I've seen raster attribute tables mentioned in several places here at GIS.SE and at ESRI websites.

However, I don't understand what is a raster attribute table (RAT). The resources I found so far mostly explains how to build/delete a RAT and errors associated in the process, rather than what it is and what it's used for.

What I've figured out so far, is that RAT is only defined if the raster in question is categorical (in other words integer valued).

What is a raster attribute table? Can it have arbitrary number of columns and as complex as vector attribute tables?

How is RAT different from the attribute table of a vector dataset, and what is RAT typically used for?

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I think Categorical != integer. But anyway a RAT ( which used to be called a VAT- value attribute table in previous ArcView/ArcInfo versions), gives you the vectorized form of the raster data. So instead of the column's and rows (matrix form), you get the data frame, where a Value column lists all distinct values in the raster and the count column gives you how many pixels/cells are available for each unique value.

1) yes you can add as many user defined columns as you want on a RAT

2) It is handy for quick area calculations. All you have to do is multiply the count of each value by cellsize^2 to get an area for any of your raster values. One of the most important RAT functions to me, is --> use the RAT table to join on Value with another table (that has a similar 'value' column' )and then use Lookup to create a completely different raster from one of the new columns from the table you joined using the value column

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