Reference: Gaussian Modeling with MIRIAD maths
2012 April 17
A note to self. Sometimes I want to use maths to generate an image with a
large Gaussian component. Let’s say that the total flux I want to capture is F
(in Janskys). Using what we know about normalized Gaussians the expression I
want is something of the form
![\frac{F}{2\pi\sigma_x\sigma_y} \exp\left(-\frac{1}{2}\left[\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2}\right]\right)](https://s0.wp.com/latex.php?latex=%5Cfrac%7BF%7D%7B2%5Cpi%5Csigma_x%5Csigma_y%7D+%5Cexp%5Cleft%28-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cfrac%7Bx%5E2%7D%7B%5Csigma_x%5E2%7D%2B%5Cfrac%7By%5E2%7D%7B%5Csigma_y%5E2%7D%5Cright%5D%5Cright%29&bg=ffffff&fg=000000&s=0 "\frac{F}{2\pi\sigma_x\sigma_y} \exp\left(-\frac{1}{2}\left[\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2}\right]\right)")
. Assuming circularity and simplifying, we get
![A\exp\left(B\left[x^2+y^2\right]\right)](https://s0.wp.com/latex.php?latex=A%5Cexp%5Cleft%28B%5Cleft%5Bx%5E2%2By%5E2%5Cright%5D%5Cright%29&bg=ffffff&fg=000000&s=0%22A%5Cexp%5Cleft(B%5Cleft%5Bx^2+y^2%5Cright%5D%5Cright)%22)
, where
and
.
For now I’ve thought through the implementation using arcsecond units. Say we have a 2047×2047 image with a pixel scale of 10 arcsec that’s uniform across the image. If I want σ to measure the ATA primary beam, with a FWHM of 3.5°/GHz, I get σ = 5350.7 arcsec/GHz. To generate a model-type image, we need to work in Jy/pixel, so A must be scaled by the arcsec²/pixel conversion ratio, 100 in this case.
Implementing this in maths is straightforward. You use the xrange and
yrange keywords to implement the Gaussian expression, ranging them from
-10230 to +10230 in the example here.