You will need an ArcGIS Organizational account to extract the values from Esri's Elevation service, since Esri's Terrain Image layer is a Subscriber content.
1. In ArcMap, sign-in to ArcGIS Online using your organizational account.
2. In ArcCatalog, GIS Servers > Add ArcGIS Server > Use GIS Services
Server URL: https://elevation.arcgis.com/arcgis/rest/services
3. Expand the Server connection, and add the Terrain Image Service to Table of Content of ArcMap. You may need to provide the login credentials again.
4. You can use this image layer Extract Values to Points Geoprocessing Tool.
Input
Output
What is subscriber content?
Subscriber content is the collection of ready-to-use map layers, analytic tools, and services published by Esri that requires an ArcGIS Online organizational subscription account to access. This includes layers from Esri such as NAIP imagery, landscape analysis layers, and historical maps. Subscriber content does not consume ArcGIS Online credits.
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