To understand how this happens, we must first look at why we visually perceive most icebergs to be white:
Icebergs come from glaciers and glaciers are formed as a result of snowfall. As water freezes it becomes crystallised. A close examination of a snowflake reveals a many-faceted crystal, not unlike a cut diamond. These facets or surfaces are capable of reflecting light. Snow appears white for this very reason. As snow accumulates, the structure of snowflakes traps a great deal of air. This is easily seen if you fill a glass to the top with snow, then bring it inside and allow it to melt. You quickly notice that most of what you thought was frozen water, was actually air. As there is a lot of trapped air in snow, light hitting the iceberg is reflected off the many internal surfaces and then visually appears white.
So, why does this iceberg not look white also?
Blue icebergs are in fact very old; they have been through possibly hundreds of thousands of years of thawing, refreezing and compression. As a result, much of the trapped air has been released, reducing the reflective capabilities of the iceberg. Now that the light is no longer being reflected as efficiently, it is instead being absorbed. The weaker wavelengths of light (towards the red spectrum) are quickly filtered out. The blue spectrum, however, has enough energy to penetrate the ice where it either; finds an internal surface to reflect back from or it manages to penetrate the whole way through; thus giving a blue colour.
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Photo courtesy of Tony Travouillon