A chain is no stronger than its weakest link. Economy dictates that all the links should be equally strong. Many broad questions merit study such as the errors associated with velocity uncertainty and with migration after, rather than before, stack. Having read this far, you are now qualified to attack these broad questions. Now we will narrow our focus and examine only the errors in downward continuation that result from familiar data processing approximations.
In the construction of a production program for wave-equation migration, weakness arises from approximations made in many different places. Economy dictates that funds to purchase accuracy should be distributed where they will do the most good. Geophysical contractors naturally become experts on accuracy/cost trade-offs in the migration of stacked data. Contractors will use the equations and program gathered below to obtain best results for the lowest cost. Users of reflection data are interested in learning to recognize imperfect migrations, so they may want to use the program to see the effect of various shortcuts.
In preparation for a big production job there are two general approaches to examining accuracy. The first approach, which gives the best insight into qualitative phenomena, is to make synthetic hyperbolas by the various methods. The synthetic hyperboloids can be be compared to the data at hand with a video movie system or by plotting on transparent paper. In the second approach you compute travel times of hyperboloids or spheroids of waves of different stepouts and frequencies for different mesh sizes, etc. Then an optimization program can be run to minimize the average error over the important range of parameters.
It is not necessary to write
a time domain 45 finite difference migration program
to see what its synthetic hyperboloids would look like.
We can simply express all the formulas in the -domain
and then do an inverse two-dimensional Fourier transformation.
To facilitate comparisons between the many migration methods
we will gather equations from different parts of the book
in the order that they are needed.
Then I'll present the program
that makes the diffraction hyperbolas for many methods.
Using the same equations you can compute travel times and solve
the optimization problem as you wish.