Doing photometric reduction (from image to brightness) includes the following steps:
* image cleaning (bias + dark + flat + etc.)
* identify/measure brightness of the object of interests from the cleaned image (using e.g. sextractor). Let's say the image was taken in U filter with the brightness (mA_U_image) in any arbitrary magnitude given any arbitrary zeropoint (ZP).
* identify/measure brightness of standard objects (which are objects that we know their brightness) using U filter and the same ZP. Let's say the brightness is mB_U_image.
* Find transformation: mB_U_real(AB) = mB_U_image(arbitrary with zeropoint = ZP) + correction where mB_U_real(AB) = magnitude of B in U filter in AB system (or in other magnitude system as identified by the database of B given U filter).
* Apply the transformation: mA_U_real(AB) = mA_U_image(arbitrary with zeropoint = ZP) + correction
Note: this method is simplified because both science objects and standards were assumed to be taken from the same U filter.
So, to answer your question:
* Typically, there are two frequently used magnitude systems: Vega and AB. You might need to get the Vega to AB correction if the transformation involve changing from one system to another.
* UGRIZ and ugriz: I found that sometimes UGRIZ is used to notify that the magnitude is expressed in Vega system and ugriz is in AB system, or vice versa. 964
For an example, https://arxiv.org/abs/1605.01054
* U and u and u' are slightly different (check their response curves from the telescope handbooks). So, transforming from one filter to another might need to include more terms in the transformation.
* If standards and science objects were taken from different filter/telescope/setting (even in the case of U and u), you might need to also include a color correction term in the transformation.
* Eq. 1 is for calculating chi-sq to be used in the optimization problem. I believe that the equation is expressed in u-g color index as an example. You should be able to use any color index pair. If you really want to use u-g index, but you have UBVR, here is what I think you should do: find u = U + correction, and g = B + correction. 964