Calculation of the seismic first-break time field and its ray path distribution using a minimum traveltime tree algorithm

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1 Geophys. J. Inr. (993) 4, 5936 Calculation of the seismic firstbreak time field and its ray path distribution using a minimum traveltime tree algorithm Shunhua Cao and Stewart Greenhalgh School of Earth Sciences, Flinders Universily of Soulh Australia, GPO Box 2, Adelaide, S.A. 5, Australia Accepted 993 February. Received 993 February ; in original form 992 October 2 SUMMARY A grid search method, analogous to Huygens Principle, is employed to find the firstbreak seismic traveltime field in a seismic model. By defining a set of directions in space, a dynamic directed graph (digraph) containing the firstbreak time information can be constructed during the minimum traveltime treesearching process. The dynamic digraph has far fewer edges than the global graph suggested by previous researchers. An efficient sorting algorithm (heapsort) is adapted to the traveltime data structure by employing an indirect heap strategy. The computational speed can thus be improved several times over other approaches. The minimum traveltime tree algorithm is an efficient and flexible method to simultaneously calculate the firstbreak time field and the corresponding ray paths. It produces a robust and global result in comparison with traditional ray tracing methods. Later seismic phases can also be handled by imposing constraints on the ray paths. A subgrid technique is better than the normal grid technique in terms of accuracy and efficiency. The construction of the digraph uses the local directional information of raypaths and therefore local anisotropic information can be naturely incorporated. The firstbreak time field and its ray paths in an anisotropic model can be calculated as easily as those for an isotropic model. Key words: minimum traveltime, rays, seismic model, tree algorithm. INTRODUCTION Both traveltimes and ray paths are vital information needed for many seismic inverse techniques. Traditionally they are obtained by either ray shooting or ray bending. In shooting, rays are traced from a source at a given initial condition by solving the general ray equations. Many trials have to be made before a ray can reach a preassigned location with an acceptable accuracy. In bending, Fermat s Principle is used in an attempt to find a ray path by minimizing the traveltime from a source to a receiver. It is well known that both methods have severe drawbacks such as slow convergence and local minimum traveltime paths. Recently, two different methods were developed to find the minimum traveltime field for a given source location. The first method is due to Vidale (988, 99) and was improved by Qin et al. (992) and Cao & Greenhalgh (992). This method directly solves the eikonal equation along an expanding wavefront. Since a wavefront is a continuous curve in 2D or surface in 3D, minimum traveltimes from a source to all nodes are found with a Downloaded from on February 28 single run. Therefore, it is extremely efficient for seismic techniques such as prestack migration and hypocentre determination, which only require traveltime information at many locations. Despite the difficulty of incorporating anisotropy into finite difference calculations, we are aware that certain research groups (e.g. Stanford University, University of Texas) are using finite difference methods for traveltime calculations in anisotropic media. However, to the best of our knowledge no details on such formulations have been published. An alternative method of calculating firstarrival times and their ray path distribution is the minimum traveltime tree (shortest path) method. The idea of using the minimum traveltime tree algorithm to track the wavefront and ray paths was originally proposed by Nakanishi & Yamaguchi (986) and further developed by Moser (99). It is based on Fermat s minimum traveltime principle for seismic rays and the algorithms developed in network theory for the construction of the shortest path. Moser (99) also showed that the heapsort algorithm is the most efficient sorting method for the minimum traveltime ray tracing. 593

2 594 S. Cao and S. Greenhalgh In this paper, we wish to show that further iniprovement on the calculation of the minimum traveltime field and the corresponding ray path distribution can be achieved with the construction of a dynamic digraph (directed graph) which is built during the ray searching process rather than a fixed graph constructed at the beginning of the searching process as used by Moser (99). The construction of the dynamic digraph is implicitly based on Huygens Principle (which states that every point on a wavefront can be viewed as a source of secondary waves) because each node in the grid is physically treated as a new source point in evolving the waves through the model. The dynamic digraph has to be constructed with the knowledge of local information about the seismic rays. SHORTESTPATH TREE ALGORITHM The minimum spanning tree algorithm can be found in many computer textbooks under the subject of algorithms (e.g. Baase 988; Sedgewick 988). For reasons of background and convenience of discussion, a brief description is necessarily included here. In this algorithm, a graph (G) is defined as a set of vertices (V) connected by edges (V) and represented as (V, E). A weighted graph or a network is defined as a graph with weighted edges (W) and represented (V, E, W). A spanning tree for graph G is a subgraph of G that is a tree and contains all the vertices of G. In a weighted graph the weight of a subgraph is the sum of the weights of the edges in the subgraph. Obviously a minimum spanning tree of a weighted graph is a collection of edges connecting all the vertices such that the sum of the weights of the edges is at least as small as the sum of the weights of any other collection of edges connecting all the vertices. When the starting vertex (source position) is chosen u priori and cumulative weights from the source are used the minimum spanning tree becomes the shortestpath tree for that particular vertex. Figure schematically illustrates the construction of a shortestpath tree. Each vertex is annotated by a capital letter with a circle. Edges are denoted by solid lines connecting vertices and the number associated with each edge is the weight of the edge. The algorithm begins by selecting an arbitrary starting vertex (a childless tree, see Fig. la). At each iteration, a new vertex and edge is chosen to join the tree constructed so far (see Fig. lb). During the course of the algorithm, the vertices may be thought of as divided into three (disjoint) categories as follows: tree vertices: in the tree constructed so far. Fringe vertices: not in the tree, but adjacent to some vertex in the tree. Unseen vertices: all others. Thus the key step in the algorithm is the selection of a vertex from the fringe vertices. The edge connecting a tree vertex to a fringe vertex is called an incident edge to the fringe vertex. For a sparse graph, priority first search is employed and it always chooses a potential incident edge with the minimum weight (Bellman 958) from all edges between a tree vertex and its associated fringe vertices (see Fig. lc). After each iterative loop of the algorithm, there may be new fringe vertices, and the set of edges from which the next selection is made will change. Fig. l(c) suggests that we (a) A weighted graph (b) The tree and fringe after the starting vertex is selected n Fringe vertices (C)After selecting an edge and vertex. BH is not shown because AH is a better choice (has lower weight) to reach H. Figure. One iteration of the shortestpath tree algorithm. (a) The graph is represented by vertices of circled capital letters and these vertices are connected by solid lines (edges) with the weights specified by the numbers. (b) The vertex A is chosen to be the starting point. (c) After selecting an edge and vertex. BH is not shown because AH is a better choice (the weight of AH, 3 versus the weight of A BH, 4). Downloaded from on February 28

3 Traveltime tree algorithm 595 need not consider all edges between tree vertices and fringe vertices. After AB was chosen, BH became a potential incident edge choice, but it is discarded because AH has lower weight and would be a better choice for reaching H. If the sum of weights from A to H via B was lower than the weight of AH, then AH could be discarded. For each fringe vertex, we need keep track of only one edge to it from the tree, i.e. the one with lowest weight. All such edges are called candidate edges. The shortestpath tree algorithm is outlined below (see also Fig. ). Note that texts in the curly brackets are explanations whereas others are pseudo computer codes. The original work on this algorithm was published in the classical paper of Dijkstra (959). Input: G = (V, E, W), a weighted graph. Output: The edges in a shortestpath tree. () {Initialization} Let x (e.g. A ) be an arbitrary vertex. lirrcc = {x}; ETrcc = {nil}; (2) {Main loop (26); x (e.g. B) has just been brought into the tree. Update fringe vertices and candidate edges. Then add one vertex and edge.} while V,,,,! = V do. (3) {Replace some candidate edges. (see Fig. lc)} for each fringe vertex y (e.g. H ) adjacent to x (i.e. B) do if [W(xy)(e.g. ABH) < W (the candidate edge e incident with y )(e.g. AH)] then xy (i.e. ABH) replaces e (i.e. AH) as the candidate edge for y ; {Note this is for the explanation purpose, and as said in the text the above comparison is not true} end if end for (4) {Find new fringe vertices and candidate edges (see Fig. Ic)} for each unseen y (e.g. C) adjacent to x (i.e. B) do y (i.e. C) is now a fringe vertex; xy (i.e. BC) is now a candidate end for; (5) {Ready to choose next edge} if there are no candidates then stop; else (6) {Choose next edge} Find a candidate edge, e (e.g. AH), with minimum cumulative weight; x (e.g. H) =the fringe vertex incident with e (i.e. AH); Add x (H) and e (AH) to the tree; {x (H) and e (AH) are no longer fringe and candidate.} end if end while The algorithm terminates with V,,,, = V, and the edges in ETrCc form a shortestpath tree. The shortestpath tree can be found in the order of [(E + V ) log V] steps. MINIMUM TRAVELTIME AND ITS RAY PATH DISTRIBUTION Nakanishi & Yamaguchi (986) and Noser (99) showed that with the vertex of the seismic source location as a starting vertex, the minimum traveltime tree gives a very Figure 2. A choice of the predefined searching directions in an 8 x 8 neighbourhood area. Directions are numbered, e.g. one representing a vertically upgoing direction. good approximation to the ray path distribution. The sum of weights from the source to each vertex in the minimum traveltime tree closely approximates the first arrival time distribution. A fixed graph is used in both papers, i.e. a global graph is considered regardless of the weight distribution, or local information. In this paper, the local information is efficiently incorporated into the construction of a dynamic directed graph (digraph). Instead of constructing a graph at the initialization stage, we employ a rule to find the edges of the dynamic digraph during the calculation process. The dynamic digraph is unknown until the minimum traveltime tree is complete. We use the word, dynamic, to refer to the fact that the digraph is growing as the construction of the minimum traveltime tree is progressing. For the convenience of discussion, we use a discrete 2D model to illustrate the methodology. 3D models can be treated in an analogous manner. Figure 2 shows a vertex with its neighbourhood vertices and potential edges. The radius of the neighbourhood is determined by the accuracy requirement, as shown by Moser (99). In this paper, an 8 x 8 square is defined as a neighbourhood for its centre. Because the accuracy is largely determined by the largest angle subtended by two neighbouring edges, we only choose 24 segments as potential edges, although more segments are available in the square. For a global graph definition, these 24 potential edges would be the real edges of the starting graph. For our dynamic digraph, the edges of the graph are selected on the basis of local information. To do this, we have to define a set of rules. First of all, edges are now defined by two parameters: weight and direction. Thus, all potential edges in Fig. 2 denote both the traveltime and direction. Directions are numbered, e.g. one representing a vertical direction. Once the direction of a potential edge and Downloaded from on February 28

4 596 S. Cao and S. Greenhalgh its tail location (i.e. starting point) is defined, the weight or traveltime for this potential edge can be calculated based on the local slowness distribution. Secondly, the edges of the dynamic digraph are determined by the local information, i.e. the direction of the incident edge and the weight distribution, which is controlled by the local slowness variation. Fringe vertices can then be selected according to the minimum traveltime principle. Figure 3 shows that for an incident edge along direction 9 linked with vertex A, edges of the growing digraph are selected based on the local information. Here it is easy to make a comparison with the ray shooting method. If only one potential fringe vertex is allowed, the best choice would be to find a vertex along a direction closest to the direction determined by Snell s Law. In doing so. the minimum :9 traveltime ray tracing degenerates into a multishooting process. To avoid this, we need to search more than one direction (see Fig. 3). There are two approaches to proceed with a number of searching directions. The first approach is to define an angular neighbourhood size according to the local slowness variation. The principle of defining the neighbourhood size is to ensure the direction defined by Snell s Law is within the neighbourhood in most cases. However, it is unnecessary to assume that the direction defined by Snell s Law must fall within the neighbourhood in all cases. If a ray could not turn an angle required by Snell s Law at one vertex, it would continue the turning process at the next vertices until Snell s Law is met. For a small slowness variation, a neighbourhood size of three would be appropriate (see Fig. 3a). Our experience suggests that a neighbourhood size of five is adequate for most situations encountered in seismology. Fig. 3(a) shows that for a smooth medium. a vertex in the growing digraph has three edges which are selected in the neighbourhood of the direction of the incident edge of the vertex. Fig. 3(c) shows that for a medium with a large velocity contrast, a vertex in the growing digraph has five edges which are also selected in the neighbourhood of the direction of the incident edge of the vertex. The second appraoch is to select a direction among the predefined directions closest to the direction defined by Snell s Law, and then search directions in the neighbourhood of the selected direction. Such a procedure provides a superior starting direction to the method above. so the neighbourhood size can be reduced. The local slowness variation in Fig. 3(b) is the same as in Fig. 3(c). but the neighbourhood size is reduced to three whereas its central direction has changed to direction 8. which is closest to the direction defined by Snell s law. The first approach has the advantage of simplicity and it generally requires slightly more computational effort. The second approach is more complicated but more efficient. For a medium having moderate velocity contrast, the first approach is preferred. For a blockwise medium, the second approach is preferred. It is obvious that a substantial computational effort can be saved with the definition of a dynamic digraph which has less edges than a fixed graph. With the definition of 24 directions. three edges are used for a smooth medium and it can perform more than seven times faster than the algorithm with the global graph definition. For a seismic problem with a large velocity contrast, five edges are used and it can still perform more than four times faster than the old algorithm. Greater speed improvement can be gained with a large number of predefined directions, in particular in three dimensions. For example, with 2 predefined spatial directions and search directions, there is approximately a fold speed up over Moser s (99) method. However, it should be noted that when more directions are defined, more searching directions are usually required to ensure that, ~.~.~..~.~~. the actual ray path is within or close to the searching area, especially in strongly heterogeneous regions. Figure 3. Schematic illustration of the choice of edges for a dynamic digraph. (a) Three edges are choen for a smooth model; NUMERICAL IMPLEMENTATION AND (b) three edges are chosen for a model with a large velocity contrast TESTING where the central edge is selected on the basis of Snell s law: and (c) five edges are chosen for a model with a large velocity contrast. To implement the minimum traveltime tree algorithm Direction numbers correspond to those specified in Fig. 2. efficiently. we need to choose a data structure that only Downloaded from on February 28

5 Traveltime tree algorithm 597 M N I J L ID 2 n Figure 4. Data structure for minimum traveltime tree algorithm. stores the necessary information and stores it in such a way that the operations required by the algorithm can be done quickly. First of all, we need three arrays to store slowness, traveltime and raypath information at all the vertices. The remaining data structure is needed to perform the minimum traveltime tree algorithm. Fig. 4 shows in a schematic form the data structure needed for our method. In this data structure, the first two arrays are used to perform indexed heapsort. Next, two (three for 3D) arrays are used to translate data positions in the heap into vertex positions. The fifth array is a status array and functions as a direction indicator which points from the parent vertex. The last one, vertex ID, assigns a sequential number to each datum (label) in the heap. Note that with the above data structure, we already assume a local heap of fringe vertices in two (or three) dimensions instead of a global heap of the whole model as adapted by other researchers with the D formulation of Knuth (968). The total size of the newly defined data structure is only a fraction of the size required by the global D formulation for a 2D model, which will be further explained below. There are two approaches to formulate the heap structure. One is a global strategy. In this approach, the whole traveltime array is organized into a heap structure by indices. At each iteration of the loop, a fringe vertex joins the minimum traveltime tree and the size of the heap decreases by one. Therefore, the size of each of the data arrays has to be of the same size as the model. It should be noted that the number of data arrays in the global approach is less than that in the local approach. A significantly improved approach can be derived from the fact that only fringe vertices are involved at each iteration of the loop. Bearing in mind that a heap structure can be easily expanded by inserting some labels (e.g. traveltime data), or shrunk by removing some labels, we can construct a heap data structure with only the fringe vertices. From iteration to iteration, the size of the data structure may increase or decrease depending on the number of fringe vertices. The maximum size of each of the arrays is determined by the maximum number of fringe vertices at any stage during the process of constructing the minimum traveltime tree. It is obvious that the maximum number of fringe vertices is much smaller than the total number of vertices, especially when working in three dimensions. The vertex ID and ray path information can be stored in the same array because the ray path information is available only after the vertex joins the minimum traveltime tree and the vertex ID is needed only when the vertex is a fringe vertex. Thus the latter approach requires considerably less computer core memory than the global approach. The traveltime spent over each edge is needed only once, and when required it is calculated on the basis of local information. Fringe vertices of each vertex already in the minimum traveltime tree in the dynamic digraph are selected based on the value of the status array at the parent vertex and the local slowness distribution. Theoreticaly, the weight or traveltime for an edge to a fringe vertex should be the minimum traveltime between the two vertices and may be obtained analytically with local slowness information. In practice, straight lines are assumed for all edges. Firstly, a ray path variation has a lower order effect on traveltime than a slowness variation. Secondly, in real applications, many more search directions (e.g. of the order of ) are employed and neighbouring edges are so close that possible discrepancies between the real ray path and the straight line are very small; their effects on traveltime are negligible. Figure S shows the wavefront and ray paths in three models. All the models are of the same size, i.e. 4 x 2 nodes with a node spacing of m. Sources are placed at (, ) in the node coordinate. Wavefronts are illustrated by isochrons with a constant time increment of ms. In Fig. 5(a), the model is homogeneous with a velocity of 5 km s.'; the model in Fig. 5(b) has a high slowness (4kms') zone embedded within a lower slowness (high velocity, i.e. 5 km sc) host material whereas the model in Fig. 5(c) has an embedded low slowness zone (highvelocity inclusion, 6 km si) within a lowvelocity (5 km s') host. The wavefronts near the source in Figs 5(a), (b) and (c) are close to being circular (24sided polygons), which gives a good indication of the homogeneity in the source region. In Fig. 5(a) the wavefronts continue to be circular further away from the source whereas their counterparts in Figs S(b) and (c) are distorted by the velocity inhomogeneity. The wavefronts in the greater slowness region in Fig. S(b) are delayed whereas those in the lower slowness region in Fig. 5(c) are advanced. The ray path distributions in the three models are drastically changed by the different slowness distributions. The minimum traveltime ray paths in Fig. 5(a) are (approximately) evenly distributed. The region having Downloaded from on February 28

6 ') 598 S. Cao and S. Greenhalgh 2 (a' 2 (b' 2 (c) 4 4 higher slowness in Fig. S(b) pushes the minimum traveltime ray paths away from itself, whereas the region with a lower slowness in Fig. S(c) attracts the minimum traveltime ray paths into itself. It is important to note that the distributions of the takeoff angles at the source are changed by the slowness variations. Seismic rays with these takeoff angles are the minimum traveltime ray paths. Ray paths corresponding to other takeoff angles may be obtained with a constrained minimum traveltime ray tracing algorithm, which is beyond the scope of this paper and will be dealt with in a separate paper. It is also important to note that the spatial distribution of minimum traveltime ray paths is not uniform due to the slowness variation; in particular there are no ray paths within the greater slowness (low velocity) zone in Fig. S(b). To facilitate the tracking of ray paths passing through a particular region, accessory interfaces have to be introduced (see e.g. Cao 988; Moser 99). In the case of Fig. S(b), if one wishes to find ray paths passing through the lowvelocity (high slowness) region, an artificial vertical interface may be introduced in the middle of the lowvelocity region of the model. Minimum traveltimes to the artificial interface from any other two points can be obtained and combined to give the time distribution along the artificial interface. If the minimum point on the artificial interface is not at the endpoints, it can then be joined with the other two points to form a ray path, which is a local minimum traveltime ray path between the two points. Fig. S(d) shows a ray tracing from A to B via the lowvelocity region of the model where a vertical accessory interface CD is introduced. (a 2 A 4 B 4 Figure 5. Wavefront and ray diagrams in various models of the same size. 4 X 2(Hl nodes with a node spacing of m. The isochronal time increment is ms. Sources are located at (. ) in the node coordinate. (a) A homogeneous reference model with a velocitv of 5 km s I: (h) an embedded lowvelocity ( km s ~ ' ) inhomogeneity: (c) an embedded highvelocity (6 kin \ ~ inhomogeneitv: and (d) conmained rav tracing from A to B via the artificial interface CD. APPLICATIONS The minimum traveltime tree algorithm is an efficient, robust and global method for tracing rays. Its robustness and global property have been demonstrated in the applications of hypocentre determination in strongly heterogeneous earth models (Mow ef a/. 992b) and in ray tomographic inversion (see e.g. Zhou, Sinadinovaki & Greenhalgh 992). In this paper, we wish to show the incorporation of local information into the technique by performing ray tracing in an anisotropic model. The minimum traveltime tree algorithm can easily incorporate the local directional information of the ray path into the traveltime calculation scheme. The slowness for an edge of the dynamic digraph can be obtained by combining the direction of the edge with the local slowness ellipticity. Fig. 6 shows traveltime contours and ray diagrams for three different models. All the three models are of the same size, i.e. 4 x 2 nodes with a node spacing of m. Sources are located at (2,2) in the node coordinate. The traveltime contour interval is ms. The model in Fig. 6(a) is isotropic with a velocity of 5. km s' and serves as a reference. Velocities for models in Figs 6(b) and (c) are elliptically anisotropic. In Fig. 6(b), the long axis of the velocity anisotropy is horizontal with a value of 6.67 km s', whereas its short axis is vertical with the same value as the reference model. In Fig. 6(c), the long axis of the velocity anisotropy is vertical with the same value as the reference model, whereas its short axis is horizontal with a value of 3.33 km s '. Wavefronts in Figs 6(b) and (c) Downloaded from on February 28

7 Traveltime tree algorithm Figure 6. Wavefront and ray diagrams in elliptically anisotropic models of the same size, 4 X 2 nodes with a node spacing of m. The isochronal time increment is ms. Sources arc located at (2,2) in the node coordinate. (a) An isotropic reference model with a velocity of 5 kms I; (b) the long axis of the velocity anisotropy (6.67 km si) is horizontally orientated, whereas the velocity for the short axis is 5 kmsi; and (c) the long axis of the velocity anisotropy (5 km s ) is vertically orientated, whereas the velocity for the short axis is 3.33 km si.. Figure 7. A choice of predefined searching directions in a staggeredgrid scheme in B 9 X 9 neighhourhood arca. accumulated from the source vertex to the other vertices. However, calculated traveltimes could be smaller than the actual traveltimes for a too coarse sampling of a velocity field, which would allow the shortest paths to bridge a lowvelocity zone (Moser rt al. 992a). While the relative error is not related to the model size, the absolute error does vary with the model size. Therefore, it is desirable to subtract from the calculated traveltime the mean error associated with the space and angle discretization and the model size. The mean errors for complicated models are difficult to obtain; an estimate can be made using a corresponding homogeneous model. clearly show the effect of the velocity anisotropy, i.e. wavefronts are elongated horizontally in Figs 6(b) and vertically in Fig. 6(c). ERROR ANALYSIS AND DISCUSSION Errors associated with the traveltimes calculated with the minimum traveltime tree algorithm are dependent upon the grid and angular spacings. They are correct up to the second order in the space and angle discretization, as shown by Moser (99). It should be pointed out that calculated traveltimes will be greater (in a statistical sense) than the actual traveltimes due to the way in which the traveltime is..! Figure 8. A subgrid technique. (a) Only filled circles are timed and open circles are ignored; and (b) potential edges for the dynamic digraph. Downloaded from on February 28

8 6 S. Cao and S. Greenhalgh....,L\. :to) Figure 6. (Confinued.) To reduce the traveltime error, either the grid spacing or the angle spacing or both have to be reduced. This requires more computational effort. In particular, a small angle spacing will require the definition of a large neighbourhood. The large size of the neighbourhood will in turn create a large number of fringe vertices during the construction of the minimum traveltime tree, which then requires more computer time. An improvement can be made by arranging the vertex distribution in a staggeredgrid fashion. Fig. 7 shows the potential edge distribution in a 9 x 9 square neighbourhood of a staggered grid space discretization. In comparison with Fig. 2, many more searching directions are available and the angular spacing is more than halved. When the size of the neighbourhood is of the order of a wavelength or less, a subgrid technique (Nakanishi & Yamaguchi 986) can be used to decrease the number of fringe vertices. Fig. 8(a) shows a 7 x 7 neighbourhood vertex distribution where a staggeredgrid technique is employed to rearrange the vertex distribution. Slowness is assumed constant in each quadrant of the neighbourhood. In the subgrid technique, only peripheral grids (filled circles) in each quadrant of the neighbourhood are considered and inside grids (open circles) are ignored based on the fact that the ray path variation has a lesser effect on traveltime than the slowness variation. Fig. 8(b) shows the potential edge distribution for the neighbourhood shown in Fig. 8(a). The total number of potential edges in the subgrid technique is obviously much smaller than that used for a normal search. With less grid points, the computational speed can be improved, especially when the number of vertices in the neighbourhood is very large. However, this subgrid technique would involve considerably more programming effort because several edges with different lengths are possible for the same spatial direcion. CONCLUSIONS A numerically efficient algorithm for the calculation of the firstbreak seismic traveltime field and its ray path distribution is obtained by using local information during the construction of the dynamic digraph. By discretizing the space in a staggeredgrid manner, better accuracy can be achieved. A subgrid technique can be used to further improve the efficiency of the algorithm when the spatial interval is of the order of a subwavelength or less. The robustness and global nature of the scheme are inherent advantages of the shortestpath tree algorithm. The capability to incorporate local information is illustrated by ray tracing through anisotropic models. The simultaneous solution of traveltimes and ray paths makes this approach very attractive for inclusion in many seismic inversion techniques. ACKNOWLEDGMENT This research was supported by a grant from the Australian Research Council. REFERENCES Baase, S., 988. Computer algorithms, AddisonWesley, Reading, MA. Bellman, R., 958. On a routing problem, Q. appl. Math., 6, 88YO. Cao, S., 988. Rays and isochrons, in PhD midterm essay. Research School of Earth Sciences, Australian National University. Cao, S. & Greenhalgh, S. A,, 992. Finitedifference solution of the eikonal equation using an efficient firstarrival wavefront tracking scheme, Geophysics, submitted. Dijkstra, E. W., 959. A note on two problcms in connexion with graphs, Numer. Math.,, Knuth, D. E., 968. The urt of computer programming, Vol. 3: sorting and searching. AddisonWesley, Reading, MA. Moser, T. J., 99. Shortest path calculation of seismic rays, Geophysics, 56, 596. Moser. T. J., Nolet, G. & Sneider, R., 992a. Ray bending revisited, Bull. seism. Soc. Am., 82, Moser, T. J., van Eck, T. & Nolet, G., 992b. Hypocenter determination in strongly heterogeneous Earth models using the shortest path method, J. geophys. Res., 97, Nakanishi, I. & Yamaguchi, K., 986. A numerical experiment on nonlinear image reconstruction from firstarrival times for twodimensional island are structure, J. Phys. Earth, 34, 952. Qin, F., Luo, Y., Olsen, K. B., Cai. W. & Schuster, G. T., 992. Finitedifference soluton of the eikonal equation along expanding wavefronts, Geophysics, 57, Sedgewick, R., 988. Algorithms, AddisonWesley, Reading, MA. Vidale. J., 988. Finitedifference calculation of travel times, Bull. seism. SOC. Am. 78, Vidale, J., 99. Finitedifference calculation of traveltimes in three dimensions, Geophysics, 55, Zhou, B., Sinadinovski, C. & Greenhalgh, S., 992. Nonlinear inversion traveltime tomography: imaging highcontrast inhomogeneitics, Expl. Geophys. 23, Downloaded from on February 28