Investigation of the convergence of the integration algorithm with the choice of the variable of integration at each step using the example of a Luneberg lens

The paper describes an integration algorithm with a choice of a variable at each step in the numerical construction of ray trajectories in a medium with a given dependence of the permittivityon coordinates. The convergence of the calculations to the exact solution is estimated using the example of the problem of calculating the trajectories of rays in a Luneberg lens. It is shown that with a decrease in the grid step, convergence to the exact solution is observed. Purpose. Assess the convergence to an exact solution of an integration algorithm with a choice of a variable at each step using the example of the problem of calculating ray trajectories in a Luneberg lens. Results. The trajectories of rays incident parallel to the ordinate axis and the trajectories of rays incident at an angle to the ordinate axis are calculated. It is shown that with a decrease in the grid step, convergence of the results to the exact solution is observed. Practical significance. It is shown that an integration algorithm with a choice of a variable at each step provides the construction of ray trajectories with an error in the coordinate not exceeding the grid step for the problem of ray propagation in a Luneberg lens.

A deterministic discretisation-step upper bound for state estimation via Clark transformations

We consider the numerical stability of discretisation schemes for continuous-time state estimation filters. The dynamical systems we consider model the indirect observation of a continuous-time Markov chain. Two candidate observation models are studied. These models are (a) the observation of the state through a Brownian motion, and (b) the observation of the state through a Poisson process. It is shown that for robust filters (via Clark's transformation), one can ensure nonnegative estimated probabilities by choosing a maximum grid step to be no greater than a given bound. The importance of this result is that one can choose an a priori grid step maximum ensuring nonnegative estimated probabilities. In contrast, no such upper bound is available for the standard approximation schemes. Further, this upper bound also applies to the corresponding robust smoothing scheme, in turn ensuring stability for smoothed state estimates.

3-D traveltimes and amplitudes by gridded rays

Seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium. They can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite‐difference methods. Unfortunately, this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used. We propose a new method for computing traveltimes and amplitudes in 3-D media that is simple, fast, unconditionally stable, and robust. Defining the slowness vector as [Formula: see text] and assuming an isotropic medium, the ray velocity v is related to the slowness vector by the relation [Formula: see text]. Rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane. The components of the slowness vector are then interpolated to gridpoints on this next horizontal plane. This is termed regridding; the process of downward propagation of rays, one vertical grid step at a time, is continued until some prescribed depth is reached. Computation of amplitudes is achieved using a method similar to that for obtaining the zero‐order approximation in asymptotic ray theory. We show comparisons with a full‐wave method on readily accessible 3-D velocity models.