Mode And Generalized Ray Theory Of Surface Waves In Horizontal Layered Media

The duality of rays and modes is a general property of waves,and for solving the classical seismic wave theory for horizontally layered media,there are two forms of wave field representation with clear physical meaning according to the form of wave field expression,one is based on mode theory,which represents the wave field as a superposition of individual modes;the other is generalized ray theory,which represents the wave field as a superposition of generalized rays of different paths,and the generalized rays contain high frequency contribution of approximate geometric seismic rays.The inversion of the Earth medium structure using the body wave arrival information of near or far earthquakes is the result under the approximation of ray theory;the inversion of the Earth medium structure at different scales based on the dispersion curves of surface waves is a typical application of the mode theory.People’s understanding of the structure of the Earth medium relies heavily on the inversion results based on surface wave dispersion curves.With the development of dense station array observation and seismic interference technology,effective multimode surface wave dispersion curves and leakage mode dispersion curves can be extracted from the active source and background noise observation data,and considering the inversion of multimode surface waves can increase the stability and improve the resolution of the inversion,while the leakage mode is more sensitive to the P-wave velocity.Therefore,the use of multiple normal modes and/or leaky modes to invert the Earth medium parameters is gradually attracting the attention of seismologists.In the case of inversion methods based on mode dispersion curves,it is required to accurately identify the mode branches of the dispersion curves,fit the theoretical dispersion curve of the prediction model to the observed dispersion curve,and find the optimal model.This has two potential risks,one is the mode misclassification and the other is the mode root leakage in the theoretical calculation.The mode misclassification is especially obvious when the "apparent intersection" of two model branches occurs,and the root leakage is related to the processing technique of the forward calculation of the dispersion curves.Both the misclassification of the observed mode and the omission of the theoretical dispersion curve will bring large errors to the inversion.Through mode analysis,understanding the eigenvalues and eigenfunctions of the normal and leaky modes,and their energy spectrum characteristics in the frequency and phase velocity domains can provide a theoretical basis for mode identification,or an inversion method based on the energy spectrum characteristics in the frequency-phase velocity domain,which can directly compare the theoretical and observed energy spectra and avoid the problems caused by the aforementioned dispersion curve-based inversion.In addition,with the development of three-dimensional array observation,it becomes possible to directly observe the energy distribution of the mode along the depth,and to study the relationship between the mode and different path rays,so as to understand the distribution of eigenfunctions at different depths and the sensitivity of the mode to different depth medium parameters.The basis for direct inference of medium parameters based on mode characteristic functions is established.Based on this,this paper mainly studies the propagation of normal modes and leaky modes in horizontally layered media based on mode and generalized ray theory,and the main innovative results of the paper are as follows:1)Based on the classical generalized reflection and transmission coefficient theoretical framework,the singularity problem of the current solution at the critical phase velocity is improved by introducing the primary functional form in the generalized solution,so that the eigenfunction of the critical modes can be calculated directly,and it is found that the eigen displacement of the critical modes remains constant with depth and does not decay.For the propagation of Rayleigh surface waves in a horizontal layered medium,if the phase velocity is the same as that of the P-wave or S-wave in each layer of the layered medium,we call it the critical phase velocity,or critical mode,of each mode of Rayleigh waves.At the critical phase velocity,the general solution of the fluctuation equation is not homogeneous plane wave(triangular function)or inhomogeneous plane wave(exponential function),but the plane wave whose amplitude varies linearly with depth(primary function).Therefore,when solving the wave propagation problem in a layered medium based on the general solution form containing only the triangular function or exponential function,the solution form does not contain the critical phase velocity.In this paper,based on the classical generalized inverse and transmission coefficient theoretical framework,we introduce the primary function form in the general solution to improve the current solution singularity problem at the critical phase velocity,so that the eigenvalues and eigenfunctions of the surface wave modes at the critical phase velocity can be calculated directly.Finally,the characteristics of the distribution of the eigen displacement corresponding to the critical phase velocity in different layered half-space models are investigated by numerical simulations.Unlike the usual eigen displacement of the simple positive mode,the eigen displacement corresponding to the critical phase velocity exhibits different characteristics,and the energy of the corresponding eigen displacement radiates into the lower half-space when the phase velocity is equal to the S-wave velocity of the underlying half-space,and remains constant with depth without decay.2)The mode-kissing phenomenon of surface waves is studied based on the mode and generalized ray theory,and the physical reasons behind the appearance of this phenomenon are explained.It is found that the mode-kissing phenomenon appears in the complex ray parameter plane,where the P pole crosses the branch point represented by 1/β(2)and basically enters the time domain of the normal mode region,and the eigendisplacement characteristics near the appearance of the mode crossing is studied before and after the contact point.In near-surface wave exploration,especially for the probing of typical sedimentary basins,mode-contact phenomena(Mode-kissing)of fundamental and first-order modes are usually observed in the energy spectrum of the frequency-phase velocity domain obtained based on the observed data.Some researchers explore whether there is a true mode-kissing at the contact point by improving the computational accuracy,which does not have more physical significance.Based on the mode theory,the surface wave modes in the frequency domain correspond to a series of poles given by the roots of the dispersion equation.Based on the generalized ray theory,the multipath in the space-time domain results in multiple modes in the frequency domain.In this paper,we combine the generalized ray theory and the normal mode theory to investigate the effect of the two surface wave poles contributions of P and S poles on the dispersion curves in the generalized ray theory to explain the actual observed mode contact phenomenon.Based on the two-layer medium model,the variation characteristics of the dispersion curves,the eigendisplacement and the polarization of the fundamental and first-order modes caused by the variation of the Swave velocity β(2)in the lower half-space with the variation of the P pole in the complex-ray parameter plane are analyzed.The mode contact phenomenon is found to occur in the complex-ray parameter plane,where the P pole crosses the branch point indicated by 1/β(2)and enters the time domain of the simple positive mode region essentially.The first-order leaky mode corresponding to this pole has a surface plasmonic trajectory of a prograde ellipse,but the transformation of the eigen displacement with depth exhibits classical surface wave characteristics,and the energy is mainly concentrated at the surface.3)The correspondence between the modes of each phase velocity partition and the generalized rays is investigated,and the interference curves and generalized ray clusters are obtained to approximate the characterized modes,and can be naturally extended to leaky modes up to infinite phase velocities,whereby the causes of the apparent intersection phenomenon between trapped modes are investigated,and the monolayer resonant modes(solotone effect)are identified from the eigenvalues of the multilayer waveguide system,which are fitted using the interference curves,and the excitation mechanism is given by the generalized ray clusters.The critical phase velocity divides each partition of the dispersion curve,and for each partitioned mode,the standing wave feature of the eigenfunction extends to different depths,which reflects the propagation of traveling waves(rays)to different depths,and therefore there is a correspondence between the modes and rays in each partition.In this paper,the family of dispersion functions is simplified by omitting the reflection-transmission coefficients with smaller energy,and the analytic interference curve is obtained,while the wave field is also simplified by the same approximation,and the generalized ray cluster is obtained,and the relationship between the phase length interference curve and the generalized ray cluster is equivalent to the relationship between the dispersion curve and the total wave field.The dispersion characteristics and excitation mechanism of trapped waves in each phase velocity partition of some typical models are investigated by using interference curves and generalized ray clusters.It is found that strong apparent intersection between energy trapped modes is also possible due to the existence of two waveguide layers with energy close to their respective independent propagation at high frequency bands,and more generally,it is also possible to observe the resonant modes of a single layer from the eigenvalues of the multilayer waveguide,which overlay the original eigenvalues and whose profiles can be predicted by the phase-length interference curves.4)Based on the Weyl integration principle and Cagniard path,ray representations of SH normal mode,cutoff mode,and leaky mode are given for Love wave in a singlesurface half-space medium model.The real ray paths related to generalized rays,including high-frequency ray paths(such as the head wave related to the branch point,the critical refraction wave path,and the reflected wave path related to the saddle point)and diffraction wave paths related to the Cagniard path at the upper edge of the branch cut or tilted close to the branch cut,are given for SH normal mode,cut-off mode,and leaky mode using the relationship between the ray parameters(horizontal slowness)of the above paths and the reciprocal of the phase velocity of the energy trap mode.After obtaining the correspondence between each phase velocity partitioning mode and the generalized ray,we try to obtain a more intuitive representation of the eigenmode,which is a geometric ray path containing reflection and transmission information at the interface obtained by power series expansion in the frequency-wave number domain,but does not fully reflect the real energy transfer process,and use the Cagniard-de Hoop method to invert it into the time-space domain,using the idea of plane wave synthesis columnar wave in Weyl integral can get a ray path corresponding to each point of branch point,saddle point,and upper edge of branch cut(tilted close to the branch cut)from the Cagniard path in the plane of complex ray parameters,combined with the relationship that horizontal slowness and phase velocity are reciprocal to obtain the ray representation of Love wave under different source and field point distributions in single surface half space.The energy transfer mechanism of the energy trapped mode is revealed.