Continuous Lensing Models: Discrete Data

[BOAS]

Hello,

I am not sure if this question is better suited to the mathematics section, but I thought it would be easier to explain the problem here.

In Schneider, Kochanek and Wambsganss's "Gravitational Lensing: Strong Weak and Micro" pages 279-280, they derive a relation for determining the azimuthally averaged mass profile of a lens, from a measurement of the tangential shear averaged over concentric circles.

$$\langle \gamma_t \rangle = \bar \kappa - \langle \kappa \rangle$$

where ##\bar \kappa## is the disc average of the convergence and ##\langle \kappa \rangle## is the ring average of the convergence.

I have some simulated data for a Singular Isothermal Sphere that I am trying to analyse. The data consists of 10,000 spatially distributed points, each with shear values.

Evidently, when I compute a ring average with my data, I am actually averaging a discrete number of data points over an annulus, not a continuous field over a ring. I'm not sure how to approach quantifying the error in this method, or adapting the analysis for discrete data. Are there standard methods to approach this kind of problem? I essentially have a discretely sampled dataset of a continuous field, but I don't think I can assume all the same relations will apply to my data as they do the fields.

I hope that my question makes sense, and if I need to provide any more information to make things clearer i'd be happy to do so.

Many thanks!

 

[kimbyd]

As it looks like you're just taking simple averages, if the data points all have the same error ##\sigma_d##, then the error on the average is simple:

$$\sigma_{\langle d \rangle} = {\sigma_d \over \sqrt{N}}$$

If the data points don't have the same errors, then you can do an error-weighted average:
https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Dealing_with_variance

Note that for doing this kind of thing in general, you may want to learn about maximum likelihood methods. These are the preferred techniques for more complicated examples where we want to use discrete data points to estimate a continuous model. This particular model seems simple enough that you can just use error-weighted averages, though.