Investigation On Hermite-Gaussian Beams In Seismic Wavefileds

The Gaussian beams method is based on ray theory. The classical ray theory iswell applicable to simulate the kinematics characteristics of seismic waves, such astravel time, ray paths, etc., when unapplicable to describe the dynamic characteristics insingular regions such as caustic regions, shadow regions,etc. There are a large numberof correction functions and new ideas suggested to overcome theses difficulties, ofwhich the widely developed approach is Gaussian beam method. The anticipated meritsof GBM were overall simplicity, fast speed, flexibility and future developmentpossiblities, which have been leading a extensive application in every aspects of seismicstudy.GBM is the high-frequency solution of elastodynamic equations concentratedclose to rays lying between wave theory and geometrical ray theory. In the classical raytheory, energy flow along the ray is limited in the ray tube, when in the GBM, energy isallowed to overflow through the lateral wall by the way of complex amplitude. Datingfrom1960s’, Gaussian beam had been applied to describe the propagation ofelectromagnetic wave. In the recent years, many lasers emit beams that approximat aGaussian profile, in which case the laser is said to be operating on the fundamentaltransverse mode. With the deeper development of optics, simple Gaussian beamstransverse mode couldn’t meet the needs of laser study which are just one possiblesolution to the paraxial wave equation. Various other sets of orthogonal solutions areused for modeling laser beams, such as Hermite-Gaussian beams, Laguerre-Gaussianbeams, Ince-Gaussian beams, Hypergeometric-Gaussian beams, etc. In the general case,if a complete basis set of solutions is chosen, any real laser beam can be desribed as asuperposition of solutions from this set.In this paper, we hopefully introduce Hermite-Gaussian beams to the computationof seismic wavefields with abundant amplitude transverse profiles in the purpose ofimproved numerical modeling. The main contents of my papers is as following:①Itlisted the detailed formulas of procedure which is based on the simulation of the seismic wavefields in3D inhomegeneous by a system of Hermite-Gaussian beams;②It isnonstrict to prove the existence of weight functions in3D and strict to deduce therepresentation in2D;③There is a detailed discuss about the selection of the HBparameter, which concentrate on how to control the physical morphology of HB andfundamental principles of selection;④There are some numerical examples inhomogeneous medium, horizontal layers, irregular layers, etc.Based on the above research, we gain the following achievements:①For theweight functions，we have gained the nonstrict proof of existence in3D by Γ functionand strict derivation of representation by recursion formula and contour integral;②Ithas been discussed in detail on the influnce of Hermite polynomial in amplitudetransverse profiles;③The effective half-width L (s)of different order beamsfollow the same principle;④The derivation of linear coefficientsd iand thedetermination of beam transerve profiles by the coefficient a;⑤we can gained theenhaced diffracted record inH0+H2order Hermite-Gaussian beam.The numerical examples in simple structures show that Hermite-Gaussian beamscontain the advantages of Gaussian beams and even gain the enhanced diffractedrecord.. However, there are still some problems to solve in urgent need. For example,some loss of precision in simple numerical examples. In a word, the HGBs is worth afuture expectations based on the good nuermical results, completeness in mathematicaland achievement in other brances.