Merging tracks.
David Adams
04apr02 1010 EST

We consider the problem of interpolating track fits to obtain an optimal
estimate between measuring surfaces. Extrapolation to surfaces before the
first measurement or after the last is trivially handled by direct
propagation.

Notation
--------

True value for both tracks is y0.

Tracks are y1 and y2
Expectation value of each is the true value:
  <y1_i> = <y0_i>
Denote difference by:
  x1 = y1 - y0
  <x1> = 0
Errors are given by:
  C1 = <(y1-y0)x(y1-y0)>
  C1_ij = <(y1_i-y0_i)(y1_j-y0_j)> = <x1_i x1_j>
  C12_ij = <x1_i x2_j> = 0 (Tracks are independent).

Uncorrelated tracks
-------------------

If the measurements y1 and y2 are uncorrelated, we find the best estimate for
the combined estimate y by minimizing the chi-square:
  CHSQ = (y_i - y1_i) W1_ij (y_j - y1_j) + (y_i - y2_i) W2_ij (y_j - y2_j)
  where W1 = inverse(C1)

Minimize.
  0 = dCHSQ/dy_i
    = 2 W1_ij (y_j - y1_j) + 2 W2_ij (y_j - y2_j)
  (W1_ij + W2_ij) y_j = W1_ij y1_j + W2_ij y2_j
  y_j = inv(W1 +W2)_ij (W1_ij y1_j + W2_ij y2_j)
  y = inv(W1 + W2) (W1 y1 + W2 y2)

Calculate the expectation value of this estimate.
  <y> = inv(W1 + W2) (W1 y0 + W2 y0)
      = inv(W1 + W2) (W1 + W2) y0
      = y0         Good--estimate is unbiased

Calculate the error matrix for the estimate.
  C_ij = <(y-y0)_i (y-y0)_j>
       = <inv(W1+W2)_ik (W1_kl x1_l + W2_kl x2_l) 
          inv(W1+W2)_jm (W1_mn x1_n + W2_mn x2_n)>
       = inv(W1+W2)_ik inv(W1+W2)_jm
         [W1_kl W1_mn <x1_l x1_n> + W2_kl W2_mn <x2_l x2_n>]
       = inv(W1+W2)_ik inv(W1+W2)_jm
         [W1_kl W1_mn C1_ln + W2_kl W2_mn C2_ln]
  C = inv(W1+W2) [W1 C1 W1 + W2 C2 W2] inv(W1+W2)
    = inv(W1+W2) [W1 + W2] inv(W1+W2)
    = inv(W1+W2)

These formulae are applicable for combining two Kalman tracks where one has
been fit forward using all preceding measurements and the other has been
fit backward using all following measurements. Both are propagated to the
intermediate surface before applying the above formulae.

Correlated tracks
-----------------

We are often interested in the case where the measurements are correlated.

Assume measurements may be expressed in terms of a common average y0+d and
independent fluctuations q1 and q2:
  x1 = d + q1
  x2 = d + q2
  C1_ij = <x1_i x1_j>
        = <d_i d_j> + <q1_i q1_j>
        = D_ij + Q1_ij
  C1 = D + Q1
  C2 = D + Q2
  <x1_i x2_j> = D_ij

Estimate chi-square using the uncorrelated pieces:
  CHSQ = (y_i - y1_i) W1_ij (y_j - y1_j) + (y_i - y2_i) W2_ij (y_j - y2_j)
  where now W1 = inv(Q1)

Minimization gives the same estimate as above (with a new meaning for W):
  y = inv(W1 + W2) (W1 y1 + W2 y2)
    = inv[inv(Q1) + inv(Q2)] [inv(Q1) y1 + inv(Q2) y2]

Calculate the error matrix for the estimate:
  C_ij = <(y-y0)_i (y-y0)_j>
       = <inv(W1+W2)_ik (W1_kl x1_l + W2_kl x2_l) 
          inv(W1+W2)_jm (W1_mn x1_n + W2_mn x2_n)>
       = inv(W1+W2)_ik inv(W1+W2)_jm
         [W1_kl W1_mn <x1_l x1_n> + W2_kl W2_mn <x2_l x2_n>
          W1_kl W2_mn <x1_l x2_n> + W2_kl W1_mn <x1_n x2_l>]
       = inv(W1+W2)_ik inv(W1+W2)_jm
         [W1_kl W1_mn C1_ln + W2_kl W2_mn C2_ln +
          W1_kl W2_mn D_ln + W2_kl W1_mn D_nl]
  C = inv(W1+W2) [W1 C1 W1 + W2 C2 W2 + W1 D W2 + W2 D W1] inv(W1+W2)
    = inv(W1+W2) [W1 (D+Q1) W1 + W2 (D+Q2) W2 + W1 D W2 + W2 D W1] inv(W1+W2)
    = inv(W1+W2) [(W1+W2) D (W1+W2) + W1 Q1 W1 + W2 Q2 W2] inv(W1+W2)
    = D + inv(W1+W2) [W1 Q1 W1 + W2 Q2 W2] inv(W1+W2)
    = D + inv(W1+W2) [W1 + W2] inv(W1+W2)
    = D + inv(W1+W2)
    = D + inv[inv(Q1) + inv(Q2)]

These result are applicable for estimating the parameters for track at a
surface between two measuring surfaces. Optimal (filtered and smoothed)
tracks from each measuring surface are each propagated to the surface of
interest. The propagation from surface a to b is described by:
  xb = f(xa)
  Cb = Fab Ca FabT + Qab
where Fab is the derivative matrix (Jacobian) and Qab is the
propagation-induced error typically due to multiple scattering.

We use Qab to approximate Q1 and Q2 in the earlier expressions.
We can estimate D from C1-Q1, C2-Q2 or an average of these.