GNU Astronomy Utilities

7.3.1 Depth and limiting magnitude

Different observations have different noise properties and different detection methods (or one method with a different set of parameters) will have different abilities to detect certain objects in an image. Therefore it is very important that there be a scale on which we can compare different observations (images) and detection methods to objectively quantify the noise and our ability to detect signal in it.

Due to the presence of correlated noise in processed images (which are used for scientific deductions), we cannot simply deduce the limiting signal properties from those of the noise. Hence a different measure is needed for each. To quantify the level of noise, we define depth and to quantify the ability to reliably detect/study objects with that methodology we define the magnitude limit. In astronomy, it is common to use the magnitude (a unit-less scale) and not physical units, see Flux Brightness and magnitude. Therefore both these measures will be in units of magnitudes, but since magnitudes don’t have units, we are just showing them like units as a place holder for clarity.

Depth

[magnitude] As we make more observations on one region of the sky and add the images over each other, we are able to decrease the standard deviation of the noise in each pixel⁹³. Qualitatively, this decrease manifests its self by making fainter (per pixel) parts of the objects in the image more visible. Quantitatively, it increases the Signal to noise ratio, since the signal is fixed but the noise is decreased. It is very important to have in mind that noise is defined per pixel (or independent data measurement unit), not per object.

You can think of the noise as muddy water that is completely covering a flat ground⁹⁴ with some hills⁹⁵ in it. Let’s assume that in your first observation the muddy water has just been stirred and you can’t see anything through it. As you wait and make more observations, the mud settles down and the depth of the transparent water increases as you wait. The summits of hills begin to appear. As the depth of clear water increases, the parts of the hills with lower heights can be seen more clearly.

The outputs of NoiseChisel include the Sky standard deviation ($\sigma$) on every group of pixels (a mesh) that were calculated from the undetected pixels in that mesh, see Tiling an image and NoiseChisel output. Let’s take $\sigma_m$ as the median $\sigma$ over the successful meshes in the image (prior to interpolation or smoothing, see Grid interpolation and smoothing). Note that even though on different instruments, pixels have different physical sizes (for example in $\mu$m), nevertheless, a pixel is the unit of data collection. Therefore, as far as noise is considered, the physical or projected size of the pixels is irrelevant. We thus define the depth of each data set as the magnitude of $\sigma_m$.

As an example, the XDF survey covers part of the sky that the Hubble space telescope has observed the most (for 85 orbits) and is consequently very small ($\sim4$ arcmin$^2$). On the other hand, the CANDELS survey, is one of the widest multi-color surveys covering several fields (about 720 arcmin$^2$) but its deepest fields have only 9 orbits observation. The depth of the XDF and CANDELS-deep surveys in the near infrared WFC3/F160W filter are respectively 34.40 and 32.45 magnitudes. In a single orbit image, this same field has a depth of 31.32. Note that a larger magnitude corresponds to less brightness, see Flux Brightness and magnitude.

Magnitude limit

[magnitude] Because of noise and methodology, no detection algorithm can be perfect and this parameter specifies the limits of the combined data and methodology used for detection. Assuming a fixed shape for all the targets, for any algorithm and its accompanying set of input parameters, there is always a certain magnitude limit below which the detections (pseudo-detections or clumps in NoiseChisel, see NoiseChisel) are no longer usable/reliable.

While adding more data sets does have the advantage of decreasing the standard deviation of the noise, it also produces correlated noise. Correlated noise is produced because the raw data sets are warped (rotated, shifted or resampled in general, see ImageWarp) before they are added with each other. This correlated noise manifests as a ‘smoothing’ or ‘blurring’ over the image. Therefore pixels in added images are no longer separate or independently measured data elements, they are ‘correlated’ and this produces a hurdle in our ability to detect objects in them.

To find the limiting magnitude, you have to use the output of MakeCatalog and plot the number of objects as a function of magnitude with your favorite plotting tool, this is called a “number count” plot. It is simply a histogram of the catalog in each magnitude bin. This histogram can be used in many ways to specify a magnitude limit, for example see Akhlaghi et al. (2015, in preparation) for one method of using multiple depth images in order to find this limit.

For any data-set and detection algorithm (with a specific set of parameters), the depth and limiting magnitudes can differ. The first is reported in the comments section of the catalog plain text file. Note that the accuracy in estimating the zero-point magnitude is a very important factor in an accurate comparison between magnitudes measured for different images, on possibly different instruments and cameras, see Flux Brightness and magnitude.

Read in other formats.

JavaScript license information

GNU Astronomy Utilities 0.1.152-9546 manual, September 2016.