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I've added a shapefile and a raster image to ArcGIS, the raster image in geotiff format spreading over some of the features.

Now, I want to create a buffer (of radius 1 mile) around the features and clip them and convert it as a Geotiff image to use it as a raster image.

migrated from stackoverflow.com Sep 25 '13 at 14:14

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You can use the Buffer tool to add buffers to create a new shapefile representing the 1 mile buffers.

Then you can use the Extract by Mask tool to clip the raster inside of the buffer shapefile features. Mind you need a Spatial Analyst extension to use this tool.

I recommend using Python or even ModelBuilder to create a loop that will step through each feature in your buffer shapefile, and clip your raster, resulting in individual rasters for each feature that was buffered. I post another link, but I do not have a high enough reputation. Simply Google "ArcGIS 10.1 ModelBuilder Iterators".

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