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When adding an 'Undefined' raster to ArcMap and defining the projection using ArcToolbox > Data Management Tools > Projections and Transformations > Define Projection, the following error message occurs:
"Failed to alter spatial reference.
The dataset already has a projection defined.
Failed to execute."
The .aux file is created when adding the image to ArcMap. The .aux file stores the coordinate system of the image and ArcMap reads 'Undefined' as an actual coordinate system.
The solution is to define the projection by using the appropriate coordinate system, as follows.