Comparison of jointcal and meas_mosaic terms
These notes are in reference to a chat I had with Naoki Yasuda of HSC, where we attempted to sort out the notational differences between jointcal and his documents describing meas_mosaic. I’ve attached said documents below, for reference.
jointcal.pdf (212.0 KB)
meas_mosaic.pdf (127.6 KB)
mosaiceq.pdf (84.0 KB)
The basics
A star s in exposure e is measured at position $x^{s,e},y^{s,e}$ (MeasuredStar $X_{\gamma,i}$) on CCD c, which corresponds to position $\alpha^s,\delta^s$ on the sky (FittedStar $F_k$), and which may be associated with a star in a reference catalog (RefStar $R_j$ with associated FittedStar $F_j$).
meas_mosiac
- Stars with no reference catalog object are internal stars, while those with reference objects are external stars.
- Sky positions $\alpha^s,\delta^s$ are mapped onto a tangent plane $\xi^{s,e},\eta^{s,e}$, taking into account the boresight $A^e,D^e$.
- The CCD positions $X_c,Y_c$ and rotations $\theta_c$ are explicitly fit parameters.
- The mapping is an nth order polynomial, with kth polynomial coefficients $a_k^e,b_k^e$ for the $\xi,\eta$ parts, respectively.
- The fit is globally for the entire focal plane.
jointcal
- A
CcdImage$\gamma$ is the catalog, initial WCS, telescope pointing, etc. from 1 CCD c and exposure e. - $M_\gamma$ maps from Pixel to Tangent plane: $x,y\stackrel{M}{\rightarrow}\xi,\eta$.
- $P_\gamma$ maps from sky to tangent plane (generally taken to be a gnomonic projection): $\alpha,\delta\stackrel{P}{\rightarrow}\xi,\eta$.
- The weight $W_{\gamma,i}$ is the measurement weight of $M_\gamma(X_{\gamma,i})$.
-
RefStars$R_j$ and their associatedFittedStars$F_j$ are mapped to the tangent plane via projection $P$ (explicitly gnomonic projection), with weight $W_j$. - CCD position/rotation are subsumed into the fitted mapping, and not dealt with explicitly.
- Fits are per-CCD per-visit, not globally over the focal plane, but a full optical distortion could be included in the $P_\gamma$ term.
Parameters
Fitted
- Mapping from Pixel to Tangent plane: if mapping $M_\gamma$ is a polynomial, it’s coefficients should be similar to $a_k^e,b_k^e$.
- $X_c,Y_c$ and $\theta_c$ are only explicitly fit in meas_mosaic: they are part of $M_\gamma$ in jointcal.
- Star positions on the sky: $F_k$ == $\alpha^s,\delta^s$ for internal stars.
Fixed
- CCD positions: $X_{\gamma,i} == x^{s,e},y^{s,e}$
- Reference star positions on the sky: $F_j = R_j == \alpha^s,\delta^s$ for external stars.
- Projected coordinates: $\alpha,\delta\stackrel{P}{\rightarrow}\xi,\eta$ for both $P$ (reference terms) and $P_\gamma$ (measured terms).
Thoughts
Though explicitly fitting the CCD positions and rotations ($X_c,Y_c,\theta_c$) and the telescope boresight ($A^e,D^e$) is a nice feature of meas_mosaic, it would probably be best to include them in the “known” set of Transforms that are corrected for before we solve for the “perturbation”, since they shouldn’t change from one exposure to the next.
Meas_mosiac Taylor expands $a_k^e,b_k^e$ series and fits the full $\chi^2$ matrix, while jointcal Taylor expands and fits the second derivative matrix $d^2\chi^2/d\theta^2$, which is much sparser.