The most important measurements for the scientific investigation of Unidentified Aerial Phenomena (UAP) using astronomical methods are presented and discussed, where results obtained in the past motivate the proposal for new observations using multiwavelength and multimodal instruments. A special emphasis is given on the techniques of magnetometry, photometry and spectroscopy, and on the importance of studying the variability of the phenomenon in order to try to understand the physical process that governs it, including a possible propulsion mechanism. The most important obtainable physical parameters are discussed in detail, with a particular emphasis on how they might be correlated together. Calculations of the integration times needed for obtaining optimum signal-to-noise-ratios in photometry and spectroscopy are presented. The idea of placing measurement instruments at areas of the world where the phenomenon is recurrent is strongly suggested. Past monitoring campaigns at such locations are briefly described together with the pertinent literature.