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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
VOLUME 14
On the Extraction of Near-Surface Index of Refraction Using Radar Phase
Measurements from Ground Targets
FRÉDÉRIC FABRY
AND
CHUCK FRUSH
National Center for Atmospheric Research, Boulder, Colorado*
ISZTAR ZAWADZKI
AND
ALAMELU KILAMBI
J. S. Marshall Radar Observatory, McGill University, Montreal, Quebec, Canada
4 October 1996 and 24 December 1996
ABSTRACT
The speed at which electromagnetic waves travel between a radar and a target is dependent on the index of
refraction of the atmosphere between the radar and the target. Modern radars can have sufficiently accurate time
bases and digitizing equipment to observe small changes in the time it takes a radar signal to travel to a fixed
target and back. These changes are related to small perturbations in the refractive index caused by changes in
humidity, temperature, and pressure. Using the phase information from ground targets and its time evolution as
a proxy for the changes in travel time of radar waves, a procedure for measuring the near-surface index of the
refraction field around the radar is demonstrated and implemented on the McGill Doppler S-band radar. This
paper describes the theory behind the measurement, and a technique used to extract refractive index data from
ground targets. Early measurements of the index of refraction field are also presented, and some of the possibilities
offered by this new radar-measured variable are identified.
1. Concept
The index of refraction has traditionally been seen in
radar meteorology as the quantity whose unusual vertical structure may cause anomalous propagation of radar waves. In parallel, it has long been recognized that
the index of refraction is strongly related to atmospheric
parameters such as pressure, temperature, and moisture
(Bean and Dutton 1968, and references therein). The
link between index of refraction and meteorological parameters makes it an interesting quantity to measure. In
recent years, considerable work has been done to measure it using the Global Positional System (GPS) on
earth–space paths and space–limb–space paths (Bevis
et al. 1992; Businger et al. 1996; Ware et al. 1996). The
link between index of refraction and the propagation of
radar waves suggests the possibility that some aspects
of this quantity may be measured by radar.
Let us consider the simplest radar equation involving
the index of refraction n: The time t taken by electro-
*The National Center for Atmospheric Research is funded by the
National Science Foundation.
Corresponding author address: Dr. Frédéric Fabry, Radar Observatory, McGill University, P.O. Box 198, Macdonald Campus, SteAnne de Bellevue, PQ H9X 3V9, Canada.
E-mail: [email protected]
q 1997 American Meteorological Society
magnetic waves to reach a target at range r and return
to the radar is
n
t 5 2r ,
c
(1)
with c being the speed of light in vacuum. This equation
is ordinarily used to determine the range of radar targets
even though this single equation has in reality two unknowns, r and n. However, n typically varies by at most
0.03% and can therefore be assumed constant for most
weather radar applications. On the other hand, for fixed
targets, r is constant and only n varies; hence, if t could
be measured precisely for such targets, the average value
of the refractive index over the path between the radar
and these targets could be determined.
2. Feasibility and sensitivity calculations
The first problem to be addressed in order to transform this theoretical concept into a practical technique
is the precision required for the time of echo return t
and for the distance r to the echo-producing target. It
is clear that neither of the two can be measured in an
ordinary radar to the needed accuracy of a few parts
per million. However, if the range to the target is fixed,
but only known to a fair accuracy, say better than 1%,
it would be enough to allow us to relate changes in t
to changes in n; the absolute calibration would then have
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NOTES AND CORRESPONDENCE
FIG. 2. Diagram illustrating the values that refractivity can take for
given temperature and moisture for a pressure of 1000 mb. In inset,
the formula used to compute N as well as some refractivity values
calculated for a dry and saturated atmosphere at a few temperatures
are shown.
FIG. 1. Illustration of the effect of index of refraction changes on
the phase of ground targets. The phases of five arbitrary targets (T1,
. . . , T5) at (a) an initial condition, and (b) its change as refractivity
increases at all ranges, or (c) increases up to a given range and
decreases beyond are presented. The dials illustrate either the current
phase of the targets (top), or the difference between the current phase
of the targets and the reference phase when n 5 no (bottom).
to be done by other means. By using this stratagem, we
are only required to determine changes in t, something
that can be done by measuring the phase of the target.
Figure 1 is helpful to understand how this works. At
an initial time (Fig. 1a), measurements of phases are
made at five targets when n 5 no. The phase of each
of these targets could be anything: It is a function of
the range to the target, the average refractive index between the radar and the target, and the shape and nature
of the target itself. If the refractive index increases everywhere (Fig. 1b), the radar wave gets compressed by
a small amount, and this results in a change of the measured phase of the targets. The magnitude of the change
is a linear function of the change in the two-way travel
time Dt of electromagnetic waves between the radar and
the targets;
4p fr
Df 5 2p fDt 5
Dn,
c
(2)
where f is the radar frequency. As a result, the phase
difference caused by changes in refractive index of air
in the path increases linearly with range and can be
easily measured to derive Dn. Note that if the index of
refraction then diminishes over a small region to a value
smaller than no (Fig. 1c), the slope of the phase difference of ground echoes between the current time and the
reference time reverses itself locally. This implies that
if enough ground clutter is available along many radials,
fields of refractive index could be retrieved by radar up
to a range of 20 to 40 km. Beyond this range, few ground
targets are visible on flat terrain.
What kind of phase changes can be expected for a
given change in n, and what does it mean in terms of
real meteorological variables? It is customary to express
the index of refraction in terms of the refractivity N,
defined as the amount that the index of refraction (in
parts per million) exceeds the value in vacuum (e.g.,
Bean and Dutton 1968);
N 5 (n 2 1) 3 106.
(3)
At microwave frequencies, the relationship between N
and the meteorological variables pressure, temperature,
and moisture is
N 5 77.6
P
e
1 3.73 3 105 2 ,
T
T
(4)
where P is pressure expressed in millibars, T is temperature in kelvins, and e is the water vapor pressure
in millibars. Equation (4) has two terms: a density term,
and an additional wet term, that makes the index of
refraction fairly sensitive to moisture. Figure 2 illustrates how refractivity varies with temperature and moisture: at cold temperatures, the air cannot hold much
moisture, and the index of refraction is mostly a function
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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
of temperature. As air temperature increases, the potential magnitude of the additional wet term increases so
that the index of refraction becomes more sensitive to
changes in moisture than to changes in temperature. As
a simple rule, a change in N of one unit corresponds
roughly to a change of 18C in temperature or 0.2 mb
(0.2 g kg21 at sea level) in water vapor. Using (2), it
can be seen that for a change of one unit in N (Dn 5
1026), a path of 1 km, and an S-band radar system (f
5 3 GHz), Df is 0.125 radians or 7.28. This implies
that a change in temperature of 18C will result in a
specific phase change (between the time when N 5 N0
and the time when N 5 N0 1 1) of 7.28 km21. This is
a significant phase change. For comparison, researchers
working on rainfall measurements techniques relying on
differential propagation (e.g., Sachidananda and Zrnic
1987) will observe such a specific differential phase
signal (two way) between vertical and horizontal polarization for downpours of over 100 mm h21. Such a
large signal when dealing with measurements of the
phase of ground targets forewarn potential aliasing problems as N can span up to 200 N units between dry and
damp days (Fig. 2), and the range between two neighboring ground targets may reach several kilometers. For
a radar wavelength of 10 cm (f 5 3 GHz), the Nyquist
interval is 625 N units for a 1-km path or 61 N unit
for a 25-km path. Note, however, that the use of the
various pathlengths between the many ground targets
around the radar offers considerable possibilities to
make dealiasing feasible.
No mention of hardware issues has been made so far
because the technique can essentially be implemented
on existing radars and data processing systems using
only additions to the processing software. The only
hardware requirement that may or may not be met in
existing systems is to ensure a stable transmit frequency
over long periods. To make proper refractive index measurements, good frequency stability is required. Although this is also the case for the phase measurements
used in the computation of Doppler velocity, the length
of time over which such stability must be achieved is
significantly different. While stability over a few hundred milliseconds is enough for Doppler processing, stability over a period of several months to a few years is
required to make accurate measurements of N. For example, a change in one part per million in frequency (3
kHz for S-band radars) over any given period will result
in a bias of 1 N unit in refractivity measurements. Since
this kind of long-term stability is not required for other
weather radar applications, the stable local oscillator
(STALO) in current operational radars may be inappropriate. Because the technique relies on the difference in
the phase of targets between the current time and a
reference time that may be months old, a drift in transmit
frequency of less than 0.5 ppm between these two times
is required to have better than 0.5-N-unit accuracy on
the refractivity measurement. Except for the stability
issue, if the radar can measure the phase of targets and
VOLUME 14
sufficient ground echoes are present, this technique can
be implemented readily.
3. Measuring fields of refractivity
The proposed technique relies on (2) to accurately
evaluate changes in index of refraction from changes in
targets phases. The first test to be made is therefore to
look at the correlation between the two quantities. For
such a test, two targets (power poles) located 20 km
away from the radar but in nearly opposite directions
were monitored for 48 h. Surface meteorological observations from the nearby Dorval airport (YUL) were
used to compute the index of refraction near the ground
using (4). The result of this test, showed in Fig. 3, shows
a remarkable correlation between the two target phases
as well as between the target phases and the computed
index of refraction. The target to the south-southwest,
generally more upstream given the southerly winds and
the movement of weather patterns from the west, often
changes in phase ahead of the northeast target. This
bodes well for attempts to extract the two-dimensional
structure of the refractivity field.
Two additional points must be made clear before trying to extract fields of N. First, whatever the elevation
pointing angle used, the refractivity measured from a
ground target at a specific azimuth and range would
always be expected to be the same. This is because this
technique uses fixed ground echoes for reference points.
Even if the antenna is raised to a higher elevation, the
target is still near the ground. One would expect only
the strength of the ground echo to change, and the path
to the target remains essentially the same. Since the
antenna is generally from a few meters to a few tens of
meters from the ground, and because most visible
ground targets are generally elevated (buildings, towers,
higher than usual terrain, etc.), the measurement of refractivity will generally be representative of conditions
a few tens of meters above the ground, with this sampling level slightly increasing with range beyond the
radar horizon. The second key point is that the measurement of fields of N is only practical in relatively
flat terrain. This is because each target phase provides
information about the average index of refraction over
the radar–target (R–T) path. In order to compute fields
of N, the contribution of the path R–T1 must be removed
from the measurement along path R–T2 in order to obtain a valid measurement for the path T1–T2. This is
only possible if the radar and the two targets are reasonably aligned. This situation generally occurs only in
flat terrain, although it is conceivable that a constantly
sloping terrain would also provide appropriate conditions. In hilly terrain, such alignment is unlikely for
many targets. On the other hand, if horizontal stratification is assumed, the echoes from various levels could
be used to retrieve an average vertical structure of N
from the level of the lowest targets to the level of the
highest targets using an inversion technique similar to
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NOTES AND CORRESPONDENCE
981
FIG. 3. Illustration of the correlation between target phase and refractivity. Two targets on power lines at
a similar range (20 km), but in opposite directions, were selected. Their position with respect to the radar
and the Dorval airport (YUL) is shown at the top-left while the top-right window shows a 0.58 PPI of received
power where the echoes selected are highlighted. In the bottom window, the time evolution of the phase of
the two targets (solid and dotted line, right scale) is contrasted with refractivity computed from YUL observations (crosses, left scale). Some weather observations from YUL are also plotted at the top of the window
using the station model; periods over which precipitation was observed are also indicated.
the one used in radiometric observations of the earth by
satellite. Since the radar used by the authors in this study
sits in the middle of a sedimentary plain, focus was put
on the flat terrain variation of the technique.
The type of ground targets also turns out to be an
important factor. Not all ground targets give suitable
returns. The ideal ground target must not move; this
may seem trivial, but this rules out all extended ground
targets that are covered by vegetation that can move
with the wind, or very tall towers that also move significantly with the wind. All areas covered by water are
also ruled out, including the radar-reflecting buoys used
for navigation. The best ground echoes are those where
the returns are dominated by a single ‘‘pointlike’’ target.
Communication towers and power poles are good examples of such targets. In these cases, the phase of the
target is also relatively insensitive to subbeamwidth inaccuracy in antenna pointing (as opposed to the case of
extended targets where complex interference may complicate matters). The easiest method to determine the
reliability of a ground echo for the purpose of this technique is to measure the phase of all targets for a period
greater than an hour in steady meteorological conditions, selecting the echoes whose phase show the least
high-frequency variability in time to eliminate capricious targets. Figure 4 shows (a) the ground targets
around the McGill radar, (b) the land usage that characterizes the targets, and (c) the targets selected by such
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FIG. 4. (a) Low-level (0.58) PPI of received power up to 45-km
range illustrating the coverage of ground echoes around the McGill
radar. (b) Sketch of the land usage and topography over the same
area. (c) Mask of the ground echoes used for the measurement of
refractivity. Coverage is good over areas covered by fixed targets
(urban areas), while much poorer over areas covered by moving targets (forests).
a process. It is seen that appropriate targets can be plentiful, especially over the city. Over several large areas,
however (e.g., tree-covered hills, shadow areas due to
obstacles), the coverage drops down significantly. One
lingering problem with the automatic selection of targets
is that echoes from sidelobes are often selected. The
sidelobe echoes are unwanted, mainly because they contaminate measurements made at a given radial with information from another radial. Ideally, one would want
to somehow prevent any computation from being performed in the contaminated cells, but this has not yet
been done in our implementation.
To extract the refractivity information from the
ground clutter data, one can proceed as follows. In
regions with limited ground echoes, individual ‘‘good’’
ground echoes are identified, and careful phase measurements are made at every scan. Each of these good
echoes is then paired with a neighboring one along the
same radial from the radar to form a path. This allows
us to measure refractivity along the path between these
two targets by using (2) at the two extremities. The
difference between the phase of these two targets is
computed to obtain a pathlength in number of radar
wavelengths modulo the closest integer number of radar
wavelengths (since only phases can be easily measured).
This pathlength measurement is then further subtracted
AUGUST 1997
NOTES AND CORRESPONDENCE
to a similar pathlength measurement made between the
same two targets during a calibration step (described
below) at a previous time. Because the pathlength
change between the calibration time and the current time
may be larger than a wavelength, a dealiasing step must
follow. Yet, there will be paths short enough so that
aliasing will not occur since any possible change in
refractive index will not be sufficient to change the pathlength by a wavelength. Hence, DN is first computed
for short paths. These DN measurements are then used
as first guesses for the next longer paths, constraining
the dealiasing problem. In theory, the best dealiasing
method is then a pyramid-like system, where short paths
are computed first, providing guidance for the next longer paths and so on until the largest path is dealiased.
In practice, given the variability in the measurements
of refractivity, a point will be reached in pathlengths
beyond which the variability in DN will result in variability in pathlength differences that becomes comparable to or exceed a Nyquist interval. For the longest
paths, accurate dealiasing may be impossible.
In clutter-rich situations, if the phases of all the targets
at the reference time are subtracted from the phases of
the same targets at the current time, one can observe
fringes in the phase difference Df field (Fig. 5b). This
image corresponds to the phase difference plotted at the
bottom of Fig. 1b. The multiple rings occur because of
the multiple folding of the phase difference as range
increases. In this case, in regions where good clutter is
present (mostly in the northeast and south for the McGill
radar) Df decreases regularly with range as the ambient
N (ø310.5 N units) is smaller than the reference N
(ø325.6 N units). Using (2), one can expect folding of
Df every 3.5 km, and this is actually observed. If the
difference between the current and reference refractivity
is uniform, fringes should make concentric circles
around the radar. If they do not, gradients in refractivity
must be present in the radar coverage area. In regions
where good ground echo coverage is large and where
such fringes appear, it is possible to measure directly
d(Df)/dr using, for example, a covariance algorithm
(such as is used in pulse-pair processing of Doppler
radar data), but in range as opposed to in time. From
this information, measurements of refractivity averaged
over the path between the targets used by the technique
(or in the regions over which the pulse-pair computation
is done) can be made (Fig. 5c).
A final optional step is to transform the path measurements of refractivity into a field (Fig. 5d). This does
not add any information but generally eases the interpretation of the data. Various approaches can be used.
In our case, we start with an initial guess field that may
be constant or the field deduced from the previous volume scan. An iterative two-step process follows. First,
the points in the field that lie along the measured paths
are forced up or down so that the field data respect the
constraints set by the path integral values. Then the
information is diffused by smoothing each point in the
983
field with its immediate neighbors. After several iterations, in data-rich regions, an equilibrium is reached as
efforts to smooth the field are exactly canceled out by
the forcing process. The final result is the smoothest
field that meets the constraints set by the path measurements. Because of the noise in the path measurements of refractivity, the resulting field shows a noticeable amount of small-scale artifacts, especially in
the form of strong–weak couplets aligned along radials.
Despite the noise, a clear larger-scale signal can generally be observed. In this example, a weak north–south
gradient can be noticed with larger refractivity values
observed to the north because of somewhat cooler and
more humid conditions there than to the south.
The main weakness of the whole technique is the need
for accurate reference or calibration information for all
the targets. Ideally, during calibration, phase measurements of targets should be made only if the refractivity
along the radar-to-target path is known using other
means. Except in the case of a very limited number of
targets, this solution is impractical. The easiest method,
and the one that has been used here, is to collect phase
data during a short period for which limited gradients
of refractivity in space and time are anticipated and
claim that these phase measurements are representative
of what should be expected for N 5 Nref everywhere,
with Nref computed from surface observations of temperature, dewpoint, and pressure. An ideal moment for
such a measurement turns out to be immediately after
a long-duration (several hours) stratiform precipitation
event, preferably in windy conditions and cool temperatures: The combined effect of the precipitation and
wind homogenizes the atmosphere, while the cool temperatures tend to narrow the possible range of values
that refractivity can take if the homogeneous conditions
are not perfectly reached.
4. Early results
Surface refractivity fields have been generated in real
time at McGill University since early June 1996 using
a calibration made in the end of April. Longer-term
comparisons made with refractivity computed from surface stations (Fig. 6) suggest that the radar measurements have definite skill.
In general, in the Montreal area, horizontal gradients
over the 30–40-km range covered by the radar are limited to about 5 N units, and except for exceptional
events, changes in time are also fairly slow (Fig. 6). As
a result, this makes refractivity one of the least contrasted of the fields that can be generated by a radar.
However, preliminary data collected with the National
Center for Atmospheric Research’s S-Pol radar suggest
that in other regions, like in the Colorado high plains
for example, considerably more small-scale variability
can be observed both in time and in space.
Even in Montreal, more rapid changes can be observed when mesoscale or synoptic fronts are within
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FIG. 5. Procedure to extract refractivity measurements from ground targets. (a) The phase of ground targets is measured; (b) the phase of
targets from the reference map are subtracted from them; (c) in regions where the phase difference data is well behaved, path-integrated
measurements of refractivity are computed using (2) and a value of 325.6 N units for the refractivity at the reference time; (d) a field is
generated using the path-integrated measurements of N. Range rings are every 10 km.
range (Fig. 7), or when surface moisture is being modified by the passage of precipitation (Fig. 8). At this
stage, the measured boundaries between regions of contrasting refractivity are somewhat fuzzy (Fig. 7) mostly
because of limitations in the current algorithm: Reluctance in the algorithm to accept phase difference measurements when strong refractivity contrasts are present,
computation of refractivity over paths of at least 1 km,
smoothing of the information during the transformation
from path information to field data, etc. Better techniques to use the phase data exist and will have to be
tried.
Coverage stays relatively constant in time except under two circumstances. The first one occurs when vertical gradients of N are significantly different than when
the calibration step was made and manifest itself by an
increase in the small-scale variance of Df. This increase
in variance is due to the fact that all the targets are not
AUGUST 1997
NOTES AND CORRESPONDENCE
985
FIG. 6. Time series of refractivity over a 60-day period (16 June–14 August 1996) from radar (solid line) and surface
stations (crosses) data. The radar refractivity data are field averages up to 45-km range measured every 5 min. The refractivity
from surface stations was computed from hourly pressure, temperature, and moisture measurements at the eight stations plotted
in Fig. 7.
exactly at the same height. Consider two neighboring
targets of different heights but similar range. Because
of the difference in target height, the actual paths from
the radar to the higher target will be slightly over the
one to the lower target. The small height difference
combined with strong vertical N gradients will result in
measurable differences in the travel time of radar waves,
resulting in a phase difference. This additional variance
in Df makes the computation of Df slopes in range
more difficult, and fewer refractivity measurements are
made. The second cause of reduction in coverage occurs
when the reflectivity from precipitation clutter becomes
strong enough to mask the ground echo signal. Another
effect of precipitation is to add an additional delay in
the travel time of electromagnetic waves between the
radar and the target. This effect is, however, relatively
minor: A sustained rainfall rate of the order of 15 mm
h21 along a path is required to cause a bias of 1 N unit
in the measured refractivity for that path.
Although the refractivity field does not often contain
structures with high contrast, the frequency with which
weaker mesoscale gradients in refractivity can be observed has been larger than we expected. Since such
gradients are not constant in time, they are unlikely to
be artifacts solely due to a bad calibration done in nonideal conditions. As a result, it seems reasonable to conclude that radar measurement of surface refractivity may
provide some additional information on surface tem-
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FIG. 7. Example of a contrasted refractivity field. (a) Surface weather observations of temperature, dewpoint, cloud cover (when available),
and winds at various stations around the radar site at 1400 UTC. The refractivity computed from these observations is indicated in brackets.
(b) Refractivity field measured by radar 10 min later. A significant gradient in refractivity is observed because of the difference in the
dewpoint temperature of the two air masses (Td 5 108C vs Td 5 148C). The two outer circles correspond to a distance of 45 km from the
radar.
FIG. 8. Example of a refractivity field in light precipitation. (a) Reflectivity from weather echoes (moving from the west-southwest) over
a 240 km 3 240 km area around the radar. (b) Refractivity field measured for the same period. Higher refractivity can be seen toward the
southwest due to an increase in surface moisture associated with the approaching rain. The two circles correspond to a distance of 45 km
from the radar.
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NOTES AND CORRESPONDENCE
perature and moisture fields that is not available from
point measurements made at conventional surface stations. Establishing the value of such information is the
subject of ongoing work.
5. Conclusions and potential
We have shown in this paper that ground echoes could
be successfully used to obtain information about the
atmosphere between them and the radar. The use of the
radar phase information from ground targets to extract
a measurement of refractivity near the surface has been
demonstrated to be theoretically feasible and practically
workable, to the point where it is now computed in real
time around the clock. While all the work presented
here involved ‘‘natural’’ ground echoes with their limitation, nothing prevents the addition of intentional targets (corner reflectors, etc.) to enhance the coverage,
usability, and reliability of the measurement.
Although the interpretation of the refractivity field
can be ambiguous, it offers the possibility of getting
some glimpse on the two-dimensional structure of the
temperature and moisture fields in a similar way that
radar reflectivity allowed us to get a better appreciation
of precipitation variability. Detailed observations of
front passages, microbursts and cold pools associated
with thunderstorm outflow, and other possible dry or
wet spots may be made using refractivity information
derived by radar.
Possible uses of refractivity will likely depend on the
sensitivity of the measurement and the coverage that
can be obtained. Examples include extraction of lowlevel moisture information for the purposes of data assimilation in models and thunderstorm initiation prediction; boundary layer studies where refractivity allows
us to see a proxy field for temperature and moisture
evolve in space and in time; and of course electromagnetic wave propagation work.
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