## Abstract

A new theory for the calculation of later seismic arrivals in heterogeneous media is presented. We introduce the concept of a ‘raylet’, which is a segment of a later arriving ray path between source and receiver defined by joint properties of forward and reciprocal traveltime fields. We show that all rays between a single source and receiver in arbitrary heterogeneous media can be divided into a unique set of raylets, any one of which can be used to construct the complete two-point path. A particularly useful property of raylets is that they correspond to stationary curves in the summed (forward and reciprocal) traveltime fields, i.e. adding the first (or second, or even later) arriving traveltime field from some point A to the first (or second, or even later) arriving traveltime field from some point B. We show that many raylets, each corresponding to a later arriving phase, require only earlier arrival traveltime fields for their construction. The theory describing the properties of raylets is the primary result of the paper. One practical consequence is that many later arrivals between source and receiver in heterogeneous media can be found from just two first-arrival traveltime fields, one from the source and the other (the reciprocal field) from the receiver.

The theory and a method for constructing later arrivals is demonstrated though numerical experiments in 2-D but holds without change in 3-D. We use a simple grid-based eikonal solver to compute forward and reciprocal first-arrival traveltime fields, and validate our results with a ray-based wave front construction (WFC) technique. In one test involving a simple wave front triplication (or swallowtail), all three arrivals are retrieved. In another example featuring severe velocity heterogeneities, 16 out of a total of 37 arrivals are found. The theory shows that combining second and higher order traveltime fields from source and receiver yields all raylets and hence all later arrivals. In practice only first-arrival traveltime fields are usually available and for this case we describe a simple procedure to find any remaining arrivals using intermediate artificial sources together with their first-arrival traveltime fields. The properties of raylets and their manifestation in the joint traveltime field appears to be a previously unrecognized feature and provides a novel new approach to the calculation of later arrivals.

The theory and a method for constructing later arrivals is demonstrated though numerical experiments in 2-D but holds without change in 3-D. We use a simple grid-based eikonal solver to compute forward and reciprocal first-arrival traveltime fields, and validate our results with a ray-based wave front construction (WFC) technique. In one test involving a simple wave front triplication (or swallowtail), all three arrivals are retrieved. In another example featuring severe velocity heterogeneities, 16 out of a total of 37 arrivals are found. The theory shows that combining second and higher order traveltime fields from source and receiver yields all raylets and hence all later arrivals. In practice only first-arrival traveltime fields are usually available and for this case we describe a simple procedure to find any remaining arrivals using intermediate artificial sources together with their first-arrival traveltime fields. The properties of raylets and their manifestation in the joint traveltime field appears to be a previously unrecognized feature and provides a novel new approach to the calculation of later arrivals.

Original language | English |
---|---|

Pages (from-to) | 1077-1092 |

Number of pages | 16 |

Journal | Geophysical Journal International |

Volume | 181 |

Issue number | 2 |

Early online date | 19 Mar 2010 |

DOIs | |

Publication status | Published - May 2010 |

## Keywords

- numerical solutions
- computational seismology
- theoretical seismology
- wave propagation