An Intelligent Seismic First-break Time Calculation Scheme for Inhomogeneous Models

Seismic first-break times can be computed by treating each node of a given model as a secondary source and applying Huygens’ Principle from the source to its surrounding nodes and successively from inner nodes to outer nodes of the source. At each node, different travel times are possible from the source via different secondary sources (virtually all the other nodes) and only the least (minimum) time is kept as the first-break time for this particular node. As the number of nodes increases, the selection of the least time for each node can be a huge computational task. However, for the first-break time only a small number of secondary sources, i.e. the immediate neighbouring nodes, need to be considered. An efficient first-break wavefront tracking algorithm is used to maintain the causality of the process.

This efficient tracking algorithm arranges the nodes of the model into a directed graph. Only neighbouring nodes are connected by edges. The weight of each edge is denoted by the travel time difference between the two connecting nodes. The local gradient of the first-break time field determines the direction of the edges. With this directed graph, a minimum travel time tree is constructed by taking the source node as its root. At any stage of this wavefront tracking, the nodes of the model can be thought of as divided into three (disjoint) categories: tree nodes in the tree constructed so far; fringe nodes which are not in the tree but adjacent to some nodes in the tree; and unseen (other) nodes. The fringe nodes roughly represent and keep track of the current first-break wavefront. Thus the key point in the construction of the minimum travel time tree is how to evolve the first-break wavefront outward from the source. To maintain causality, the fringe node of least travel time is always selected as a secondary source to illuminate (forward) nodes along the local wave propagation direction. Once the travel times for forward neighbouring nodes are updated, the current node of least travel time is converted to a tree node and the first-break wavefront advances one node away from the source. This updating process is repeated until all the nodes are converted into tree nodes.

An efficient sorting method, the heap sort method, is employed to find the least travel time node from the current fringe. A special data structure is adopted for the implementation of an intelligent indirect heap sort. At any stage of the wavefront tracking, only the fringe nodes are in this special data structure and thus only local sorting is required.