Joseph H. Taylor, K1JT
272 Hartley Ave, Princeton, NJ 08540: k1jt@arrl.net

High-Accuracy Prediction andMeasurement of Lunar Echoes
K1JT describes a series of recent lunar echo measurements at
144 and 432 MHz, and tests his new EMEcho software to
predict and measure Doppler shift, frequency spread,and polarization of EME signals using amateur equipment.
1. Introduction
For me, one of the fascinations of amateur
Earth-Moon-Earth (EME) communicationis the range of interesting physics that
accompanies it. Motivated in part by aneed to test a new software program called
EMEcho, and also by a desire to see how wellI could predict and measure the phenomenaof Doppler shift, frequency spread, andpolarization of EME signals using amateur
equipment, I recently made an extensive
series of lunar echo measurements at 144
and 432 MHz. This paper describes how themeasurements were made and presents a
selection of results.
Doppler shifts of EME signals arecaused by continuous changes in the total
line-of-sight distance between a transmitter,reflecting or scattering spots on the lunar
surface, and a receiver. The relevant rates
of change are usually dominated by Earthrotation, which at the equator amounts toabout 460 m/s. As a consequence, two-way
Doppler shifts can be as large as ± 440 Hz
at 144 MHz, ± 4 kHz at 1296 MHz, and± 30 kHz at 10 GHz. Different reflection
points on the lunar surface produce slightly
different Doppler shifts, so the echo of a
monochromatic signal is spread out over asmall and predictable frequency range. The
full range of spread can be as large as 4 Hz at
144 MHz and 300 Hz at 10 GHz. However, at
VHF and UHF a majority of reflected power
is returned from a region near the center ofthe lunar disk, so the observed half-power
Doppler spread is always considerably lessthan the full limb-to-limb amount.

A smooth moon would produce a
specular reflection that preserves linear
polarization and reverses the sense of circularpolarization. A rough moon (on the scaleof one wavelength) would produce diffuse
echoes and significant depolarization;
cross-polarized return echoes might bejust a few dB weaker than the dominantpolarization. At VHF and UHF frequencies
the circumstances are closer to the specular
limit. Received echoes should be almost fullypolarized, and with linear polarization theyshould have a polarization angle that depends
on geographic locations of the transmitterand receiver and the amount of ionospheric
Faraday rotation.
Together with our knowledge of solarsystem dynamics, the relevant physics
is such that EME Doppler shifts can becalculated with high accuracy (parts in 1010,
or better) for any time and any terrestrial
location. Maximum Doppler spread acrossthe full lunar disk is also predictable. Faraday
rotation depends on latitude, moon elevation,
time of day, solar activity, and ionospheric“weather”; the resulting effects are generally
not predictable in detail. For optimum
efficiency, EME operators must know
about and take account of this full range ofphenomena, both predictable and otherwise.
2. Equipment Setup
I used single-station echo tests tomeasure Doppler shift, frequency spread,and polarization during the moon pass ofJanuary 2-3, 2015. My equipment was thatof the 144 MHz EME station at K1JT andthe 432 MHz station at W2PU, the Princeton
University Amateur Radio Club. The twostations are configured in a similar way.
Both have four dual-polarization Yagis —4×2Mxp28’s at 144 MHz1 and 4×15LFAJT’s at 432 MHz.2 Both stations use asingle low-loss feed line for transmitting
and separate LNAs and receive feed lines
for each polarization. The receivers use
dual-channel down-converters to produce
four baseband signals, I and Q (in-phaseand quadrature) for each polarization.
WSE converters by SM5BSZ3 were used at
144 MHz, and the IQ+ receiver by HB9DRI4
at 432 MHz. Four-channel sound cards(M-Audio Delta44) digitize the I/Q signals
at 96000 samples per second, and in normal
EME operation all further processing takes
place in the computer programs Linrad5and MAP656. For this echo experiment my
new program EMEcho was used in placeof MAP65. Transmitter power was about
500 W at the antenna, at each station.
2.1 EMEcho Software
EMEcho is a new program designed tomake reliable tests of lunar echoes froman amateur EME station. It goes beyond
the Echo mode available in WSJT in two
important ways. Doppler calculations aredone with state-of-the-art accuracy, basedon the Jet Propulsion Laboratory’s DE405
planetary ephemeris.7 In addition, EMEchotakes full advantage of a dual-polarization
system by measuring the polarization as wellas the frequency and strength of echo signals.
The basic echo-testing cycle is similar
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+10Frequency Offset (Hz)144 MHz, Matched Polarization

0-10Frequency Offset (Hz)+10144 MHz, Cross-Polarized

0

-10
21

22

23

0

1

2

3
4
UTC

5

6

7

8

9

10

Figure 1 — Measured echo power at 144 MHz as a function of frequency offset and time. Upper:
linear polarization angle matched to that of the echo. Lower: orthogonal linear polarization.

Figure 2 — Frequency structure of echoes at 144 MHz near meridian transit (solid line) and near
2130 UTC, a time of minimum libration rate (dashed line). The upper panel uses an expanded
vertical scale to show the weakest spectral features. Horizontal bars in the lower panel indicate
the full range of predicted Doppler spread. The pair of curves with intensity a few percent of
maximum show the measured cross-polarized power, which in this case may be mostly a
result of minor alignment imperfections in the K1JT antenna. Horizontal bars in the lower panel
indicate the full limb-to-limb ranges of predicted Doppler spread.

2 QEX September/October 2016

to that used in program WSJT. The cycle
repeats every six seconds, starting at 0, 6,
12, ... seconds of a UTC minute. A fixedfrequency tone is transmitted for 2.3 s,
the echo is received and recorded about
2.5 s later, and the spectrum is computed,
displayed, and (if desired) recorded in adisk file. In a dual-polarization MAP65compatible system, spectra can be displayedfor both the matched linear polarization andthe orthogonal polarization.
3. Measurements
A six-second measurement cycle meansthat some 8400 2.3-second pulses weretransmitted at each station over the fullmoon pass. A few of the return echoes were
rejected for failing a simple interference test;
the remainder were averaged in groups of
10 to produce about 800 sets of polarized
spectra. Measurements are reported here
for both 144 and 432 MHz. They include
frequency profiles of the echoes with bin
spacing 0.37 Hz, Doppler shifts accurate
to around 0.1 Hz, and polarization angles
accurate to a few degrees.
3.1 Doppler Shift and Doppler Spread
Figure 1 is a grayscale plot showing
matched-polarization and cross-polarizedecho strengths as a function of frequency
and time. No ad hoc frequency adjustments
have been made; the plotted “Frequency
Offset” is that of the received spectrum
relative to the predicted Doppler shift. The
Doppler shift, in turn, is based on station
location, UTC according to the internetsynchronized computer clock, and the JPL
DE405 planetary ephemeris. The grayscalechosen for Figure 1 is logarithmic, so as toemphasize the weakest features.
It’s easy to see that the frequency width ofreturn echoes is greater in the middle of the
run than near either end. These differences
are consistent with the predicted dependenceof Doppler spread during the course of amoon pass. Further details of this effect canbe seen in Figure 2, where spectra have been
averaged over about an hour near the timesof minimum and maximum libration rate.
Dashed curves represent the echo profilearound 2130 UTC, shortly after moonrise,
while solid curves are the average spectral
profile around 0330 UTC, near lunar
culmination.
Figures 3 and 4 display the corresponding
results obtained at 432 MHz. Again themeasured frequency offsets are essentiallyzero (within the measurement uncertainties).
Doppler spreads are rather more than 3 times
larger than at 144 MHz, owing to the higher
frequency and the somewhat larger size ofthe lunar reflecting region. Cross-polarizedecho signals are essentially undetectable in+10
Frequency Offset (Hz)432 MHz, Matched Polarization

0

-10
+10
Frequency Offset (Hz)the grayscale plot and only barely visible in
the expanded view (upper panel) of Figure 4.
It is interesting to see that at both frequencies
the weak wings of the spectral profile
extend out to the full calculated limb-tolimb Doppler spread — as they should, withadequate sensitivity. These effects have been
noted before by EME operators.8,9
I consider the Doppler calculations
used in this paper to be the best achievablewith today’s knowledge of solar system
dynamics. The JPL DE405 ephemeris
represents a numerically integrated modelof the solar system based on several
hundred years of astronomical observations,radar observations of planets out as far as
Saturn, and tracking observations of many
interplanetary spacecraft. Positions andvelocities of the Earth and Moon (and manyother solar system objects) are tabulated in adata file suitable for numerical interpolation.
Doppler calculations in EMEcho include thefollowing steps:
1. Convert geodetic coordinates of
the antenna to geocentric coordinates,
accounting for the Earth’s oblateness.
2. Convert UTC to UT1 and to LAST
(local apparent sidereal time). Note that UTC
runs at an essentially constant rate definedby the average of many atomic clocks, plus
occasional leap seconds. UT1 represents theactual measured rotation of the Earth. UTC
and UT1 can differ by up to ± 0.9 s.
3. Compute 3-dimensional position andvelocity of the antenna with respect to centerof Earth.
4. Convert UTC to ET (Ephemeris Time),
accounting for all leap seconds up to thepresent.
5. Interpolate the DE405 ephemeris
to obtain the 3-dimensional position andvelocity of center of moon relative to centerof Earth.
6. Combine results of items 3 and 5 to get
position and velocity of antenna with respect
to moon.
7. Calculate Doppler shift from the lineof-sight component of velocity obtained initem 6.
Achieving the accuracies required to
produce the results presented in Figures 1
through 4 requires knowledge of antenna
coordinates to better than one km andclock accuracy better than one second.
The transmitter frequency in the antenna’sreference frame must also be specified
accurately: for example, 144.118 MHz
rather than 144 MHz. The Doppler
calculations must avoid certain shortcuts andapproximations that have typically been usedin amateur EME-related software, including
my own programs WSJT and MAP65.
Figure 5 illustrates some potential
consequences of ignoring one or more of the

432 MHz, Cross-Polarized

0

-10
21

22

23

0

1

2

3
4
UTC

5

6

7

8

9

10

Figure 3 — Measured echo power at 432 MHz as a function of frequency offset and time. Upper:
linear polarization angle matched to that of the echo signal. Lower: the orthogonal linear
polarization.

Figure 4 — Frequency structure of echoes at 432 MHz near meridian transit (solid line) andnear 2130 UTC, a time of minimum libration rate (dashed line). The upper panel uses an
expanded vertical scale to show the weakest features. Horizontal bars in the lower panel
indicate the full limb-to-limb ranges of predicted Doppler spread.

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3

Figure 5 — Examples of contributions that might affect the accuracy of computed Doppler shifts.

Figure 6 — Measured polarization angles at 144 MHz (left scale) and 432 MHz (right scale).

4 QEX September/October 2016

warnings mentioned above. To produce thisgraph I used GPS-measured coordinates ofthe W2PU antenna and calculated Doppler
shifts at a nominal frequency of 1.0000 GHz,
at frequent intervals from moonrise to moon
set on January 2-3, 2015, the date of my echotest observations. The horizontal straight line
at zero represents my supposed state-of-theart calculation. The ten numbered curves
show the differential effects of various
changes, omissions, or assumptions in the
Doppler calculation, as follows:
1. Antenna moved 1 km East.
2. Antenna moved 1 km North.
3. Antenna moved to 1 km higher
elevation.
4. UTC clock error +1 s.
5. Frequency changed to 1.0001 GHz.
6. Time difference UT1‑UTC ignored.
7. Nutations ignored.
8. Calculation of moon position based ona closed-form series expansion rather thaninterpolation of the DE405 ephemeris.
9. Doppler calculation made by program
EME Planner, by VK3UM.12
10. Doppler calculation made by program
EME System, by F1EHN.11
I call particular attention to several points
relating to Figure 5. First, the Doppler
calculations in EME Planner and EME
System are very good — in this example,
accuracies better than 1 Hz at 1 GHz,
through a full moon pass. Note that in orderto obtain this accuracy you must start withvery accurate station coordinates: a six-digit
Maidenhead locator is not good enough.
Your computer clock must be synchronized
to UTC and your software must be updatedwith the latest leap seconds. And you must
use the actual transmitter frequency, not justthe frequency of the band edge, for example.
Careful use of EMEcho should give Doppler
predictions even better than those of EME
Planner and EME System, accurate to about
1 Hz at 10 GHz.
3.2 Polarization
In addition to Doppler shift and Doppler
spread, my experiment yielded polarization
measurements at both 144 and 432 MHz.
The polarized spectra recorded on disk wereaveraged over five-minute intervals. The
resulting polarization angles are plotted in
Figure 6. A solid line connects sequential
measurements at 144 MHz. As shown by thescale at left, these angles increased graduallyfrom about 60 degrees to 200 degrees in thetwo hours around local sunset. The angles
then decreased through some 1080 degrees— three full turns — over the next five hours.
Subsequent angles decreased only slightly
more from 0400 UTC (an hour before local
midnight) until moon set, about two hours
before sunrise.
Polarization angles measured at 432 MHz

are plotted as filled triangles in Figure 6,
using the scale at right. Note that the left
and right scales are in the ratio of 9 to 1. The
close tracking of the scaled results at thetwo frequencies is an excellent confirmationthat Faraday rotation scales inversely as thesquare of frequency.
4 Conclusion
Program EMEcho requires a MAP65compatible EME station and was written
mainly for my personal use. However, allof its features except the dual-polarization
capability have been incorporated into the
WSJT-X, the latest program version in the
WSJT project.12 WSJT-X also offers a numberof features such as automatic rig control andDoppler tracking, which make it especiallyattractive for amateur EME communication
on any band.
Joe Taylor was first licensed as KN2ITP
in 1954, and has since held call signs K2ITP,WA1LXQ, W1HFV, VK2BJX and K1JT. He
was Professor of Astronomy at the Universityof Massachusetts from 1969 to 1981 andsince then Professor of Physics at Princeton
University, serving there also as Dean of theFaculty for six years. He was awarded theNobel Prize in Physics in 1993 for discovery of
the first orbiting pulsar, leading to observations
that established the existence of gravitational
waves. After retirement he has been busy
developing and enhancing digital protocols forweak-signal communication by Amateur Radio,
including JT65 and WSPR. He chases DX from
160 meters through the microwave bands.
Notes
www.m2inc.com.
Joe Taylor, K1JT, and Justin Johnson,GØKSC. Dubus, 4/2014, p 53.
3
www.sm5bsz.com/linuxdsp/optrx.htm.
4
www.linkrf.ch/.
5
www.sm5bsz.com/linuxdsp/linrad.htm.
6physics.princeton.edu/pulsar/K1JT/
map65.html.
7
ftp://ssd.jpl.nasa.gov/pub/eph/planets/
fortran/userguide.txt.
8
www.sm5bsz.com/sm5frh/sm5frh.htm.
9physics.princeton.edu/pulsar/K1JT/
EME2010_K1JT.pdf.
10
www.vk3um.com/eme%20planner.html.
11
www.f1ehn.org/.
12
physics.princeton.edu/pulsar/K1JT/wsjtx.
html.
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