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int full_improvement( OBSERVE FAR *obs, int n_obs, double *orbit,
const double epoch, const char *limited_orbit,
const int sigmas_requested, const double epoch2);
Simply put, this function (found in 'orb_func.cpp') does a leastsquares
fit of observations, doing a single iteration. However, as time has gone
by and Find_Orb's capabilities have expanded, this function has _really_
expanded. It's not actually as scary/complicated as it may appear, but the
level of math and jargon is not for the faint of heart.
I won't describe the mathematics of leastsquares fitting; you can get a
good description of that in any introductory orbital mechanics text. But
there are some specific things in 'full_improvement' that need explaining.
In leastsquares fitting an orbit, one has to compute the derivatives
of the residuals relative to changes in the orbit parameters. Usually,
that means six parameters, which can be the state vector elements x, y,
z, vx, vy, vz (which is what Find_Orb uses), or six classical orbital
elements, or classical elements with q and Tp used instead of M and a,
or several variations on this theme.
However, we aren't always trying to fit six parameters. If the object
is small and light enough to be affected by solar radiation pressure, we
may be fitting its area/mass ratio. Or we may need to fit the mass of an
asteroid perturber. In either case, n_params = 7. Comet nongravitational
effects are fitted as two or three parameters, resulting in n_params = 8
or 9.
Occasionally, people will do tricks such as assuming e=1 or e=0 (parabolic
or circular orbit) and fitting only the remaining five orbital elements,
leading to n_params = 5. Find_Orb doesn't do this, mostly because it's
fitting the state vector rather than orbital elements. Find_Orb handles
"constrained" orbits somewhat differently; it just pretends that you have
"observed" that e=0 or e=1 with low error. This has its benefits; you
can tell it that you have simultaneously "observed" that e=.1 and a=2.5,
for example, making for multiple constraints.
The reason the code works in terms of the state vector is to avoid
mathematical singularities. If the leastsquares fit includes the
inclination or eccentricity, for example, you'll get very singular
behavior when those quantities are nearly zero... which is a common
occurrence; when i is near zero, the argument of perihelion and longitude
of the ascending node become illdefined or singular, and when e is near
zero, the mean anomaly, time of perihelion, and argument of perihelion
all become singular. Since there are plenty of objects with e or i near
zero, that's a Very Bad Thing indeed.
Anyway. The code figures out how many parameters will be fitted 
usually six, sometimes more  and initializes the leastsquares fitting
code, telling it how many parameters to expect. The state vectors will be
varied along the six 'unit_vectors'; these could, in theory, be any
six linearly independent vectors; I'll explain the choice later. We also
do a check to make sure that there are at least three observations and that
the orbit isn't totally bizarre, bombing out with a negative return value
if either is the case.
We may be computing sigmas and covariances for a desired epoch 'epoch2'.
The idea is that our observations may cover, say, 1 Mar 2013 to 13 Mar 2013,
and we're therefore working with an orbit of epoch 7 Mar 2013. But we're
really interested in sigmas/covariances for the epoch 19 October 2014,
because that's when the object will be a possible impactor. In this case,
'epoch' would be 7 Mar 2013 and 'epoch2' would be 19 October 2014. Similarly,
the constraints are for 'epoch2'; if we set, say, q=1.2, that would
mean the osculating perihelion distance would be 1.2 AU for 19 October 2014.
I mention all this because if there are no constraints and we aren't
looking for sigmas/covariances, there is no need to compute orbits for
'epoch2'. In that case, the variable 'need_second_epoch' will be zero.
In doing leastsquares fitting, we need to compute the derivatives
of zillions of quantities relative to the six (or more) parameters.
(The 'zillions of quantities' include orbital elements at epoch2 and
the computed RA/decs for each observation.) Computing these derivatives
is a slightly more complex version of the textbook standard methods
df(x)/dx = (f(x + h)  f(x)) / h + O(h)
df(x)/dx = (f(x + h)  f(x  h)) / 2h + O(h ^ 2)
...i.e., "tweak" our nominal orbit x by some small quantity h, and
compare the results to those from the nominal orbit. (The second method
is more accurate, but requires twice as many function evaluations; we
can't just reuse f(x) each time. Find_Orb can do this, if the
'use_symmetric_derivatives' variable is nonzero. But I've never found
that it did anything except slow things down.)
So. For each of the n_param parameters being varied, we take the
current nominal orbit 'orbit' and create 'orbit2', same thing with a
small amount added in the direction of one of the unit vectors. We
then compute RA/decs for each observation, and an orbit for the epoch
'epoch2' if needed, based this slightly tweaked orbit.
The amount of the 'tweak' can be difficult to determine. Too large,
and that O(h) term causes error; too small, and you get roundoff error.
Find_Orb tries to keep the tweak size such that the maximum change in
computed RA/decs (or computed range or rangedot, for radar data) is
1.5 sigmas. (Though a large range can work and will be used.) If the
change is too big or small, then we try again with a suitably
resized tweak.
Eventually, though, we're all done: we've computed RA/decs for
each observation and (usually) orbital elements at 'epoch2', and done
so for the nominal orbit and for six (or more) tweaked orbits. We can
then do a standard, textbook, linear leastsquares improvement.
We may also write out all our sigma/covariance data to the file
'covar.txt'. In addition, the eigennvalues and eigenvectors of the
covariance matrix are computed. I did this for a couple of reasons.
First, as Milani and Gronchi point out in their book _Theory of Orbit
Determination_, the direction of the line of variations is that of
the eigenvector with lowest eigenvalue, with length 1/sqrt( lambda)
where lambda=lowest eigenvalue. This provides a quick and easy way
to compute ephemeris uncertainties and impact probabilities, and for
achieving better linkages between arcs.
But the other advantage I've found is that it turns out that, if
your tweaks are applied in the direction of the eigenvectors, you get
somewhat better stability. So Find_Orb now stores the eigenvectors
in the 'unit_vectors' for use on the next full_improvement.
The reason this helps is that depending on how you parameterize the
orbit, the uncertainty area can look like an ellipsoid, or like a
"bananaoid." In the latter case, finding a minimum point can be a lot
harder. Consider the Rosenbrock "banana" function :
https://en.wikipedia.org/wiki/Rosenbrock_function
f(x, y) = (ax)^2 + b(yx^2)^2
This is sometimes used to test the stability/convergence of
minimization routines. It can be a pretty brutal test. But if,
instead of looking for a minimum x and y, we look for a minimum
in x and z=yx^2, the function becomes
f(x, z) = (ax)^2 + bz^2
Almost any minimization routine, including least squares, will
then converge quickly on x=a, z=0, and thereby y=a^2.
It's also for this reason that the initial unit vectors are set up
using the set_fixed_unit_vectors( ) function. For a short arc, the
uncertainty ellipsoid (and therefore the eigenvectors) will have two
long axes and four shorter ones. The long axes/lowest eigenvalues
will be in the radial velocity and distance direction; that is, we've
a decent idea of the tangential velocity, but the distance and radial
velocity are poorly determined. So we set up two vectors that run
along those directions, with four more somewhat arbitrarily set up
orthogonal to them; it's really getting those two long axes of the
ellipsoid set up that matters most. It appears to really improve the
stability of the initial leastsquares fit.
