Measurement equation

The measurement equation (also known as the Radio Interferometer Measurement Equation; RIME) used in hyperdrive's calculations is:

\[ V_{s,f}(u_f,v_f,w_f) = \int\int S_{s,f}(l,m) e^{2 \pi i \phi} \frac{dl dm}{n} \]

where

• \( V_{s,f}(u_f,v_f,w_f) \) is the measured visibility in some Stokes polarisation \( s \) for some frequency \( f \) at baseline coordinates \( u_f, v_f, w_f \);
• \( S_{s,f} \) is the apparent brightness in the direction \( l, m \) at the same frequency \( f \);
• \( i \) is the imaginary unit;
• \( \phi = \left(u_fl + v_fm + w_f(n-1) \right) \); and
• \( n = \sqrt{1 - l^2 - m^2} \).

As we cannot ever have the true \( S_{s,f} \) function, we approximate with a sky-model source list, which details the expected positions and brightnesses of sources. This effectively turns the above continuous equation into a discrete one:

\[ V_{s,f}(u_f,v_f,w_f) = \sum S_{s,f}(l,m) e^{2 \pi i \left(u_fl + v_fm + w_f(n-1) \right)} \]

hyperdrive implements this equation as code, either on the CPU or GPU (preferred), and it is a good example of an embarrassingly parallel problem.