Measuring relative abundances in the solar corona with optimised linear combinations of spectral lines

13 pages, 14 figuresContext. Elemental abundances in some coronal structures differ significantly from photospheric abundances, with a dependence on the first ionization potential (FIP) of the element. Measuring these FIP-dependent abundance biases is important for coronal and heliospheric physics. Aims. We aim to build a method for optimal determination of FIP biases in the corona from spectroscopic observations in a way that is in practice independent from Differential Emission Measure (DEM) inversions. Methods. We optimised linear combinations of spectroscopic lines of low-FIP and high-FIP elements so that the ratio of the corresponding radiances yields the relative FIP bias with good accuracy for any DEM in a small set of typical DEMs. Results. These optimised linear combinations of lines allow retrieval of a test FIP bias map with good accuracy for all DEMs in the map. The results also compare well with a FIP bias map obtained from observations using a DEM-dependent method. Conclusions. The method provides a convenient, fast, and accurate way of computing relative FIP bias maps. It can be used to optimise the use of existing observations and the design of new observations and instruments