference effects.Itcanbedirectlyappliedtocomplexvelocitymodels. Velocity-model smoothing and Fresnel-zone smoothing are not required. Different illumination measurements derived from this method can be applied to target-oriented or volumetric illumination analyses. This new method is flexible and practical for illumination analysis in complex 2D and 3D velocity models with nontrivial acquisition and target

Extracting Angle Domain Information From Migrated Wavefield

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.

Heterogeneity spectrum and scale-anisotropy in the upper crust revealed by the German Continental Deep-Drilling (KTB) Holes

Crustal heterogeneities and their statistical characteristics bear important information about the dynamic processes and evolution of the crust. The velocity well-log data from the German Continental Deep Drilling Project (KTB) offer a rare opportunity to measure directly the properties of crustal heterogeneities. In this study, we first estimated the power spectrum of crustal heterogeneities from the P-velocity well-logs of the two holes. For the first time, a power-law spectrum of crustal heterogeneities was directly observed in the spatial wavelength range from a few meters to a few kilometers. The slope of the vertical 1D power spectrum of the random velocity perturbations is about 1.1, corresponding to a flicker-noise random medium. The scale-anisotropy ratio (aspect ratio) and the horizontal spectral slope of crustal heterogeneities were also estimated by cross-correlation analysis between these two holes to be about 1.8 and 2.0, respectively.

Introduction: Seismic Wave Scattering in Three-dimensionally Heterogeneous Earth

The classical earth model of spherical symmetry ( or layered structure) is undergoing a revolution. The earth has been revealed to be laterally heterogeneous everywhere from the crust, mantle to the core with scale from the grain size of rocks to the lowest orders of global spherical harmonics. Figure I shows the strength and scale length of heterogeneities in the crust and mantle of the earth, in terms of ii, the perturbation index of seismic (P or S) wave speed, defined as the r.m.s. fractional variations of wave speed over the measured region. The scale length of heterogeneities revealed by seismic waves, not including the laboratory measurements of rock samples, spans 8 orders of magnitude. These heterogeneities with different scales have different effects to seismic waves. The velocity and density heterogeneities can cause the change in waveform, phase (or travel-time) and amplitude fluctuation, as well as apparent attenuation of the direct arrivals. They can also generate coda waves such as the P-coda, S-coda, and Lg-coda caused by the lithospheric heterogeneities and precursory waves such as the scattered PKP waves as the precursors to PKIKP caused by the heterogeneities near the core-mantle boundary. The near-source or near-receiver structures can modify the seismic waveforms by resonance and other effect. Rough topography or rough interface can cause the coupling between body wave and surface wave. Aligned cracks in the crust can produce the effective anisotrophy. A great complexity arises when heterogeneities have interaction with anisotrophy and nonlinearity. The modification of seismic wave caused by the three-dimensional heterogeneities is broadly called seismic wave scattering. In order to gain some perspective on this extremely complex phenomena, let us try to classify the seismic wave scattering analogous to the classification made for waves in other branches of physics. We can discuss the effects of regional and local heterogeneities on seismic waves, i.e. different scattering phenomena in terms of different propagation regimes. With heterogeneities of scale a and strength (perturbation index ii, the wave propagation regime can be characterized by three dimensionless numbers: ka = 2na/A, L/a and ii, where k is the wave number and A is the wavelength in the medium, L is the propagation length or the extent of the heterogeneous region. Therefore, ka is the normalized wave number or the inverse of the normalized

Directional illumination analysis using beamlet decomposition and propagation

We evaluate directional illuminationDIand acquisitionaperture efficacy through wave theory-based beamlet decomposition of the wavefield. Beamlet decomposition wavelet transform along spatial axis provides localizations in both space and direction of a wavefield. We introduce the image conditions in beamlet domain and local angle domain and then define the local image matrix LIM. We calculate the DI in the image space for a given source or a group of sources by decomposing Green’s functions into local angle domainatimagepoints.Acquisition-apertureefficacyAAE matrixandacquisitiondip-responseADRvectorcanbedefined to quantify the efficacy of an acquisition configuration foragivensubsurfacepoint.Asnumericalexamples,wecalculatetheDImapsandADRmapsforhigh-andlow-velocity lens models and for the SEG-EAGE 2D salt model. We further investigate the influences of acquisition geometry and overlaying structures on the quality of prestack depth migration image for the subsalt area of the SEG-EAGE model.We find that the ADR maps for different dip angles have good correlation with the image qualities of the corresponding reflectors.DIanalysiscanbeusedintheaperturecorrectionfor image amplitude in local angle domain for wave theorybasedmigrationmethods.

Extended local Born Fourier migration method

A migration approach based on a local application of the Born approximation within each extrapolation interval contains a singularity that can make direct application unstable. Previous authors have suggested adding an imaginary part to the vertical wavenumber to eliminate the singularity. However, their method requires that the reference slowness must be the maximum slowness of a given layer; consequently, the slowness perturbations are larger than those when the average slowness is selected as a reference slowness. Therefore, its applicability is limited. We develop an extended local Born Fourier migration method that circumvents the singularity problem of the local Born solution and makes it possible to choose the average slowness as a reference slowness. It is computationally efficient because of the use of a fast Fourier transform algorithm. It can handle wider angles (or steeper interfaces) and scattering effects of heterogeneities more accurately than the split‐step Fourier (SSF) method, which acco...