Requirements for Single-Dish Holography Parameter Specification Goal Measurement error <10  m rms <5  m rms Transverse resolution <0.1 m <0.1 m Measurement time 60 min 30 min (per observing frequency) Primary frequency, f GHz Secondary freq, f2 f2/f1>1.2 or <0.8 (at least 20% separation) Tuning about f1 and f2 Minimum step size <5 MHz <1 MHz Range >130 MHz >200 MHz Settling time <60 sec <1 sec

Design Parameters Primary frequency104.0 GHz Secondary frequency 78.9 GHz Tuning about each frequency >130 MHz range, <1 MHz steps Range300 m Transmitter height above ground 50 m Transmitter height above elevation axis 43 m Nominal polarization (xmtr and rcvrs)vertical Receiver processing bandwidth10 kHz Integrating time per measurement msec (nominally 48 msec) Derived: Scan angle for 0.1 m resolution 2.18 deg at 78.9 GHz (  1.09) 1.65 deg at GHz (  0.83) Minimum scan angle 2.29 deg due to near field geometry Transmitter beamwidth at ‑ 3 dB4.6 deg (twice antenna Transmitter antenna gain33 dB Transmitter EIRP >20  W Transmitter power to antenna >10 nW Reference antenna beamwidth, -3 dB4.6 deg (twice scan range) Main antenna feed beamwidth, -3 dB128 deg (-3 dB edge taper)

Error Budget Thermal noise <5  m rms (xmtr pwr, integ time) Feed phase pattern knowledge <5  m pk (critical) Reference antenna pattern knowledge <10  m (insensitive) Multipath interference 0.4  m (-20 dB, random phase) Frequency error 0 Near field correction error unknown RSS (except near field correction)7.2 microns

Transmitter Power Calculation, inputs Assumed hardware parameters: Receiver pre ‑ correlation bandwidthB10 kHz System temperature, each receiverT3200 K Frequency f92 GHz ( =3.26mm) Antenna diameterD12 m  3dB = o RangeR300 m Transmitter EIRPPTBD Measurement requirements: Transverse resolution  0.1 m Surface displacement accuracy  z 5  m Total measuring time (OTF scanning)t<30 min Derived Paramters: Scan angle  o (+ ‑ o )  = (l/D)(D/D) Number of measurementsK180 2 K = (sD/D) 2, oversampling factor s=1.5 Integrating time per measurementt27.8 msec t = tK + overhead, allowing 100% overhead. Reference antenna diameter d50 mm ‑ 3dB beam = 2q = l/d

Transmitter Power Calculation, outputs Reference antenna power rcvdP r (1.736e ‑ 9 P) P r = (1/16)(d/R) 2 P Main antenna power rcvd on boresight P s (0)(1.000e ‑ 4 P) P s = (1/16)(D/R) 2 P Receiver noise powerkTB4.42e ‑ 16 W On ‑ boresight noise  0 [(1.59e ‑ 22W)(P)] {1/2} Off ‑ boresight noise (P r term)  1 [(2.76e ‑ 27W)(P)] {1/2} Generally,  2 = [kTB + P r + P s (  )] kT/  P s (  ) = P s (0)[J 1 (  D/ )/(  D/2 )] 2 where  is scan angle, ‑  /2 < a <  /2. Can neglect kTB term for any P>1  W. Noise dominated by P s term until power is down by 50 dB. That happens when J 1 (2x)/x > 3e ‑ 3 => x > 31.7 =>  > 20.2l/D. Outside there, the P r term dominates.

Average Noise Over Map Very conservative estimate: With sampling every 0.75l/D (s=1.5), there are about p 27 2 = 2270 samples where the P s term dominates. Thus, there are ‑ 2270 = 30,130 samples where the P r term dominates. Of those where P s dominates, take the inner 4 to be s 0, the next 25 to be 10x lower, and the rest to be 100x lower, in accordance with the envelope of J 1 (x)/x.  avg 2 = [(4+25/ /100)        = 9.084e ‑ 4  0 2 = (1.444e ‑ 25W) P This is actually the noise from a single, real correlator. For complex correlation, we need to increase this by 2 times. From D’Addario (1982) eqn (30):  z =.044 ( /sD) 2 D 2 K {1/2}  avg / ( M 0 ) =.044 /s 2 K {1/2} (  avg / M 0 ) =.044(3.26mm)/ {2(1.444e ‑ 25W)/P} (1/2) /4.167e ‑ 7 = (2.754e4 m) {(2.888e ‑ 25W) / P} (1/2) For  z = 5  m, this implies P = 8.76  W.

Observing Strategy Ground-based single-dish holography –Transmitter on tower –OTF raster scanning max 0.5d/sec ~12 min minimum scan time –Multiple frequencies for multipath mitigation: repeat raster at each Ground-based interferometric holography –Transmitter on tower –Astronomy receivers –Test correlator Astronomical interferometric holography –Astronomy receivers, test correlator –Natural sources, primarily SiO masers