# III. 2MASS Details

## 3. Data Processing

### e. Position Reconstruction

#### Position Reconstruction Technique

Table 3: Formulation of Least-Squares Problem

The total sum "*S*" of weighted differences squared can be broken down
into three components:

(a) frame-to-frame differences

where
*n*_{f} = number of frames in scan

*n*_{kj} = number of points common to frames *k* and
*j*

*We*_{kji} = inverse variance weighting for
*i*^{th} common point between frames *k* and
*j*

*USx*_{ki} = *U*-scan *x* coordinate of frame
*k* source associated with point *i* of the *kj* set of
common points

*USx*_{ji} = *U*-scan *x* coordinate of frame
*j* source associated with point *i* of the *kj* set of
common points

*USy*_{ki} = *U*-scan *y* coordinate of frame
*k* source associated with point *i* of the *kj* set of
common points

*USy*_{ji} = *U*-scan *y* coordinate of frame
*j* source associated with point *i* of the *kj* set of
common points;

(b) frame-to-reference differences

where
*n*_{f} = number of frames in scan

*r*_{k} = number of reference stars matched to frame *k*

*Wr*_{ki} = inverse variance weighting for *i*^{th}
reference match to a frame *k* source

*USx*_{ki} = *U*-scan *x* coordinate of frame
*k* source associated with *i*^{th} reference match within
frame

*USy*_{ki} = *U*-scan *y* coordinate of frame
*k* source associated with *i*^{th} reference match within
frame

*USxR*_{ki} = *U*-scan *x* coordinate of
*i*^{th} reference star matched within frame *k*

*USyR*_{ki} = *U*-scan *y* coordinate of
*i*^{th} reference star matched within frame *k*;

and,

(c) relative *a priori* frame-to-frame differences

where
*n*_{f} = number of frames in scan (only J2 frames used)

*n*_{p} = number of artificial pegs in each frame-to-frame overlap

*W*_{ap} = arbitrarily small weighting factor

*USxP*_{ki} = *U*-scan *x* coordinate of peg from frame
*k* associated with *i*^{th} common point of the
*(k*-1*, k)* set

*USyP*_{ki} = *U*-scan *y* coordinate of peg from frame
*k* associated with *i*^{th} common point of the
*(k*-1*, k)* set

*USxP*_{(k}_{-1)i} = *U*-scan *x*
coordinate of peg from frame *k*-1 associated with *i*^{th}
common point of the *(k*-1*, k)* set

*USyP*_{(k}_{-1)i} = *U*-scan *y*
coordinate of peg from frame *k*-1 associated with *i*^{th}
common point of the *(k*-1*, k)* set.

The third set of differences is weighted very lightly
and is intended to bridge gaps. Three artificial stars or "pegs" are added
between each J2 frame and the next. The "pegs" are placed in a triangular
pattern within the *a priori* frame-to-frame overlaps, with one set
between J2 frames 1 and 2, a second set between frames 2 and 3 and so on.
With all frames in their *a priori* positions, these differences are
all zero.

In order to formulate the required set of these simultaneous equations,

where
*S*_{FF} = frame-to-frame sums

*S*_{FR} = frame-to-reference sums

*S*_{AP} = *a priori* J2-frame-to-J2-frame sums

*P*_{i} = parameter *i*

*n*_{P} = total number of parameters,

partial derivatives of *S* with respect to each of the 563 adjustment
parameters are set to zero.

Return to Section IV.6.