ref~ is shorthand for ref~1 and means the commit's first parent. ref~2 means the commit's first parent's first parent. ref~3 means the commit's first parent's first parent's first parent. And so on.
ref^ is shorthand for ref^1 and means the commit's first parent. But where the two differ is that ref^2 means the commit's second parent (remember, commits can have two parents when they are a merge).
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P. Burdziakowski, N. Razmjooy, V. Estrela, and J. Hemanth. Imaging and Sensing for Unmanned Aircraft Systems: Volume 2: Deployment and Applications, Institution of Engineering and Technology, (April 2020)