The Effect of the Differential Antenna Height Term in the Delay
Model
ALMA Technical Note Number: 7
Status: FINAL
Prepared by:

Organization:

Date:

Ed Fomalont (NRAO)

ALMA31 July 2014

THE EFFECT OF THE DIFFERENTIAL ANTENNAHEIGHT TERM IN DELAY MODELE Fomalont
July 31, 2014
SUMMARY: This memo describes the systematic residual phase offsets versus
elevation for ALMA data that are now occurring for configurations larger thanabout 400 m. It is shown that the systematics are caused by the lack of the height
difference delay term and the Earth-curvature term that are not included in theCalc model used to obtain the correlator delay model. The effects will produce
very large phase offsets between even close calibrators and targets during thelong-baseline campaign if not compensated for. The algorithms needed aredescribed, and it is hoped that these can be added to the ALMA correlator
model before the campaign begins in early September.

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Systematic Phase Residuals after Baseline Corrections:

The positions of the ALMA antennas are determined with observations of quasars
consisting of approximately 100 20-sec scans of 30 quasars distributed over thesky. These are called baseline runs. From these data, the position of the antennas (Bx, By, Bz) can be determine to < 0.1 mm accuracy, using software intelcal that operates on the ASDM, or by a python script (written by M Sugimoto and E Fomalont), that operates on the measurement set. The solutions
can be obtained with no or with wvr-delay corrections. This position accuracy
is needed to decrease the systematic phase difference between a calibrator andtarget from a typical science experiment. Often, the limit of the target imagequality is not from the short term phase fluctuations, but from the systematic
phase differences between calibrator and target. Pad position errors are thelargest, but not the only contribution, to this systematic phase difference.
As the array has expanded out beyond the 0.4 km size, the results of thebaseline runs (both with and without the wvr correction) show that there isa clear systematic phase residual with source elevation that persists even after
accurate antenna positions have been derived in good weather conditions. Thissystematic error is proportional to the pad height difference between the antenna
and a reference antenna that is taken within 100 m of the center of the array
which is flat to about 2.0 meters.
Fig. 1 (top) shows the residual phase for two antenna pads that have been
occupied for two months and are on pads with the maximum height difference
from the array center, about 9 m up and down. All ten baseline runs show
the same residual trend, although the phase noise from general atmosphericconditions change the rms noise level. The systematic phase error is almost
completely removed. The phase residual shown is for 100 GHz. It scaleswith frequency and should already be a concern at bands 7 and 9 forthe present configurations.
The expected additional delay term caused by the height difference among
the antennas was added to the data off-line, and then the antenna positions
were determined. As shown in the bottom of Fig. 1, the systematic phase error
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is almost completely removed, indicating that the delay term associated withthe antenna height difference should be included. The rms of the phase scatterwith time (this plot is not shown, only the elevation dependence) decreases
significantly when the height term is included.
The difference between the baseline parameters with and without the heightterm can differ by over 4 mm, as shown in the table below. The differences indicate that the baseline parameters can ’mimic’ some, but not all of the height
difference phase effects. [As an aside, the differences for runs with and without
wvr correction are about 1mm.] The most accurate and stable baseline parameters will be those obtained with the wvr correction and the additional delay
model terms.
BASELINE CORRECTIONS BEFORE AND AFTER HEIGHT CORRECTIONANTENNA
BX
BY
Bz
rms
(millimeters)(deg)
DA60 before
0.66
-1.04
-0.29 24.6
DA60 after
-0.71
2.24
0.98 10.7
DV15 before
0.24
-0.39
-0.38 18.8
DV15 after
1.59
-3.88
-1.57
9.3

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The Effect on the Long Baselines:

The systematic phase introduced between a calibrator and target, separated
by five degrees for the two antennas, illustrated above, range from about +10◦
to −20◦ , depending on their relative positions in the sky. This is not serious
problem for the 10-m height differences. But, these residuals are for 100 GHz
and scale with frequency.
The following table shows the height differences for some of the long baseline
pads that will be occupied in September, and range from -16m to +255m. Since
the effect scales linearly with height difference, the residual systematic phaseerror between the antennas on pad S301 and S305, for example, will increasesystematic phase errors by about 25-fold, to well over 100◦ for calibrator target
separations of 5◦ in the sky at 100 GHz. Multi-calibrator techniques can remove
some of this, but will not be the main mode of phase referencing observations
unless this is absolutely necessary which would be the case if this known delay
term is not removed.
HEIGHT DIFFERENCES FOR SELECTED PADSANTPAD
HEIGHT(m)
DA60 A090-9.1
DV15 A089
9.1
P403
-140
S301
-16
S305
-255
W204
-191
W210
172

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The Dry Height Term

Calc was written mainly for VLBI arrays in which each antenna is at a different location and the accurate delay above the antennas are calculated based on
local conditions. For small diameters arrays such as the EVLA and ALMA, theconditions at each antenna are assumed the same and this is a good approximation. However, two delay terms associated with small-diameter arrays are notincluded in Calc (although they could be): The differential delay between twoantennas that are at different heights above the mean site level. This delay isalso proportional to the refractivity of air. The other term is associated withthe curvature of the earth across the array; i.e. that antennas observe a source
at slightly different zenith angles. Both of these terms are added to the Calc
delay model used at the EVLA. A full description of the history of the heighteffect and comments from various people is given in ALMA JIRA ICT-272.
The delay associated D with the height term is:
D = N ∗ DH ∗ sec(z)(1)

where DH is the height difference, z is the zenith angle and N is the refractive
index on the ground (anywhere in the array to first order). The units of D are
those of DH. N ≈ 77.6 ∗ P/T ∗ 10−6 with P in mbars and T in K.
For the ALMA site, nominal values of P=555, T=270, giving N=0.159mm.
Hence, for a 10 meter height difference at 100 GHz, the phase is about 190◦ ∗sec(z). This large phase is not as serious as it appears, since it is the phasedifference for two close sources (say separated 5◦ in elevation) which is in theorder of 10◦ at 100 GHz.
The wet component of the delay associated with the height difference between antennas is taken out directly with the application of the wvr correction;
hence application of the wvr correction, besides removing short-term fluctuations, also removes the systematic phase difference from the ambient watervapor density near ground level.

4Earth Curvature Term:

The other delay term that is probably not included in the Calc model is thatcaused by the earth curvature between antenna elements, which means thateach antenna observes any source at a slightly different zenith angle. This termis
E = (c ∗ τ )/re ∗ L ∗ sec(z)2(2)where E = path length difference due to earth curvature, c ∗ τ is the geometricdelay in meters, re is the earth radius = 6370000 meters, z is the zenith angle,
and L = total atmospheric path delay at the zenith. L (in cm) ≈ 0.228 ∗P (millibars)
For the ALMA site P=555 mbar; for 1000 m baselines c∗τ /re = 1.57×10−4 .
At 100 GHz this produces a curvature phase difference between two antennas
up to 24◦ sec2 (z).

35

Testing the Additions:

When these additional delay models are added to the current Calc model thatis used by the ALMA correlator, a 30-min baseline run can check whether theresidual elevation-dependent phase has been largely removed.

Figure 1: Top Row: The residual phase versus elevation for the two antennas,
DA60 on A090 and DV15 on A089, with the maximum height difference inconfiguration C32-5. Each point represents the phase residual for a 20-sec scan,
after removing the best baseline parameters for the antenna. Bottom Row:shows the residual phase versus elevation after correcting the phase for thedelay associated with the antenna height difference.. The elevation dependenceis largely removed.

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