## Monday, 17 September 2007

### Bursa-Wolf transformations explained

.. courtesy of Adrian Custer on the OpenJUMP mailing list:

> > just one question .. what is this bursa wolf parameter option?

> My impression is that this is scary math I never quite understood.

Well, Bursa was a 9 year old bicyclist from the Alps and...no, no, no, i
lie. Actually it's not particularly scary math and quite easy to
understand. All you really need to remember is that no one has ever been
to the center of the earth.

So everyone started surveying (mostly so the repressive central
governments could exploit taxes from people and have lots of jolly wars
where people could slog through the mud and kill each other so they'd be
blood and suffering for all). Each group started from some random place
on the surface of the earth. Right away, it becomes obvious to everyone
that euclidean rules don't work so well. Some didn't care so much since
taxes are basically arbitrary anyway and getting serious about it means
you'd have to walk through fields and woods and get lots of mud on your
shoes. Others kept at it and resorted to spherical geometry. Once you
start doing that precisely and at continental scales you realize that
doesn't really work either so you decide to try the next hardest thing,
an ellipsoid of rotation. Now how do you know which one to choose? Well
you pick one that minimizes your squared errors. All good and nice but
(1) you are surveying the ground which is anything but an ellipsoid
since it has all those ditches you keep falling into and that keep
getting your clothes covered in mud and (2) you are not perfect
especially with all that mud on your paper. So you have a bunch of
errors. Well everyone that does this comes up with lots of different
ellipsoids that work really nice for their data and everyone is sure
they clearly have found the 'one true ellipsoid' and they decide to use
that for all their work. Then everyone guesses where they actually are
on each of their particular ellipsoids which involves lots of going
outside at night and looking up from the mud at the stars. But then it's
not like the edges of each survey was nice and level on these ellipsoids
either --- think of the eastern USA. You can start nice and clean and
warm and dry at an inn in Boston on the edge of the sea drinking clam
chowder and having a good time but a few months later it will be bitter,
bitter cold in that tiny town of Denver because you are somewhere like a
mile high up in the air and you're wet and covered in mud from slogging
through the plains in a snowstorm. So you've got a pretty good idea that
your data is on a major slant but, well, you'll do your best to make up
for it but it really doesn't help the effort any, especially what with
all that mud that's still itching in your hair. So your errors may be a
wee bit big but hey it's all right: it's good enough to wage lots of
good wars with lots of mud and blood and to keep collecting lots of
taxes so no one cares too much.

Fast forward to more recent times where some people want to talk to lots
of different governments and work with lots of different data. They take
everyone's guess and try to line them up. Well it turns out, when you
try to line everything up, that the center points of all the different
ellipses aren't really the same points and even the orientation of the
three axes are all a bit off because of how everyone guessed where their
were on their ellipsoids. So now, to go from one data set to another so
they line up "the best," you need estimates of how much to rotate each
of the axes and how to shift the center point around; all this beyond
even the obvious stuff of changing between the different definition of
all those "one true" ellipsoids.

When you do this mathematically, you need a bunch of parameters: these
now have the names of the wolf and the bursa. Generally, you can only
come up with good parameters if you have lots of data to compare and
some good software to do the comparing. That's what the EPSG did for
everyone. The guys in the pickup trucks that went out looking for oil
kept falling into ditches along the way and getting mud on their faces
but when they got back to the office they had a good sense of what lined
up with what and could say: "yep, that hill there is the same as this
squiggle here and there's this big ditch right here that cost us our
third flat tire and..." So they collected as much data as they could and
compared it and came up with a database of parameters by which you go
from one data set to another. So that's it. That's why we use their
data; we don't have to fall in any ditches and can avoid getting mud on
our clothes. They give us their parameters and we can mostly line up
data from one survey against data from another. But you do need some
good parameters because the earlier folk had a harder time of the mud
and the data they created don't just line up the way we would like them
to.

Actually doing the math is a bit harder but the concept is pretty
straight forward: geographic data all ultimately gets tied into points
on the earth surface and that requires estimating where the points
really are and how they line up on the estimated ellipsoid being used.
That in turn means none of ellipsoids quite line up and we need
parameters to move between them.