Data Misfit

The observations $d^{obs}$ encode our knowledge about the model obtained from the survey. Any candidate solution $m$ must produce simulated (or predicted) data $d$ that “fit” the observations. Central to this goal is the choice of a “criterion” for measuring misfit between observed and predicted data, and a “tolerance” for deciding what value constitutes an acceptable fit. This is a subject on which an enormous amount of literature has been written but a good reference, written for the inverse problem, is Parker (1994). When errors are Gaussian with zero mean and known standard deviation, then the $L_2$ -norm, is an appropriate choice. We define the misfit to be

where $d_j^{obs}$ is the observation, $\epsilon_j$ is its estimated standard deviation, and $d_j$ is the predicted datum. The quantity

is a random variable with zero mean and unit standard deviation and $\phi_d$ , which is the sum of squares of these variables, is the well known $\chi^2$ statistical variable. It has an expected value ((3)) and variance ((4)).

Thus if we are attempting to find a model $m$ that acceptably fits the data, then models with $\phi_d \simeq N$ are good candidates. For many problems we often denote a target misfit $\phi_d^*$ to be

but this must only be regarded as a reasonable estimate and some flexibility should be entertained.

In times past, it was felt that getting an acceptable fit to the data was a sufficient criterion for having a successful inversion. The observed data $\mathbf{d}_{obs}$ are inverted to produce a model $\mathbf{m}$, which is used to forward model the predicted data $\mathbf{d}$. The observed and predicted data are then compared using the data misfit measure. If $\phi_d<\phi_d^*$ the model $\mathbf{m}$ is accepted, but if not, the inversion parameters are adjusted and the process is repeated until an acceptable model is reached. The workflow procedure is delineated below.

Finding a model that fits the data is a necessary component of an inversion algorithm, but as shown in the next section, it is not sufficient.