The effect of noise and Tikhonov's regularization in inverse problems. Part I: The linear case

Abstract In this paper we analyze the roles of noise in data and that of the regularization for nonlinear inverse problems, generalizing the analysis shown in a first accompanying paper for linear inversion. As in the linear case, the noise shifts the solution found by nonlinear optimization, but in the nonlinear case it also deforms non-homogenously the topography of the cost function. Under certain conditions, the noise might decrease the size of the regions of very low misfits, and at the same time increase the size of the regions of medium misfits. As a result of that the inverse problem solution becomes more difficult to be found by search methods, but locating the region (or regions) of medium misfits is easier. Also, with respect to the use of local optimization methods the noise might act as a certain kind of regularization, since it serves to decrease the size of the region of equivalence. The effect of the regularization in the linearized region of equivalence is similar to the linear case; nevertheless this region does not coincide with the nonlinear region of equivalence. Although the use of regularization techniques has a positive effect in decreasing the uncertainty of the solution that has been found, unfortunately it does not provoke the disappearance of the nonlinear equivalent models. This manuscript, and its accompanying paper, provides new insights about of the role of noise and that of the regularization in the linear and nonlinear uncertainty analysis of inverse problems. This knowledge is very important, especially in applied geophysics, because noise in data is always present, and the possibility of having at disposal good prior information is rare. Therefore, the nonlinear uncertainty analysis and appraisal of the solution is always needed.