Houze Robert A., Jr. Cloud Dynamics

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4.1 General Characteristics of Meteorological Radars

109

ments made by transmitting and receiving at various polarizations. In Sees, 4.2-

4.4, we will examine the physical basis of the parameters

Z, and V

and how they

may be used in the analysis of precipitating clouds.

Most meteorological radars measure only the range and reflectivity. They are

called

noncoherent or conventional radars. A radar that is instrumented to deter-

mine the velocity of the target is called a

Doppler radar. A radar that provides

information on the polarization of the signal is called a

polarimetric or multiple-

polarization

radar. These radars generally operate at wavelengths A = 1-30 em,

with 3, 5, and 10 em being the most commonly used. Radiation at these wave-

lengths is scattered primarily by precipitation-sized particles, and the main use of

meteorological radars is for mapping and analyzing the precipitation field and its

associated physics and dynamics. Useful meteorological radar signals can also be

obtained at these wavelengths from strong turbulent fluctuations in clear air, chaff

particles intentionally introduced into the air, and insects. These data help deter-

mine the structure of the atmosphere surrounding and just preceding the occur-

rence of precipitating cloud systems.

Radar signals emitted at the shorter end of the 1-30-cm wavelength range are

sensitive to more weakly reflecting targets, and they do not require a large-diame-

ter antenna to focus the beam; however, they are subject to attenuation of the

signal in heavy rain. The lO-cm wavelength (S-band) is the shortest at which

attenuation by rain is essentially eliminated. However, the antenna required to

focus the lO-cm wavelength beam is very large (a parabolic dish about 8 m in

diameter is required for a 1° beamwidth), and the radar equipment is therefore

bulky and expensive. Nonetheless, primary land-based radar facilities are typi-

cally 10em in wavelength. Shipborne and airborne radars, on the other hand, tend

to be 5 em (C-band) or 3 em (X-band) systems because of space limitations.

Although satellite-borne meteorological radars have yet to be employed, they are

envisioned as having a wavelength of about 2 em

(K,

band)-also

because of size

Iimitations." One must be cautious about the effects of attenuation when using the

data from these shorter-wavelength systems. However, when proper care is

taken, the data are useful.

Most meteorological radars have scannable antennas, which allow the radar

beam to be pointed in a specified direction. By continuously scanning in azimuth

and elevation, radars can obtain data in three spatial dimensions with high time

resolution throughout a volume of space surrounding the radar. Depending on the

size of the sector to be scanned, a set of three-dimensional data can be obtained

about every 2-10 min, although more rapidly scanning antennas will be used on a

few radars in the near

future."

Most S-, C-, and X-band weather radars can detect

precipitation at horizontal ranges up to 200-400 km. However, quantitative mea-

surements can usually be obtained only within about 100-200 km

ofthe

radar site.

Nonetheless, this capability allows precipitation to be mapped over mesoscale

regions and data are obtained with high resolution. Each radar pulse extends over

81 See TRMM: A Satellite Mission to Measure Tropical Rainfall, edited by Simpson

(1988).

82 For example, see Joss and Collier (1991).

110 4 Radar Meteorology

a very small

(-100

m) increment of range. The more restrictive limit on spatial

resolution is the beamwidth (angular distance between half-power points), which

for most precipitation radars is between 1 and

83 As a result of the beam

resolution limits, data are obtained with resolution of less than a kilometer close to

the radar and a few kilometers at the outer range limits.

Meteorological radars with very short wavelength (bands centered around 1

and 8 mm) are useful for detecting cloud particles. However, these bands are too

highly attenuated by rain to be useful in heavy precipitation. Lidar uses laser light

(at

A -

0.3-10

/Lm) for observing atmospheric aerosol particles and cloud struc-

ture.P' Radars with wavelengths in the

UHF

(75 em) and VHF (6 m) bands are

useful for clear air sounding of the atmosphere; these radars are referred to as

profilersP The discussions in Sees. 4.2 and 4.3 apply mainly to centimetric wave-

length radars, which detect precipitation particles. In general, these microwave

radars are the most widely used for precipitation physics and cloud dynamics.

However, many

ofthe

Doppler radar techniques discussed in Sec. 4.4 are applica-

ble to radars at any wavelength.

4.2 Reflectivity Measurements

4.2.1 Obtaining Reflectivity from Returned Power

Precipitation particles (rain, snow, graupel, and hail) detected by meteorological

radar are called distributed targets, since many scattering elements are simulta-

neously illuminated by the radar beam. The volume containing the illuminated

particles is called the

resolution volume of the radar. This volume is determined

by the beamwidth and duration of the transmitted pulse of radiation. Since the

distributed targets are generally moving relative to each other, as a result of

differing fall speeds and sheared and/or turbulent winds, the power (as well as

phase) returned from the resolution volume centered at a range

r fluctuates in

time. The instantaneous power returned depends upon the arrangement of scatter-

83 Usually, the horizontal and vertical beamwidths are the same. However, some airborne systems

require a restricted antenna dimension and hence broader beamwidth in the vertical. For example, the

U.S. National Oceanographic and Atmospheric Administration WP-3D research aircraft have a C-

band radar with a

1.5° horizontal beamwidth and 4° vertical beamwidth. The antenna is mounted below

the fuselage.

84 Lidar is an acronym for

Light

Intensity Detection

And

Ranging. In a lidar, the returned laser light

is collected in a telescope and focused on a photomultiplier detector, after which it is amplified,

digitized, and recorded. The lidar is used essentially like a radar and is often called a

"laser

radar."

Its

beam can be used to scan a volume

ofthe

atmosphere, and it can be instrumented to make Doppler and

multiple-polarization measurements. See p.

412 of Stull (1988)for a brief nontechnical discussion and

some examples of the types of measurements that can be made.

85 These long-wavelength radars do not usually scan but rather operate with several fixed beams:

for example, with one beam vertical and two

-15°

off vertical. Their utility derives from their ability to

receive reflections from refractive index variations of the air as well as from particles suspended in the

atmosphere (see Rottger and Larsen, 1990for a review of this type of radar).

4.2 Reflectivity Measurements

111

ers. However, a time average over

-0.01-0.1

s averages out the fluctuations, and

the average returned power

is given by

pp2

)}OHOv'CpCoTJr

p = (4.1)

r 512(2 In 2)1r

where PI is the transmitted power, G is the antenna gain (a dimensionless number

allowing for the focusing effect of the antenna),

is the duration of the emitted

radar pulse,

Co is the speed of light, ()H and ()v are the horizontal and vertical

beamwidth angles (Le., the angles between power points expressed in radians),

respectively, and

"fir is called the radar reflectivity per unit volume of air.

is the

effective scattering cross section of a unit volume of air, and it is what is deduced

from the measurement of average returned power, since all the other terms on the

right-hand side of (4.1) are known. The radar equation (4.1) is obtained from

geometrical considerations and by assuming that there is no attenuation by inter-

vening targets and that the resolution volume is uniformly filled with targets.

There are no losses of signal in the radar system itself, and the antenna gain is

uniform across the beam width.

86 The term 2 In 2 is a factor that adjusts the half-

power beamwidths to an effective beamwidth, which has a constant distribution of

power. This adjustment is necessary since the antenna gain in (4.1) is assumed to

be constant across the beam.

The radar reflectivity

"fir is the sum of the scattering cross sections of the

individual scatterers within the resolution volume.

is given by the summation

L<T,

where <T is the effective backscatter cross section of one particle within the

resolution volume and the summation is over all the particles in the volume. The

backscatter cross section is a factor introduced to account for the fact that the

target may not scatter radiation isotropically.

is the cross-sectional area an

isotropic scatterer (which scatters all impinging radiation equally in all directions)

would need in order to produce the scattering by the actual target. For a single

spherical target whose diameter

D is small compared to the radar wavelength,

Rayleigh" scattering theory applies, in which case

(4.2)

where

IKI2 is a function of the complex index of refraction of the substance of

which the scatterer is composed."

Precipitation particles are usually small enough that the particle diameter

D <

0.IA.

Consequently, Rayleigh scattering theory can be used to approximate the

characteristics of signals reflected back to the radar. Substitution of

L<T

for "fir in

(4.1), with

<T given by (4.2), and some algebraic rearrangement

ofterms

show that

86 See Chapter 4 of Battan (1973) or Chapters 4 and 5 of Rinehart (1991).

87 The same Lord Rayleigh who derived the theory of convection described in Sec. 2.9.3.

88 See Chapter 4 of Battan (1973) or Chapters 4 and 5 of Rinehart (1991) for further discussion.

89 This condition holds for drizzle, as well as for most rain, snow, and graupel, but not for hail-

stones, which are

-1-15

em in diameter, and often not for very large raindrops or aggregated snow-

flakes, especially at shorter wavelengths.

(4.3)

112

4 Radar Meteorology

the returned power P

from scatterers of known

IKI2

located within a resolution

volume

res

centered at a range r from a radar yields the radar reflectivity

factor

Z =

_1_:L

= r Pr;R

res

IKI

where

(4.4)

and

(

r ) 2

co'rp

res

1r(JH(JV"2

-2-

(4.5)

The term

2-D6

in (4.3) is the sum 'of the sixth powers of the diameters of all of the

scatterers in the volume

res

•

Note that since 'V

res

is proportional to r

, C

is a

constant depending only on the characteristics of the particular set of radar equip-

ment being used.

Usually there is no way to be certain of the value of

IKI2.

The scatterers could

be composed of liquid, ice, melting ice, insects, turbulent eddies, or chaff. The

convention therefore is to use the measured

and

rand

(4.3) to calculate the

equivalent radar reflectivity factor,

r PrC

Z = (4.6)

e 0.93

where 0.93 is the value of

IKI2

for liquid water.

thus would be the value that the

reflectivity factor of the particles producing the returned power P

detected at

range

r would have

ifthey

were composed purely of liquid water. The value of IKI2

for ice is usually set to 0.197. If it were known that the reflectors were actually ice

particles, then the true reflectivity factor could be obtained simply by multiplying

Z, by the ratio (0.93)/(0.197). However, since the composition of the scatterers is

not normally known with certainty, radar data are usually expressed as

Ze, in

units of

mm"

-3,

which refer to particle size to the sixth power per unit volume of

air. Typically, the numbers are given in decibel units:

asz, =

101og

z, (4.7)

Typical values of dB[Ze(mm

) ]

are

- 30 to 0 in marginally detectable precip-

itation;

-0-10

in drizzle, very light rain, or light snow;

-10-30

in moderate rain

and heavier snow;

-30-45

in melting snow;

30-60 in moderate to heavy rain;

and

-60-70

or more in hail.

4.2.2 Relating Reflectivity to Precipitation

One of the most important uses of meteorological radar is to estimate the precipi-

tation content of the air, the precipitation rate, and the fall speed of the precipita-

4.2 Reflectivity Measurements

113

tion at the

earth's

surface. In this section, we will outline the techniques used to

make these estimates from measurements of radar reflectivity.

4.2.2.1 Particle Size

Method

Since the radar reflectivity factor Z is related to particle size, there is a physical

basis for a quantitative relationship between the precipitation content of the air

and the received radar echo intensity. According to the definition (4.3), the radar

reflectivity factor is the sixth moment of the particle size distribution.

the

number of particles in a radar-sampled volume is very large, a continuous version

of (4.3) can be used:

(4.8)

(4.9)

(4.10)

where

N(D)

is the particle size distribution function defined such that

N(D)dD

the number of particles of diameter

D to D + dD per unit volume of air. If the

scatterers are liquid water,

Z = Ze. The particle size distribution also determines

the mixing ratio of rainfall,

qr = nPL

(00

D3N(D)dD

where PL is the density of liquid water, P is the density of air, and the rainfall rate

mis given by

m= nPL

(00

V(D)D

3N(D)

where

V(D)

is the fall velocity related empirically or theoretically to drop of

diameter

D [Eq.(3.68)].

is evident from (4.8)-(4.10) that measurements of the

drop size distribution

N(D)

can be used to determine a set of values

(Z,q"m).

For

rainfall of a given type in a given climatological setting, the curves of log

Z vs. log

mand log Z vs. log

pq,

are usually linear and therefore yield empirical relation-

ships of the form

Z =

Z = a

(pqr)b

(4.11)

(4.12)

where

d, b,

iiI,

and hi are positive constants derived from the slopes and inter-

cepts of the log-log plots. From (4.9) and (4.10), it is evident that the ratio

fft./(pqr)

is the mass-weighted particle fall speed V defined by (3.69). The relationships

(4.11) and (4.12) imply that this ratio is a function of Z of the form

V =a

(4.13)

where ii

and h

are constants. Sometimes a factor accounting for the variable

density of air through which the particle is falling is included in the fall speed

114 4 Radar Meteorology

formula.P' The relationships (4.11)-(4.13) are used for estimating rainfall rate, rain

mixing ratio, and rain fall speed from radar reflectivity measurements. Similar

relationships can be obtained for snow. The values of the constants should be

determined from particle size measurements made within the particular type of

precipitation being studied.

To be able to use a Z-ffi relation like (4.11), the equivalent radar reflectivity

factor

obtained from the observed radar data must first be converted to an

appropriate value of Z. Several types of sampling problems typically make this

conversion difficult: (i) The composition of the particles (liquid or ice) must be

assumed. This problem is hard to overcome when the beam is partially filled with

liquid and ice particles, contains melting ice, or is partially filled by ground tar-

gets.

(ii) Because of both the curvature of the earth and the elevation angle of the

beam, the center

the

radar

beam is typically found at higher altitude the greater

the range. The radar thus indicates a value of Z for a radar resolution volume

"If

res

located some distance above the

earth's

surface, whereas the drop size distribu-

tion measurements are usually made at a point on the surface. The particle size

distribution may undergo significant evolution as a result of diffusional and collec-

tional growth, evaporation, breakup, or sedimentation, which may occur in the

population of raindrops between the time the drops are in the radar resolution

volume and when they finally reach the

earth's

surface [recall Eq. (3.52)]. This

problem is greatly exacerbated if the radar beam at low elevation angles is blocked

by an intervening mountain. In such a case, the precipitation at lower altitudes

behind the mountain cannot be observed, and a vertical profile of reflectivity must

be assumed for the precipitation below the lowest elevation angle that is not

blocked." (iii) According to (4.6),

is computed from the returned power P

under the assumption that the distributed targets completely fill the resolution

volume

"If

res

at a given range. The returned power may therefore be underesti-

mated if the resolution volume is not filled completely. This problem becomes

more likely the greater the range since the beamwidth angles in (4.5) are fixed.

These sampling problems lead to an uncertainty of about a factor of 2 (or more in

the case of topographic shielding) in estimating rain from Z-ffi relations based on

drop size distribution measurements. Some cancellation of errors can be obtained

by integrating radar measurements over long times or large

areas."

The accuracy

Z-q,

and

Z-

V relations is less known but probably similar.

4.2.2.2 Rain Gauge Method

Although the particle size methodology described above for determining pre-

cipitation rates from radar data is elegant in that it is developed around the

physical relationship between particle size distribution and radar reflectivity, it

is limited by the uncertainties mentioned at the end of the preceding subsection.

90 See Foote and DuToit (1969) or Beard (1985).

91 The assumed profile could be based on a climatology of the vertical profile of reflectivity.

For

discussion of the problem of shielding by mountains, see Joss and Waldvogel (1990).

For

further details see Joss and Waldvogel (1990) or Austin (1987).

4.2 Reflectivity Measurements

115

To avoid these problems, one can directly relate surface rain gauge and radar-

measured values of

Ze.

Rain gauge

data

can be used to obtain the probability density function

P(ffi),

which is defined such that

P(ffi)

dffi is the probability that if rain is falling, its rate

(in mm

) is between ffi and ffi + dffi.

For

the same population of rain-producing

clouds, radar

data

can

be used to determine another probability density function

P(Ze), which is defined such that P(Ze)

is the probability that if rain (and

hence

radar

echo) is present at a given range from the radar, the equivalent radar

reflectivity is between

and

The probability density function for the

equivalent

radar

reflectivity

can

be determined for an interval of range centered at

Let

the sum of the

area

covered by echo within this annular region be AE(r). Let

the area covered by those echoes with reflectivity between

Z, and

+ dZ

A(Ze

,r).

Then

the probability density function P(Ze) for the region centered at r is

given by

(4.14)

can be shown'? that since Z; and ffi are functionally related, the correct transfor-

mation of one into the other will produce equal probabilities, such that

(4.15)

where the pairs

(Zei ,ffi

define the Zei-ffii relationship.

follows also that

(4.16)

The

expression in (4.14) can be substituted in the right-hand integral. If rain

gauges located within the annular zone centered at range

r from the radar are used

to derive

P(ffi)l"

which is the probability density function for mthat applies

specifically in this zone, then this empirical probability density can be substituted

for the integrand on the left-hand side of (4.16). The

Ze-ffi

relationship is then

obtained by finding the pairs of lower limits (Z»;

ffi;) that make the integrals equal.

Thus, the empirical probability density functions for

Z, and ffi can be used to

obtain a

Ze-ffi

relationship that applies at a given range from the radar. By repeat-

ing the process for various ranges,

Ze-ffi

relationships can be found that apply

throughout the field of view of the radar. This technique thus allows one to relate

Ze and ffi without having to account for radar-beam geometry or differences be-

tween

Ze and Z.

4.2.3 Estimating Areal Precipitation from Radar Data

One important use

radar data is to estimate rainfall over an area covered by

radar observations. A straightforward way to make this estimate is to apply a

93 See Calheiros and Zawadzki (1987).

(4.17)

116

4 Radar Meteorology

Ze-ffi relationship to all the low-altitude reflectivity

data

obtained

over

a region in

the

x-y

plane to estimate the rainfall rate as a function ffi(x,y). Then the area-

integrated rain rate

ffi

area

(mass of

water

per

unit time) is obtained by integrating as

follows:

ffi

area

ffi(x,y)dxdy

where AT is the total

area

covered by rain. Although up to now this approach has

been the standard, it depends on having accurate rain rates everywhere within

AT.

An alternative method for obtaining the area-wide rain rate, which is outlined

below,"

allows an estimate of this rate to be made by using radar

data

to identify

the

area

covered by rain

and

using a statistical knowledge of the rainfall character-

istics to estimate the amount of precipitation falling within the area. This tech-

nique can be useful in the case of observations with a satellite-borne radar, which

operates at a wavelength

-2

em. Because of attenuation at this wavelength

and

other difficulties of making spaceborne radar observations, this type

radar may

accurately estimate only the lower rain rates. The rain rates at every point within

the area, especially where the rain is heavy, may be difficult to estimate. How-

ever, the measurement

the lower rates will allow the areas in which the rain

exceeds a given threshold to be outlined accurately. Airborne

radar

measure-

ments also tend to describe best the areas covered by rain but do not usually

accurately describe the rain rate at all the interior points of the area. The following

technique

can

be useful in estimating the areal rain rates in these cases.

Let

A(ffi

)

be the

area

covered by rain with rates exceeding a given threshold

rain rate

ffi

•

The

ratio of

A(ffi

)

to the total area covered by rain is then

(4.18)

where, as in Sec. 4.2.2.2,

P(ffi)

is the probability density function for rainfall rate.

F(ffi

)

is thus

the

fraction of the total

area

covered by rain with rates exceeding the

threshold rain rate

ffi

•

The

term

on the right-hand side of (4.18) expresses the fact

that the ratio of areas is equivalent to the cumulative probability of rain with rates

>ffi

•

The average rain

rate

within regions where the threshold is exceeded is

(ffi)T

foo

ffip(ffi)dffi/foo

P(ffi)dffi

!fiT !fiT

The overall average rain rate is then

(4.19)

(4.20)

94 For further details, see Doneaud et al. (1981, 1984), Atlas et al, (1990), and Rosenfeld et al,

(1990),

4.3 Polarization Data

117

Multiplying the right-hand sides of (4.18) and (4.19), and substituting from (4.20),

we obtain

F(!Jl,

)(!Jl

), =

(!Jl)"

[J~,

!JlP{!Jl)

d!Jl

/ r

!JlP(!Jl)

d!Jl

]

(4.21)

Both of the terms in brackets and

(ffi>T

are determined by the empirical probability

density function

p(ffi).

follows that the ratio of

(ffi>T

to the term in brackets is

also determined solely by

P(ffi).

If we call this ratio

~(ffiT)'

then the area-wide

average rain rate is given by

(4.22)

Thus, the area-wide average rain rate can be determined by using radar to mea-

sure the fractional area covered by rates above the threshold

ffi

provided the

probability density function

P(ffi)

[and hence the factor

~(ffiT)]

is known empiri-

cally from other sources of information, such as rain gauge records. The area-

integrated rain rate

ffi

area

(mass of water per unit time) is obtained by multiplying

(ffi>o by the total area covered by rain:

ffi

area

(ffi)oA

(4.23)

The limitation of this method of estimating the areal rainfall is that the probability

density function

P(ffi)

must be determined for the particular climatological set-

ting.

might have more value in determining averages of large populations of

storms than the rainfall in individual cases. However, it may sometimes (e.g., with

satellite or airborne radar) be the only method available.

4.3 Polarization Data

To interpret the returned radar signal, atmospheric scientists have four fundamen-

tal properties of electromagnetic waves at their disposal: amplitude, phase, fre-

quency, and polarization.

For

precipitation, the polarization characteristics are

related to the mean values and distributions of size, shape, and spatial orientation

of the particles filling the radar resolution volume, their thermodynamic phase

state, and fall behavior. To take advantage of polarization properties, polarimetric

radars are instrumented to transmit and receive radiation polarized in several

specified orientations." Circularly polarized polarimetric radars transmit an elec-

tric field vector whose direction normal to the path of the radiation rotates with

time. Linearly polarized polarimetric radars transmit and receive pulses of radia-

tion polarized in two orthogonal directions.

95 For further discussions of multiple polarization in meteorological radar, see Bringi et al.

(1986a,b), Wakimoto and Bringi (1988), Bringi and Hendry (1990), Jameson and Johnson (1990), and

Herzegh and Jameson (1992).

118 4 Radar Meteorology

A parameter that can be derived from the data of a circularly polarized radar is

the

circular depolarization ratio (CDR).

is defined as

(

Zpar )

CDR

10 log

Zorth

(4.24)

(4.25)

(4.26)

where

Zpar

is the reflectivity factor received in circular polarization of the same

sense as that transmitted (called the parallel component) and

Zorth

is that received

in the opposite (or orthogonal) polarization.

the transmitted pulse is right-hand

circularly (RHC) polarized, the ratio in (4.24) would be the ratio of the right-hand

to the left-hand (counterclockwise) polarized received reflectivities. CDR indi-

cates the shapes

the particles producing the signal. When circularly polarized

waves are scattered by a spherical particle, the backscattered signal is also circu-

larly polarized but in the opposite sense, because the direction of propagation is

reversed. In this case, CDR

.-00.

A typical value of CDR measured in rain is

-20

-35.

On the other hand, if the target is infinitely long and thin, the

parallel and orthogonal components are equal in magnitude, and CDR

The

value of the CDR, on a scale of

-00

to 0, thus indicates the relative sphericity of

the particles producing the radar echo.

may therefore be regarded as an indica-

tor

ofthe

shapes

ofthe

detected precipitation particles. However, as will be noted

below, the rather small index of refraction of ice reduces the magnitude of the

expected shape effects.

Linearly polarized radars usually transmit and receive radiation in the horizon-

tal and vertical planes, although, in principle, the two orthogonal planes of polar-

ization could be at any chosen orientation. The motivation for choosing horizontal

and vertical planes is that raindrops tend to become oblate as they increase in

size,"

and they fall with the major axis oriented rather horizontally (Fig. 4.2). Two

useful parameters that can be obtained from linearly polarized radars that operate

in the horizontal and vertical are the

differential reflectivity ZDR and the linear

depolarization ratio

LDR. They are defined as

(

ZHH )

ZDR

10glO

Zvv

and

LDR

10glO(ZHV)

ZHH

where

ZHH,

Zvv,

and ZHV are the horizontally transmitted/horizontally received,

vertically transmitted/vertically received, and horizontally transmitted/vertically

received reflectivity factors, respectively.

Since raindrops tend to be oblate and horizontally oriented, positive values of

ZDR are produced, generally in the range

0-5,

as shown in Fig. 4.2. The larger

drops produce the larger values since the degree of oblateness increases with drop

size. Regions of ice particles tend to produce values of

ZDR near zero. These

96 The oblateness of raindrops is discussed by Pruppacher and Klett (1978).