I'm not trying to be offensive; but, have you ever been in the desert for more than a few hours? The axe is unnecessary weight, and how much water does a canteen hold? 1qt, 2qts? That's not enough water to last half a day if you're doing more than just casual walking, most times of the year.
If you're digging, cutting brush, building shelter, constantly changing elevation, walking through huge spots of soft sand, or any of the other activities that would be considered normal in a lot of desert environments you're going to need a lot of water. What is a lot, and how do you plan on carrying it? When I was in the Army, here's what I carried in the desert:
2, 2qt canteens
2, 1qt canteens
3 liter water bladder
That's about 1.5 gallons of water, and it was not enough to last the day. We always had a water buffalo that was accessible at least once a day, and usually carried several 5 gallon jerry cans of water in the vehicle.
Besides water, the other useful items were food, poncho, 550 cord, cravats, fleece to keep nights semi-comfortable, lighter/matches/fire starting stuff, machete, 3-4" fixed blade knife, swiss-army knife, and e-tool.
If I wasn't digging fox holes and fighting positions, I would loose the e-tool. You can chop everything with a machete, and do some digging if you have to.
The desert is strange; you need to keep cool and protect yourself from the sun/heat/dehydration/wind-wind is underestimated in the desert, and at night keep warm.