Skip to content
Permalink
Browse files

Merge pull request #4 from lsst-dm/tickets/DM-14574

DM-14574 add magnitude photometry section
  • Loading branch information
parejkoj committed Jul 23, 2018
2 parents 393a702 + 4503a27 commit a0d78cafca38ad13052c49ebb580e6f34a6868cc
Showing with 256 additions and 64 deletions.
  1. +186 −20 DMTN-036.lyx
  2. +70 −23 DMTN-036.tex
  3. +0 −21 jointcal.bib
@@ -1,5 +1,5 @@
#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
@@ -31,6 +31,8 @@
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
@@ -70,6 +72,7 @@
\suppress_date false
\justification true
\use_refstyle 0
\use_minted 0
\index Index
\shortcut idx
\color #008000
@@ -82,7 +85,10 @@
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
@@ -211,6 +217,7 @@ sky
\begin_inset CommandInset citation
LatexCommand citep
key "2016A&A...595A...2G"
literal "true"

\end_inset

@@ -223,6 +230,7 @@ SCAMP
\begin_inset CommandInset citation
LatexCommand citep
key "2006ASPC..351..112B"
literal "true"

\end_inset

@@ -293,6 +301,7 @@ Photographic Plates
\begin_inset CommandInset citation
LatexCommand citep
key "1960AN....285..233E"
literal "true"

\end_inset

@@ -304,6 +313,7 @@ SDSS übercal
\begin_inset CommandInset citation
LatexCommand citep
key "2008ApJ...674.1217P"
literal "true"

\end_inset

@@ -315,6 +325,7 @@ Pan-Starrs übercal
\begin_inset CommandInset citation
LatexCommand citep
key "2013ApJS..205...20M"
literal "true"

\end_inset

@@ -326,6 +337,7 @@ DECam WcsFit
\begin_inset CommandInset citation
LatexCommand citep
key "2017PASP..129g4503B"
literal "true"

\end_inset

@@ -1328,17 +1340,17 @@ The sums present in this expression can be performed using matrix algebra;

matrices into a single large, sparse matrix (the Jacobian),
\begin_inset Formula
\[
J\equiv\left[\{K_{\gamma i},\forall\gamma,i\},\{K_{j},\forall j\}\right]
\]
\begin{equation}
J\equiv\left[\{K_{\gamma i},\forall\gamma,i\},\{K_{j},\forall j\}\right]\label{eq:Jacobian}
\end{equation}

\end_inset

and thus we simply have
\begin_inset Formula
\[
\frac{1}{2}\frac{d^{2}\chi^{2}}{d\theta^{2}}=J^{T}J
\]
\begin{equation}
\frac{1}{2}\frac{d^{2}\chi^{2}}{d\theta^{2}}=J^{T}J\label{eq:gradient_equation}
\end{equation}

\end_inset

@@ -1486,7 +1498,15 @@ Photometry example

\begin_layout Standard
As an illustrative example, we will work through a particular photometry
mapping, consisting of a constant zero-point per CCD (
mapping taking on-chip fluxes and positions
\begin_inset Formula $S_{\gamma i}=(f_{\gamma i},x,y)$
\end_inset

to on-sky fluxes
\begin_inset Formula $\phi_{\gamma i}$
\end_inset

, consisting of a constant zero-point per CCD (
\begin_inset Formula $f_{0}$
\end_inset

@@ -1509,9 +1529,11 @@ th order 2-D Chebyshev polynomial (
on a given CCD) per visit.
Thus, the mapping will be
\begin_inset Formula
\[
M_{\gamma i}(\eta,S_{\gamma i})=M_{CCD}(f_{0}^{-1},f_{\gamma i})M_{visit}(a_{j,k},x_{\gamma i},y_{\gamma i})=f_{\gamma i}[f_{0}]^{-1}\sum_{j=0}^{j=n}\sum_{k=0}^{k=m}a_{j,k}T_{j}(u_{\gamma i})T_{k}(v_{\gamma i})=\phi_{\gamma i}
\]
\begin{eqnarray}
M_{\gamma}(\eta,S_{\gamma i}) & = & M_{CCD}(f_{0}^{-1},f_{\gamma i})M_{visit}(a_{j,k},x_{\gamma i},y_{\gamma i})\label{eq:photometry-mapping}\\
& = & f_{\gamma i}[f_{0}]^{-1}\sum_{j=0}^{j=n}\sum_{k=0}^{k=m}a_{j,k}T_{j}(u_{\gamma i})T_{k}(v_{\gamma i})\nonumber \\
& = & \phi_{\gamma i}\nonumber
\end{eqnarray}

\end_inset

@@ -1546,14 +1568,157 @@ reference "eq:photometry_residual_vector_measurement"

),
\begin_inset Formula
\begin{eqnarray}
\frac{\partial D_{\gamma i}}{\partial f_{0}^{-1}} & = & f_{\gamma i}M_{visit}(a_{j,k}x_{\gamma i},y_{\gamma i})\label{eq:photometry-derivatives}\\
\frac{\partial D_{\gamma i}}{\partial a_{j,k}} & = & f_{\gamma i}f_{0}^{-1}T_{jk}(u_{\gamma i},v_{\gamma i})\nonumber \\
\frac{\partial D_{\gamma i}}{\partial F_{i}} & = & -1\nonumber
\end{eqnarray}

\end_inset

and for the reference terms (recall eq.

\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:photometry_residual_vector_reference"

\end_inset

),
\begin_inset Formula
\begin{equation}
\nabla D_{j}=\frac{\partial D_{j}}{\partial F_{j}}=1\label{eq:photometry-reference-derivatives}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
This model is degenerate to multiplying by a scale factor:
\begin_inset Formula $M_{CCD}\rightarrow aM_{CCD},M_{visit}\rightarrow a^{-1}M_{visit}$
\end_inset

.
This degeneracy is not removed by the reference catalog.
To break this degeneracy, we hold fixed one CCD's
\begin_inset Formula $f_{0}^{-1}$
\end_inset

(chosen to be the CCD closest to the center of the focal plane), and fit
all other CCD's relative to that.
\end_layout

\begin_layout Subsection
Magnitude-based photometry example
\end_layout

\begin_layout Standard
As an illustrative example, we will work through a particular photometry
mapping taking on-chip fluxes and positions
\begin_inset Formula $S_{\gamma i}=(f_{\gamma i},x,y)$
\end_inset

to on-sky magnitudes
\begin_inset Formula $m_{\gamma i}$
\end_inset

, with the transform consisting of a constant zero-point per CCD (
\begin_inset Formula $f_{0}$
\end_inset

: the CCD's filter response) and an
\begin_inset Formula $(n+m)$
\end_inset

th order 2-D Chebyshev polynomial (
\begin_inset Formula $\sum a_{j,k}T_{j}(u)T_{k}(v)$
\end_inset

: the optics+sky response, where
\begin_inset Formula $(u(x,y),v(x,y))$
\end_inset

are the focal plane coordinates of pixel
\begin_inset Formula $(x,y)$
\end_inset

on a given CCD) per visit.
Thus, taking
\begin_inset Formula $m_{CCD}$
\end_inset

,
\begin_inset Formula $m_{visit}$
\end_inset

, as the respective magnitude components, the mapping is
\begin_inset Formula
\begin{eqnarray}
M_{\gamma}(\eta,S_{\gamma i}) & = & m_{\gamma i}+m_{CCD}+m_{visit}\label{eq:magnitude-mapping}\\
& = & -2.5\log_{10}(M_{CCD}(f_{0}^{-1},f_{\gamma i}))-2.5\log_{10}(M_{visit}(a_{j,k},x_{\gamma i},y_{\gamma i}))\\
& = & m(f_{\gamma i})+m(f_{0}^{-1})+\sum a_{j,k}T_{j}(u)T_{k}(v)\nonumber \\
& = & m_{\gamma i}\nonumber
\end{eqnarray}

\end_inset

Making the visit component an additive polynomial of position makes the
derivatives simpler, while adding complexity to the resulting
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
class{PhotoCalib}
\end_layout

\end_inset

(it becomes an exponential term
\begin_inset Formula $10^{M_{visit}(x,y)/-2.5}$
\end_inset

).
Computing the derivatives with respect to
\begin_inset Formula $\eta=\left(f_{0},a_{j,k}\forall j,k\right)$
\end_inset

gives us,
\end_layout

\begin_layout Standard
\begin_inset Formula
\begin{eqnarray*}
\frac{\partial D_{\gamma i}}{\partial f_{0}^{-1}} & = & f_{\gamma i}M_{visit}(a_{j,k}x_{\gamma i},y_{\gamma i})\\
\frac{\partial D_{\gamma i}}{\partial a_{j,k}} & = & f_{\gamma i}f_{0}^{-1}T_{jk}(u_{\gamma i},v_{\gamma i})\\
\frac{\partial D_{\gamma i}}{\partial F_{i}} & = & -1
\nabla D_{\gamma i} & = & (\frac{\partial D_{\gamma i}}{\partial f_{0}^{-1}},\frac{\partial D_{\gamma i}}{\partial a_{0,0}},\ldots,\frac{\partial D_{\gamma i}}{\partial a_{n,m}},\frac{\partial D_{\gamma i}}{\partial F_{i}})
\end{eqnarray*}

\end_inset

where, for the measurement terms we have (recall eq.

\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:photometry_residual_vector_measurement"

\end_inset

),
\end_layout

\begin_layout Standard
\begin_inset Formula
\begin{eqnarray}
\frac{\partial D_{\gamma i}}{\partial f_{0}^{-1}} & = & 1\label{eq:magnitude-derivatives}\\
\frac{\partial D_{\gamma i}}{\partial a_{j,k}} & = & T_{jk}(u_{\gamma i},v_{\gamma i})\nonumber \\
\frac{\partial D_{\gamma i}}{\partial F_{i}} & = & -1\nonumber
\end{eqnarray}

\end_inset

and for the reference terms (recall eq.

\begin_inset CommandInset ref
@@ -1564,9 +1729,9 @@ reference "eq:photometry_residual_vector_reference"

),
\begin_inset Formula
\[
\nabla D_{j}=\frac{\partial D_{j}}{\partial F_{j}}=1
\]
\begin{equation}
\nabla D_{j}=\frac{\partial D_{j}}{\partial F_{j}}=1\label{eq:magnitude-reference-derivatives}
\end{equation}

\end_inset

@@ -1575,7 +1740,7 @@ reference "eq:photometry_residual_vector_reference"

\begin_layout Standard
This model is degenerate to multiplying by a scale factor:
\begin_inset Formula $M_{CCD}\rightarrow aM_{CCD},M_{visit}\rightarrow a^{-1}M_{visit}$
\begin_inset Formula $M_{CCD}\rightarrow aM_{CCD},M_{visit}\rightarrow-aM_{visit}$
\end_inset

.
@@ -2407,6 +2572,7 @@ In the LSST stack
\begin_inset CommandInset citation
LatexCommand citep
key "LDM-151"
literal "true"

\end_inset

0 comments on commit a0d78ca

Please sign in to comment.
You can’t perform that action at this time.