The chests in the picture are labelled by number. The number related to the location number here.
Location 1 (White Chest)
Just east of the transition point
. The chest is on top one of the huts. Jump onto the three white crates, then to the chest.
Location 2 (White Chest)
This chest is in the Hyperion encampment where you encounter the Hyperion 'Super-Bad' Soldier. It is in the northeast corner of the encampment. You may need to crawl through an opening in the wall to reach this chest.
Location 3 (Red Chest)
This chest is southeast of the Hyperion camp and past the clap trap spawns. It is inside a half-house like structure.
Location 4 (White Chest)
This chest is right before the large cylinder structure to the south of the Hyperion camp. It is located to your right when looking at the structure from its north side.
Location 5 (Red Chest)
Just west of the previous chest, there is a ramp leading up to a giant Hyperion sign. Shoot the part of the ramp that is raised to make it fall, allowing you to reach the chest.
Location 6 (White Chest)
In front of the previous chest, there is a gap in the metal ramp. Drop through into the fenced off area. The chest is in the south end of this area.
Location 7 (Red Chest)
West of the previous chest, there are more bandits. The chest is behind one of the houses, near where you get one of the tires for Marcus in Burnin' Rubber.
Location 8 (White Chest)
There is a ramp leading up to a broken clap trap, where you collect one of the components for Tannis in "Are You From These Parts?". The chest is at the top.
Location 9 (Red Chest)
Heading south, the chest is just before going into the next area with bandits. It is behind the first ramp.
Location 10 (Red Chest)
East of the bandit camp in the south area of the Hyperion Dump, there are clap traps and burning tires. The chest is on top of a house just before the tires. Jump on the rock surface nearby and onto the roof.
Location 11 (Red Chest)
Heading west from the bandit camp, there are Skags in a small valley. The chest is by one of the dens.
Location 12 (Red Chest)
The chest is on top of another house, in a raised area with more bandits.
Location 13 (White Chest)
To the east, there will be another cylinder structure with more bandits. The chest is located at the top. Use the walkway to get to the top.
Location 14 (Red Chest)
Northeast of the entrance from Tartarus Station and west-northwest of the first white chest (Location 1) on top of a building, easily spotted.