EikonalSolvers.jl

A library to perform seismic traveltime computations by solving the eikonal equation in two (2D) and three dimensions (3D) with the possibility of computing the gradient of a misfit function with respect to velocity and source location. The coordinate system can be either regular Cartesian or spherical. The forward algorithm is based on a fast marching (FMM) method (2nd order) with a refinement of the grid around the source location. The computation of the gradients relies on the discrete adjoint method.

Both forward and gradient (adjoint) computations can be run in parallel using either Julia's distributed computing functions for distributed memory or threads for multicore processor. The parallelisation scheme is "by source", distributing calculations for different seismic sources to different processors.

This code is part of a larger project G⁻¹Lab (a superset of HMCLab) targeting probabilistic geophysical inverse problems. Please cite the following papers if you use this code:

• Andrea Zunino, Scott Keating, Andreas Fichtner (2025), A discrete adjoint method for deterministic and probabilistic eikonal-equation-based inversion of traveltime for velocity and source location, Geophysical Journal International, Volume 243, Issue 1, October 2025, ggaf283, https://doi.org/10.1093/gji/ggaf283

• Andrea Zunino, Lars Gebraad, Alessandro Ghirotto, Andreas Fichtner (2023), HMCLab: a framework for solving diverse geophysical inverse problems using the Hamiltonian Monte Carlo method, Geophysical Journal International, Volume 235, Issue 3, Pages 2979–2991, https://doi.org/10.1093/gji/ggad403

Example of forward calculations in 2D

Here below an example of how to calculate traveltimes at receiver stations in 2D, given a grid geometry and positions of sources and receivers.

# load the package
using EikonalSolvers

# create the Grid2DCart struct
grd = Grid2DCart(hgrid=0.5,cooinit=(0.0,0.0),grsize=(300,220))  # create the Grid2D struct
nsrc = 4 # number of sources
nrec = 10 # number of receivers
# coordinates of the sources (4 sources)
coordsrc = [grd.hgrid.*LinRange(10,290,nsrc)  grd.hgrid.*200.0.*ones(nsrc)] # coordinates of the sources (4 sources)
# coordinates of the receivers (10 receivers)
coordrec = [ [grd.hgrid.*LinRange(8,294,nrec) grd.hgrid.*20.0.*ones(nrec)] for i=1:nsrc] # coordinates of the receivers (10 receivers)
# velocity model
velmod = 2.5 .* ones(grd.grsize...)   # velocity model
# increasing velocity with depth
for i=1:grd.grsize[2] 
  velmod[:,i] = 0.034 * i .+ velmod[:,i] 
end

# run the traveltime computation
ttimepicks = eiktraveltime(velmod,grd,coordsrc,coordrec)

Optionally it is possible to retrieve also the traveltime at all grid points.

ttimepicks,ttimegrid = eiktraveltime(velmod,grd,coordsrc,coordrec,returntt=true)

Finally, functions are provided to back-trace rays from receivers to sources.

<p align="center"><img src="docs/src/images/rays2.png" width="70%" /></p>

Example of gradient calculations in 2D

The gradient of the misfit functional (see documentation) with respect to velocity and source location can be calculated as following. A set of observed traveltimes, error on the measurements and a reference velocity model are also required, see the documentation for a detailed example.

# calculate the gradient of the misfit function w.r.t. velocity
gradvel,misf = eikgradient(vel0,grd,coordsrc,coordrec,dobs,stdobs,:gradvel)

The gradient of the misfit function w.r.t. the source position can also be computed, e.g., as

# calculate the gradient of the misfit function w.r.t. source location
gradsrcloc,misf = eikgradient(vel0,grd,coordsrc,coordrec,dobs,stdobs,:gradsrcloc)

# now print the partial derivatives
@show(gradsrcloc)

4×2 Matrix{Float64}:
  0.139602    0.000915479
 -0.28355     0.408047
 -0.152851    0.072085
 -0.0200304  -0.249815

Calculations in 3D

Calculations in 3D for both Cartesian and spherical coordinates are analogous to the 2D function, see the documentation.

<img src="docs/src/images/examplegrad3Dcarsph.png" alt="Example gradient 3D" height="300"/>