Finite difference calculation of traveltime in rectangular topology mesh

Finite difference methods have been widely employed in solving the eikonal equation, to calculate first arrival time. Most previous researchers used regular grids, which did well in small scale problems, but the regular grids required denser meshes to sample the irregular interfaces, such as the Moho discontinuity, the 660km discontinuity, and it is difficult to solve large scale problems that the spherical surface effect must be taken into consideration. We propose a new finite difference method to solve the eikonal equation in rectangular topology grids. This method can accept rectangular topology mesh with slight distortion. And we also provided a reverse timing procedure to deal with head waves. After adopting appropriate mesh, we tested this method to calculate the first arrival times in a two layer model with an irregular interface, and the result showed it has similar accuracy with ten times grid spacing in regular mesh. Then we compared result obtained by our method with ray tracing's result in IASP91 model, and the traveltime curves at free surface are very close. If we add a rolling 660 km discontinuity, the traveltime curve shows an obvious perturbation while delta ranges from 24 degree to 34 degree, which indicates the effect caused by topography of 660 km discontinuity. This algorithm is also suitable to quickly calculate the first arrival travel time at local distances where a laterally varying Moho plays an important role or at upper mantle triplication distances where the 410km and 660 km discontinuities show substantial topographic variation. But our method may fail when the topography is too difficult to be defined with a mesh consisting of well-behaved quadrangles (aspect ratio too large or the angles too different from 90 degree).