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This methodological paper is part of a short series dedicated to thelong-standing astronomical problem of de-projecting the bi-dimensional,apparent morphology of a three-dimensional distribution of gas. We focus on thequantification and spatial recovery of turbulent motions in planetary nebulae(and other classes of expanding nebulae) by means of long-slit echellogramsover a wide spectral range. We introduce some basic theoretical notions,discuss the observational methodology, and develop an accurate proceduredisentangling all broadening components of the velocity profile in all spatialpositions of each spectral image. This allows us to extract random, non-thermalmotions at unprecedented accuracy, and to map them in 1-, 2- and 3-dimensions.We present the solution to practical problems in the multi-dimensionalturbulence-analysis of a testing-planetary nebula (NGC 7009), using thethree-step procedure (spatio-kinematics, tomography, and 3-D rendering)developed at the Astronomical Observatory of Padua. In addition, we introducean observational paradigm valid for all spectroscopic parameters in all classesof expanding nebulae. Unsteady, chaotic motions at a local scale constitute afundamental (although elusive) kinematical parameter of each planetary nebula,providing deep insights on its different shaping agents and mechanisms, and ontheir mutual interaction. The detailed study of turbulence, its stratificationwithin a target and (possible) systematic variation among different sub-classesof planetary nebulae deserve long-slit, multi-position angle, wide-spectralrange echellograms containing emissions at low-, medium-, and high-ionization,to be analyzed pixel-to-pixel with a straightforward and versatile methodology,extracting all the physical information stored in each frame at best.

The gas turbulence in planetary nebulae: quantification and multi-D maps from long-slit, wide-spectral range echellogram