The Theory Of Normal Modes Of Surface Waves And Earth’s Free Oscillations

Normal modes reflect the intrinsic properties of a physical eigen-system, and thus provide a unique tool for us to understand and invert the structural parameters of the oscillating system. When studying a given eigen-system, we firstly need to solve its eigenfrequencies and the associated eigenfunctions. However, even for the 1-D models, some modes may be difficult to be accurately calculated or even lost in the course of root-searching because of the special properties of the models such as the inclusion of low-velocity zones, fluid regions or simply displaying velocity-decreasing behavior with depth. Though the formulisms of surface waves and free oscillations differ in the governing equations and the involved mathematical treatments due to the differences of the geometric scales they aim at and of the consequent physical conditions, they have remaining problems and numerical methods in common. For example, numerical instability causing loss of precision at high frequencies and the inaccurate calculations of some modes whose energies are trapped in a certain region of the model are present for them. As with surface waves, many variants based on the original Thomson-Haskell method were proposed to remedy the numerical instability. And when resorting to numerical integration to solve the eigenvalue problems of surface waves and free oscillations, one adopts "minors" approach, from a purely mathematical point of view, to transform the original system of ordinary differential equations into one of larger dimension, to alleviate the numerical problems. The key point is that, the conventional methods use a sole dispersion equation to search all the modes without taking into account the different propagating characteristics of various modes.Built upon the generalized reflection and transmission coefficients (GRTC), we can construct secular equations adaptively, on the condition that the analytical solution in a homogeneous layer or the high-frequency asymptotic solution in an inhomogeneous layer is given, to suit the various propagating characteristics of modes, so that the eigenvalue problem posed can be solved in an accurate, fast and stable way. For surface waves, based on the proposed concept of family of secular functions, we further demonstrate by numerical experiments that, the secular equations corresponding to all the low-velocity layers and/or to all the fluid-solid interfaces of a model, together with the usually employed secular equation determined on the free surface, are capable to find all the physically existent modes. Therefore, it guarantees that the synthesized surface wave seismograms at arbitrary depth are accurate, which is very important to express the seismic response of a deep event. We extend the computation of normal modes to leaky modes. Then when inverting velocity structures by dispersion curves of surface waves, we may avoid the errors associated with misidentification of leaky modes as normal modes, and in the case of a model with velocities decreasing with depth, the gap in the dispersion curve in some frequency range may be filled with leaky modes via the natural transition from normal modes to leaky modes. We have further studied some special modes associated with the particularity of a model, such as fundamental modes, trapped modes, and Stoneley modes, and have inadvertently identified radiation modes which do not satisfy the radiation condition as the normal modes do. We have reconsidered the definition of the fundamental mode and discussed how to define a fundamental mode of physical significance. It is also demonstrated that no matter how close a trapped mode may be to its adjacent modes at the same frequency, they remain still distinct and would not coincide, which may be seen from their eigenfunctions, justifying the avoided-crossing character of dispersion curves. Finally, it is pointed out that, in the theoretical framework of GRTC and family of secular functions, we may synthesize complete seismograms with body-wave phases, in terms of sum of normal modes and a integral of radiation modes, which is an alternative among the current methods to synthesizing complete seismograms. Thus, the accurate computation of normal modes in 1-D case forms the basis for dealing with lateral heterogeneities in the medium based on existent mode-coupling methods.We then proceed to apply GRTC and the concept of family of secular functions to the high-frequency free-oscillation eigenvalue problems. After deriving the zeroth order asymptotic solution in terms of Langer approximation in an inhomogeneous spherical shell, the surface wave formulism is then applied to generate the numerical algorithm of solving toroidal and spheroidal modes. By selecting suitable combination of airy functions to adjust the solution behaviors, we have successfully computed asymptotic eigenfrequencies of toroidal modes. Our results not only have no the discontinuous jump of a dispersion branch, which is present in the WKBJ’s results due to the inherent limitation of WKBJ method, but also show a noticeable improvement in the precision compared with WKBJ’s results. And our results are in good agreement with the accurate results produced by Mineos, and the differences of them become even smaller particularly at high frequencies. Therefore, it is very promising to compute in high precision the modes like Stoneley modes whose energies become very weak when reaching the ground surface. The strategy of adding more radial knots used in Mineos to improve the precision of the modes may be also avoided. Therefore, we have provided an alternative method of computing eigensolutions of Earth’s free oscillations.With the eigenfunctions of modes of 1-D case prepared for basis functions, it is convenient to represent the eigensolutions of a 3-D Earth model by Rayleigh-Ritz method. The relevant theories are developed gradually from the non-rotating, elastic Earth model to a general one incorporating Earth’s rotation and anelasticity. The perturbation theory which treats the asphericities deviating from spherical symmetry is then summarized briefly and completely, with the complex surface spherical harmonics consistently utilized. We then investigated the applicability of self-coupling, group coupling and full coupling, these different coupling approximations, by applying them to some low-order modes and comparing with the observations, and qualitatively evaluated the effect of 3-D lateral heterogeneities on the mode splitting and multiplet coupling. Besides, by analyzing Stoneley modes it is shown that the high-precision computation of eigenfunctions of the spherically symmetric case is necessary to simulate accurately the 3-D theoretical seismograms. Finally, the alternatives to diagonalizing large coupling matrix are suggested, so that it becomes more practical to perform the needed wide-band coupling calculations for studies such as the effect of lateral density variations on the synthetic spectra.