If you add the ImageryDEM layer file to you TOC, and enter the properties of the service, you will find this.
The highlighted part shows the REST URL to the service. If go to the URL:
Folder: Imagery
... you will see that you loaded the MapService. What you want to use is this ImageService:
Imagery/ChathamDEM_2009 (ImageServer) ... which references the same data but now as an image service. If you connect to the server as a user like this:
You can add the image service "ChathamDEM_2009" found in the folder Imagery. You can see it has a different icon than the ElevationDEM you used:
This layer can be used for calculations. For instance after setting the proper extent and resolution, you can apply a simple calculation with the Raster Calculation:
Which will create a local raster dataset you can use for anything you want.
Do check if there are any restrictions that apply to the usage of the information.