The Murray-Davies equation was originally invented to model the color of hard dots. It has long served as a process control device to monitor dot gain on a web offset press. More recently, it has migrated to alternate forms of printing where the dots are no longer crisp, with well-defined edges. The calculation received a new name when this happened: TVI. Unfortunately, the equation itself was not updated to reflect the wide range of dot sharpness. The equation was never all that accurate at predicting the actual color of a halftone. Furthermore, as dots soften and blend together to form more of a continuous tone, the prediction gets worse.
The limitations of TVI and the Murray-Davies formula are described in this paper. Two alternate mathematical models are considered: one based on the assumption that a dot thins as it spreads, and the other the Yule-Nielsen equation. These two models yield results that are surprisingly similar. The fact that both equations are very close to linear in CIELAB leads to a third mathematical model which is strictly empirical, but computationally simpler.
This model is used to characterize print from numerous types of presses. In most cases, the prediction error is less than 1 DE*ab, as compared with prediction errors which are often greater than 5 DE*ab when the Murray-Davies equation is used.