Data Reduction

The term reduction means the calibration of a raw image to account for the unwanted signal from bias and dark counts as well as from non-uniformities in the detector or optical system. Reducing the

• Dark Current
• Bias Level
• Non-uniformities

# Background We can represent each $i$ and $j$-th pixels in a light frame as the sum of the contributions from the following signals: $$L(\lambda)_{ij} = \Bigl[\epsilon(\lambda)_{ij}I(\lambda)_{ij} + T(\lambda)_{ij}\Bigr]t + B(\lambda)_{ij}$$

$L(\lambda)_{ij}$ is measured in counts and is the signal shown in SharpCap and saved in the data portion of the FITS file. This is the raw data in any light frame. While not super important for this data reduction example, it's otherwise important to note that these counts are rather limited in their numerical precision and therefore in their ability to precisely represent continuous physical values, i.e., $L(\lambda)_{ij}\in \mathbb{Z}<2^{\tiny{BITPIX}}-1$.

$\epsilon(\lambda)_{ij}$ is the pixel-by-pixel homogeneity. This is measured as a fraction with each pixel carrying a (to computer precision) continuous value in the range $\epsilon(\lambda)_{ij}\in \mathbb{R}_{[0,1]}$ Broadly, this is the pixel-by-pixel sensitivity to incident photons. “Baked-in” is more than the detector quantum efficiency, it “bakes-in” the reduced efficiency due to any filter, any inhomogeneity due to hardware or software amplification via GAIN modulations, as well as any other imperfections in the optical system that impact the ability for the detector to record photons incident on the primary mirror (all of that schmutz in our system!).

$I(\lambda)_{ij}$ is the source signal. It is measured in counts, and precision in the counts recorded come at the expense of signal contributions from the Dark and Bias signals (see below). The aim of any reduction pipeline is to take the signals in a raw light frame and reduce the contributions of other signals such that only this source signal is present. The result is called a science frame or science image and hypothetically is the closest thing to $I(\lambda)_{ij}$ we can muster with our system and reduction pipeline.

$T(\lambda)_{ij}$ is the thermal signal, it is a component of the dark signal along with the Bias (below). Thermal signal is a measure of the electrons that gained enough thermal energy to liberate themselves from the valence band of the CMOS semiconductor substrate and be read-off as signal by the Analog-Digital-Converter (ADU). As such, the thermal signal is time-dependent. I.e., the longer you give electrons in the valence band time to explore microstates made available by some heat bath, the more likely some will find the microstates that allow them to get to the conduction band. Since the thermal signal is not generated by photons and the quantum efficiency of the pixels play no part, we can safely drop any implied $\lambda$ dependence. $T(\lambda)_{ij} \rightarrow $T_{ij}$$ $t$ is the exposure or integration time. “Integration” time is favored by me because it reminds the reader that we are integrating radiance or flux over time. This integration also couples time with the counts generated by the source and thermal signals.

$B(\lambda)_{ij}$ is the bias signal, it is a component of the dark signal along with the thermal signal (above). Bias signal is a measure of the electrons that were liberated by the internal voltages that are constantly and non-uniformly applied across all pixels. These internal voltages are responsible for 'storing' liberated electrons (either by photons or thermal excitation) until read-time by the ADC. These internal voltages are responsible for some initial 'jolt' of extra elections, but after this initial 'jolt' electrons that trickle into the storage circuits are liberated by thermal agitation. In this way, the Bias signal is not time dependent. Further, since the bias signal is not generated by photons and the quantum efficiency of the pixels play no part, we can safely drop any implied $\lambda$ dependence. $$B(\lambda)_{ij} \rightarrow B_{ij}$$

## Dark Signal It's likely clear from the above that the Dark Signal is built from a time-integrated thermal signal as well as constant (but non-uniform) bias signal.