The Logarithmic Derivative in Scientific Data Analysis

The logarithmic derivative is shown to be a useful tool for data analysis in applied sciences because of either simplifying mathematical procedures or enabling an improved understanding and visualization of structural relationships and dynamic processes. In particular, spatial and temporal variations of signal amplitudes can be described independently on their sign by one and the same compact quantity, the inverse logarithmic derivative. In the special case of a single exponential decay function, this quantity becomes directly identical to the decay time constant. Generalized, the logarithmic derivative enables to flexibly describe local gradients of physical or non-physical system parameters by using exponential behavior as a meaningful reference. It can be applied to complex maps of data containing multiple superimposed and alternating ramping or decay functions as, e.g., in the case of time-resolved plasma spectroscopy, multiphoton excitation or spectroscopy. Examples of experimental and simulated data are analyzed in detail together with reminiscences on early activities in the field. Further emerging applications are addressed.