Air

Compressing the air is about increasing the air pressure inside our body. There are several ways to do this. The problem is that some ways tend to restrict the air passage. By being relaxed at the same time as we create pressure, most of this air pressure can reach the embouchure without hindrance.

More about the use of air can be found in books by : Arnold Jacobs, Claude Gordon, Donald Reinhart : Encylopedia of the pivot system

Compression and relaxation

For the higher register, some more pressure is necessary. When coughing, there is a movement of the stomach inwards. By exploiting this movement in the playing, the required air compression can be created.

It seems especially important to relax the upper part of the body. When playing, one should feel the compression in the stomach area, and possibly in the lower back and the buttocks. By concentrating on the lower part of the body, the upper body have a greater chance of remaining relaxed.
Pushing out the stomach while exhaling, is a long taught technique. Unfortunately this is not very effective, can be painful, and it can cause hernia. Therefore, this should be avoided.

If trying too hard, the throat will close. The result is the Valsalsa maneuver, the natural body reaction which comes form fear, body waste and childbirth.
Very many players are plagued with this problem. Especially when approaching the upper register.
The difficult part is generating the maximum amount of air pressure possible and still being relaxed. If the throat is relaxed and the embouchure closed, it will expand when playing. This will happen, even at very little air pressure. A good test is playing low notes at pianissimo to verify a relaxed, open throat, and looking in the mirror to see the throat expanding.
The position of the head does have an impact on tension in the throat. By moving the head forward, the Adams apple will fall down, leaving the throat open.
One exercise which has proven successful for some, is singing a low pitch while playing ascending scales. This secures the Adams apple at the lowest possible position, leaving a very open throat.
With the tongue too far back in the mouth, there is a danger of the tongue closing the throat. The best exercises I have found, are breath attacks. By not using the tongue, I become more aware of when this problem occur. It also has a positive effect on the embouchure, as it "remembers" pitches better, without interference from the tongued attack. When breath attacks work well, I later add the tongue. Exercising this way seem to open up the throat. 

Air intake

If one has a problem of learning "timed breathing", it is possibly one's way of playing or one's equipment which is to blame.
If one plays with a too open aperture, the tendency will be to always run out of air. This player will usually overbreathe, becuase he can never get enough air to finish the phrases. Playing a horn too big for the individual, migh result in the same problem.
If the player has the opposite problem of too little air flow, tension will manifest itself in an exhale required before starting each new phrase. The reason for the problem might simply be that the player is overbreathing. But another possible cause is that the combination of embouchure and equipment is too tight for that person.

Too much air can tense up the body, often referred to as overbreathing. Also, it can make the player dizzy, possibly with the result of falling into the floor.

Shallow breaths are no good either, resulting in too little compression. The player will also have problems with long phrasings.

The instinct while playing high, is to take in as much air as possible. Unfortunately, this results in overbreathing. High notes takes very high compression, and very little air volume is necessary. To find out what is required, try to take in different amounts of air before playing a high C at forte. 

Yoga breathing

The essence of yoga breathing is filling up with air from the bottom of the lungs, starting with the abdominal area and expanding upwards.
Exhaling is done in the exact reverse order, emptying from the top and downwards.

While many breathing methods concentrate on the intake, yoga breath concentrates on the exhale.

TIP: Learning breathing, empty yourself completely before inhaling. 

Shew breathing technique

1. Intake (small), abdomen moves outwards slightly, but relaxed
2. Intake (large), abdomen moves inwards (horizontally) to create wedge position (still relaxed.)
3. Intake (large), abdomen holds position (not tense) shoulders lift straight up.
4. Grip (isometrically) abdominal muscles, maintaining innermost position (lock wedge tension.)
5. Relax and lower shoulders to comfortable playing position
6. Blow (as if spitting rice.)

These steps should be executed in a flowing manner. Learning is most effective starting slowly and increasing speed as feels natural.

Shew is inspired by sports research. By repeating a muscular pattern many times every day for at least 3 weeks, this pattern will turn into a reflex. Shew recommends that the yoga breathing is repeated 60 times each day for 21 days. If one day is skipped, the body will forget, so one has to start over again at day one.

The Norwegian Trumpet Forum Bobby Shew Clinic.

I use a combination of yoga breathing and Shew's wedge. I only use the wedge for the high register and try never to rise the shoulders.