How do barrier reefs and lagoons form around sinking islands?

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The formation of barrier reefs and lagoons around sinking islands is explained by the subsidence theory originally proposed by Charles Darwin.12

As a volcanic island slowly sinks or the ocean floor subsides, coral reefs that initially fringe the shoreline grow upward to remain near the surface, where conditions are optimal for growth. Over thousands of years, the reef structure continues to grow upward and outward, eventually becoming separated from the receding volcanic shoreline by an increasing expanse of water. This intervening stretch of water forms a lagoon, and the reef is then classified as a barrier reef.123

If the volcanic island continues to subside until it is completely submerged, the reef remains as a ring-shaped structure known as an atoll, which encloses a central lagoon. While subsidence is a primary driver, modern understanding also incorporates the effects of sea level fluctuations and plate tectonics as significant factors in the development of these features.124