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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−1rcos 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 rkea,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 =eα(273.16 −β)

(Td−β)2δT , (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

T3δT+

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.

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