Extension of the kinematic approximations to the multilayered elastic orthorhombic medium

Understanding the kinematics of horizontally-layered reservoir rocks is important to their proper characterization and to accomplish this it is necessary to specify the explicit model for these kinematic properties. The accurate approximations for traveltime and relative geometrical spreading in an elastic homogeneous orthorhombic (ORT) have been investigated with different forms: Shifted Hyperbola Form (SHF), Taylor Series (TS) and the Rational Form (RF). This paper extends these approximations to the multi-layered ORT model by adopting composite coefficients and effective model parameters. The multi-layered model is characterized without and with the azimuthal variation among layers. There is an overdetermined problem when the azimuthal variation exists; and to address that case, the Least Squares Method (LSM) is adopted. To check the feasibility of the expansion, we select the SHF (Shifted Hyperbola Form) approximation specified in the homogeneous elastic ORT model for the calculation in the numerical example. Four groups of examples are analyzed to investigate the influence on the accuracy of the approximation with the change in rotation angle, degree of anisotropy, and the direction of the orientation. The results indicate that, for the multi-layer, the accuracy of the approximation is proportional to the degree of anisotropy and the value of the angle of rotation. The relative errors in travel time and relative geometrical spreading in this multi-layered extension are very small and can be implemented in practical applications

Approximations for traveltime, slope, curvature, and geometrical spreading of elastic waves in layered transversely isotropic media

Each seismic body wave, including quasi compressional, shear, and converted wave modes, carries useful subsurface information. For processing, imaging, amplitude analysis, and forward modeling of each wave mode, we need approximate equations of traveltime, slope (ray-parameter), and curvature as a function of offset. Considering the large offset coverage of modern seismic acquisitions, we propose new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis (VTI). The proposed approximation for traveltime is a modified version of the extended generalized moveout approximation that comprises six parameters. The proposed direct approximations for ray-parameter and curvature use new, algebraically simple, equations with three parameters. We define these parameters for each wave mode without ray tracing so that we have similar approximate equations for all wave modes that only change based on the parameter definitions. However, our approximations are unable to reproduce S-wave triplications that may occur in some strongly anisotropic models. Using our direct approximation of traveltime derivatives, we also obtain a new expression for the relative geometrical spreading. We demonstrate the high accuracy of our approximations using numerical tests on a set of randomly generated multilayer models. Using synthetic data, we present simple applications of our approximations for normal moveout correction and relative geometrical spreading compensation of different wave modes.

Weak-anisotropy approximation of P-wave geometrical spreading in horizontally layered anisotropic media of arbitrary symmetry: TTI specification

Understanding the role of geometrical spreading and estimating its effects on seismic wave propagation play an important role in several techniques used in seismic exploration. The spreading can be estimated through dynamic ray tracing or determined from reflection traveltime derivatives. In the latter case, derivatives of non-hyperbolic moveout approximations are often used. We offer an alternative approach based on the weak-anisotropy approximation. The resulting formula is applicable to P-waves reflected from the bottom of a stack of horizontal layers, in which each layer can be of arbitrary anisotropy. At an arbitrary surface point, the formula depends, in each layer, on the thickness of the layer, on the P-wave reference velocity used for the construction of reference rays, and on nine P-wave weak-anisotropy (WA) parameters specifying the layer anisotropy. Along an arbitrary surface profile, the number of WA parameters reduces to five parameters related to the profile. WA parameters represent an alternative to the elastic moduli, and as such can be used for the description of any anisotropy. The relative error of the approximate formula for a multilayered structure consisting of layers of anisotropy between 8% and 20% is, at most, 10%. For models including layers of anisotropy stronger than 20%, the relative errors may reach, locally, even 30%. For any offset, relative errors remain under a finite limit, which varies with anisotropy strength.

Attenuation and Basin Amplification Revealed by the Dense Ground Motions of the 12 July 2020 Ms 5.1 Tangshan, China, Earthquake

On 12 July 2020, an Ms 5.1 moderate earthquake occurred on the north segment of the Tangshan fault in North China, which was the seismogenic fault of the 1976 Ms 7.8 Tangshan earthquake and numerous small-to-moderate earthquakes in recent decades in the Tangshan seismic zone. The Ms 5.1 event was well-recorded by dense ground-motion observation stations, including the national strong-motion stations and seismic intensity stations. This many ground-motion recordings, obtained for such a moderate event in North China for the first time, provided a rare opportunity to investigate the attenuation and site effects on ground motion. The distance decay in the Tangshan seismic zone was first explored using the spectral amplitudes from the vertical component. The strong anelastic attenuation and weak geometrical spreading effects were clearly found. The hinged trilinear form may be more effective at describing the geometrical spreading. No geometrical spreading decay was visible at medium distances (60–100 km). Anomalous areas with extraordinary high amplitudes occurred in the spatial distribution of peak ground accelerations and peak ground velocities that we attribute to significant basin amplification effects, which was confirmed by the wideband and high amplifications on the standard spectral ratio and the later-arriving, long-period surface waves observed in waveforms in the Ninghe–Baodi area and south of Beijing. The basin-induced surface waves in the 2–5 s period were most prominent in the Ninghe–Baodi area. We further inferred that basin effects may be responsible for the high-intensity anomaly areas observed in the 1976 Ms 7.8 Tangshan earthquake.

Robust Empirical Time–Frequency Relations for Seismic Spectral Amplitudes, Part 2: Model Uncertainty and Optimal Parameterization

ABSTRACT In a companion article, Safarshahi and Morozov (2020) argued that construction of distance- and frequency-dependent models for seismic-wave amplitudes should include four general elements: (1) a sufficiently detailed (parametric or nonparametric) model of frequency-independent spreading, capturing all essential features of observations; (2) model parameters with well-defined and nonoverlapping physical meanings; (3) joint inversion for multiple parameters, including the geometrical spreading, Q, κ, and source and receiver couplings; and (4) the use of additional dataset-specific criteria of model quality, while fitting the logarithms of seismic amplitudes. Some of these elements are present in existing models, but, taken together, they are poorly understood and require an integrated approach. Such an approach was illustrated by detailed analysis of an S-wave amplitude dataset from southern Iran. The resulting model is based on a frequency-independent Q, and matches the data closer than conventional models and across the entire epicentral-distance range. Here, we complete the analysis of this model by evaluating the uncertainties and trade-offs of its parameters. Two types of trade-offs are differentiated: one caused by a (possibly) limited model parameterization and the second due to statistical data errors. Data bootstrapping shows that with adequate parameterization, attenuation properties Q, κ, and geometrical spreading parameters are resolved well and show moderate trade-offs due to measurement errors. Using the principal component analysis of these trade-offs, an optimal (trade-off free) parameterization of seismic amplitudes is obtained. By contrast, when assuming theoretical values for certain model parameters and using multistep inversion procedures (as commonly done), parameter trade-offs increase dramatically and become difficult to assess. In particular, the frequency-dependent Q correlates with the distribution of the source and receiver-site factors, and also with biases in the resulting median data residuals. In the new model, these trade-offs are removed using an improved parameterization of geometrical spreading, constant Q, and model quality constraints.

Amplitude and phase changes for reflected and transmitted waves from a curved interface in anisotropic media

Summary It is well known that seismic data that have been recorded in complex geological environments must be compensated for geometrical spreading before AVO/AVA analysis, in order to avoid erroneous imaging interpretation. By investigating analytically both the effect of the geometrical spreading and the effect of the reflector curvature on amplitude and phase changes for reflected and transmitted waves between anisotropic media, using ray theory, we show that these data should be compensated for interface effects as well. In order to gain insight more specifically in the focusing effect of the interface, the special case of homogeneous isotropic media separated by a curved interface of syncline type is discussed and compared to the case of a plane interface. 3D numerical simulations of wave reflection from curved interfaces using a Spectral-Element Method validate our analytical derivations. In particular, numerical seismograms obtained at a vertical receiver array highlight that the effect of interface curvature on the reflected events is much more pronounced in a restricted area associated with the existence of caustics, which is consistent with our analytical predictions. Moreover, comparisons between the numerical and the analytical results confirm the fact that using plane-wave reflection coefficients without correction for the interface effect may lead to wrong interpretation of AVA/AVO analysis.

Testing a Local-Distance Rg/Sg Discriminant Using Observations from the Bighorn Region, Wyoming

ABSTRACT This study explores the effectiveness of local-distance (<200 km) seismic discriminant to distinguish between surface mine blasts, single-shot borehole explosions, and earthquakes in the Bighorn Mountains region, Wyoming. We focus on the ratio between local-distance fundamental-mode surface waves (Rg) and the crustal shear-wave (Sg) signals. The observed spectral amplitude measurements are fit to propagation models that account for distance-dependent geometrical spreading and attenuation, and site amplification factors. The results support previous observations that Rg attenuates rapidly, is amplified in sedimentary basins, and has suppressed amplitudes in isolated mountainous terrain. Sg attenuates less rapidly than Rg but exhibits a similar spatial site amplification pattern. We compute an Rg/Sg source discriminant by taking the ratio between site- and distance-corrected Rg and Sg amplitude measurements. The results suggest that the site- and distance-corrected Rg/Sg ratios can distinguish events larger than ML∼1.5 (in the Bighorn region). The discriminant may also be sensitive to explosion emplacement conditions, where the ratios are higher for borehole shots in sedimentary strata and lower for explosions within the basement. The analysis shows that the Rg/Sg discriminant is effective for events in the Bighorn region for events larger than ML∼1.5 if proper considerations are made to account for event size and near-source material.

Exact Rayleigh-Wave Solution from a Point Source in Homogeneous Elastic Half-Space

ABSTRACT A Rayleigh wave, often the most visible component in the far-field seismograms, is an important type of seismic-wave motion associated with the Earth’s surface. In this study, we explore some of the general properties of the Rayleigh wave in a homogeneous elastic half-space. Starting from the displacement expressed in the form of a wavenumber integral in the frequency domain, we extract the contribution from the pole in the complex wavenumber plane to obtain the excitation formulae of the Rayleigh wave by the residue theorem for complex integrals. Numerical results are compared with the full wavefield solutions to validate our solutions. By examining the analytical expressions obtained, we explore some basic properties of Rayleigh waves such as the particle motion and geometrical spreading. We also demonstrate that these properties of the Rayleigh wave excited by a point source are slightly different from but mostly consistent with the well-known classical properties of plane Rayleigh waves.