Background

Many types of geophysical data consist of measurements of a given quantity, collected at arbitrary locations near the earth's surface. The problem is to infer the value of this quantity at all locations in the study area - in other words, to estimate the earth model of the quantity that gave rise to the collected data. These ideas are embodied in the following simple linear relationship  

$$\bf B m = d.$$ (1)

$\bf d$ and $\bf m$ are the measured data and estimated model, respectively, while $\bf B$ is a linear operator that carries out the ``experiment'' by sampling the model at the measurement locations and mapping these values to the data vector. However, the reverse - an estimate of the model, given the data - is usually more useful. In this case, $\bf B$ must be ``inverted'' in some sense:  

$$\bf \hat{m} = \bold B^{\dagger} d.$$ (2)

The simplest choice is $\bold B^{\dagger} = \bold B^T$, but any choice so that the model honors the known data has a measure of validity. $\bold B^{\dagger}$ may also be cast as a least squares inverse. Depending on the acquisition geometry of the experiment, the least squares problem may be underdetermined, overdetermined, or more commonly, both Menke (1989). For purely overdetermined problems, the least squares inverse is $\bold B^{\dagger} = (\bold B^T \bold B)^{-1} \bold B^T$, but if some model points are undetermined, $\bold B^T \bold B$ is singular.