This question specifically relates to Windows 10 and not earlier versions of Windows, through which I have found the same problem, but without any solutions.
The Problem: The System Image Backup fails as follows:
1. When using the Backup and Restore (Windows 7) from within the Control Panel and using a USB external drive, with which I successfully backed up on 7/30/2015, shortly after installing Windows 10, I receive a backup error stating "The system cannot find the file specified (0x80070002). I am set up to include the System Image Backup as part of the backup.
2. When attempting to only do the System Backup and Restore I have seen two pop-up error windows. The first showed 0x807800C5, failure preparing backup image, and 0x80780081, backup disk cannot be found.
3. After starting the Volume Shadow Copy, the first error remains at 0x807800C5 and the 2nd error changed to 0x8078004F.
4. After each failure, I have found an error message in the log file on the backup drive stating: "Backup of volume \\?\GLOBALROOT\Device\HarddiskVolume2\ has failed. The mounted backup volume is inaccessible. Please retry the operation." These error log messages are under the directory "Windows ImageBackup\PC-Name\Logs".
5. As it turns out the specified volume, in #4, is the UEFI System Boot sector on the hard drive. I found this out by using DISKPART from within an Administrative Command Window.
After trying several of the solutions from earlier versions of Windows (mostly Windows 7), I have been unable to get the System Image Backup to work in any form and would greatly appreciate a solution to the problem.