The idea behind a contingency anchor is to provide a releasable system in case someone gets stuck on the rope during rappel. When rappelling in a waterfall, landing in current or in a hydraulic at the base of a waterfall, it is always advisable to use a releasable system.
The photo illustrates a contingency anchor using a munter-mule combination. The munter is the hitch that is tied onto the carabiner. The mule is the hitch tied onto the rope just below the munter. In an emergency, the mule can be untied like the bow on a shoe lace. Once the mule is released, it is important to maintain control of the rope so you can lower the person to the bottom. The munter provides the friction necessary for a controled lower.
Once everyone else is down, the last person unties the munter-mule, retrieves the carabiner and rappels on both strands or sets up a block and raps on one strand.
IMPORTANT:
Note how the rope is passed through the rappel ring. This is done before tying the munter-mule, but only so the rope is pre-threaded and ready to go for the last person down. Once the rope is threaded through the ring, make sure you tie the munter-mule on the rappel strand of the rope. Focus on the two strands of rope coming out of the bottom of the carabiner -- the rappel strand and the release strand. Ignore the strand coming out the other side of the rappel ring.
Safety Knot
The loop created by the mule hitch should be secured to avoid accidental release. One simple method is to tie an overhand knot onto the rappel strand of rope using the loop. In an emergency, untie the overhand, then release the mule hitch.
Safety Carabiner
Another method of securing the loop is to use a "safety biner". Clip a carabiner onto the anchor, then clip the loop into the carabiner. It is not necessary to lock the carabiner. In an emergency, remove the loop from the carabiner, then release the mule hitch.
Two Half Hitches
Some people say that using a mule hitch to back up a munter might create a problem. When the rappel rope is weighted, the loop of the mule hitch might roll as it is released. If it does, it might create a tight twist that cannot be pulled through.
I have never had any difficulty releasing a mule, so I suspect the problem might have something to do with how the sheath of the rope is braided. Some ropes may be more prone to this problem than others.
If you discover that you have difficulty releasing a mule hitch with your rope, you can use two half hitches instead as illustrated in the photo. While one mule hitch is secure, one half hitch is not. You must use two.