Term, frequently used within inverse theory, to indicate the difference between observations, \mathbf{d}, and forward modeled observations. These forward modeled observations are obtained by inverting the original observations using the governing equations, Q, of the system, which relates observations to parameters, \bm{\hat{m}}, of a model.

An estimation of the model parameters, \bm{\hat{m}}, is obtained by:

\bm{\hat{m}} = Q^\dagger\mathbf{d}

with Q^\dagger being the pseudo inverse of Q. The residual, \mathbf{r}, is calculated by subtracting the observations from the inverted observations, obtained by calculating the observations caused by the estimated model parameters, or:

\mathbf{r} = Q\bm{\hat{m}} -\space \mathbf{d}