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I have an x (easting), y (northing) and z (height) values associated with my point shapefile. I would like to calculate the z values from the DTM I have - how do I do this in QGIS? I am fairly new to QGIS. I can't do it in ArcGIS as don't have the extensions.

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You may use the Point sampling tool, see links below.

http://pvanb.wordpress.com/2010/02/15/sampling-raster-values-at-point-locations-in-qgis/

http://www.digital-geography.com/qgis-plugins-point-sampling-tool/

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