Relative‐error‐based non‐linear inversion: application to seismic traveltime tomography

SUMMARY The aim of geophysical inversion is to infer information about the subsurface model parameters subject to certain constraints and plausibility considerations. It is achieved through a systematic adjustment of model parameters along a solution path to the minimum of an objective function. The objective function is related to both the observations (data) and the a priori information about the subsurface model parameters (model). When different statistics are assumed for the data and the a priori information about the model, different objective functions are constructed and different solution models become inevitable. Traditionally, data residuals are used directly in constructing the objective function. If the residuals are statistically proportional to the data magnitudes, a least‐squares (LS) method often leads to a solution model biased towards (i.e. which emphasizes) observations of large magnitude, whereas a least absolute deviation (LAD) method can reduce this data‐magnitude‐dependent effect. An alternative is to formulate the objective function in terms of relative differences. It completely eliminates the possible solution bias due to the intrinsic magnitude‐dependent discrepancies in the objective function. The solution path is altered in such a way that data residuals are reduced in a more uniform manner. The idea is illustrated through a seismic traveltime tomographic inversion. Traveltime residuals are often correlated with ray‐path lengths. Greater residuals of long ray paths dominate the objective function in the traditional tomographic formulation. The solution path is thus strongly affected by those long ray paths, and short ray paths play a reduced role in the inversion. The final solution is biased toward longer ray paths. The new alternative formulation eliminates this bias by exploiting the relative‐error measure to give equal weights to data of the same quality regardless of their ray‐path lengths.