Least-squares inversion of seismic arrivals can provide remarkably detailed models of the Earth's subsurface. However, cycle skipping associated with these oscillatory arrivals is the main cause for local minima in the least-squares objective function. Therefore, it is often difficult for descent methods to converge to the solution without an accurate initial large-scale velocity estimate. The low frequencies in the arrivals, needed to update the large-scale components in the velocity model, are usually unreliable or absent. To overcome this difficulty, we propose a multi-objective inversion scheme that uses the conventional least-squares functional along with an auxiliary data-domain objective. As the auxiliary objective effectively replaces the seismic arrivals by bumps, we call it the bump functional. The bump functional minimization can be made far less sensitive to cycle skipping and can deal with multiple arrivals in the data. However, it can only be used as an auxiliary objective since it usually does not provide a unique model after minimization even when the regularized-least-squares functional has a unique global minimum and hence a unique solution. The role of the bump functional during the multi-objective inversion is to guide the optimization towards the global minimum by pulling the trapped solution out of the local minima associated with the least-squares functional whenever necessary. The computational complexity of the bump functional is equivalent to that of the least-squares functional. In this paper, we describe various characteristics of the bump functional using simple and illustrative numerical examples. We also demonstrate the effectiveness of the proposed multi-objective inversion scheme by considering more realistic examples. These include synthetic and field data from a cross-well experiment, surface-seismic synthetic data with reflections and synthetic data with refracted arrivals at long offsets.
- Numerical solutions
- Inverse theory
- Non-linear differential equations
- Seismic tomography