Why so many AI Sires?
Why is a great number of AI sires in an Icelandic sheep's geneology so desireable?   
The reason for using an AI ram in the first place is to improve your flock. When an AI
ram is used on a ewe, he contributes 50% of the genetic material that makes up the
lamb, and the other half of the genetic material donated by the ewe is usually of
(assumably) inferior quality. But in actuality, it will
not be that half of the genetics in
the pairing are good and half are less good,
MORE THAN HALF of the DNA strand and
the chromosome strands could use improvement; because
some of the genetic
material contributed by the AI ram will be less than optimal as well, so with about
three billion bits of information on the DNA strand, and not all of it having been
improved by using the AI ram; some of the less desireable qualities of BOTH ewe
and ram are still present and will reassert themselves at some point in future
generations.
What many generations of AI sires accomplishes, is that each of the excellent AI
sires in a lambs family tree helps to replace an additional little bit of DNA that is
inferior. With proper culling practices and wise breeding choices on the part of the
shepherd, many excellent AI rams in a lambs geneology potentially means that
much of the inferior genetics have been replaced with genetics of higher quality.
Looking at the situation with a little more detail; each chromosome has many
locuses, and each locus has a number of alleles, of which each allele lends its own
particular bit of influence to the physical attribute which that particular locus is
controlling. Some of the alleles in any given locus are going to be more excellent
than the other alleles in that particular group. Many AI sires adding their genetics
over a number of generations gives the opportunity for the alert shepherd to replace
inferior alleles in any given locus of each individual chromosome, so the ability to
sustain excellence is reinforced because the alleles themselves are of a more
consistent quality.
 But it takes a continual influx of superior genetics over many generations to
achieve that level of consistency.  The greater the number of AI sires in a lambs
background, the higher the likelihood of it being an excellent animal
itself, while
also retaining a greater ability to reliably pass on it's excellence to future
generations.
In an animal with a dozen or more AI ancestors in it's family tree, by the time the
most recent lamb gets to you, many of the inferior alleles will have been replaced
with better genetics, and when paired with a
comparable mate, will be more reliable
in it's ability to pass on it's excellence to the next generation.
 All that hard work and planning on your part, however,
can be UNDONE by the
injection of inferior genes by mating with an inferior animal; so, to MAINTAIN
CONSISTENCY, you would do well to strive for a high level of excellence in your
whole herd, and CULL those animals who display inferior attributes as soon as
possible! If you do not cull in order to raise the general quality of the animals that
remain, as well as continually introduce superior genetics into the flock, you will
gradually
go backward in the level of quality of your herd.
 The advice of George Washington, our nation's first President (who was also a
sheep breeder) is a useful rabbit trail to go down at this point. He said (in essence)
"The quickest way to most profoundly improve the quality of your whole flock is to
buy a superb ram!"
 Obviously, a superb
ewe can only contribute one to three excellent lambs per year,
while a superb ram may potentially produce two or three
hundred!  But you will go
even further and faster if you possess not only a superb ram, but have developed
many superb ewes to breed him to!
  Interestingly enough, I find myself in an unusual situation; as in, some of my ewes
are so excellent, I really have to be careful as to whom they are bred to, lest I dilute
or corrupt their superb characteristics with an inferior ram! Tell me , if you would,
who would
you breed with the "Tuligirl?"