In this video guide I will show you how to get Davis's memory echos. Now this may be at a slightly different time than me, because Davis might offer the option earlier to search the Depths. I got the option after beating the first three bosses, then talked to Davis. Just be sure to talk to Davis often to see if he offers the option sooner.
When exploring the Depths you will only be limited to a portion of the map for now. But you will still be able to find the memory fragment needed for the first part, and the achievement. The memory will be called
Hermes Vestige 1. Once you pick it up head back to base and talk to Io to view it. Once the memory is done the achievement will pop.