Sentiment Analyses are widely used approaches to understand and identify emotions, feelings, and opinion on social media platforms. Most sentiment analysis systems measure the presumed emotional polarity of texts. While this is sufficient for some applications, these approaches are very limiting when it comes to understanding how social media users actually use language resources to make sense of extreme events. In this paper, the authors apply a Sentiment Analysis based on the Appraisal System from the theory of communication called Systemic Functional Linguistics to understand the sentiment of event-driven social media communication. A prototype was developed to code and visualise geotagged Twitter data using the Appraisal System. This prototype was applied to tweets collected during and after the Sydney Siege, a hostage situation in a busy café in Sydney's inner city at the 15th of December 2014. Because the Appraisal System is a theorised functional communication method, the results of this analysis are more nuanced than is possible with traditional polarity based sentiment analysis.