Generalized non‐linear elastic inversion with constraints in model and data spaces

SUMMARY Seismic data are non‐linearly related to model parameters such as seismic velocities. However, seismic inversion is usually considered in a linear approximation. Such techniques as the Born inversion were recently applied to seismic data. Non‐linear inversion is more complicated and involves extensive calculations. Non‐linear inversion was developed in the frame work of an unconstrained optimization procedure. It uses as a priori information an initial model and probability distribution functions in the data and model spaces (This a priori information is called ‘soft’ bounds). In this paper, we propose a new technique for solving a constrained non‐linear inversion. This technique will allow us to use a priori information not only in terms of ‘soft’ bounds, but ‘hard’ bounds as well (usually giving more stable and accurate solutions). Non‐linear inversion is considered as an iterative procedure which involves a dual transform at each iteration. A dual transform allows for considering the problem in terms of the Lagrangian multipliers. The number of Lagrangian multipliers is equal to the number of available data and thus, significantly reduces the dimension of the problem (this is true for underdetermined problems only). However, the most important property of the dual transform is that it allows us to consider a constrained problem as an unconstrained problem. Another important property is that proper constraints incorporate small‐wave numbers in the generalized inversion. It is shown that conventional (unconstrained non‐linear inversion) is a special case of the constrained non‐linear inversion developed in this paper if the truncation operator is represented by the identity matrix.