I have read the article [Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets](http://link.springer.com/article/10.1007/s00190-005-0011-y) and I don't know how to interpret Equation (10) on
page 630, because this equation modified matrix $\Phi\_2$ to vector and I don't know how. 
Authors comment to this equation is "This is done by dividing the $\Phi\_2(v\_p,v\_q)$ matrix into classes according to $\sqrt{v\_p^2+v\_q^2}$ and computing class mean values."


The equation (10) has the form
$$
 \Phi\_2(v)=\Phi\_2(\sqrt{v\_p^2+v\_q^2}),
$$
where $\Phi\_2(v\_p,v\_q)$ is the matrix of a Fourier transformation computed by
$$
F(v\_p,v\_q)=\sum\_{k=0}^{K\_x-1}\sum\_{l=0}^{K\_y-1}f(x\_k,y\_l).e^{-i2\pi(pk/K\_x+ql/K\_y)},
$$
with $p=0,1,...,K\_x-1$ and $q=0,1,...,K\_y-1$. The corresponding 2D-power spectral densities is
$$
\Phi\_2(v\_p,v\_q)=|F(v\_p,v\_q)|^2
$$



Problem with understanding an equation
$$\Phi\_2(\nu)=\int\_0^{2\pi}\Phi\_2(\nu\cos\phi,\nu\sin\phi)\nu d\phi$$


in words, you start from a function defined on the two-dimensional plane, and you average over the angular coordinate to arrive at a function that depends only on the radial coordinate.


