Antenna Position

Fundamentals

A synthesis imaging radio instrument consists of a number of radio elements (radio dishes, dipoles, or other collectors of radio emission), which represent measurement points in u,v,w space. We need to describe how to convert an array of dishes on the ground to a set of points in u,v,w space.

E, N, U coordinates to x, y, z

The first step is to determine a consistent coordinate system. Antennas are typically measured in units such as meters along the ground. We will use a right-handed coordinate system of East, North, and Up (E, N, U). These coordinates are relative to the local horizon, however, and will change depending on where we are on the spherical Earth. It is convenient in astronomy to use a coordinate system aligned with the Earth's rotational axis, for which we will use coordinates (x, y, z) as shown in Figure 1. Conversion from (E, N, U) to (x, y, z) is done via a simple rotation matrix:

Fig. 1: The relationship between E, N, U coordinates and x, y, z coordinates, for a latitude $\lambda$. The direction of z is parallel to the direction to the celestial pole. The directions y and E are the same direction.

${\begin{bmatrix}x\\y\\z\\\end{bmatrix}}={\begin{bmatrix}0&-\sin \lambda &\cos \lambda \\1&0&0\\0&\cos \lambda &\sin \lambda \\\end{bmatrix}}{\begin{bmatrix}E\\N\\U\\\end{bmatrix}}$

which yields the relations:

${\begin{aligned}x&=-N\sin \lambda +U\cos \lambda \\y&=E\\z&=N\cos \lambda +U\sin \lambda \end{aligned}}$

Baselines and Spatial Frequencies

Note that the baselines are differences of coordinates, i.e. for the baseline between two antennas we have a vector:

Fig. 2: Geometry of an interferometer baseline where a delay $\tau$ is inserted in one antenna, in order to steer the phase center to a direction $\theta _{o}$ from the vertical $\lambda$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{B} = (B_x, B_y, B_z) = (x_2 - x_1, y_2 - y_1, z_2- z_1)}

This vector difference in positions can point in any direction in space, but the part of the baseline that matters in calculating u,v,w is the component perpendicular to the direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta _o} (the phase center direction), which we called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{proj}} in Figure 2. Let us express the phase center direction as a unit vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s_o}} $=(h_{o},\delta _{o})$, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_o} is the hour angle (relative to the local meridian) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_o} is the declination (relative to the celestial equator). Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{proj} = \vec{B} \cdot \vec{s_o}= B \cos \theta _o} .

Recall that the spatial frequencies u,v,w are just the distances expressed in wavelength units, so we can get the u,v,w coordinates from the baseline length expressed in wavelength units from the following coordinate transformation (see Thompson 1999 for details):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} u \\ v \\ w \\ \end{bmatrix} = \vec{B_{\lambda}} \cdot \vec{s_o} = (1/\lambda) \begin{bmatrix} \sin h_o & \cos h_o & 0 \\ - \sin \delta _o \cos h_o & \sin \delta _o \sin h_o & \cos \delta _o\\ \cos \delta _o \cos h_o & - \cos \delta _o \sin h_o & \sin \delta _o \\ \end{bmatrix} \begin{bmatrix} B_x \\ B_y \\ B_z \end{bmatrix} } $(1)$

How baseline errors can contribute to the error in phase

The geometric phase difference at the phase center (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} term in (1)) is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_g = 2\pi\tau_g\nu = (2\pi/\lambda)[B_x\cos \delta_o \cos h_o - B_y\cos \delta_o\sin h_o + B_z\sin \delta_o]}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau _g = \vec{B} \cdot \vec{s} /c} , geometric delay. We can see what can affect the geometric phase by taking the differential of this expression:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} d\phi_g = 2\pi\nu d\tau_g = (2\pi/\lambda)[&dB_x\cos \delta_o \cos h_o - dB_y\cos \delta_o \sin h_o + dB_z\sin \delta_o \\ +\ &d\alpha_o\cos \delta_o (B_x\sin h_o + B_y\cos h_o) \\ +\ &d\delta_o (-B_x\cos h_o \sin \delta_o + B_y\sin h_o \sin \delta_o + B_z\cos \delta_o)]\end{align}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2)}

where we have used the relation between right ascension and hour angle: $h_{o}=LST-\alpha _{o}$, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dh_o = -d\alpha_o} . Equation (2) shows how baseline errors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (dB_x, dB_y, dB_z)} and source position errors (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_o} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_o} ) will affect the error in group delay $d\tau _{g}$ (or yield an error in phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\phi_g} ). Note that a clock error is equivalent to a source position error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\alpha_o} .

If we have a source whose position is known, we can use Equation (2) to find the location of the antennas (this is called baseline determination). The error in antenna position is largely independent of the baseline lengths. For example, say that we can measure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\phi_g} to within 1 degree at 5 GHz (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} = 6 cm). Then we can measure $dB_{x}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dB_y} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dB_z} to a precision of order (1 / 360) 6 cm ~ 1 / 60 cm even though Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = (B_x^2 + B_y^2 + B_z^2)^{1/2}} = 5000 km or more (VLBI).

The time of day and location of the antennas must be known to relatively high accuracy -- needed for determining the geometric delay. A clock error of 1 s, or a baseline error of a few cm, will cause a serious phase shift of the source over, say, 10 minutes. At OVRO, using a GPS clock and measuring baselines with cosmic source calibration, we get a time accuracy of << 1 ms, and baseline errors of about 3 mm. Therefore, these effects are not serious over a short time interval, but may still be problematic over 8 hours. This is one reason that we do phase calibration observations every ~ 2 hours.

EOVSA Antenna Position Calibration

The positions of EOVSA antennas are determined using observations by the 27-m (Ant 14) low-frequency receiver (S and C band) of celestial radio sources during several observation runs in fall 2016. This document describes the procedure followed and the final? calibrated antenna positions.

For calibrator sources with locations with sufficient accuracy (we use caibrators from the VLA Calibrator Manual), and a good time-keeping accuracy at EOVSA (what is our time-keeping accuracy? --Bchen 19 November 2016) , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\alpha_o} and $d\delta _{o}$ in Eq. 2 can be neglected. Hence Eq. 2 can be simplified to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_g = \phi_o + (2\pi/\lambda)(dB_x\cos \delta \cos h - dB_y\cos \delta \sin h + dB_z\sin \delta)} ,

where $\phi _{o}$ is the intrinsic instrumental phase at the given baseline.

We use a two-step calibration to solve for the EOVSA baseline error as following:

1. Determine baseline errors in X and Y

Observing one strong and point-like calibrator for a sufficiently long time (at least several hours). Note it is important to observe for a long time in order to have sufficient variation of the phase vs. hour angle curve as determined by sin(h) and cos(h). We use a function of the following form to fit the observed phases at a baseline involving antenna i and j:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{ij} = (2\pi/\lambda)(c_0 + c_1 \cos h + c_2 \sin h) }

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} c_0 & = \phi_{oij}/(2\pi/\lambda) + \sin \delta (dB_{zi}-dB_{zj}) \\ c_1 & = \cos \delta (dB_{xi}-dB_{xj}) \\ c_2 & = -\cos \delta (dB_{yi}-dB_{yj}) \end{align} } (4)

In a usual case, visibilities are measured at many baselines (e.g., for N antennas one would normally have N(N-1)/2 unique baselines). In that case, one can solve for the antenna-based phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{i}} as a function of hour angle for each antenna i. The resulted fit parameters c₁ and c₂ then only involve the absolute position error dB_i for antenna i. For EOVSA, we only have one 27-m antenna in the array, so we have to use the 13 baseline-based phases $\phi _{i-14}$ to solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dB_{(x, y, z)i}-dB_{(x, y, z)14}} . For simplification, I will omit the subscripts (i-14) In the following discussions.

For each antenna i-14 baseline pair, we have two unique polarization measurements. To take advantage of both polarization measurements, we fit the following equations separately:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi_{XX} + \phi_{YY} & = (2\pi/\lambda)(c_{0XX} + c_{0YY} + 2c_1 \cos h + 2c_2 \sin h) \\ \phi_{XX} - \phi_{YY} & = (2\pi/\lambda)(c_{0XX} - c_{0YY}) \end{align} }

and obtain the four parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{0XX}} , $c_{0YY}$, $c_{1}$, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2}} . The baseline error for antenna i (relative to antenna 14) is then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} dB_{x} & = \frac{c_1}{\cos \delta } \\ dB_{y} & = -\frac{c_2}{\cos \delta} \\ \end{align} }

After the baseline errors in X and Y (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dB_x and dB_y} ) are determined, these values can be applied to the visibility date to correct for the sinusoidal variation of the phase vs. hour angle. A fit example is given in Fig. 3.

Fig. 3: Phase vs. hour angle for Antennas 9, 10, 11, 13 w.r.t. Antenna 14 at both XX and YY polarizations. Circular symbols are measured phases and curves are the corresponding sinusoidal curves. Different colors represent measurements/fits at different frequency channels