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--- wiki:astronomy:observational_astronomy:snr [2024/11/04 20:29] – Roy Prouty
+++ wiki:astronomy:observational_astronomy:snr [2024/11/04 21:06] (current) – Roy Prouty
@@ Line 60: / Line 60: @@
 ==== Noise ====
-For a Normal Distribution, we can call the $\sigma$ (i.e., the uncertainty) the noise in the signal ($\mu$--the average value measured in the stack). So then, any light frame pixel can be said to have an average value with some signal ($\mu$) along with some noise ($\sigma$). If there is high confidence in the measured signal (meaning low uncertainty), then $\mu \gg \sigma$.
+For a Normal Distribution, we can call the $\sigma$ (i.e., the uncertainty or the standard of deviation) the noise in the signal ($\mu$--the average value measured in the stack). So then, any light frame pixel can be said to have an average value with some signal ($\mu$) along with some noise ($\sigma$). If there is high confidence in the measured signal (meaning low uncertainty), then $\mu \gg \sigma$.
+Further, for the sources of unwanted signal, they each come with their own signal and noise -- trusted to be Normally Distributed. So in the construction of the calibrated frame, only the *signal* from unwanted signal sources can be removed. The uncertainty associated with these signals cannot be separated -- they were measured at the same time! All one can hope to do here is either (a) not collect unwanted signal & noise in the first place or (b) acknowledge the uncertainty in the final result.
+{{ :wiki:astronomy:observational_astronomy:lightframe.png?600 |}}
+{{ :wiki:astronomy:observational_astronomy:calibratedframe.png?600 |}}
+In the above, despite the average Dark Signal being removed, the uncertainty associated with the collection of the Dark cannot be disentangled from the uncertainty in the source signal -- so it remains. The distribution of dark signal is therefore centered on $0$, but with some spread about $0$.
+==== Signal to Noise Ratio ====
+In returning to the ultimate goal of claiming one hypothesis true over another ... The Null Hypothesis is that the signal is indistinguishable from the noise. The Alternate Hypothesis is that the signal is distinguishable from the noise. We can construct the statistical model of the Alternate Hypothesis with $\mathcal{Norm}(\mu, \sqrt{\mu^2 + \mu_D^2 + \mu_U^2 + ... })$. And the Null Hypothesis with $\mathcal{Norm}(0, \sqrt{\mu_D^2 + \mu_U^2 + ... })$ -- where the uncertainties are just the sum of the uncertainties in quadrature. \\ \\
+So the figure of merit here is the Signal-to-Noise Ratio (SNR). In short, this quantity is the number of standard deviations the average calibrated signal is from the uncertainty in that average signal *collected*.\\ \\
+**At the UMBC Observatory, we aim for an SNR > 10 to confidently reject the Null Hypothesis.**