The Least-Squares Criterion in Inverse Problems

The full expression for the updated model \(\tilde{m}\) is given with priors and posterior covariance matrices \(C_D, C_m\) : \[\tilde{m} = m_{prior} + C_M G^T \Big(G C_M G^T + C_D \Big)^{-1} \Big(d_{obs} - Gm_{prior} \Big)\] \[\tilde{C_M} = C_M - C_M G^T \Big(G C_M G^T + C_D\Big)^{-1} G C_M\]

References and further reading :

Inverse Problem Theory and Methods for Model Parameter Estimation. Albert Tarantola. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2004. ISBN 0898715725. Available here.