appendix.rst

Appendix

The derivation of the segment-specific readnoise variance (var_s^R) is shown here. This pertains to both the 'conventional' and 'weighted' readnoise variances - the only difference being the number of groups in the segment. This derivation follows the standard procedure for fitting data to a straight line, such as in chapter 15 of Numerical Recipes. The segment-specific variance from read noise corresponds to σ_b² in section 15.2.

For read noise R, weight w = 1/R², 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, = (1, 2, 3, ...n + 1) ⋅ t

${S_1 = \sum_{k=1}^n w}$

${S_x = \sum_{k=1}^n (w \cdot x_k) t}$

S_xx = ${\sum_{k=1}^n (w \cdot x_k)^2 t^2}$

D = S₁ ⋅ S_xx- S_x²

Summations needed:

${\sum_{k=1}^n k = n \cdot (n+1) / 2 = n^2 /2 + n/2 }$

${\sum_{k=1}^n k^2= n \cdot (n+1) \cdot (2 \cdot n+1) / 6 = n^3/3 + n^2/2 +n/6 }$

The variance due to read noise = var_s^R = S₁/D = S₁/(S₁ ⋅ S_xx − S_x²)

= ${ \dfrac {w \cdot n} { [w \cdot n \cdot \sum_{k=1}^n (w \cdot x_k^2 \cdot t^2)] - [\sum_{k=1}^n (w \cdot x_k \cdot t)] ^2}}$

= ${ \dfrac {n} { w \cdot t^2 \cdot [ n \cdot ( n^3/3 + n^2/2 +n/6 ) - (n^2/2 + n/2 )^2 ] }}$

= ${ \dfrac {1} { ( n^3/12 - n/12 ) \cdot w \cdot t^2 }}$

= ${ \dfrac{12 \cdot R^2} {(n^3 - n) \cdot t^2}}$

This is the equation used in the step code and listed in the segment-specific computations section of the step Description.