Propagation Of High-frequency Rayleigh Waves In3D Medium With Topography

Rayleigh waves propagate along the free surface and vanish exponentially in vertical direction, which are formed under the constructive interference of P-and Sv-waves near the free surface, and were first discovered in theory by Lord Rayleigh in1887. Since they have strong energy, Rayleigh waves are the most damaging wave type during earthquakes, and regarded as strong noise in petroleum exploration. However, because the dispersive characteristic of Rayleigh waves in layered earth model was found in1950’s, they have been used as effective waves in several geophysical problems, for example in deducing inner structure of earth and soil mechanic parameters, material composition researching of the crust and the mantle, near-surface structure survey and non-destructive engineering detecting. At present, there are some classical literatures in describing the propagation characteristics of Rayleigh waves in elastic media. And Rayleigh waves have been widely used in several multi-scale geophysical researches, and the dispersion characteristics study of Rayleigh waves becomes a hot issue of geophysical researches. In recent years, the surface wave method based on Rayleigh waves propagation characteristics in elastic media, has been considerable advancing.At present, researches of Rayleigh-waves dispersive characteristics are focused on flat layered media. However, the actual earth is not completely flat layered media. Especially the earth surface is not wholly horizontal. There are extensive complex areas on the earth surface, such as mountains, seas, deserts, marshes, lakes. Hence, research of propagation of Rayleigh waves under complex topographic surface becomes significantly important in detecting the interior structure of the earth. In addition, many researches about the dispersive characteristics and numerical simulation technique of Rayleigh wave are carried out in two dimension (2D) media. However, the real earth is three-dimension (3D). To let the models approach the real underground media, our researches should be proposed in3D. Some complex wave fields can not be well explained by2D numerical simulation results. However, the3D numerical simulation results can be used to quantitatively explain a lot phenomenon of complex wave fields. Hence, the research of propagation of Rayleigh waves in3D media with topography is an imperative work. To study the propagation of Rayleigh waves in3D media with topography, a reasonable free-surface condition approach for the staggered-grid finite-difference method is applied in numerical simulations. A series of typical topographic models are designed, and the propagations of Rayleigh waves in these models are numerical simulated. Some propagation regularities of Rayleigh waves on topographic surface are obtained. In our study, we focused on the following issues.(1) Methods for simulation of high-frequency Rayleigh waves propagation in three-dimensional media with topography. We propose a feasible finite difference scheme incorporating the acoustic/elastic interface approach (AEA) into a ’stair-case’ mesh for modeling surface wave propagation with the topographic free-surface.(2) To study the propagation of Rayleigh waves under the topographic surface, a series of typical three-dimensional half-space models are designed, such as a slope free surface, a horst case free surface, valley case free surface, independent mountain case free surface, and independent pot case free surface. With the numerical results of Rayleigh waves on these models, we analyze the propagation characteristics of Rayleigh waves under topographic surfaces.(3) To study the true3D surface-wave method, we design a fan-case arbitrary acquisition geometry to get the synthetic seismograms. High-resolution linear Radon transform method is used to obtain the dispersive energy curves from the synthetic seismograms. With the dispersive energy curves, we analyze the effects of the true3D surface-wave method in eliminating influences of small topographies for Rayleigh-wave dispersion. The one dimensional S-wave profiles for the orthocenter of the fan areas are got from the inversion of the dispersive curves.By the analysis of the numerical simulation results of Rayleigh-wave propagation in the topographic surface models we designed, we conclude that:(1) A series of half-space model with slope free surface are numerically investigated with the slope angles from0degree to90degree with10degree increment. By the analysis of the numerical results, we further check the modeling precision of our2D numerical simulation approach for the topographic models. The numerical results show that our AEA method is superior to stress image method (SIM) in numerical Rayleigh wave simulation on the slope free-surface models, especially for the half-space model with45degree dipping planar free surface. It is noted that the error increases, for the dip angel increases from0to90degree, more harsh for a coarser grid spacing. The error of the AEA decreases more rapidly with increasing grid pints than the SIM. However, to achieve good accuracy for all dip angles with a fine sampling for the wavefields, approximately more than60ppw is required (also see Bohlen and Saenger,2006). And we get similar accuracy as the rotated staggered grid (RSG) finite difference method when the same ppw is used. However, it is obviously that our method is much easier in implementation.(2) Propagation and dispersion of Rayleigh waves in2D layered models with slope free surface are numerical analyzed. The results demonstrate that the Rayleigh-wave propagation along the slope free-surface is dominated by the real thickness of the layer. Therefore, the inverted1D S-wave velocity profile of the multichannel analysis of surface waves method (MASW) should be the S-wave velocity vs. depth in the direction perpendicular to the slope surface. To our knowledge, this conclusion points out the important characteristics of the Rayleigh-wave propagation along a slope free surface. The simulation results of a wedged layer underlying a half space model also demonstrate this important characteristics. And the numerical comparisons further prove the middle-of-receiver-spread assumption which is an implication to guide us in Rayleigh-wave exploration in practice.(3) Three2D linear arrays and two fan-case acquisition geometries are designed in numerical simulations. Numerical results of the3D two layer model with flat and slope free surfaces demonstrate the feasibility of the true3D surface wave method. The dispersive energy images of synthetic seismograms of linear array and arbitrary fan acquisition geometries generated by high-resolution linear Radon transform. The coincidence of the dispersive energy images with the analytical dispersion curves of a layered model demonstrates the feasibility of the true3D surface wave method.(4) Numerical results of3D half-space models with topographic surface as a horst case, a valley case, a single hill case and a single pit case show the influence of topographic surface for the propagation of Rayleigh waves. When Rayleigh waves pass the topographic areas, they are strongly influenced by the topographic surface, the wave form distorts, and the energy is reflected and scattered from the edge of the topographic surface. Numerical comparisons show that the influence of the valley case topography is more distinct than the horst case, and the influence of the single pit case topography is more distinct than the single hill case. Due to the existence of reflection and scattering of Rayleigh waves, energy bifurcations appear in the dispersive energy images generated from the synthetic seismograms of2D linear arrays. However, the energy bifurcations are suppressed obviously in the energy images of the true3D surface method. These results demonstrate the superiority of the true3D surface wave method. With the numerical results of Rayleigh wave propagation along the valley case topography and the single pit case topography, we find that the energy of Rayleigh waves is diffracted beside the valley or the single pit. And when Rayleigh waves pass the single hill topographic surface, a part of energy is trapped in the hill and reflected by the cliff again and again. This phenomenon can explain the fact when an earthquake occurs, the constructions in the mountaintop are damaged more seriously than the constructions on the bottom of the hill.(5) The multichannel analysis of surface wave (MASW) method has been effectively used to determine near-surface S-wave velocity. The assumption of the dispersion curves determined by the geophysical structure within the geophone spread of MASW method has been demonstrated by Luo in2009with synthetic and real-world examples. These tests also show the fact that a dispersion curve generated from the MASW spread is an average effect of the underneath geophysical structure of the receiver spread. Therefore, the MASW method possesses an average effect for underneath structure. With the numerical results of a series of2D and3D models with typical topographic free surfaces, we find that the average effect of MASW method can eliminate the harmful influences of topographic surfaces in Rayleigh-wave dispersion energy images generating. The numerical results of two2D half-space models with a single hill or a pit topographic free surface reveal the effects of2D MASW method in eliminating the harmful influences of topographic surfaces for generating Rayleigh waves dispersive energy images. By the analysis, we find that for the single-hill-case topography, when spread length is3times of the width of the hill, the harmful influences of topography can be eliminated by the MASW method. However, for the single-pit-case topography the spread length should be more than5times of width of the pit. We also analyze the average effects of the true3D surface method in eliminating the harmful influences of a3D single hill or pit topography for generating the dispersive energy images. We conclude that for the single-hill-case topography, when the arbitrary acquisition area is1.5times of the hill area, the harmful influence can be eliminated. However, for the single-pit-case topography, the acquisition area should be more than twice of the pit area. All the results point out simple but important guidelines of the MASW method.(6) The middle-of-receiver-spread assumption has been proved in the numerical and real-world examples by many researchers. Similar to the2D MASW method, we have an orthocenter assumption for the true3D surface wave method. The dispersion curves we get from the seismograms of an arbitrary acquisition geometry can be used to invert the underneath S-wave structure of the orthocenter of the arbitrary acquisition geometry area. The numerical results of a3D wedged layer model proved the orthocenter assumption. Based on the middle-of-receiver-spread assumption, a series of dispersion curves are generated from the numerical results of a2D two layer fault model and a3D two layer fault model with linear spread receivers. And by allying the inversion results of these dispersion curves, we get a pseudo-2D S-wave velocity profile and a pseudo-3D S-wave velocity body. As is shown on the pseudo-2D and pseudo-3D S-wave velocity results, the layer interface and the fault are reconstructed very well. Meanwhile, based on the orthocenter assumption, a series of dispersion curves are generated from numerical results of a3D two-layer topographic-layer-interface model with a series of arbitrary acquisition geometries. By allying the inversion results of these dispersion curves, we get a pseudo-3D S-wave velocity body. On the pseudo velocity body, the topographic-layer interface is reconstructed well. These inversion results prove the orthocenter assumption further and have an implication to guide us in performing Rayleigh-wave exploration in practice.