Geometric theory of inversion and seismic imaging

The goal of inversion is to estimate the model which generates the data of observations with a specific modeling equation. One general approach to inversion is to use optimization methods which are algebraic in nature to define an objective function. This is the case for objective functions like minimizing RMS of amplitude, residual traveltime error in tomography, cross correlation and sometimes mixing different norms (e.g. L1 of model + L2 of RMS error). Algebraic objective function assumes that the optimal solution will come up with the correct geometry. It is sometimes difficult to understand how one number (error of the fit) could miraculously come up with the detail geometry of the earth model. If one models the earth as binary rock parameters (only two values for velocity variation), one could see that the geometry of the rugose boundaries of the geobodies might not be solvable by inversion using algebraic objective function.