Shortest Path Ray Tracing Research Articles

Incorporating topographic effects in surface wave tomography based on shortest-path ray tracing

SUMMARY We propose a new method to obtain 3-D shear wave velocity structures for regions with undulating topography based on surface wave dispersion data. In our method, we assume that surface waves propagate along a curved free surface, and the dispersion effects of these waves can be modelled as a frequency-dependent ray tracing problem. We use the shortest-path method to address off-great-circle propagation arising from inhomogeneous elastic parameters and topographic variations. We then apply our method to both synthetic and real-data inversions and demonstrate that ignoring topographic effects may significantly distort the inverted images. Finally, we analysed the accuracy of our method and provided a rule-of-thumb principle to quantitatively assess the need to account for topographic effects in surface wave tomography for a region.

Shortest-path ray tracing on self-adapting random grids

SUMMARY A new ray tracing algorithm based on the shortest path method was designed and optimized to model seismic rays. The workflow is based on Dijkstra’s algorithm to find the shortest path iteratively on self-adapting random grids. These random grids adapt from one iteration to the next, storing in memory those nodes that minimize the traveltime in each iteration, consequently automatically densifying the grid in areas with significant velocity variations. Different data structures and grid geometries were studied, and it was confirmed that using a combination of a binary heap and random grids prevents systematic errors generated by using regular grids while keeping the computational times low. Since the nodes that minimize the traveltime are stored in each iteration, the memory requirements increase logarithmically, with the initial iterations having the most significant impact on the error minimization but plateauing afterwards. It was found that, albeit this increase in memory requirements, by focusing the densification of nodes in areas close to the actual trajectory of the rays, the error minimization requires fewer iterations than other implementations based on multiple realizations of random grids (requiring less than a tenth of the number of iteration of other methods). A function to handle extreme topographic variations was also designed, which uses a boolean test to reject rays crossing the topography, making the algorithm suitable for first arrival modelling in complex topography areas, such as the required for tomographic inversion of first arrivals in thrust-belt land seismic.

Image Human Thorax Using Ultrasound Traveltime Tomography with Supervised Descent Method

The change of acoustic velocity in the human thorax reflects the functional status of the respiratory system. Imaging the thorax’s acoustic velocity distribution can be used to monitor the respiratory system. In this paper, the feasibility of imaging the human thorax using ultrasound traveltime tomography with a supervised descent method (SDM) is studied. The forward modeling is computed using the shortest path ray tracing (SPR) method. The training model is composed of homogeneous acoustic velocity background and a high-velocity rectangular block moving in the domain of interest (DoI). The average descent direction is learned from the training set. Numerical experiments are conducted to verify the method’s feasibility. Normal thorax model experiment proves that SDM traveltime tomography can efficiently reconstruct thorax acoustic velocity distribution. Numerical experiments based on synthetic thorax model of pleural effusion and pneumothorax show that SDM traveltime tomography has good generalization ability and can detect the change of acoustic velocity in human thorax. This method might be helpful for the diagnosis and evaluation of respiratory diseases.

3-D simultaneous inversion for velocity and reflector geometry using multiple classes of arrivals in a spherical coordinate frame

The two key requirements in conducting 3-D simultaneous traveltime tomography on real data at the regional and global scale with multiple classes of arrival time information are (1) it needs an efficient and accurate arrival tracking algorithm for multiply transmitted, reflected (or refracted) and converted waves in a 3-D variable velocity model with embedded velocity discontinuities (or subsurface interfaces), and (2) a subdimensional inversion solver is required which can easily search for different types of model parameters to balance the trade-off between the different types of model parameter updated in the simultaneous inversion process. For these purposes, we first extend a popular grid/cell-based wavefront expanding ray tracing algorithm (the multistage irregular shortest-path ray tracing method), which previously worked only in Cartesian coordinate at the local scale, to spherical coordinates appropriate to the regional or global scale. We then incorporated a fashionable inversion solver (the subspace method) to formulate a simultaneous inversion algorithm, in which the multiple classes of arrivals (including direct and reflected arrivals from different velocity discontinuities) can be used to simultaneously update both the velocity fields and the reflector geometries. Numerical tests indicate that the new inversion method is both applicable and flexible in terms of computational efficiency and solution accuracy, and is not sensitive to a modest level of noise in the traveltime data. It offers several potential benefits over existing schemes for real data seismic imaging.

Shortest path ray tracing in cell model with a second-level forward star

SUMMARY The high-level forward star is routinely applied in seismic ray tracing using graph theory (sometimes referred to as the shortest path method) with a grid model. For a cell model, the forward star is often restricted to nodes at the same cell (i.e. first-level forward star). The performance of a cell model with second-level forward stars is found to be comparable in both computation time and accuracy to that of a doubly dense cell model with first-level forward stars. Moreover, the cell model with second-level forward stars has the advantage of halving the required computer storage. An optimization of the secondary node geometry leads to a further 20 per cent improvement in accuracy. Concepts derived from grid models for analytical error estimation are found to be less applicable to cell models. An empirical approach works better in the optimization of the secondary node geometry.

A Global Minimum Traveltime Ray Tracing Algorithm of Wavefront Expanding with Interface Points as Secondary Sources

AbstractThe shortest path ray tracing presented by Moser is a kind of fast and stable method to calculate global minimum traveltimes and raypaths. But it has two drawbacks. One is that most derived zig‐zag raypaths are longer than true paths and thus mispositioned. The other is that there may be several equally short paths, especially in flat areas with low velocity variations. The wavefront expanding method with interface points as secondary sources studied in the paper aims to solve the problems. Different from Moser's method, the method only set secondary sources on interfaces between different media. Straight rays from a secondary source can reach any node as long as they don't penetrate interfaces. So waves propagate along accurate paths in homogeneous layers or blocks. Obviously, there are much more ray directions from a secondary source in the method than in Moser's method. The main tracing errors of the method only come from interface discretization. As a result, ray tracing with the method can get much more accurate results than Moser's method. At the same time, the method is efficient as there are much less secondary sources than those in Moser's. Numerical examples show that the method is effective.

不規則四面体分割されたモデル空間への最短経路波線追跡法の適用

A way to apply the shortest path ray tracing (SPR) method to irregularly tetrahedralized velocity structure and its efficiency are described. SPR is the ray tracing method utilizing Dijkstra's shortest path algorithm in network graph theory. The tetrahedralization by Delaunay tessellation is one of the irregular parameterization of 3D velocity structure in travel time tomography. The irregular parameterization has great advantage of flexible expression of velocity structure though the computational costs in ray-tracing are relatively high. The hybrid ray tracing between SPR (for initialization) and pseudo-bending (for optimization) might be more economic. It has been known as the well proven scheme in the sense of the simplicity and robustness. Though SPR was originally developed under regular parameterization, it can easily be expanded to irregular parameterized structure without any theoretical modification. We propose a simple design of network for SPR that conforms the shape of tetrahedron. Generally, the cost and accuracy of SPR depends on the network design. In this paper, we have carried out a synthetic examination with a complex velocity model to assess the cost and accuracy of SPR with irregular network. Result reveals that the proposed ray tracing method is practically economic and accurate (almost the same accuracy as SPR under the regular cubic cell model). Further, it has potential to obtain more accuracy by lower costs due to the economical network integrated with flexible element size.

Shortest path ray tracing with sparse graphs

A technique for improving the efficiency of shortest path ray tracing (SPR) is presented. We analyze situations where SPR fails and provide quantitative measures to assess the performance of SPR ray tracing with varying numbers of nodes. Our improvements include perturbing the ray at interfaces according to Snell’s Law, and a method to find correct rays efficiently in regions of low velocity contrast. This approach allows the investigator to use fewer nodes in the calculation, thereby increasing the computational efficiency. In two‐dimensional (2-D) cross‐borehole experiments we find that with our improvements, we need only use 2/3 as many nodes, saving up to 60 percent in time. Savings should be even greater in three dimensions. These improvements make SPR more attractive for tomographic applications in three dimensions.