There a are globally two perspectives
the discrete perspective uses a categorization system. There are many different systems, with more or less core emotions and sub-emotions. As the one shown in your post.
the dimensional perspective considers one, mainly two, sometimes more, scales to identify an emotional value. Valence (happy/sad) and arousal (sleepy/excited) are quite frequent for those models.
It's not so easy to choose between the two perspectives. And there are different models for each perspective. Discrete based models are easier to handle by anyone because they make use of usual names. But it suffers a bias of interpretation of those names meaning.
In this way, indicating a level of valence or arousal is maybe more accurate, but less user-friendly.
For the dimensional perspective, Scherer's Component Process Model version (2005) is quite comprehensive, based on the updated model of Russell (2003).
Depending of the model, some tools are developed to assess the emotion.
Scherer, K. (2005) What are emotions and how can they be measured?
Russell, J. A. (2003). Core affect and the psychological construction of emotion. Psychological Review, 110, 145-172.