Document calculation of readnoise variance (JP-2670) (#6924)

* Document calculation of readnoise variance

* Document calculation of readnoise variance

* minor corrections based on review suggestions

* minor corrections based on review suggestions

CHANGES.rst

@@ -6,6 +6,13 @@ general
 
 -
 
+
+ramp_fitting
+------------
+
+- Added documentation for the calculation of the readnoise variance [#6924]
+
+
 resample
 --------
 

docs/jwst/ramp_fitting/appendix.rst

@@ -0,0 +1,46 @@
+Appendix
+========
+
+The derivation of the segment-specific readnoise variance (:math:`{ var^R_{s}  }`) is shown here.  This derivation follows the standard procedure to fitting data to a straight line, such as in chapter 15 of Numerical Recipes.  The segment-specific variance from read noise corresponds to :math:`{\sigma_b^2}` in section 15.2. 
+
+For read noise R, weight w = :math:`{1 / R^2}`, which is a constant. 
+
+n = number of groups (`ngroups` in the text)
+
+t = group time (`tgroup` in the text)
+
+x = starting time for each group, = :math:`{(1,2,3, ... n+1) \cdot t}`
+
+
+:math:`{S_1 = \sum_{k=1}^n w}`
+
+:math:`{S_x = \sum_{k=1}^n (w  \cdot  x_k) t}`
+
+S\ :sub:`xx`\  = :math:`{\sum_{k=1}^n (w \cdot x_k)^2 t^2}`
+
+D = :math:`{S_1 \cdot S}`\ :sub:`xx`\ - :math:`{S_x^2}`
+
+
+Summations needed:
+
+:math:`{\sum_{k=1}^n k = n \cdot (n+1) / 2 = n^2 /2 + n/2 }`
+
+:math:`{\sum_{k=1}^n k^2= n \cdot (n+1) \cdot (2 \cdot n+1) / 6 = n^3/3 + n^2/2 +n/6 }`
+
+
+The variance from read noise 
+= :math:`{var^R_{s} = S_1 / D = S_1 / (S_1 \cdot S_{xx} - S_x^2)}` 
+
+
+= :math:`{ \dfrac {w \cdot n} { [w \cdot n \cdot \sum_{k=1}^n (w \cdot x_k^2 \cdot t^2)] - [\sum_{k=1}^n (w \cdot x_k \cdot t)] ^2}}`     
+
+
+= :math:`{ \dfrac {n} { w \cdot t^2 \cdot [ n \cdot ( n^3/3 + n^2/2 +n/6 ) - (n^2/2 + n/2 )^2 ] }}`
+
+
+= :math:`{ \dfrac {1} { ( n^3/12 - n/12 ) \cdot w \cdot t^2 }}`
+
+
+= :math:`{ \dfrac{12 \cdot R^2}  {(n^3 - n) \cdot t^2}}` 
+
+This is the equation in the code and in the segment-specific computations section of the Description.

docs/jwst/ramp_fitting/description.rst

@@ -224,7 +224,9 @@ The variance of the slope of a segment due to read noise is:
 
 where :math:`R` is the noise in the difference between 2 frames, 
 :math:`ngroups_{s}` is the number of groups in the segment, and :math:`tgroup` is the group 
-time in seconds (from the keyword TGROUP).  
+time in seconds (from the keyword TGROUP). The derivation of this equation is given in 
+the appendix of this section, at `readnoise variance derivation. <https://jwst-pipeline.readthedocs.io/en/latest/jwst/ramp_fitting/appendix.html>`_.
+
 
 The variance of the slope in a segment due to Poisson noise is: 
 

docs/jwst/ramp_fitting/index.rst

@@ -10,6 +10,6 @@ Ramp Fitting
    description.rst
    arguments.rst
    reference_files.rst
-
+   appendix.rst
 
 .. automodapi:: jwst.ramp_fitting