I tried using Extract Multi Values to Points in ArcGIS Pro with a query layer for the points layer and a local geotiff for the raster. I received the error "ERROR 010560: Cannot create table or feature class". When I used the same query layer and raster with Extract Values to Points to create a new layer the tool completed successfully.

Can Extract Multi Values to Points ever be used with a query layer (pending one having the proper write permissions to the database underlying the query layer), or does it only work with local feature classes and shapefiles?

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No, you cannot update the values in a query layer, nor add writable fields to a query layer. (In general... technically this is not always true, but for my understanding of this case, it is true.) The documentation for ArcGIS Query layers states, "Query layers are read only when accessed directly from a database connection."

The main difference between the 'Multi' and non-Multi 'Extract' to Points tools is that the (single) 'Extract Values to Points' tool writes its output to a new output feature class, but the 'Extract Multi Values to Points' tool updates the input feature layer (adding field(s) to it and writing the values to those field(s)). It even displays the warning "This too modifies the input data" at the top of the tool pane.

Query layers cannot be modified, therefore they cannot work with the 'Multi' tool. You could use the underlying feature class (that the query layer uses as its spatial component) as the input, rather than the query layer itself.

Query layers are simply a read-only (spatial) database view. They are not like a feature layer, which has a simple feature class as its data source.

Note that "proper write permissions to the database" are not the real issue here. The issue is that query layers are not writable. You could use a feature layer (feature class) which is stored in the same database, and then, so long as you had write permission to that feature class, it should work.

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