Objective measure of anisotropy

Question

If we know a function $f(\phi, \theta)$ in $\mathbb R^3$ only over a convex surface (which for simplicity let's assume a sphere of radius $r$), is there any measure for the degree of anisotropy over this surface? That is, given two functions, $f_1, f_2$ is there any objective measure $A(f)$ where we can make qualitative comparisons like $A(f_1) > A(f_2)$?

Thoughts:

• For an electrostatic field produced by a centered monopole we should get $A(f_{\text{monopole}})=0$.
• For sufficiently large $r$, the field produced by a possibly non-centered monopole versus that of a possibly non-centered dipole should be $A(f_{\text{dipole}}) > A(f_{\text{monopole}})$.
• An expansion in spherical multipole moments seems like a good start, but how would you compare the two expansions?
• Would the comparison generalize to a non-spherical but concave surface provided you normalized for the surface area?
• This may be related to the calculation of the degree of anisotropy of the CMB, but I'm not familiar with how they define anisotropy.

Answer 1

I do this sort of work in UHECR anisotropy. There are numerous techniques (which I am currently working on advancing). Expansion in spherical harmonics is popular (and what I am working on). For that we use the fact that the spherical harmonics are complete to write $$f=\sum_{\ell,m}a_{\ell m}Y_{\ell m}$$ and then from orthogonality we can write (assuming a given normalization) $$a_{\ell m}=\int d\Omega fY_{\ell m}$$ Once we have the coefficients $a_{\ell m}$ we have a number of approaches. One obvious one is to follow the approach of the CMB people and write the power spectrum as $$C_\ell=\frac1{2\ell+1}\sum_ma_{\ell m}^2$$ which is axis independent.

Then you can compare this to that from isotropy. In the pure case you can write for monopoles that $C_0>0,C_\ell=0\forall\ell>0$ where the value depends on the normalization. The remaining terms all show deviations from anisotropy. For the sum of discrete events a MC simulation is useful here.

Another common standard definition that is quite simple is to define $$\alpha=\frac{\max f-\min f}{\max f+\min f}$$ Isotropy gives $\alpha=0$ as desired.

In addition, for the simple dipole case, we can write $f=1+A\cos\theta$ where $A$ measures the strength of the dipole. $A=0$ is obviously isotropy. $A=1$ gives $f=2$ at $\theta=0$ (the "direction" of the dipole) and $f=0$ at $\theta=\pi$. Note that the 1 (and the implicit requirement $A\le1$) is to ensure that $f$ is positive-definite (which may or may not be necessary for your situation). Then, at $A=1$, we get $\alpha=1$ (and in fact it is easy to show that $A=\alpha$). Note that the three $(\ell,m)=(1,m)$ spherical harmonics are all degenerate.

A similar thing can be done for the quadrupole case, at least for the $m=0$ case, with $f=1-B\cos^2\theta$ where $B$ and $\alpha$ have a more complicated (but still simple) relationship.

For more complicated shapes you can try to match your geometry with spherical harmonics with what is known as the $K$-matrix approach presented here (arxiv abs) in section 3. In that example they consider a sphere with non-uniform exposure. I suspect that the same (or a similar) approach could be used for your case. I should warn though that the reconstruction power falls off very quickly with $\ell_{\rm{max}}$ in the $K$-matrix, so don't go any higher than you have to.

Answer 2

I agree that a good start is expanding into spherical multipole moments. I have used spherical multipole moment expansion to quantify anisotropy in my own research (pertaining to photons escaping from a simulated sphere of gas). I determine the coefficients of multipole expansion of both the gas distribution in the simulated gas sphere and the escaping photon flux. I then take the coefficients from the expansion of the escaping photon flux with the coefficients of the expansion of the gas distribution and compare the coefficients that relate to the same multipole. This unfortunately does not give a sort of "general" measure of anisotropy that you are looking for, but it does provide a direct comparison of anisotropy along specific multipoles. I don't know if this can be generalized past specific multipoles (please let me and the questioner know if it can be generalized).

As for the CMB, this figure from the Wikipedia article referenced in the question does use the coefficients of multipole expansion as their measure of the anisotropy (the $C_l$ listed on the y-axis are the coefficients of multipole expansion).