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<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>

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<LI><UL>
<LI><UL>
<LI><UL>
<LI><A NAME="tex2html677"
  HREF="node10.html#SECTION001000010000000000000">Precision and Accuracy.</A>
<LI><A NAME="tex2html678"
  HREF="node10.html#SECTION001000020000000000000">Noise</A>
</UL>
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<BR>
<LI><A NAME="tex2html679"
  HREF="node10.html#SECTION001010000000000000000">CCD Noise Sources</A>
<LI><A NAME="tex2html680"
  HREF="node10.html#SECTION001020000000000000000">Noise of a Pixel Value</A>
<LI><A NAME="tex2html681"
  HREF="node10.html#SECTION001030000000000000000">Dark Frame Subtraction</A>
<LI><A NAME="tex2html682"
  HREF="node10.html#SECTION001040000000000000000">Flat Fielding</A>
<LI><A NAME="tex2html683"
  HREF="node10.html#SECTION001050000000000000000">Instrumental Magnitude Error of a Star</A>
</UL>
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<HR>

<H1><A NAME="SECTION001000000000000000000"></A><A NAME="ap:noise"></A>
<BR>
Noise Modelling
</H1>

<P>
The issue of noise modelling is essential in any photometric endeavour. Measurement values 
are next to meaningless if they aren't acompanied by a measure of ther uncertainty. 

<P>
One can assume that the noise and error modelling only applies to deriving an error figure. 
This in true only in extremely simple cases. In general, the noise estimates will also affect 
the actual values. For instance, suppose that we use several standards to calibrate a field. 
From the noise estimate, we know that one of the standards has a large probable error. Then, 
we choose to exclude (or downweight) that value from the solution--this will change the
calibration, and directly affect the result (not just it's noise estimate).

<P>

<H4><A NAME="SECTION001000010000000000000">
Precision and Accuracy.</A>
</H4> The precision of a measurement denotes the degree to 
which different measurements of the same value will yield the same result; it measures the 
repeatability of the measurement process. A precise measurement has a small <EM>random error</EM>.

<P>
The accuracy of a measurement denotes the degree to which a measurement result will represent 
the true value. The accuracy includes the <EM>random error</EM> of the measurement, as well as 
the <EM>systematic error</EM>.

<P>
Random errors are in a way the worst kind. We have to accept them and take into account, but 
they cannot be calculated out. We can try to use better equipment, or more telescope time
and reduce them. On the other hand, since random errors are, well, random in nature (they 
don't correlate to anything), we can in principle reduce them to an aribitrarily low level 
by averaging a lerge number of measurements.

<P>
Systematic errors on the other hand can usually be eliminated (or at least reduced) by 
calibration. Systematic errors are not that different from random errors. They differ
fundamentally in the fact the they depend on <EM>something</EM>. Of course, even random
errors ultimately depend on something. But that something changes incontrollably, and 
in a time frame that is short compared to the measurement time scale.

<P>
A systematic error can turn into a random error if we have no control over the thing that 
the error depends on, or we don't have something to calibrate against. We could treat this 
error as ``random'' and try to average many measurements to reduce it, but we have to make 
sure that the something that the error depends on has had a change to vary between the 
measurements we average, or we won't get very far.

<P>

<H4><A NAME="SECTION001000020000000000000">
Noise</A>
</H4> is the ``randomest'' source of random errors. We have no way to 
calibrate out noise, but it's features are well understood and relatively easy to model.
One doesn't have a good excuse not to model noise reasonably well.

<P>
We will generally talk about ``noise'' when estimating random errors that derive from 
an electrical or optical noise source. Once these are combine with
other error  sources (like for instance 
expected errors of the standards), we will use the term ``error''. Of course, there are two
ways of understanding an error value. If we know what the true value should be, we can talk 
about and <EM>actual error</EM>. If we just consider what error level we can expect, we talk 
about an estimated, or <EM>expected error</EM>.

<P>

<H1><A NAME="SECTION001010000000000000000">
CCD Noise Sources</A>
</H1>

<P>
There are several noise sources in a CCD sensor. We will see that in the end they can 
usually be modeled with just two parameters, but we list the main noise contributors 
for reference.

<OL>
<LI><EM>Photon shot noise</EM> is the noise associated with the random arrival of photons 
at any detector. Shot noise exists because of the discrete nature of light and 
electrical charge. The time between photon arrivals is goverened by Poisson 
statistics. For a phase-insensitive detector, such as a CCD, 
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ph} = \sqrt{S_{\rm ph}}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="91" HEIGHT="36" BORDER="0"
 SRC="img129.png"
 ALT="\begin{displaymath}
\sigma_{\rm ph} = \sqrt{S_{\rm ph}}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img130.png"
 ALT="$S_{\rm ph}$"></SPAN> is the signal expressed in electrons. Shot noise is sometimes called 
``Poisson noise''.
</LI>
<LI><EM>Output amplifier noise</EM> originates in the output amplifier of the sensor. 
It consists of two components: thermal (white) noise and flicker noise. Thermal noise 
is independent of frequency and has a mild temperature dependence (is proportional to 
the square root of the absolute temperature). It fundamentally originates in the thermal 
movement of atoms. Flicker noise (or <SPAN CLASS="MATH"><IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img131.png"
 ALT="$1/f$"></SPAN> noise) is strongly dependent on frequency. 
It originates in the existance of long-lived states in the silicon crystal (most notably
``traps'' at the silicon-oxide interface). 

<P>
For a given readout configuration and speed, these noise sources contribute a constant 
level, that is also independant of the signal level, usually called the <EM>readout noise</EM>. 
The effect of read noise can be reduced by increasing the 
time in which the sensor is read out. There is a limit to that, as
flicker noise will begin to kick in. For some cameras, one has the option of trading readout
speed for a decrease in readout noise.

<P>
</LI>
<LI><EM>Camera noise.</EM> Thermal and flicker noise are also generated in
the camera electronics. the noise level will be independent on the signal. 
While the camera designer needs to make a distiction between the 
various noise sources, for a given camera, noise originating in the camera and the ccd 
amplifier are indistinguishable.

<P>
</LI>
<LI><EM>Dark current noise.</EM> Even in the absence of light, electron-hole pairs are generated 
inside the sensor. The rate of generation depends exponentially on temperature (typically
doubles every 6-7 degrees). The thermally generated electrons cannot be separated from
photo-generated photons, and obey the same Poisson statistic, so
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm dark} = \sqrt{S_{\rm dark}}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="113" HEIGHT="30" BORDER="0"
 SRC="img132.png"
 ALT="\begin{displaymath}
\sigma_{\rm dark} = \sqrt{S_{\rm dark}}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
We can subtract the average 
dark signal, but the shot noise associated with it remains. The level of the dark current 
noise depends on temperature and integration time.

<P>
</LI>
<LI><EM>Clock noise.</EM> Fast changing clocks on the ccd can also generate spurious charge. 
This charge also has a shot noise component associated. However, one cannot read the sensor 
without clocking it, so clock noise cannot be discerned from readout noise. The clock noise 
is fairly constant for a given camera and readout speed, and independent of the signal level.
</LI>
</OL>

<P>
Examining the above list, we see that some noise sources are independent of the signal level. 
They are: the output amplifier noise, camera noise and clock noise. They can be combined in
a single equivalent noise source. The level of this source is called <EM>readout noise</EM>, and
is a characteristic of the camera. It can be expressed in electrons, or in the camera output
units (ADU). 

<P>
The rest of the noise sources are all shot noise sources. The resulting value will be:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm shot} = \sqrt{\sigma_{\rm ph}^2 + \sigma_{\rm dark}^2}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="159" HEIGHT="36" BORDER="0"
 SRC="img133.png"
 ALT="\begin{displaymath}
\sigma_{\rm shot} = \sqrt{\sigma_{\rm ph}^2 + \sigma_{\rm dark}^2}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm shot} = \sqrt{S_{\rm ph} + S_{\rm dark}} = \sqrt{S}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="209" HEIGHT="36" BORDER="0"
 SRC="img134.png"
 ALT="\begin{displaymath}
\sigma_{\rm shot} = \sqrt{S_{\rm ph} + S_{\rm dark}} = \sqrt{S}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img135.png"
 ALT="$S$"></SPAN> is the total signal from the sensor above bias, expressed in electrons. So to calculate the 
shot noise component, we just need to know how many ADUs/electron the camera produces. This
is a constant value, or one of a few constant values for cameras that have different gain
settings. We will use <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN> to denote this value. 

<P>

<H1><A NAME="SECTION001020000000000000000">
Noise of a Pixel Value</A>
</H1>
We will now try to model the level of noise in a pixel value. The result of reading one pixel 
(excluding noise) is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
s = s_b + A ( S_d + S_p)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="150" HEIGHT="32" BORDER="0"
 SRC="img137.png"
 ALT="\begin{displaymath}
s = s_b + A ( S_d + S_p)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img138.png"
 ALT="$s_b$"></SPAN> is a fixed bias introduced by the camera electronics, <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img139.png"
 ALT="$S_d$"></SPAN> is the number of 
dark electrons, and <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img140.png"
 ALT="$S_p$"></SPAN> is the number of photo-generated electrons (which is the number 
of photons incident on the pixel multiplied by the sensor's quantum efficiency).

<P>
Now let's calculate the noise associated with this value.
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma^2 = \sigma_r^2 + A^2 (S_d + S_p) = \sigma_r^2 + A(s - s_b)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="304" HEIGHT="32" BORDER="0"
 SRC="img141.png"
 ALT="\begin{displaymath}
\sigma^2 = \sigma_r^2 + A^2 (S_d + S_p) = \sigma_r^2 + A(s - s_b)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">6</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Where <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$\sigma_r$"></SPAN> is the readout noise expressed in ADU, and <SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img135.png"
 ALT="$S$"></SPAN> is the total signal
expressed in electrons. Note that we cannot calculate 
the noise if we don't know the bias value. The bias can be determined by reading 
frames with zero exposure time (bias frames). These will contribute some read noise though. 
By averaging several bias frames, the noise contribution can be reduced. Another approach is 
to take the average across a bias frame and use that value for the noise calculation of all
pixels. Except for very non-uniform sensors this approach works well. <SMALL>GCX </SMALL>supports both 
ways.

<P>
Note that a bias frame will only contain readout noise. By calculating the standard 
deviation of pixels across the difference between two bias frames we obtain <SPAN CLASS="MATH"><IMG
 WIDTH="28" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img143.png"
 ALT="$\sqrt{2}$"></SPAN> times the
readout noise.

<P>

<H1><A NAME="SECTION001030000000000000000">
Dark Frame Subtraction</A>
</H1>

<P>
A common situation is when one subtracts a dark frame, but doesn't use bias frames. 
The noise associated with the dark frame is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_d^2 = \sigma_r^2 + A^2 S_d
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="118" HEIGHT="31" BORDER="0"
 SRC="img144.png"
 ALT="\begin{displaymath}
\sigma_d^2 = \sigma_r^2 + A^2 S_d
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">7</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The resulting pixel noise after dark frame subtraction will be:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ds}^2 = 2\sigma_r^2 + A^2 (2 S_d + S_p)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="193" HEIGHT="32" BORDER="0"
 SRC="img145.png"
 ALT="\begin{displaymath}
\sigma_{\rm ds}^2 = 2\sigma_r^2 + A^2 (2 S_d + S_p)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">8</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
while the signal will be
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
s_{\rm ds} = AS_p
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="74" HEIGHT="32" BORDER="0"
 SRC="img146.png"
 ALT="\begin{displaymath}
s_{\rm ds} = AS_p
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">9</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Using just the camera noise parameters, we cannot determine the noise anymore. 
We have to keep track of the dark subtraction and it's noise effects. We however
rewrite the dark-subtracted noise equation as follows:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ds}^2 = \left(\sqrt{2\sigma_r^2 + 2 A^2 S_d}\right)^2 + A^2 S_p
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="248" HEIGHT="48" BORDER="0"
 SRC="img147.png"
 ALT="\begin{displaymath}
\sigma_{\rm ds}^2 = \left(\sqrt{2\sigma_r^2 + 2 A^2 S_d}\right)^2 + A^2 S_p
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">10</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
If we use the notation <!-- MATH
 $\sigma_r' = \sqrt{2\sigma_r^2 + 2 A^2 S_d}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="157" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img148.png"
 ALT="$\sigma_r' = \sqrt{2\sigma_r^2 + 2 A^2 S_d}$"></SPAN>, we get: 
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ds}^2 = \sigma_r'^2 + A^2 S_p
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="127" HEIGHT="32" BORDER="0"
 SRC="img149.png"
 ALT="\begin{displaymath}
\sigma_{\rm ds}^2 = \sigma_r'^2 + A^2 S_p
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">11</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
This is identical in form to the simple pixel noise equation, except that the true 
camera readout noise is replaced by the equivalent read noise <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img150.png"
 ALT="$\sigma_r'$"></SPAN>. What's more, the
bias is no longer an issue, as it doesn't appeear in the signal equation anymore. We can 
derive the pixel noise from the signal directly, as:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ds}^2 = \sigma_r'^2 + As_{\rm ds}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="123" HEIGHT="31" BORDER="0"
 SRC="img151.png"
 ALT="\begin{displaymath}
\sigma_{\rm ds}^2 = \sigma_r'^2 + As_{\rm ds}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">12</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The same parameters, <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img150.png"
 ALT="$\sigma_r'$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN> are sufficient to describe the noise in the 
dark-subtracted frame.

<P>

<H1><A NAME="SECTION001040000000000000000">
Flat Fielding</A>
</H1>

<P>
To flat-field a frame, we divide the dark-subtracted pixel value <SPAN CLASS="MATH"><IMG
 WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img152.png"
 ALT="$s_{\rm ds}$"></SPAN> by the flat field
value <SPAN CLASS="MATH"><IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img153.png"
 ALT="$f$"></SPAN>. The noise of the flat field is <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img154.png"
 ALT="$\sigma_f$"></SPAN>. The resulting signal value is
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
s_{\rm ff} = \frac{1}{f}AS_p
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="85" HEIGHT="45" BORDER="0"
 SRC="img155.png"
 ALT="\begin{displaymath}
s_{\rm ff} = \frac{1}{f}AS_p
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">13</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
If we neglect second-order noise terms, the noise of the flat-fielded, dark subtracted pixel 
is: 
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ff}^2 = f \sigma_r'^2 + A^2 S_p + \left(\frac{\sigma_f}{f}AS_p\right)^2
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="239" HEIGHT="49" BORDER="0"
 SRC="img156.png"
 ALT="\begin{displaymath}
\sigma_{\rm ff}^2 = f \sigma_r'^2 + A^2 S_p + \left(\frac{\sigma_f}{f}AS_p\right)^2
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">14</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm ff}^2 = f^2 \sigma_r'^2 + A f s_{\rm ff} + \left(\sigma_f s_{\rm ff}\right)^2
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="222" HEIGHT="32" BORDER="0"
 SRC="img157.png"
 ALT="\begin{displaymath}
\sigma_{\rm ff}^2 = f^2 \sigma_r'^2 + A f s_{\rm ff} + \left(\sigma_f s_{\rm ff}\right)^2
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">15</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The problem with this result is that f is not constant across the frame. So in general, the noise
of a flat-fielded frame cannot be described by a small number of parameters. In many cases though, 
<SPAN CLASS="MATH"><IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img153.png"
 ALT="$f$"></SPAN> doesn't vary too much across the frame. We can then use it's average value, <SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img158.png"
 ALT="$\widetilde{f}$"></SPAN> 
for the noise calculation. This is the approach taken by the program.

<P>
We can identify the previous noise parameters, <!-- MATH
 $\sigma_r'' = 
\widetilde{f}\sigma_r'$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="74" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img159.png"
 ALT="$\sigma_r'' =
\widetilde{f}\sigma_r'$"></SPAN> and <!-- MATH
 $A' = A\widetilde{f}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="70" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img160.png"
 ALT="$A' = A\widetilde{f}$"></SPAN>. For specifing the
effect of the flat-fielding, we introduce a new parameter, 
<SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img154.png"
 ALT="$\sigma_f$"></SPAN>.

<P>
Without reducing generality, we can arrange for <!-- MATH
 $\widetilde{f} = 1$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="47" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img161.png"
 ALT="$\widetilde{f} = 1$"></SPAN>. This means that the 
average values on the frames don't change with the flatfielding operation, and is a common 
choice.

<P>
In this case, <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$\sigma_r$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN> aren't affected by the flatfielding operation, while the 
third noise parameter becomes <!-- MATH
 $\sigma_f/\widetilde{f}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="43" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img162.png"
 ALT="$\sigma_f/\widetilde{f}$"></SPAN>, which is the reciprocal of the SNR of 
the flat field.

<P>
<SMALL>GCX </SMALL>models the noise of each pixel in the frame by four parameters: <SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$\sigma_r$"></SPAN>, <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN>, 
<!-- MATH
 $\sigma_f/\widetilde{f}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="43" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img162.png"
 ALT="$\sigma_f/\widetilde{f}$"></SPAN> and <!-- MATH
 $\widetilde{s_b}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img163.png"
 ALT="$\widetilde{s_b}$"></SPAN>. The noise function <SPAN CLASS="MATH"><IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img164.png"
 ALT="$n(s)$"></SPAN> of each pixel 
is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
n^2(s) = \sigma^2 = \sigma_r^2 + A |(s-\widetilde{s_b})| + 
\left(\frac{\sigma_f}{\widetilde{f}}\right)^2(s-\widetilde{s_b})^2
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="367" HEIGHT="58" BORDER="0"
 SRC="img165.png"
 ALT="\begin{displaymath}
n^2(s) = \sigma^2 = \sigma_r^2 + A \vert(s-\widetilde{s_b})\...
...t(\frac{\sigma_f}{\widetilde{f}}\right)^2(s-\widetilde{s_b})^2
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">16</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<SPAN CLASS="MATH"><IMG
 WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$\sigma_r$"></SPAN> comes from the <TT>RDNOISE</TT> field in the frame header. <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN> is the
reciprocal of the value of the <TT>ELADU</TT> field. <!-- MATH
 $\sigma_f/\widetilde{f}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="43" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img162.png"
 ALT="$\sigma_f/\widetilde{f}$"></SPAN> comes from
<TT>FLNOISE</TT>, while <!-- MATH
 $\widetilde{s_b}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img163.png"
 ALT="$\widetilde{s_b}$"></SPAN> comes from <TT>DCBIAS</TT>. 

<P>
Every time frames are processed 
(dark and bias subtracted, flatfielded, scaled etc), the noise parameters are updated.

<P>

<H1><A NAME="SECTION001050000000000000000">
Instrumental Magnitude Error of a Star</A>
</H1>

<P>
Once we know the noise of each pixel, deriving the expected error of an instrumental magnitude 
is straightforward. Let <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img166.png"
 ALT="$N_b$"></SPAN> be the number of pixels in the sky annulus, and <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img167.png"
 ALT="$s_i$"></SPAN> the level
of each pixel. The noise of the sky estimate is:<A NAME="tex2html68"
  HREF="footnode.html#foot1103"><SUP>A.<SPAN CLASS="arabic">1</SPAN></SUP></A>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_b^2 = \frac{1}{N_b}\sum_{i=1}^{N_b}n^2(s_i)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="137" HEIGHT="59" BORDER="0"
 SRC="img168.png"
 ALT="\begin{displaymath}
\sigma_b^2 = \frac{1}{N_b}\sum_{i=1}^{N_b}n^2(s_i)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">17</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Now let <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img169.png"
 ALT="$N_s$"></SPAN> be the number of pixels in the central aperture. The noise from these pixels is:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_s^2 = \sum_{i=1}^{N_s}n^2(s_i)
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="109" HEIGHT="58" BORDER="0"
 SRC="img170.png"
 ALT="\begin{displaymath}
\sigma_s^2 = \sum_{i=1}^{N_s}n^2(s_i)
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">18</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The total noise after sky subtraction will be:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_n^2 = \sigma_s^2 + N_s \sigma_b^2.
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="122" HEIGHT="31" BORDER="0"
 SRC="img171.png"
 ALT="\begin{displaymath}
\sigma_n^2 = \sigma_s^2 + N_s \sigma_b^2.
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">19</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
The program keeps track and reports separately the photon shot noise, the sky noise, 
the read noise contribution and the scintillation noise.

<P>
Scintillation is an atmospheric effect, which results in a random variation of the
received flux from a star. We use the following formula for scintillation noise:

<P>
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\sigma_{\rm sc} = 0.09 F \frac{A^{1.75}}{D^\frac{2}{3}\sqrt{2t}}
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="145" HEIGHT="51" BORDER="0"
 SRC="img172.png"
 ALT="\begin{displaymath}
\sigma_{\rm sc} = 0.09 F \frac{A^{1.75}}{D^\frac{2}{3}\sqrt{2t}}
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">20</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Where <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img44.png"
 ALT="$F$"></SPAN> is the total flux received from the star, <SPAN CLASS="MATH"><IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img136.png"
 ALT="$A$"></SPAN> is the airmass of the observation, 
<SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img11.png"
 ALT="$D$"></SPAN> is the telescope aperture in cm, and <SPAN CLASS="MATH"><IMG
 WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img6.png"
 ALT="$t$"></SPAN> is the integration time. Scintillation varies
widely over time, so the above is just an estimate.

<P>
Finally, we can calculate the expected error of the instrumental magnitude as
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">

<!-- MATH
 \begin{equation}
\epsilon_i = 2.51188 \log\left(1 + \frac{\sqrt{\sigma_n^2 + \sigma_{\rm sc}^2}}{F}\right).
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="264" HEIGHT="54" BORDER="0"
 SRC="img173.png"
 ALT="\begin{displaymath}
\epsilon_i = 2.51188 \log\left(1 + \frac{\sqrt{\sigma_n^2 + \sigma_{\rm sc}^2}}{F}\right).
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(A.<SPAN CLASS="arabic">21</SPAN>)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>

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2005-11-27
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