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Source apportionment studies have struggled to quantitatively link secondary organic aerosols (SOAs) to their precursor sources due largely to instrument limitations. For example, aerosol mass spectrometer (AMS) provides quantitative measurements of the total SOA fraction but lacks the chemical resolution to resolve most SOA sources. In contrast, instruments based on soft ionisation techniques, such as extractive electrospray ionisation mass spectrometry (EESI, e.g. the EESI time-of-flight mass spectrometer, EESI-TOF), have demonstrated the resolution to identify specific SOA sources but provide only a semi-quantitative apportionment due to uncertainties in the dependence of instrument sensitivity on molecular identity. We address this challenge by presenting a method for positive matrix factorisation (PMF) analysis on a single dataset which includes measurements from both AMS and EESI-TOF instruments, denoted "combined PMF" (cPMF). Because each factor profile includes both AMS and EESI-TOF components, the cPMF analysis maintains the source resolution capability of the EESI-TOF while also providing quantitative factor mass concentrations. Therefore, the bulk EESI-TOF sensitivity to each factor can also be directly determined from the analysis. We present metrics for ensuring that both instruments are well represented in the solution, a method for optionally constraining the profiles of factors that are detectable by one or both instruments, and a protocol for uncertainty analysis. As a proof of concept, the cPMF analysis was applied to summer and winter measurements in Zurich, Switzerland. Factors related to biogenic and wood-burning-derived SOAs are quantified, as well as POA sources such as wood burning, cigarette smoke, cooking, and traffic. The retrieved EESI-TOF factor-dependent sensitivities are consistent with both laboratory measurements of SOA from model precursors and bulk sensitivity parameterisations based on ion chemical formulae. The cPMF analysis shows that, with the standalone EESI-TOF PMF, in which factor-dependent sensitivities are not accounted for, some factors are significantly under- or overestimated. For example, when factor-dependent sensitivities are not considered in the winter dataset, the SOA fraction is underestimated by ∼25 % due to the high EESI-TOF sensitivity to components of primary biomass burning such as levoglucosan. In the summer dataset, where both SOA and total OA are dominated by monoterpene oxidation products, the uncorrected EESI-TOF underestimates the fraction of daytime SOA relative to nighttime SOA (in which organonitrates and less oxygenated C x H y O z molecules are enhanced). Although applied here to an AMS and EESI-TOF pairing, cPMF is suitable for the general case of a multi-instrument dataset, thereby providing a framework for exploiting semi-quantitative, high-resolution instrumentation for quantitative source apportionment. [ABSTRACT FROM AUTHOR]