Add frozen raspberries, water and sugar to a saucepan.  Bring to a simmer over medium-high heat then reduce to medium. Simmer, stirring occasionally, for 5 minutes until berries are very soft and mostly dissolved. Stir in the mint sprigs and remove pan from heat. Let cool, allowing mint to steep in hot liquid, then discard mint. For a thicker beverage, puree in a blender or using an immersion blender. For a smoother texture, strain and discard solids. Stir in lime juice.

Divide juice among 6 glasses then top with soda water and mint sprigs.