I have a single band raster, but I want to export it with the same colors as in ArcMap.
How can I do this?
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You can create and save a color map to achieve what you want. Keep in mind that for geotiffs, colormaps are supported only with certain bit depths. ArcGIS Help 10.2 has excellent instructions on creating a color map (.clr) file:
The colors you see in ArcMap are merely a graphical representation of your data. When you load a new data file into ArcMap, it will randomly assign a color scheme to it. If what you want is to load the data with the same color scheme every time, you will want to look into Saving a Layer File. Your other option, if this is not what you are looking to do, is to export the graphical representation of the data as an image (map export).
If you want to export the Image, such that it will be exported with the same render as you have in ArcMap, you need to check the 'Use Renderer' check box on the Export Raster Data dialog. Fore more details, see this Article: Exporting a raster in ArcMap
If the symbology of your source TIFF is defined by colormap, and you extract a sub region of this TIFF, and want to keep the same symbology.
You might want to (go to ArcGIS)
I did such operation using ArcGIS with CDL (crop data layer) dataset.