Right at the start, drop down to the water and follow a small pathway east from the elevation you started in. Enter the small alcove and step onto the spot containing the double-barreled shotgun to register the secret.
After going down the first lift, you will see four rooms numbered 1 to 4 (the numbers are engraved in red on the ground right in front of their respective doors). The wall with a lightning bolt symbol next to the first door can be opened to obtain a megaarmor.
At the very end of the map, walk along the thin white ledge "framing" around the exit portal to find a soulsphere right behind it.