Pre-stack depth migration overcomes some of the major issues (okay, approximations!) with time migration.
Time migration doesn't really handle refraction, either within a rock layer (as a result of the seismic velocity gradient increasing through effects like compaction) or at an unconformity, especially with a layer with a much higher seismic velocity than sediment, like salt or basalt.
For this to work, however, you need an accurate velocity model, that has all of the sub-surface variations that you want to take into account included. This model is in depth, and usually a lot more accurate than the veloicities you use for stacking, or pre-stack time migration.
In some ways this means you need to have the velocity model to image the data, but you also need to interpret the data in some way to make the model - as a result, models are often built iteratively, and need to have some kind of sturctural information (ie an interpretation), ideally constrained by well data.
The results are almost always better than time-migrated data, especially when there is complex structure, faulting, or strong lateral and vertical changes in seismic velocity.
Its pretty much essential for areas that have salt domes (Gulf of Mexico, parts of the North Sea, Netherlands and Germany, offshore Brazil for example), or under thick basalt sheets (West of the Shetland Islands)
The need to build a complex model, and run migrations to find out if the model is any good, makes it expensive.
A lot of data is pre-stack time migrated as part of building the velocity model, and then depth imaging run on a subset. In some areas, the improvment from depth imaging doesn't justify the expense on for exploration, but it is applied after discovery (and a hence well data being created) to help define a prospect for production.
The need for in depth geological knowledge and integration with wells, and its use of sensitive projects, means that some oil companies do some or all of this type interpretative processing in house. The data might well prepared - and often prestack time migrated to aid with model building - externally.
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