Noise attenuation by filtering and modeling

Chapter tackles the noise problem generated by modeling uncertainties. Using the mathematical framework of least-squares inversion, it is theoretically possible to eliminate the noise present in the data by estimating the data covariance operator Tarantola (1987). The data covariance operator can be seen as a filtering operator that operates in the data space. Building on Nemeth (1996), it is also possible to separate noise and signal by incorporating a noise modeling operator within the inversion. In both techniques, the residual components become independent and identically distributed (IID). In Chapter , I review both approaches and show that multidimensional prediction-error filters (PEFs) can approximate covariance and modeling operators. In addition, I unravel the mathematical links between the two approaches and establish that the noise modeling technique can be algebraically reduced to a filtering method with projection operators.