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

• Analysis of the effect of noise in the topography of the cost function in linear inversion
• Analysis of the effect of regularization in the topography of the cost function in linear inversion
• Comparison of both effects
• Impact on the uncertainty analysis in linear inverse problems

Inverse problems are a special kind of optimization problems because the cost function involves data, always affected by noise, that in the case of ill-conditioning is amplified back into the model parameters through the generalized inverse operator. Then, the inversion might provide spurious solutions if no regularization techniques are used. For a given misfit tolerance the models that fit the observed data are called equivalent, and are located in a region of the model space that is bounded by the linear hyper-quadric surface. This paper analyzes in detail the role of noise in data in linear inverse problems, providing a geometrical interpretation for the role of the regularization. The noise shifts the solution found by least squares methods and deforms homogeneously the topography of the cost function, while Tikhonov's regularization transforms the linear hyper-quadric from an elliptical cylinder to a very oblong hyper-ellipsoid in the directions that originally spanned the kernel of the linear forward operator. Furthermore, in the case of the regularization, this deformation is anisotropic affecting differently the axes of the linear hyper-quadric. The model of reference informs the coordinates of the solution that originally resided in the kernel of the forward operator. The differences with nonlinear inversion are highlighted in the second accompanying paper. This knowledge, although theoretical at this stage, might impact how the uncertainty analysis is performed in geophysical inversion, since noise in data is always present, and good prior models are not always available.