Variation of radio refractivity with respect to moisture and temperature and influence on radar ray path

Abstract

In this study, the variation of radio refractivity with respect to temperature and moisture is analyzed. Also, the effects
of vertical gradients in temperature and moisture on the propagation paths of electromagnetic waves of weather radar are examined
for several sites across the United States using several years of sounding data from the National Weather Service. The ray
path is important for identifying storm characteristics and for properly using the radar data in initializing numerical weather
prediction models. It is found that during the warm season the radio refractivity gradient is more sensitive to moisture gradients
than to temperature gradients. Ray paths from the commonly accepted vertical ray path model are compared to a ray path computed
from a stepwise ray tracing algorithm using data from actual soundings. For the sample of about 16 000 soundings examined,
we find that only a small fraction of the ray paths diverge significantly from those calculated using a ray path model based
on the US Standard Atmosphere. While the problem of ray ducting in the presence of a temperature inversion is fairly well
known, we identify the presence of a strong vertical moisture gradient as the culprit in the majority of the cases where significant
deviations occurred.

Full text

ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 25, NO. 6, 2008, 1098–1106
Variation of Radio Refractivity with Respect to Moisture
and Temperature and Influence on Radar Ray Path
Jidong GAO∗1, Keith BREWSTER1, and Ming XUE1,2
1Center for Analysis and Prediction of Storms
2School of Meteorology, University of Oklahoma, Norman, Oklahoma 73072,USA
(Received 16 August 2007; revised 11 March 2008)
ABSTRACT
In this study, the variation of radio refractivity with respect to temperature and moisture is analyzed.
Also, the effects of vertical gradients in temperature and moisture on the propagation paths of electromag-
netic waves of weather radar are examined for several sites across the United States using several years of
sounding data from the National Weather Service. The ray path is important for identifying storm charac-
teristics and for properly using the radar data in initializing numerical weather prediction models. It is found
that during the warm season the radio refractivity gradient is more sensitive to moisture gradients than to
temperature gradients. Ray paths from the commonly accepted vertical ray path model are compared to a
ray path computed from a stepwise ray tracing algorithm using data from actual soundings. For the sample
of about 16 000 soundings examined, we find that only a small fraction of the ray paths diverge significantly
from those calculated using a ray path model based on the US Standard Atmosphere. While the problem
of ray ducting in the presence of a temperature inversion is fairly well known, we identify the presence of
a strong vertical moisture gradient as the culprit in the majority of the cases where significant deviations
occurred.
Key words: radio refractivity, radar ray path
Citation: Gao, J. D., K. Brewster, and M. Xue, 2008: Variation of radio refractivity with respect to
moisture and temperature and influence on radar ray path. Adv . At mos. Sci.,25(6), 1098–1106, doi:
10.1007/s00376-008-1098-x.
1. Introduction
The United States operational WSR-88D Doppler
radar network (NEXRAD) is a vital tool for the
real-time detection and warning of hazardous weather
(Crum and Albert, 1993; Crum et al., 1998; Serafin
and Wilson, 2000). It is also an essential observing sys-
tem for initializing non-hydrostatic, storm-resolving
(i.e., horizontal grid spacing on the order of 1 km) nu-
merical weather prediction (NWP) models (e.g., Lilly,
1990; Droegemeier, 1990, 1997). Attempts to demon-
strate such capability began early in the past decade
(e.g., Sun et al., 1991), and subsequent efforts have
been notably successful (e.g., Gao et al., 1998; Sun
and Crook, 2001; Weygandt et al., 2002a,b; Crook and
Sun, 2002; Xue et al., 2003; Brewster, 2003; Gao et al.,
2004; Hu et al., 2006a,b).
To utilize the radar reflectivity and radial veloc-
ity data in real-time warning and quantitative pre-
cipitation estimation and to assimilate the data into
NWP models, it is necessary to accurately determine
the spatial locations of individual radar measurements.
Because the propagation path of the electromagnetic
waves can be affected by the refractivity of the atmo-
sphere, the propagation path or the ray path is usu-
ally not a straight line. A suitable ray path equation
is needed; the local direction of the path also affects
the radial velocity forward operator that projects the
Cartesian velocity components on the model grid to
the local radial direction in data assimilation systems.
Most early radar data assimilation studies used rel-
atively simple ray path equations in the forward op-
erator formulation, with the simplest one being based
on the Cartesian flat-earth geometry (e.g., Gao et al.,
1998, 2004; Weygandt et al., 2002a,b; Shapiro et al.,
2003). The next level of sophistication is to use the
∗Corresponding author: Jidong GAO, jdgao@ou.edu

NO. 6 GAO ET AL. 1099
four-thirds earth radius model (see, e.g., Doviak and
Zrnic, 1993; Gao et al., 2006) for the radar ray path
calculations (e.g., Brewster, 2003). This model takes
into account the curvature of the earth but assumes
that the atmosphere has a constant vertical gradient
of refractivity in the lower troposphere as determined
from the U.S. Standard Atmosphere. In reality, the
gradient of the refractivity is seldom constant and sig-
nificant departures from the assumption exist when
there are strong temperature inversions and/or large
vertical moisture gradients. A better understanding
of the variation of the ray path due to the gradient
of refractivity as well as a better understanding of the
frequency of occurrence of significant departures from
the path prediction of simple models is valuable to
radar data quality control and quantitative precipita-
tion estimation (Bech et al., 2003). Serious contami-
nation of radar data can occur in atmospheric condi-
tions that cause anomalous propagation of the radar
beam (Moszkowicz et al., 1994; Pamment and Conway,
1998). In this study, the variation of radio refractivity
with respect to gradients in temperature and moisture
is first analyzed; the influence of atmospheric environ-
mental conditions on the ray paths at locations rep-
resenting four different climate regions of the United
States is then examined using sounding data over sev-
eral years from the US National Weather Service.
The rest of this paper is organized as follows: In
section 2, the four-thirds earth radius model for radar
ray path calculations is briefly reviewed. An analysis
of the variation of refractivity with respect to temper-
atureandmoistureisgiveninsection3.Insection4,
a stepwise ray trace method is introduced. In section
5, the influence of atmospheric refractivity on the ray
path at different geographical locations in the United
States is examined using historic sounding data from
the National Weather Service. Finally, a summary and
conclusions are given in section 6.
2. Ray path equations based on the four-
thirds earth radius model
Under the assumption that temperature and hu-
midity are horizontally homogeneous, so that refrac-
tivity is a function of height above ground only, a for-
mula can be derived (e.g., Doviak and Zrnic, 1993)
that expresses the ray path in terms of a curve follow-
ing a sphere of effective radius
ae=a
1+a(dn/dh)=kea, (1)
where ais the earth’s radius and keis a multiplier that
depends on the vertical gradient of refractive index of
air, dn/dh.Herehis height above ground. When the
U.S. Standard Atmosphere is considered, it is found
that keis approximately equal to 4/3. This is often
referred to as the “four-thirds earth radius model”.
The refractive index of air, n, is a function of air tem-
perature, pressure and humidity, and is usually taken,
subject to certain assumptions, as (?),
N=(n−1) ×106=77.6p
T+3.73 ×105e
T2,(2)
where pis the air pressure (including the water vapor
pressure, in hPa), eis the water vapor pressure (hPa),
and Tis the air temperature (K). It is convenient to
use the quantity N, defined as the atmospheric radio
refractivity, instead of n. N represents the departure
of nfrom unity in parts per million. Ntypically has
a value of about 300 near the surface and its varia-
tion with the height, dN/dh, can be considered more
conveniently.
The following two equations relate the height above
ground, h, and the surface range (distance along the
earth’s surface from radar), s, to radar-measurable pa-
rameters, the slant path, rand radar elevation angle,
θe(Doviak and Zrnic, 1993),
s=aesin−1rcos θe
ae+h,(3)
h=(r2+a2
e+2raesin θe)1/2−ae.(4)
In Doviak and Zrnic (1993), it is also shown that
if rkea,andthecoordinatesx, y and zare related
to the radar range, elevation and azimuth coordinates
(r, θe,φ)by,
x≈rcos θ
esin φ, (5a)
y≈rcos θ
ecos φ, (5b)
z=h=(a2
e+r2+2rkesin θ
e)1/2−kea, (5c)
where θ
e, the angle between the radar beam and the
earth’s tangent plane below the data point, is the sum
of two terms expressed as the following:
θ
e=θe+tan
−1[rcos θe/(ae+rsin θe)] .(6)
From Eq. (5a) and Eq. (5b), one can easily derive
the distance along the earth’s surface as,
s≈rcos θ
e.(7)
Equation (7) is an approximation of the ray path Eq.
(3). Equation (5c) uses the effective earth radius beam
height Eq. (4).
3. Variation of refractivity with respect to
temperature and dewpoint
In Eq. (2), the first term on the right hand side
is known as the dry term, the second term the moist

1100 VARIATION OF RADIO REFRACTIVITY AND INFLUENCE ON RADAR RAY PATH VOL. 25
term. The value of radio refractivity Ncan be com-
puted from measurements of pressure, p, temperature,
T, and water vapor pressure, e. In the troposphere
the fractional decrease in pwith height is larger than
that in T, so the variation of radio refractivity N
with height, dN/dh, is usually negative. For the U.S.
Standard Atmosphere, dN/dh is about −39.2 km−1.
If Ndecreases more (less) rapidly with height than
the Standard Atmosphere, the beam may be refracted
more (less), and in such cases, the height of a target
may be overestimated (underestimated) by the four-
thirds earth radius model. In an extreme condition,
e.g., in the presence of a sharp refractivity gradient of
about −150 km−1below 100 m AGL (above ground
level), a ray sent at a small positive elevation angle
may actually decrease in height with range and even-
tually strike the earth surface.
Because the air pressure usually makes a rather
stable contribution to the variation of N, we will only
analyze the variation of Nwith respect to tempera-
ture and moisture. The amount of moisture in the air
can be expressed in many forms. Four commonly used
moisture variables are dewpoint temperature, Td,wa-
ter vapor pressure, e, relative humidity, and specific
humidity. To facilitate comparisons with the response
to temperature variations, we choose the dewpoint as
the moisture variable for our sensitivity study. A com-
monly used approximate relation between dewpoint
and water vapor pressure is Teten’s formula (Krish-
namurti, 1986):
e=6.11 exp α(Td−273.16)
Td−β,(8)
where for water α=17.26,β =35.86 and for ice
α=21.87,β=7.66. Taking the leading-order varia-
tion of Eq. (8) with respect to dewpoint gives
δe =eα(273.16 −β)
(Td−β)2δT , (9)
where δe is the variation of water vapor, e,andδTdis
the variation of dewpoint.
By taking the leading-order of variation of the re-
fractivity equation [Eq. (2)] with respect to tempera-
ture and water vapor pressure, we have
δN =−77.6P
T2+2×3.73 ×105e
T3δT+
3.73 ×105
T2δe , (10)
where δN is the variation of refractivity, and δT is the
variation of temperature. Substituting Eq. (9) into
Eq. (10), and letting
A≡∂N
∂T =−77.6p
T2+2×3.73 ×105
T3,(11)
B≡∂N
∂Td
=3.73 ×105(273.16 −β)αe
T2(Td−β)2,(12)
we have
δN =AδT +BδTd.(13)
It is obvious from Eqs. (11) and (12), A<0and
B>0. These two terms reflect the response of re-
fractivity to temperature and dewpoint variations re-
spectively. Figures 1a and 1b show the variations of A
and Bas a function of base variables temperature and
dewpoint, respectively, within a temperature range of
−10◦Cto40
◦C, and a dewpoint range of −32◦Cto
40◦C (or from about 10% to 100%, in terms of relative
humidity) at a constant pressure of 1000 hPa. Figure
1c shows the absolute ratio between the variation with
dewpoint (term B) and that of temperature (term A).
It is clear from Fig. 1c that refractivity is more sensi-
tive to dewpoint than temperature when the base tem-
perature is high. Note especially that when the base
temperature is at or above 30◦Candthebasedew-
point is greater than −16.0, the variability of refractiv-
ity with respect to dewpoint is 5 to 6 times greater in
magnitude than with respect to temperature. When
the low-level temperature is between 10◦Cto30
◦C,
the temperatures typically found in mid-latitudes from
spring to early fall, the variability of refractivity with
respect to dewpoint is 2 to 4 times greater than to
temperature. When the temperature is around 0◦C,
the winter situation, the change of refractivity with
respect to temperature and moisture variables are of
similar magnitudes (see the low-left corner of Figs. 1a
and 1b). Since Aand Bhave opposite signs, the gradi-
ent of refractivity is often nearly constant in the upper
levels of the atmosphere and during winter, when and
where air temperature is low. When the base pressure
is set to 700 hPa, the pattern of variation is very sim-
ilar to that shown in Fig. 1, though values of term A
and Bare slightly smaller (not shown). Therefore, the
above discussion is applicable for the entire depth of a
typical planetary boundary layer, where the humidity
has a significant influence on the atmosphere.
Dividing (13) by δh,weget
δN
δh =AδT
δh +BδTd
δh .(14)

NO. 6 GAO ET AL. 1101
Fig. 1. The variation of refractivity with respect (a)
to temperature (◦C) indicated by the contours of A≡
∂N/∂T in (11), and (b) to dewpoint (◦C) indicated by
the contours of B≡∂N/∂Tdin (12), and (c) the absolute
ratio between Band A.
Normally, both temperature and dewpoint de-
crease with the height, i.e., δT /δh < 0andδTd/δh <
0. So, the temperature term makes a positive contri-
bution to the rate of decrease in Nbut the moisture
term makes a negative contribution. To satisfy the
condition that the decrease in Nwith height exceeds
a critical value (i.e., δN/δh < −157 km−1), and so that
electromagnetic beams are bent toward the surface of
the earth, i.e., for them to be trapped, either δT /δh
should be greater than zero, which happens in the in-
version layers, or δTd/δh should be much less than
zero, which happens when a very dry layer overlays a
relatively moist layer.
To further quantify our analysis, given a basic
state with relative humidity RH=60%, temperature
T=17◦C and pressure p=1000 hPa, we can calcu-
late the values of the other base variables Td=11.7◦C,
e=13.7 hPa and N=328.25. Substituting these values
into Eqs. (11) and (12), we get
A≡∂N
∂T =−1.34
and
B≡∂N
∂Td
=4.02 .
These values indicate that a 1◦C change in tempera-
ture causes a 1.34 unit change in refractivity N; while
a1
◦C change in dewpoint causes a 4.02 unit change
in refractivity. Since variability on the order of few
degrees is typical of both temperature and dewpoint
in the lower atmosphere, we can therefore say that the
radio refractivity is about three times more sensitive
to dewpoint than to temperature near the surface for
the above typical condition. This point will be further
demonstrated in section 5. Among a large number of
soundings that we examine in section 5, many of the
most extreme deviations of ray paths from the four-
thirds earth model are caused by large vertical mois-
ture gradients; usually when a very dry layer is present
above a moist boundary layer. From our discussion,
it can also be concluded that it is easier to retrieve
moisture variations than temperature from refractiv-
ity if it is observed by radars or satellites during the
warm season. Weckwerth et al. (2005) showed an in-
teresting result that under most daytime summertime
conditions, refractivity from the radar measurements
was representative of an atmospheric layer about 250
m deep layer and could be useful for detecting low-level
moisture boundaries.
4. A stepwise ray tracing method
In the last two sections, we presented a review on
the ray path equations based on the four-thirds earth
radius model, and analyzed the variation of radio re-
fractivity with respect to temperature and dewpoint.

1102 VARIATION OF RADIO REFRACTIVITY AND INFLUENCE ON RADAR RAY PATH VOL. 25
In this section, the influence of different environmen-
tal thermodynamic profiles on the radar ray path is
examined by using actual observed sounding data. To
accurately estimate the radar ray path based on arbi-
trary sounding data, a stepwise ray tracing method is
employed whose steps of calculation are as follows:
(1) Starting from the second range gate from the
radar, for each radar measurement, calculate the re-
fractivity Ni−1for the previous gate according to Eq.
(2) based on the given thermodynamic profile where
iis the index of the gates. Calculate the gradient of
refractive index according to the differential of Eq. (2)
with respect to beam height h,
dn
dh 


i−1
=10
−6dN
dh 


i−1
.(15)
(2) Calculate ae,i−1=ke,i−1aaccording to Eq. (1)
using the gradient of refractive index from step (1) at
the previous gate, i−1;
(3) Calculate the angle between the radar beam
and the tangent plane below the data point, θ
e,i−1us-
ing Eq. (6) for each radar gate;
(4) Finally, calculate the radar beam height hand
the surface range sfor gate iusing formulas
⎧
⎨
⎩
hi=hi−1+∆rsin(θ
e,i−1),
si=si−1+∆rcos(θ
e,i−1),
(16)
where ∆ris the gate spacing, which is 250 m for U.S.
operational WSR-88D radial velocity. Variables hiand
sibare the beam height and surface distance for each
gate, respectively. Steps (1) through (4) are repeated
for successive gates until the last gate of each beam.
Note that within this study a single observed sound-
ing is used for each case, thus, we assume that the
sounding profile is representative of the vertical struc-
ture of the atmosphere within the entire radar range,
which may not be true for some cases. For the data
assimilation purposes, three dimensional gridded fields
of temperature, moisture and pressure can be used to
determine the local values of refractivity and further
used in ray tracing calculations. The gridded fields
can be from the analysis background or from a prelim-
inary analysis that has already incorporated sounding
and other non-radar observations.
5. Ray paths as determined from observed
soundings
In this section, we examine the influence of radio
refractivity on the ray path and its climatology in rep-
resentative geographical regions of the United States
by calculating the ray paths for historic sounding data
during a six-year period from 1 January 1998 to 31
December 2003 at four locations; namely, Oakland,
California (OAK), Key West, Florida (EYW), Dulles
Airport, Virginia (IAD) and Topeka, Kansas (TOP).
These sites were chosen to represent the West Coast,
Tropical Southeast, East Coast and Great Plains re-
gions of the United States, respectively. Quality-
controlled soundings were obtained from the online
database of the NOAA Global Systems Division of the
Earth System Research Laboratory.
For each sounding, the radar beam heights at all
range gates, for the lowest 0.5◦elevation, are calcu-
lated, using the four-thirds earth model and the step-
wise ray-tracing method with the actual observed at-
mospheric profiles. The difference of the beam heights
using these two methods is then divided by the corre-
sponding beam width at the corresponding range, as-
suming a 0.93 degree beam width for the United States
WSR-88D network radars. The result can be regarded
as relative beam height error, and is relevant because
the radar observation is taken to be representative of
the volume described by the beam width rather than
a point measurement. This metric is chosen because,
in relation to the use of the data in numerical mod-
els, there is smoothing in the vertical corresponding
to the beam shape and thus there are changes in posi-
tion that are small relative to the beam width and will
have a small effect on the integrated forward model re-
sult. Table 1 shows the distribution of errors among
six error intervals for the locations 50 km from a hypo-
thetical radar site at the location and elevation of the
radiosonde launch site. Among more than 4000 sound-
ings for each site over the 6 year period, we find that,
most of the time, the ray paths determined from the
four-thirds earth radius model are in good agreement
with the stepwise ray tracing method. More than 90%
of the soundings result in relative beam height errors
of less than 0.2. The ray paths determined from the
Table 1 . Distribution of relative beam height errors among 6 error intervals for locations 50 km from the radar site.
Raob Sites Obs. No. Error Distributions (%)
[0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, ∞]
OAK 4234 94.31 4.65 0.92 0.12 0.00 0.00
EYW 4202 94.10 5.02 0.79 0.05 0.00 0.05
IAD 4088 97.31 0.86 1.10 0.05 0.44 0.24
TOP 4253 93.63 3.74 1.27 0.75 0.19 0.42

NO. 6 GAO ET AL. 1103
Table 2 . Distribution of relative beam height errors among 6 error intervals for locations 120 km from the radar site.
Raob Sites Obs. No. Error Distributions (%)
[0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, ∞]
OAK 4234 76.17 4.82 7.58 10.01 1.42 0.00
EYW 4202 71.68 11.61 7.38 8.02 1.26 0.05
IAD 4088 91.34 3.69 1.71 1.32 1.10 0.83
TOP 4253 86.93 5.48 1.98 2.66 0.59 2.37
200
300
400
500
600
700
800
900
1000
-80 -60 -40 -20 0 20 4
0
Pressure(hPa)
Temperature ( C)
o
(a)
0
1
2
3
4
5
6
-800 -600 -400 -200 0 200 400 600
height (km)
The gradient of refractivity ( km )
-1
(b)
0
1
2
3
4
5
6
0 50 100 150 200
height (km)
beam range (km)
(c)
Fig. 2. (a) The temperature (solid) and dewpoint
(dashed) profiles, (b) the refractivity gradient profiles
(km−1) calculated from the U.S. Standard Atmosphere
(solid) and from the observed sounding (dashed), and (c)
the radar ray paths calculated for the 0.5◦elevation us-
ing the Standard Atmosphere (solid) and actual sounding
and ray tracing method (dashed, only clear in between
0–50 km), for 0000 UTC 3 May 1999 at Topeka, Kansas
(TOP).
four-thirds earth radius model are, on average, more
accurate with the soundings from Oakland, California
(OAK) than those from soundings of other sites; no
relative errors greater than 0.8 are observed with the
Oakland sounding profiles. The ray paths are less ac-
curate with the soundings from Topeka, Kansas, with
0.4% of soundings having relative beam height errors
greater than unity.
Table 2 shows the distribution of errors for the lo-
cations 120 km away from the virtual radar sites. For
this distance, over 70% of soundings result in relative
beam height errors of less than 0.2 for OAK and EYW.
The number of soundings having beam height errors
less than 0.2 is 91.3% and 86.9% for IAD and TOP,
respectively, which are better than those for OAK and
EYW sites. However, the numbers of soundings which
result in relative beam height errors of above 1 are
larger, at 0.8% and 2.4%, at IAD and TOP, respec-
tively. As we might expect, range gates further away
from the radar sites are more likely to have larger beam
height errors using the four-thirds earth radius model
due to the accumulation of error over a distance and
a greater chance of encountering a layer with an ex-
treme refractivity gradient. Thus, radar data far away
from radar sites are more prone to have location errors
compared to data closer to the sites.
From Table 2, we also notice that more than 20%
of soundings result in beam height errors of between
0.2 and 0.8 for OAK and EYW, but less than 10% for
IAD and TOP lie in the same range. No soundings
from the OAK site and only 0.05% of soundings from
EYW had relative errors greater than 1. However,
0.83% for IAD and 2.37% for TOP result in errors of
1 or greater. This indicates that while more soundings
from IAD and TOP result in accurate ray path calcu-
lations based on the simple model, they also give rise
to more cases which have very large relative errors, in-
dicating more variability in the vertical refractivity at
these sites.
Figure 2 shows the sounding, refractivity profile
and the calculated ray path for one of the worst cases
from the TOP site. It is clear that the very strong
moisture gradient found in this sounding is responsi-
ble for the large vertical refractivity gradient (Figs. 2a,
b). The radar beam refracted downward toward the
earth surface due to the layer of sharp refractivity gra-
dient below the 1-km level. In this case, the gradient
of radio refractivity is largely caused by the vertical
variations in humidity. For this site, we also examined

1104 VARIATION OF RADIO REFRACTIVITY AND INFLUENCE ON RADAR RAY PATH VOL. 25
(a)
Temperature ( C)
o
height (km)
(b)
The gradient of refractivity ( km )
-1
height (km)
beam range (km)
(c)
0
1
2
3
4
5
6
-1000 -800 -600 -400 -200 0 200 400 600
0
1
2
3
4
5
6
0 50 100 150 200
-80 -60 -40 -20 0 20 40
200
300
400
500
600
700
800
900
Pressure(hPa)
Fig. 3. As Fig. 2, but for 1200 UTC 8 June 2005 at
Amarillo, Texas (AMA).
many other cases having large ray path deviations.
In most of those cases a large moisture gradient was
found at the low levels which caused the beams to
be refracted to the ground at a close radar range (as
seen in Fig. 2c). Therefore, within the period ex-
amined, the beam ducting phenomena occurred more
often in the Great Plains and East Coast areas of the
U.S. than in the West Coast or Tropical Southeast ar-
eas because large moisture gradients near the surface
occurred more frequently. In the Great Plains, this
situation can be caused by high boundary-layer mois-
ture from local sources or due to advection from the
Gulf of Mexico that is overlaid with dry air coming
from the Rockies to the west. Similarly for Virginia,
dry air advected from the Appalachians, or with a his-
tory of subsidence, can often be found above a shallow
layer of moist air near the ground with origins from
the Atlantic Ocean.
As an illustration of a case that for which the envi-
ronmental thermodynamic structure had a clear effect
on the radar observations, Fig. 3 shows a recent case
study for Amarillo, Texas. A large moisture gradient
and a temperature inversion (roughly 100 m AGL) are
quite pronounced. The calculated beam is trapped in
a layer just 100 m above the ground (Fig. 3c). To show
the effect on the radar data in this case, Fig. 4 shows
the radar image at 1347 UTC 8 June 2005 from WSR-
88D radar at Amarillo, Texas (KAMA). The beam
might be partially or completely hitting ground tar-
gets in places where colors of orange, red or white are
found in the figure, as indicated within the area de-
noted by the circle. In addition, many pixels within
the areas showing high reflectivity have been removed
by the automated clutter filter (black adjacent to red
or white areas).
Suppose that we require that the error in the beam
height relative to the beam width be no more than 0.5
for the location estimate to be accurate enough for
data assimilation purposes, then we can see from Ta-
bles 1 and 2 that the number of soundings which qual-
ify for the use of the four-thirds earth radius model
for the ray path calculation is well above 90%. How-
ever, beam ducting and strong departures from the
four-thirds earth model do occur at small percent-
ages, especially in the Great Plains and Eastern United
States. Some of these could be found in situations pre-
ceding severe weather events—situations for which the
WSR-88D radar data are most needed. Most of these
phenomena are caused by large moisture gradients in
the lower atmosphere. For this reason, quality control
and data assimilation systems should check for such
situations, and, when present, use the more accurate
ray-trace method for beam height calculation and/or
discard low-level data that may be contaminated by
ground targets due to beam ducting and/or inaccu-
rate height determination.
6. Summary and conclusions
Radar ray path equations are used to determine
the physical location of each radar measurement for
data display, radar-data based automated detection
algorithms, quality control and data assimilation. To
best use radar data, the accuracy of ray path calcula-
tions and the assumptions involved need to be exam-
ined thoroughly to see if significant deviations from the
typically used Standard-Atmosphere-based four-thirds
effective earth radius model occur frequently. Such
large deviations occur when a strong temperature in-
version and/or large vertical moisture gradient exist.
In this study, we first analyzed the variation of radio
refractivity with respect to atmospheric temperature

NO. 6 GAO ET AL. 1105
Fig. 4. Radar reflectivity (dBZ) image at 1347
UTC 8 June 2005 for KAMA at Amarillo,
Texas.
and moisture. It is shown that radio refractivity
gradient is more sensitive to the moisture than to tem-
perature; therefore moisture has a more significant in-
fluence on the radar ray path calculation than temper-
ature.
To accurately calculate the radar ray path based
on general sounding profiles, a stepwise ray tracing
program is developed. The influence of atmospheric
refractivity on the ray path is examined using a large
number of historic soundings from four sites repre-
senting different geographical regions of the United
States. For the soundings examined, 90% result in
small relative beam height errors when calculated us-
ing the standard-atmosphere-based four-thirds earth
radius model and only a small fraction of ray paths
thus calculated diverge significantly from those cal-
culated based on true soundings using the ray trac-
ing method. But these small fractions of deviations
could occur more often in situations preceding severe
weather. For many of the problematic cases exam-
ined, the vertical moisture gradient is found to be a
more significant contributor. The results of this paper
may provide a useful guidance to radar data quality
control, as well as the assimilation of radar data into
numerical weather prediction models.
Acknowledgements. This work was supported by
U. S. NSF Grant Nos. ATM-0331756, ATM-0331594,
ATM-0530814 and EEC-0313747, and by DOT-FAA Grant
NA17RJ1227-01. The first and third authors were also
supported partially by the National Natural Science Foun-
dation of China under Grant Nos. 40620120437 and
40505022.
REFERENCES
Bean, B. R., and E. J. Dutton, 1968: Radio Meteorology.
Dover Publication, 435pp.
Bech, J., B. Codina, J. Lorente, and D. Bebbington, 2003:
The sensitivity of single polarization weather radar
beam blockage correction to variability in the verti-
cal refractivity gradient. J. Atmos. Oceanic Techno l.,
20, 845–855.
Brewster, K. A., 2003: Phase-correcting data assimilation
and application to storm scale numerical weather
prediction. Part II: Application to a severe storm out-
break. Mon. Wea. Rev.,131, 493–507.
Crook, N. A., and J. Sun, 2002: Assimilating radar, sur-
face, and profiler data for the Sydney 2000 Forecast
Demonstration Project. J. Atmos. O ceani c Techn ol. ,
19, 888–898.
Crum, T. D., and R. L. Albert, 1993: The WSR-88D
and the WSR-88D operational support facility. Bull.
Amer. Meteor. Soc.,74, 1669–1687.
Crum, T. D., R. E. Saffle, and J. W. Wilson, 1998: An up-
date on the NEXRAD Program and future WSR-88D
support to operations. W ea. Forecasting,13, 253–
262.
Doviak, R. J., and D. S. Zrnic, 1993: Doppler Radar
and Weather Observations. Academic Press, 2nd ed.,
562pp.
Droegemeier, K. K., 1990: Toward a science of storm-

1106 VARIATION OF RADIO REFRACTIVITY AND INFLUENCE ON RADAR RAY PATH VOL. 25
scale prediction. Preprint, 16th Conf. on Severe Local
Storms, Kananaskis Park, Alberta, Canada, Amer.
Meteor. Soc., 256–262.
Droegemeier, K. K., 1997: The numerical prediction of
thunderstorms: Challenges, potential benefits, and
results from real time operational tests. WMO Bul-
letin,46, 324–336.
Gao, J., M. Xue, Z. Wang, and K. K. Droegemeier, 1998:
The initial condition and explicit prediction of con-
vection using ARPS forward assimilation and adjoint
methods with WSR-88D data. Preprints, 12th Conf.
Num. Wea. Pred., Phoenix, AZ, Amer. Meteor. Soc.,
176–178.
Gao,J.,M.Xue,K.Brewster,andK.K.Droegemeier
2004: A three-dimensional variational data assimila-
tion method with recursive filter for single-Doppler
radar. J. Atmos . O ceani c Tech nol .,21, 457–469.
Gao, J., K. Brewster, and M. Xue, 2006: A comparison
of the radar ray path equation and approximations
for use in radar data assimilation. Ad v. Atm os. S ci. ,
23(2), 190–198, 10.1007/s00376-006-0190-3.
Hu, M., M. Xue, and K. Brewster, 2006a: 3DVAR and
cloud analysis with WSR-88D level-II data for the
prediction of Fort Worth tornadic thunderstorms.
Part I: Cloud analysis and its impact. Mon. Wea.
Rev.,134, 675–698.
Hu, M., M. Xue, J. Gao, and K. Brewster, 2006b:
3DVAR and cloud analysis with WSR-88D level-II
data for the prediction of Fort Worth tornadic thun-
derstorms. Part II: Impact of radial velocity analysis
via 3DVAR. Mon. Wea. Rev.,134, 699–721.
Krishnamurti, T. N., 1986: Workbook on numerical
weather prediction for the tropics for the training of
class I and class II meteorological personnel. Pub-
lished by World Meteorological Organization, No.
669, Geneva, Switzerland.
Lilly, D. K., 1990: Numerical prediction of
thunderstorms—Has its time come? Quart. J.
Roy. Me teor. Soc .,116, 779–798.
Moszkowicz, S., G. J. Ciach, and W. F. Krajewski, 1994:
Statistical detection of anomalous propagation in
radar reflectivity patterns. J. Atmo s. Ocea nic Tech-
nol.,11, 1026–1034.
Pamment, J. A., and B. J. Conway, 1998: Ob jective Iden-
tification of Echoes Due to Anomalous Propagation
in Weather Radar Data. J. Atmos. Oceanic Technol.,
15, 98–113.
Serafin, R. J., and J. W. Wilson, 2000: Operational
weather radar in the United States: Progress and
opportunity. Bull. Amer. Meteor. Soc.,81, 501–518.
Shapiro, A., P. Robinson, J. Wurman, and J. Gao,
2003: Single-Doppler velocity retrieval with rapid-
scan radar data. J. Atmos. Oceanic. Technol.,20,
1758–1775.
Sun, J., D. W. Flicker, and D. K. Lilly, 1991: Recovery of
three-dimensional wind and temperature fields from
simulated single-Doppler radar data. J. Atmos. Sci.,
48, 876–890.
Sun, J., and N. A. Crook, 2001: Real-time low-level
wind and temperature analysis using single WSR-
88D data. Wea. Forecas tin g,16, 117–132.
Weckwerth, T. M., C. R. Pettet, F. Fabry, S. Park, M. A.,
LeMone, and J. W. Wilson, 2005: Radar refractivity
retrieval: Validation and application to short-term
forecasting. J. Appl. Meteor.,44, 285–300.
Weygandt, S. S., A. Shapiro, and K. K. Droegemeier,
2002a: Retrieval of initial forecast fields from single-
Doppler observations of a supercell thunderstorm.
Part I: Single-Doppler velocity retrieval. Mon. Wea.
Rev.,130, 433–453.
Weygandt, S. S., A. Shapiro, and K. K. Droegemeier,
2002b: Retrieval of initial forecast fields from single-
Doppler observations of a supercell thunderstorm.
Part II: Thermodynamic retrieval and numerical pre-
diction. Mon. Wea. Rev.,130, 454–476.
Xue, M., D.-H. Wang, J. Gao, K. Brewster, and K. K.
Droegemeier, 2003: The Advanced Regional Predic-
tion System (ARPS), storm-scale numerical weather
prediction and data assimilation. Meteorology and
Atmospheric Physics,82, 139–170.