Optimized nonhyperbolic moveout correction equation of long-offset seismic data for TI media with an arbitrary spatial orientation

The conventional long-offset nonhyperbolic moveout equation is derived for the transverse isotropic media with a vertical symmetric axis（VTI）.It cannot be extended to the transverse isotropic media with an arbitrary spatial orientation of symmetry axis（ATI）.In this paper,we optimize a modified long-offset nonhyperbolic moveout equation for ATI media based on the conventional nonhyperbolic moveout equation and the exact analytical solution of the quartic moveout coefficient（A_4） and NMO velocity for ATI media that were derived in our previous work.Compared with the exact traveltimes of the ray-tracing algorithm for anisotropic media,this optimized equation can be used to calculate the traveltime varying with survey line azimuth in arbitrary strong or weak ATI media.It can replace the time-consuming, multi-offset,and multi-azimuth ray tracing method for forward modeling of long-offset reflection traveltimes in ATI media,which is useful to further anisotropic parameter inversion using long-offset nonhyperbolic moveout.

BIFURCATIONS TO A HETEROCLINIC MANIFOLD WITH NONHYPERBOLIC EQUILIBRIA IN R^n

The authors study bifurcations from a heteroclinic manifold connecting two non-hyperbolic equilibrium P-0 and P-1 for a n-dimensional dynamical system. They show that under some assumptions, each equilibrium P-i splits into two equilibria <(P)over tilde (i)> and P-i(alpha), i = 0, 1, and find the Melnikov vector conditions assuring the existence of a heteroclinic orbit from P-1(alpha) to P-0(alpha) along directions that are tangent to the strong unstable (resp.strong stable) manifold of P-1(alpha) (resp.P-0(alpha)). The exponential trichotomy and the unified and geometrical method are used to prove their results.

Local and Global Bifurcations With Nonhyperbolic Equilibria

The normal forms of coupling functions governing local and global bifurcations are studied for a generic (d+1) -parameter family of three-dimensional systems with a heteroclinic orbit connecting a hyperbolic saddle and a nonhyperbolic equilibrium occurring in the saddle-node,transcritical and pitchfork bifurcations,respectively.Singularity theory and a version of Melnikov function are used in this paper.

Automatic estimation of traveltime parameters in VTI mediausing similarity-weighted clustering

Compared with hyperbolic velocity estimation methods,nonhyperbolic methods(such as shifted hyperbola)are better choices for large offsets or vertical transverse isotropy(VTI)media.Since local seismic event slope contains subsurface information,they can be used to estimate zero-offset two-way traveltime and normal moveout velocity.The traditional velocity estimation methods require a great deal of manual work and are also prone to human error.In order to estimate the traveltime parameters for VTI media automatically,in this paper,we propose to use predictive painting and similarity-weighted clustering to obtain traveltime parameters.The predictive painting is used to estimate zero-offset two-way traveltime,and the shifted-hyperbola traveltime equation is used to obtain velocity and anisotropy attributes.We first map local slopes to zero-offset two-way traveltime and moveout-parameters domain and then use similarity-weightedk-means clustering to find the maximum likelihood anisotropy parameters of the main subsurface structures.In order to demonstrate that,we apply the similarity-weighted clustering method to synthetic and field data examples and the results are of higher accuracy when compared to the ones obtained using multiparameter semblance-based method.From estimation error section,it can be seen that the estimation error of multiparameter semblance-based method is about 3-5 times that of the proposed method.

Elementary Bifurcations of Non-Critical but Non-Hyperbolic Invariant Tori

Consider the time-periodic perturbations of n-dimensional autonomous systems with nonhyperbolic but non-critical closed orbits in the phase space. The elementary bifurcations, such as the saddle-node, transcritical, pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space (x, t), are studied. Some conditions which depend only on the original systems and can be used to determine the bifurcation structures of these problems are obtained. The theory is applied to two concrete examples.