We present methods for obtaining numerical and analytic solutions of the complex eikonal equation in inhomogeneous acoustic VTI media (transversely isotropic media with a vertical symmetry axis). The key and novel point of the method for obtaining numerical solutions is to transform the problem of solving the highly nonlinear acoustic VTI eikonal equation into one of solving the relatively simple eikonal equation for the background (isotropic) medium and a system of linear partial differential equations. Specifically, to obtain the real and imaginary parts of the complex traveltime in inhomogeneous acoustic VTI media, we generalize a perturbation theory, which was developed earlier for solving the conventional real eikonal equation in inhomogeneous anisotropic media, to the complex eikonal equation in such media. After the perturbation analysis, we obtain two types of equations. One is the complex eikonal equation for the background medium and the other is a system of linearized partial differential equations for the coefficients of the corresponding complex traveltime formulas. To solve the complex eikonal equation for the background medium, we employ an optimization scheme that we developed for solving the complex eikonal equation in isotropic media. Then, to solve the system of linearized partial differential equations for the coefficients of the complex traveltime formulas, we use the finite difference method based on the fast marching strategy. Furthermore, by applying the complex source point method and the paraxial approximation, we develop the analytic solutions of the complex eikonal equation in acoustic VTI media, both for the isotropic and elliptical anisotropic background medium. Our numerical results demonstrate the effectiveness of our derivations and illustrate the influence of the beam widths and the anisotropic parameters on the complex traveltimes.

The traditional algorithm for computing the complex travel time, e.g., dynamic ray tracing method, is based on the paraxial ray approximation, which exploits the second-order Taylor expansion. Consequently, the computed results are strongly dependent on the width of the ray tube and, in regions with dramatic velocity variations, it is difficult for the method to account for the velocity variations. When solving the complex eikonal equation, the paraxial ray approximation can be avoided and no second-order Taylor expansion is required. However, this process is time consuming. In this case, we may replace the global computation of the whole model with local computation by taking both sides of the ray as curved boundaries of the evanescent wave. For a given ray, the imaginary part of the complex travel time should be zero on the central ray. To satisfy this condition, the central ray should be taken as a curved boundary. We propose a nonuniform grid-based finite difference scheme to solve the curved boundary problem. In addition, we apply the limited-memory Broyden-Fletcher-Goldfarb-Shanno technology for obtaining the imaginary slowness used to compute the complex travel time. The numerical experiments show that the proposed method is accurate. We examine the effectiveness of the algorithm for the complex travel time by comparing the results with those from the dynamic ray tracing method and the Gauss-Newton Conjugate Gradient fast marching method.

The sound speed of seawater is not constant. It varies with season, day time, depth and range. This variation was not considered in marine seismic data processing and imaging before deep-water seismic acquisition became a routine activity. As a result, non-ignorable errors may be contained in the final migrated-image of deep-water seismic data. To eliminate such errors, we propose here a scheme for inverting the seawater velocity under the condition that the seabed has a complex topography. In this scheme, the seabed topography is represented by step grids constrained by measured water depth data, and the sound profile is assumed to be the one satisfying the Munk formula. Thus, only the parameters appearing in the Munk formula need to be inverted. This is quite different from the conventional inversion schemes that take the velocity as the inversion target. We use the conjugate gradient method to minimize the objective function given as the sum of squares of traveltime differences. Tests on synthetic common-shot data set show that our scheme presented here works as expected. Applications to real marine data set further validates the reasonability and feasibility of our scheme under complex seabed topography. To investigate the reliability of our inversion scheme, we migrate the real 2D data set using the Kirchhoff migration with the inverted Munk profile as the velocity model. The corresponding results show that the seawater velocity obtained by our inversion scheme improves the imaging quality of underwater structures.

This paper aims at the problem that in deep water the migration quality is poor, and starts with the influence that velocity model accuracy has on migration, studying influence that variable seawater velocity makes on migration effect. At first, variable seawater velocity influenced by temperature, pressure and salinity is defined to replace the true seawater velocity. Then variable seawater velocity’s influence on interface migration location, layer sickness and migration energy focusing degree are analyzed in theory. And finally a deep water layered medium model containing variable seawater velocity, a syncline wedge shape model and a complex seafloor velocity model are constructed. By changing the seawater velocity of each model and comparing migration results of constant seawater-velocity model and variable seawater-velocity model, we can draw the conclusion: Under the condition of deep water, variable seawater-velocity’s impact on the quality of seismic migration is significant, which not only can change the location of geologic body migration result, but also can influence the resolution of geologic interface in the migration section and maybe can cause migration illusion. Investigación de los efectos de la variación en la velocidad del agua marina sobre las imágenes de migración en la geología de aguas profundasResumenEste artículo se enfoca en el problema de la baja calidad de la migración en aguas profundas. Se analiza la influencia que tiene el modelo de precisión de velocidad en la migración y se estudia el impacto que la variación de velocidad del agua marina tiene en el efecto de movimiento. En primera instancia, se define la variación de la velocidad del agua marina afectada por la temperatura, la presión y la salinidad para reemplazar la velocidad del agua marina actual. Luego se analiza la teoría de la influencia de la velocidad del agua marina sobre la interfaz de la ubicación de migración, el grosor de la capa y la energía de movimiento de acuerdo con la inclinación. Además, finalmente, se construyó un modelo medio por capas de aguas profundas que contiene las variaciones de velocidad del agua marina, un modelo de forma de cuña sinclinal, y un modelo de complejo de velocidades del lecho marino. Al cambiar la velocidad del agua marina en cada modelo y comparar los resultados de migración del modelo de velocidad de agua marina constante con los del modelo modelo de velocidad variable se encontró la conclusión. Bajo las condiciones de agua profunda, el impacto de la variación de la velocidad del agua es significativo en la migración sísmica, lo que no solo puede cambiar el resultado de migración en la ubicación del cuerpo geológico sino que además puede influir en la resolución de la interconexión geológica en la sección de movimiento y podría causar la ilusión de migración.