RIME

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Radio Interferometer Measurement Equation abbreviates as RIME.[১]

RIME of a single source

Case of a single antenna and source

Consider quasi-monochromatic signal coming from a single point source in the sky. The signal at a fixed point in space and time is described by the complex vector <math>\mathbf{e}</math> and for a xyz coordinate with z along the LOS it can be written as,

<math>\mathbf{e} = \begin{pmatrix}

e_x \\ e_y \end{pmatrix} </math>

Assumption: all transformations along the signal path w.r.t. <math>\mathbf{e}</math> are linear.

All linear transformations of a 2-vector can be represented by a matrix multiplication,

<math>\mathbf{e}' = \mathbf{J} \mathbf{e}</math>

where <math>\mathbf{J}</math> is the <math>2\times 2</math> complex matrix known as Jones matrix. There are many effects along the signal path each of which can be represented by a different Jones matrix forming the so called Jones chain, and also all the Jones matrices can be collapsed to form a cumulative Jones matrix. The order in the Jones chain should be meaintained as matrix multiplication does not commute in general.

<math>\mathbf{e}' = \mathbf{J}_n \mathbf{J}_{n-1}....\mathbf{J}_1 \mathbf{e} = \mathbf{J} \mathbf{e}</math>

This signal hits our antenna and converted to voltage by our antenna feeds. Ler's assume we have two feeds named a and b. Than,

<math>\mathbf{v} = \begin{pmatrix}

v_a \\ v_b \end{pmatrix} = \mathbf{J}\mathbf{e}</math>

So there is a linear relationship between the signal and the voltage measured by the antenna feeds. Here <math>\mathbf{J}</math> obviously represents all the effects along the propagation path. This is the total Jones matrix.

Case of a single source and an interferometer

In case of an interferometer we will get the visibility matrix. Let's say 2 antennas p and q each containing 2 feeds, a and b form an interferometer and measure 2 independent voltages <math>v_p</math> and <math>v_q</math>. These 2 voltages are fed into a correlator that produces 4 pairwise correlations. The correlations are arranged in a certain form to give the visibility matrix,

<math>V_{pq} = 2\begin{pmatrix}

\langle v_{pa} v_{qa}^* \rangle & \langle v_{pa} v_{qb}^* \rangle \\ \langle v_{pb} v_{qa}^* \rangle & \langle v_{pb} v_{qb}^* \rangle \end{pmatrix} = 2 \left\langle \begin{pmatrix} v_{pa} \\ v_{pb} \end{pmatrix} (v_{qa}^*\ v_{qb}^*) \right\rangle = 2\langle \mathbf{v}_p \mathbf{v}_q^H \rangle </math>

where H is the conjugate transpose or Hermitian transpose.

The measurement equation

The visibility matrix represents the voltages measured from the correlator output. But to get the actual signal from this matrix we need to take into account the Jones matrices. That's when RIME comes into the picture.

For 2 paths of the signal leading to the 2 antennas we have 2 respective total Jones matrices. Thus the visibility matrix with the Jones matrices becomes,

<math>V_{pq} = 2\langle \mathbf{J}_p \mathbf{e} (\mathbf{J}_q\mathbf{e})^H \rangle = 2\langle \mathbf{J}_p (\mathbf{e}\mathbf{e}^H) \mathbf{J}_q^H \rangle</math>

The average is taken over some small time and frequency bins. Assuming that the Jones matrices are constant over the averaging interval they can be taken out,

<math>V_{pq} = 2\mathbf{J}_p \langle (\mathbf{e}\mathbf{e}^H) \rangle \mathbf{J}_q^H = 2\mathbf{J}_p \begin{pmatrix}

\langle e_x e_x^* \rangle & \langle e_x e_y^* \rangle \\ \langle e_y e_x^* \rangle & \langle e_y e_y^* \rangle \end{pmatrix} \mathbf{J}_q^H</math>

Here we can define the brightness matrix,

<math>2 \begin{pmatrix}

\langle e_x e_x^* \rangle & \langle e_x e_y^* \rangle \\ \langle e_y e_x^* \rangle & \langle e_y e_y^* \rangle \end{pmatrix} = \begin{pmatrix} I+Q & U+iV \\ U-iV & I-Q \end{pmatrix} = B</math>

Thus we get the final RIME for a single source,

<math>V_{pq} = \mathbf{J}_p B \mathbf{J}_q^H</math>

which establishes a relation between the observed visibility matrix (that we derived before) and the actual sky brightness matrix via some Jones matrices. In expanded form it can be written as,

<math>\begin{pmatrix}

v_{aa} & v_{ab} \\ v_{ba} & v_{bb} \end{pmatrix} = \begin{pmatrix} j_{11p} & j_{12p} \\ j_{21p} & j_{22p} \end{pmatrix} \begin{pmatrix} I+Q & U+iV \\ U-iV & I-Q \end{pmatrix} \begin{pmatrix} j_{11q} & j_{12q} \\ j_{21q} & j_{22q} \end{pmatrix}^H</math>

This equation is independent of coordinate systems. Jones chains can be written instead of total Jones matrices,

<math>V_{pq} = \mathbf{J}_{pn}(...(\mathbf{J}_{p2}(\mathbf{J}_{p1}B\mathbf{J}_{q1}^H)\mathbf{J}_{q2}^H)...)\mathbf{J}_{qm}^H</math>

where <math>n \neq m</math>.

RIME of a single corrupted point source

The visibility given by the RIME of a single un-corrupted point source is called source coherency or just coherency,

<math>V_{pq} = K_p B K_q^H = B e^{-2\pi i[u_{pq}l+v_{pq}m+w_{pq}(n-1)]} = X_{pq}</math>

If there is corruption due to antenna gain, the RIME becomes,

<math>V_{pq} = G_p X_{pq} G_q^H</math>

Multiple discrete sources

The RIME equation for different sources s, becomes,

<math>V_{pq} = \sum_s J_{sp} B_s J_{sq}^H</math>
<math>J_{sp} = J_{spn}.....J_{sp2}J_{sp1} = G_p E_{sp} K_{sp}</math>

Here contamination 1 is in the source end, n is in the antenna end. We have separated gain (G) and phase delay (K) terms and put everything else inside E. Thus,

<math>V_{pq} = G_p \left( \sum_s E_{sp} K_{sp} B_s K_{sq}^H E_{sq}^H \right) G_q^H = G_p \left( \sum_s E_{sp} X_{spq} E_{sq}^H \right) G_q^H</math>

where <math>X_{spq}</math> is the coherency of the source s, as defined in the Jones matrix page. Here <math>G_p</math> represents the direction-independent effects (DIE) or the uv-Jones terms and <math>E_sp</math> the direction-dependent effects (DDE) or the sky-Jones terms.

This summation in theory should be done over all the bright sources. But our FOV is limited by the voltage beam pattern or the horizon in case of full sky instruments like LOFAR. Beam pattern or beam gain can be expressed as yet another Jones term. This term will ensure that <math>E_{sp} \rightarrow 0</math> for all sources outside the beam.

Full-sky RIME

In practice sky consists of a continuous brightness distributions <math>B(\sigma)</math> where we have DDEs for every direction <math>\sigma</math>,

<math>V_{pq} = \iint\limits_{4\pi} \mathbf{J}_p(\sigma) B(\sigma) \mathbf{J}_q^H(\sigma) d\Omega</math>

We perform a sine projection of the sphere onto the plane of sky, i.e. on the l,m plane,

<math>V_{pq} = \iint\limits_{lm} \mathbf{J}_p(l) B(l) \mathbf{J}_q^H(l) \frac{dl dm}{n}</math>

where l and (l,m) are being used interchangeably. Now the Jones term can again be separated,

<math>\mathbf{J}_p(l) = G_p \tilde{E}_p(l) K_p(l) = G_p \tilde{E}_p(l) e^{-2\pi i[u_pl+v_pm+w_p(n-1)]}</math>

similarly,

<math>\mathbf{J}_q(l) = G_q \tilde{E}_q(l) e^{-2\pi i[u_ql+v_qm+w_q(n-1)]}</math>

Thus the visibility can be written as,

<math>V_{pq} = G_p \left( \iint\limits_{lm} \frac{1}{n} \tilde{E}_p B \tilde{E}_q^H e^{-2\pi i[u_{pq}l+v_{pq}m+w_{pq}(n-1)]} dl dm \right) G_q^H</math>

This is the full-sky RIME. It's a 3D Fourier transform with a non-coplanarity term in the exponent. If this term can be separated it may easily become a much simpler 2D FT. The non-coplanarity term can be separated for two antennas as <math>w_{pq}=w_p-w_q</math> and each one of them can be considered as DDE,

<math>W_p = \frac{1}{\sqrt{n}} e^{-2\pi i w_p(n-1)} \text{ and } E_p = \tilde{E}_p W_p</math>

Now if the apparent sky seen by the baseline <math>pq</math> is <math>B_{pq} = E_p B E_q</math> than,

<math>V_{pq} = G_p \left( \iint\limits_{lm} \frac{1}{n} B_{pq} e^{-2\pi i[u_{pq}l+v_{pq}m+w_{pq}(n-1)]} dl dm \right) G_q^H</math>

Sky coherency

From the final all-sky RIME equation we see that we measure the 2D FT of the sky. But sky is diffreent for different baselines as apparent from the term <math>B_{pq}</math>. But self-calibration assumes that we are measuring the 2D FT of a common sky, not a collection of skies for different baselines. This is contradictory. We better make our visibility same for all baselines.

It can be done if all DDEs are identical across all antennas, <math>E_p(l) \equiv E(l)</math> or at least when <math>B(l) \neq 0</math>. As DDEs are same for all antennas apparent sky observed by different baselines will be same, which we can name <math>B_{app}</math>.

<math>B_{pq}(l) = B_{app}(l) = E(l) B(l) E^H(l)</math>

Than the full-sky RIME can be written as,

<math>V_{pq} = G_p X_{pq} G_q^H</math>

where <math>X_{pq}=X(u_{pq},v_{pq})</math> and the matrix function <math>X(\mathbf{u})</math> is just the element-by-element 2D FT of <math>B_{app}</math>

<math>X(\mathbf{u})=\mathcal{F}B_{app} = \iint\limits_{lm} \frac{1}{n} B_{app} e^{-2\pi i[u_{pq}l+v_{pq}m+w_{pq}(n-1)]} dl dm</math>.

See that, this is very similar to the RIME for a single corrupted point source derived in the Jones matrix page. In case of single source we were calling X as source coherency. Now we will call X as sky coherency. The Van Cittert–Zernike theorem (FT of the mutual coherence function of a distant, incoherent source is equal to its complex visibility) has been derived in the meantime.

Time variability

Our interferometer effectively measures the sky coherency (X) at different points in the uv-plane which is same for all baselines at a certain time. This is called snapshot measurement. Snapshot measurements give a limited sampling of the uv-plane. For better sampling we rely on the rotation of the Earth, i.e. the synthesis imaging. In this imaging technique Earth's rotation swings each baseline vector through an arc on the uv-plane over several hours.

For fulfilling the fundamental assumption of self-calibration the DDEs must remain constant over the whole observation time also,

<math>\mathbf{E}_p(t,l) \equiv \mathbf{E}_p(l) \equiv \mathbf{E}(l)</math> for all t,p

The DDEs that satisfy this equation are called trivial DDEs. They effectively replace the true sky B with the <math>B_{app}</math>. The calibration can be done entirely in the image plane. This can be true for the following cases:

  1. narrow FOV. E_p rapidly goes to zero away from the field centre
  2. small arrays, small <math>w_p</math>, all stations see through the same atmosphere
  3. higher frequencies: narrow FOV, less ionospheric effects
  4. coplanar arrays: <math>w_p=0 \Rightarrow W_p=1</math>

References

  1. This article is based on "O.M. Smirnov, 2011" papers