Transit SNR Methodology

Definition of SNR used here

In this context, transit SNR is the significance (in sigma units) of detecting a nonzero transit depth \( \delta \) in the presence of per-point photometric scatter \( \sigma \) (RMS), given a finite number of samples in transit and out of transit.

A white-noise approximation is used: each data point has independent noise with standard deviation \( \sigma \). The transit is treated as a “two-level” signal: baseline (OOT) vs. depressed flux (in transit).

Inputs

\( \delta \): transit depth (ppt)
\( \sigma \): per-point scatter (ppt). Can use RMS of residuals.
\( N_{\mathrm{in}} \): number of in-transit points
\( N_{\mathrm{out}} \): number of out-of-transit points

(Depth and RMS are both in ppt, so the ratio \( \delta/\sigma \) is unitless.)

Methodology

The depth estimate is treated as a difference of two means:

\[ \hat{\delta} = \bar{y}_{\mathrm{out}} - \bar{y}_{\mathrm{in}} \]

If each point has standard deviation \( \sigma \) and points are independent, then the standard error of each mean is:

\[ SE(\bar{y}_{\mathrm{in}})=\frac{\sigma}{\sqrt{N_{\mathrm{in}}}}, \qquad SE(\bar{y}_{\mathrm{out}})=\frac{\sigma}{\sqrt{N_{\mathrm{out}}}} \]

The variance of a difference of independent quantities adds:

\[ \mathrm{Var}(\hat{\delta}) = \mathrm{Var}(\bar{y}_{\mathrm{out}}) + \mathrm{Var}(\bar{y}_{\mathrm{in}}) = \frac{\sigma^2}{N_{\mathrm{out}}} + \frac{\sigma^2}{N_{\mathrm{in}}} \]

So the uncertainty on the depth is:

\[ \sigma_{\delta} = \sqrt{\mathrm{Var}(\hat{\delta})} = \sigma \sqrt{\frac{1}{N_{\mathrm{in}}}+\frac{1}{N_{\mathrm{out}}}} \]

And the SNR is “signal divided by its uncertainty”:

\[ \mathrm{SNR} = \frac{\delta}{\sigma_{\delta}} = \frac{\delta}{\sigma\sqrt{\frac{1}{N_{\mathrm{in}}}+\frac{1}{N_{\mathrm{out}}}}} \]

Equivalent algebraic form (sometimes easier to interpret):

\[ \mathrm{SNR}=\left(\frac{\delta}{\sigma}\right)\sqrt{N_{\mathrm{eff}}} \]

This introduces an “effective number of points”:

\[ N_{\mathrm{eff}}=\frac{N_{\mathrm{in}}N_{\mathrm{out}}}{N_{\mathrm{in}}+N_{\mathrm{out}}} \quad\Rightarrow\quad \mathrm{SNR}=\left(\frac{\delta}{\sigma}\right)\sqrt{N_{\mathrm{eff}}} \]

The “Potential SNR”

If \(N_{\mathrm{out}}\) is small, the baseline mean is noisy, and that uncertainty propagates into the depth estimate. That’s why the SNR calculated here can be noticeably smaller than the common shortcut.

The program also prints:

\[ \mathrm{SNR}_{\mathrm{potential}}=\left(\frac{\delta}{\sigma}\right)\sqrt{N_{\mathrm{in}}} \]

This comes from the full expression in the limit \(N_{\mathrm{out}}\to\infty\):

\[ \mathrm{SNR} = \frac{\delta}{\sigma\sqrt{\frac{1}{N_{\mathrm{in}}}+\frac{1}{N_{\mathrm{out}}}}} \xrightarrow{\,N_{\mathrm{out}}\to\infty\,} \frac{\delta}{\sigma\sqrt{\frac{1}{N_{\mathrm{in}}}}} = \left(\frac{\delta}{\sigma}\right)\sqrt{N_{\mathrm{in}}} \]

So it is correctly interpreted as: “Potential SNR (if OOT → ∞)” = the maximum SNR achievable for the same depth, RMS, and \(N_{\mathrm{in}}\), if you could make the OOT baseline arbitrarily well measured by adding many more OOT points (with the same per-point RMS).

The key choices embedded in this calculation

Two-level (in vs out) model

The signal is treated as a depth relative to baseline, not a full-shape matched filter. That’s why the “difference of means” method applies.

White-noise assumption

Independent errors with a single per-point scatter \( \sigma \) is assumed. Therefore the uncertainty scales as \(1/\sqrt{N}\). If residuals have time-correlated (“red”) noise, this formula can overstate SNR.

Using RMS as \( \sigma \)

RMS of residuals is a reasonable proxy for per-point scatter after detrending/model fit. It’s a pragmatic choice that makes the calculation easy and consistent with common aperture/detrending optimization workflows.

Units

By assuming depth and RMS are in ppt, the computation stays dimensionless and avoids unit mistakes.