August 2014 

NASA/CR–2014-218510 

Investigation of Advanced Radar Techniques 

for Atmospheric Hazard Detection with 

Airborne Weather Radar 

Andrew L. Pazmany 

ProSensing, Inc., Amherst, Massachusetts 
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August 2014 

NASA/CR–2014-218510 

Investigation of Advanced Radar Techniques 

for Atmospheric Hazard Detection with 

Airborne Weather Radar 

Andrew L. Pazmany 

ProSensing, Inc., Amherst, Massachusetts 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
Contents 

1 INTRODUCTION ......................................................................................................................................2

2 RADIOMETRY ..........................................................................................................................................4

Radiometric Brightness Temperature Seen From Aircraft ...........................................................................................5

The Radiometer Measurement ...................................................................................................................................7

Radiometer for Severe Weather Avoidance ...............................................................................................................8

Atmospheric Attenuation Details ................................................................................................................................9

Simulated Radiometer Measurements in Severe Weather .......................................................................................10

Retrieving Total Atmospheric Attenuation from Radiometer Data ...........................................................................13

Retrieving Rain-Rate and Rain-Depth from Radiometer Data ...................................................................................15

Brassboard Radiometer Description .........................................................................................................................17

Brassboard Radiometer Experiment .........................................................................................................................18

Combined Radar/Radiometer Retrieval ....................................................................................................................19

Adding Radiometer Mode to a Weather Radar System ............................................................................................22

Radiometer Summary and Conclusions ....................................................................................................................24

3 POLARIMETRY .....................................................................................................................................25

Review of Weather Radar Polarimetry .....................................................................................................................25

3.1.1 DIFFERENTIAL PHASE SHIFT (ΦDP) .....................................................................................................................26

3.1.2 SPECIFIC DIFFERENTIAL PHASE SHIFT (KDP) .....................................................................................................26

3.1.3 DIFFERENTIAL REFLECTIVITY (ZDR) .....................................................................................................................27

3.1.4 CORRELATION COEFFICIENT (ρHV) .....................................................................................................................27

Rain Rate Estimation from X-band Polarimetric Radar Parameters ..........................................................................29

3.1.5 Rain Rate – Z Estimator ......................................................................................................................................29

3.1.6 Rain Rate – Kdp Estimator ...................................................................................................................................30

3.1.7 Rain Rate – Z, Zdr Estimator ..............................................................................................................................30

3.1.8 Rain Rate – Z, Zdr, Kdp Estimator .........................................................................................................................31

3.1.9 Comparison of various rain rate estimators .....................................................................................................31

Dual Polarized Aircraft Weather Radar Antenna. ......................................................................................................32

3.1.10 Antenna requirements ....................................................................................................................................33

4 MULTI-FREQUENCY RADAR (MFR) .............................................................................................42

iii
MFR Measurement Concept .....................................................................................................................................42

4.1.1 Point Scattering ..................................................................................................................................................42

4.1.2 Volume Scattering ..............................................................................................................................................43

4.1.3 Attenuation .........................................................................................................................................................47

4.1.4 MRF Measurement Parameters ........................................................................................................................47

4.1.5 MFR Information Content ..................................................................................................................................48

MFR Components – Current Technology ..................................................................................................................50

4.1.6 Antennas .............................................................................................................................................................50

4.1.7 Transmitters ........................................................................................................................................................51

MFR Detection ..........................................................................................................................................................54

4.1.8 Cloud and Precipitation Model ..........................................................................................................................55

4.1.9 Detection to 10 and 20 km Range .....................................................................................................................63

MFR Differential Attenuation ...................................................................................................................................69

5 SENSITIVITY ..........................................................................................................................................76

6 REFERENCES .........................................................................................................................................79

iv
Contents 

1 INTRODUCTION ......................................................................................................................................2

2 RADIOMETRY ..........................................................................................................................................4

Radiometric Brightness Temperature Seen From Aircraft ...........................................................................................5

The Radiometer Measurement ...................................................................................................................................7

Radiometer for Severe Weather Avoidance ...............................................................................................................8

Atmospheric Attenuation Details ................................................................................................................................9

Simulated Radiometer Measurements in Severe Weather .......................................................................................10

Retrieving Total Atmospheric Attenuation from Radiometer Data ...........................................................................13

Retrieving Rain-Rate and Rain-Depth from Radiometer Data ...................................................................................15

Brassboard Radiometer Description .........................................................................................................................17

Brassboard Radiometer Experiment .........................................................................................................................18

Combined Radar/Radiometer Retrieval ....................................................................................................................19

Adding Radiometer Mode to a Weather Radar System ............................................................................................22

Radiometer Summary and Conclusions ....................................................................................................................24

3 POLARIMETRY .....................................................................................................................................25

Review of Weather Radar Polarimetry .....................................................................................................................25

3.1.1 DIFFERENTIAL PHASE SHIFT (ΦDP) .....................................................................................................................26

3.1.2 SPECIFIC DIFFERENTIAL PHASE SHIFT (KDP) .....................................................................................................26

3.1.3 DIFFERENTIAL REFLECTIVITY (ZDR) .....................................................................................................................27

3.1.4 CORRELATION COEFFICIENT (ρHV) .....................................................................................................................27

Rain Rate Estimation from X-band Polarimetric Radar Parameters ..........................................................................29

3.1.5 Rain Rate – Z Estimator ......................................................................................................................................29

3.1.6 Rain Rate – Kdp Estimator ...................................................................................................................................30

3.1.7 Rain Rate – Z, Zdr Estimator ..............................................................................................................................30

3.1.8 Rain Rate – Z, Zdr, Kdp Estimator .........................................................................................................................31

3.1.9 Comparison of various rain rate estimators .....................................................................................................31

Dual Polarized Aircraft Weather Radar Antenna. ......................................................................................................32

3.1.10 Antenna requirements ....................................................................................................................................33

4 MULTI-FREQUENCY RADAR (MFR) .............................................................................................42

iii
MFR Measurement Concept .....................................................................................................................................42

4.1.1 Point Scattering ..................................................................................................................................................42

4.1.2 Volume Scattering ..............................................................................................................................................43

4.1.3 Attenuation .........................................................................................................................................................47

4.1.4 MRF Measurement Parameters ........................................................................................................................47

4.1.5 MFR Information Content ..................................................................................................................................48

MFR Components – Current Technology ..................................................................................................................50

4.1.6 Antennas .............................................................................................................................................................50

4.1.7 Transmitters ........................................................................................................................................................51

MFR Detection ..........................................................................................................................................................54

4.1.8 Cloud and Precipitation Model ..........................................................................................................................55

4.1.9 Detection to 10 and 20 km Range .....................................................................................................................63

MFR Differential Attenuation ...................................................................................................................................69

5 SENSITIVITY ..........................................................................................................................................76

6 REFERENCES .........................................................................................................................................79

iv
INVESTIGATION OF ADVANCED HAZARD 

DETECTION CAPABILITIES WITH 

MULTIFREQUENCY RADAR 

Andrew L. Pazmany, Ph.D. 

ProSensing Inc., Amherst, MA 

EXECUTIVE SUMMARY 

Current state of the art Aircraft Weather Radars (AWRs) are scanning, X-band, single polarization (H-pol) and 

coherent with Doppler measurement capability. The measured radar parameters include reflectivity and Doppler 

velocity mean and standard deviation. The X-band (~9.4 GHz) operating frequency offers the optimum compromise 

between available transmit power, antenna gain with a moderate size aperture, receiver sensitivity (front-end losses 

and LNA noise figure) and acceptable attenuation in moderate liquid precipitation. The Doppler velocity standard 

deviation measurements are effective in identifying turbulence in precipitation. 

However, current X-band AWRs have the following shortcomings: 

1. Insufficient sensitivity in low reflectivity hazard conditions, such as: 

a. Super-cooled, high liquid water content drizzle and clouds. 

b. High ice water content, high altitude clouds. 

2. Inability to correct for rain attenuation, resulting the under estimation of severe weather cores. 

3. Limited measurement range in highly attenuating heavy liquid precipitation. 

There are several possible design options that can be considered in the next generation of AWRs, but unfortunately 

there is no single design solution to improve detection in all the above listed hazard conditions. The following are 

the recommendations of this report, in order of increasing complexity and cost. 

Recommendation #1: Add Radiometer Measurement Capability to AWRs 

The addition of radiometric measurement capability to future AWRs should cost very little and can have potentially 

significant impact on detecting and avoiding inflight icing hazard, especially in super-cooled liquid clouds and 

drizzle that are often below the detection threshold of AWRs. The radiometric brightness temperature measurements 

can be used directly to identify the “coolest” direction with the least amount of liquid water path. Or if some 

assumptions can be made on drop size distribution (e.g., stratified precipitation), the radiometer measurements can 

be combined with the radar reflectivity profiles to quantify some cloud and precipitation parameters and thus the 

potential hazard. 
2

Recommendation #2: Support the Development of Dual-Polarized AWR Antennas to Enable Polarimetric 

Measurement Capability 

The use of polarimetry for characterizing precipitation (rain rate) and severe weather (hail detection) and correcting 

for attenuation has been well documented ([1-8]). The cost of upgrading the current AWRs to dual-polarized 

capability is relatively simple and inexpensive. The transmitted pulse has to be equally divided to the Vertical and 

Horizontally polarized antenna ports and a second receiver has to be added to the radar. The signal processing is 

computationally modest and polarimetric radar parameters, such as differential reflectivity (Zdr) and specific 

differential phase shift (Kdp) can be effectively used to correct for attenuation in moderate to heavy rain conditions. 

The only missing (enabling) technology is a dual-pol antenna, similar in quality to the current standard flat plate 

waveguide array. 

Recommendation #3: Develop Dual-Frequency (X/C-band) AWRs for Severe Weather Prone Regions 

In severe weather conditions X-band attenuation can be severe, resulting in underestimating dangerous weather 

cores, and worse, diminishing the received signal below the detectable threshold. By shifting to a lower frequency, 

attenuation quickly decreases and thus improves the probability of detection. A dual-frequency X/C-band system 

would have the higher sensitivity to smaller scatterers at X-band but the ability to penetrate highly attenuating (high 

rain rate) regions. 

 

1 Introduction 

Most AWRs (aircraft weather radars) operate at X-band (8-12 GHz) using magnetron tube or solid state amplifier 

transmitters and slotted waveguide array antennas. These X-band radars provide an optimal tradeoff between 

sensitivity, size, weight and cost, but are susceptible to severe attenuation in heavy rain and have limited or no 

ability to help pilots detect and quantify icing potential. This vulnerability of AWRs to attenuation and their inability 

to characterize and quantify precipitation has been a contributing factor in numerous aircraft incidents. 

 

Undetected peak convective cells, hidden by attenuation, led to the forced landing and resulting fatalities of 

Southern Airways flight 242 on April 4, 1977. If attenuation information was available from AWR, or if the radar 

signal was less attenuated, the peak convective region would have been correctly identified and likely avoided. 

Newer AWRs with Doppler capability attempt to work around this attenuation problem by measuring the radar 

signal spectrum width. Hazardous convective severe weather cores are almost always highly turbulent and thus 

cause a broadening of the Doppler signal spectrum. Furthermore spectrum width measurements are theoretically 

unbiased by attenuation, as long as the received signal is above the detection threshold. So the spectrum width 

technique can effectively detect convective severe weather, but attenuation can still diminish measurement range to 

impractically short distances. Since attenuation of atmospheric water droplets is approximately proportional to the 

square of the radar frequency, a potential solution to severe attenuation is multi-frequency AWR with a lower 

frequency channel. A second radar channel at C-band for example, would have a significantly lower rate of 

attenuation in heavy rain than at X-band, and consequently could better penetrate and more reliably detect severe 

weather. The multi-frequency radar technique is investigated in Section 4. Other methods to improve measurement 

sensitivity are discussed in Section 5. 

Some hazardous atmospheric conditions however do not have elevated turbulence or wind sheer, and radar 

backscatter intensity is not necessarily a good metric of hazard potential, such as freezing drizzle and rain. On 

October 31, 1993, undetected or under estimated freezing rain caused the crash of American Eagle flight 4184. 

Other icing related aircraft incidents are listed in http://www.skybrary.aero/index.php/In-Flight_Icing. Icing hazard 

potential is related to the total volume of supercooled liquid along the flight path and drop size (larger drops can 

cause enhanced surface roughness and can have larger areas of impingement, sometimes beyond the ice protection 

system). A technique not yet utilized with AWRs for better measurement of severe weather precipitation or freezing 

rain is radar polarimetry. The quantitative measurement of rain with S-band (~ 3 GHz) polarimetric ground-based 

weather radars was successfully developed in the 1990s and by 2010 the polarimetric upgrade of the US network of 

159 ground-based weather radars began and is scheduled to be completed by the end of 2013 [1][2]. Moreover, new 

ground based radars are almost never built without polarimetric measurement capability. The initial low-frequency 

polarimetric rain retrieval techniques have been modified and validated for C and X-band [10] [15], but have not yet 
3

been implemented with AWRs. One of the recommendations of this report is to upgrade future AWRs with 

polarimetric measurement capability for better characterization of rain in convective severe thunderstorms and in 

freezing rain. Weather radar polarimetry and its implementation with AWRs are discussed in Section 3. 

Supercooled clouds and drizzle can contain a large amount of liquid water comprised of small droplets, making 

them undetectable by most weather radars, yet still an icing threat to smaller aircraft. Higher frequency radars 

have better sensitivity to small droplets but also have a higher rate of attenuation in high liquid water 

conditions, making them actually less sensitive at the ranges that are needed from an aircraft, than the current 

X-band AWRs (see Section 0). Radiometers however have been effectively used to measure drizzle and cloud 

liquid for many decades and AWRs already have a receiver that can be used for brightness temperature 

measurements. This “passive”, receive-only measurement concept involves scanning ahead of the aircraft in 

azimuth while pointing in elevation above the horizon. The observed brightness temperature (or measured 

noise power) is the attenuation weighted, physical temperature along the antenna beam can be approximated as 

(see Eq. (5))

(1) 

  T

effB

eTT

.4 34

1





where 

T

is the total atmospheric attenuation along the antenna beam in dB, and 

Teff

is the effective physical temperature of the attenuating cloud or precipitation layer. 

Since microwave attenuation due to small (relative to the wavelength) water droplets is directly proportional to 

the total liquid water, when the total liquid water along the beam is low, very low brightness temperature 

would be observed. If the total attenuation is high, the radiometer measurement will saturate, observing the 

physical temperature of the cloud or precipitation. But the total path attenuation can be controlled to some 

degree by pointing the beam up in elevation, to cut through a highly attenuating region at a steeper angle, 

thereby reducing the attenuating distance, and thus reduce the chance of saturation (Figure 1). If the 

propagation length through the cloud or precipitation can be approximated, then the average liquid density can 

be also estimated. But even without any knowledge of the propagation distance, by scanning the beam in 

azimuth at a constant elevation angle, the “coolest” direction can be found that corresponds to the safest path 

with the least amount of total liquid (Figure 2). This radiometric measurement technique is presented and 

analyzed next, in Section 0. 
4

2 Radiometry 

The addition of radiometric measurement capability to future AWRs should cost very little and can have a 

potentially significant impact on detecting and avoiding inflight icing hazard, especially in super-cooled liquid 

clouds and drizzle that are often below the detection threshold of AWRs. The radiometric brightness 

temperature measurements can be used directly to identify the “coolest” direction with the least amount of 

liquid water path. Or if some assumptions can be made on drop size distribution, like in stratified precipitation, 

the radiometer measurements can be combined with the radar reflectivity profiles to quantify some cloud and 

precipitation parameters and thus the potential hazard. 

Earth (~300 K)

Space (~ 2.7 K) Observed Brightness Temperature

120 K 280 K 285 K

265 K 275 K

Figure 1. Side view of an aircraft radiometer measuring a cloudy scene. As long as the antenna is 

pointed slightly above the horizon, the background is the cold space, making the observed 

brightness temperature highly sensitive to attenuation and therefore to the total liquid along the 

antenna beam. 
5

70 K 250 K 150 K Observed Brightness Temperatures

Figure 2. Top view of an aircraft radiometer measuring the brightness temperature of a cloud-filled 

scene. The radiometer can scan in azimuth, slightly above the horizon, to look for the “coolest” 

direction with the least amount of total (supercooled) liquid. 

Radiometric Brightness Temperature Seen From Aircraft 

Figure 3 shows an aircraft flying at altitude, , above the earth. Located in the nose of the aircraft is a 

radiometer that senses the brightness temperature, in degree Kelvin (K), of Planck emission from the 

atmosphere and from objects in front of the aircraft. In the direction defined by vector, ̅, the radiometric 

brightness temperature, , of the atmosphere can be computed as [3],

(2) ∫ 

( ) ( ) ( ) 

 

 

where, 

 (see Figure 3),

 elevation angle associated with ̅,

 distance from the aircraft along the vector, ̅, 

 maximum distance along ̅ thru the atmosphere (see text below), 

( ) ( ) ( ) ( ), 

total extinction ( ) at point ,

 

( ) extinction ( ) due to liquid water droplets (e.g., clouds and rain), 

 

( ) extinction ( ) due to water vapor, 

 ( ) extinction ( ) due to oxygen, 

 ( ) air physical temperature (K), 

 background temperature (K) (see text below), 

and,
6

(3) ( ) ∫ 

( ) 

 

 

(4) ( ) ∫ 

( ) 

 

 

Figure 3. Geometry of aircraft flying at altitude, A, above the Earth. 

Note that ( ) is the total atmospheric attenuation along the path defined by ̅. Sometimes, (2) is 

approximated as [3],

(5) [ ] , 

where in degree K, is an effective physical temperature for the atmosphere. 

The integration in (2) along the path defined by ̅ generally intersects many layers of the spherically-stratified 

Earth atmosphere. When ̅ is tilted up from horizontal, this integration ends at the top of the atmosphere, and 

 is the distance from the aircraft (along ̅) to the top of the atmosphere. However, when ̅ is tilted down 

from horizontal, the integration path can end either at the top of the atmosphere or on the Earth surface. When 

the integration path ends at the top of the atmosphere then is equal to the cosmic background temperature 

(~2.7 K). When the integration path ends on the Earth surface then is equal to the Earth brightness 

temperature, and is the distance from the aircraft (along ̅) to the Earth surface. Earth brightness 

temperature varies over roughly 100-300 K, depending on surface composition, observation angle, and 

radiometer measurement frequency/polarization. 

Clearly, the proposed radiometer measurements should be conducted with the cold 2.7 K cosmos as the 

background to be sensitive to atmospheric path attenuation. So the antenna has to be pointed above the 

horizon. But, by how much? The next section (0) formulates a model for the gain pattern of a typical antenna 

and presents an example of measured earth surface bias as a function of elevation angle with a 1’ diameter 

AWR antenna in section 0. 

Earth Center

Z

X

H

R

θP

P

E

A
7

The Radiometer Measurement 

The radiometer measurement is called antenna brightness temperature, ( ). It is defined [3] as a spatially 

weighted average of with weights derived from the antenna power-receive pattern, . Here, the following 

calculation for will be used. 

(6) ∫ ∫ 

 

 

( 

 

 ) 

 

 

 

where, ( ) , ∫ ∫ 

 

 

 

 

 , and ( ) are spherical coordinate system angles associated 

with the axes shown in Figure 3. Here, it is also assumed that, 

(7) [

 ( 

)

 

 

 ( 

)

 

]

 

 

 

 

 

 

 

 

 ,

where ( ) is the antenna half-power beamwidth. 

Antenna beamwidth causes the radiometer field-of-view (FOV) to be a conical region. For the airborne 

radiometers shown in Figure 4, this conical FOV is depicted by the red-colored lines. Note that FOV for the 

radiometer on aircraft #1 makes visible the of trees and soil on the Earth’s surface. For the radiometer on 

aircraft #2, the FOV does not contain the Earth but does make visible the sun, aircraft #3, and radar 

transmissions from aircraft #3. Here, the associated with Earth, sun, and other aircraft are considered 

interference, since only atmospheric is of interest. As suggested by Figure 4, there are strategies for 

avoiding this interference. For example, flying higher or pointing the radiometer antenna above horizontal can 

reduce interference from the Earth and from man-made transmissions (radars) located on the Earth. Solar 

interference can be avoided by pointing the radiometer antenna away from the sun. Software is available for 

determining sun location and can be used when the sun is obscured by clouds. Hence, with knowledge of 

aircraft altitude, heading, and pitch/roll, much of the radiometer interference can be greatly reduced via 

repointing of the radiometer antenna or mitigated by flagging the data. Finally, radiometers themselves can 

determine when the measurement is corrupted by man-made signals. These interference detection 

techniques are described in [4] and [5]. 

Figure 4. Field-of-view (red boundaries) for radiometers aboard three aircraft. 

Altitude Effect and Other Factors (ie. sun, other aircraft)

Earth

2

1

3
8

Radiometer for Severe Weather Avoidance 

With the background of the previous sections in-hand, the radiometer’s role in severe weather avoidance can 

now be explained. First, note that radars can sometimes fail to detect severe weather because of intervening 

water-laden atmosphere that is both heavily attenuating (i.e., large ) and weakly reflective. This radar 

problem is shown pictorially in Figure 5 where the heavily attenuating atmosphere and severe weather (shown 

in red) has obscured radar returns from areas further in range (indicated by pink arrows). However, as 

described below, the radiometer views the same scene much differently than the radar and would probably not 

be “blind” to the areas indicated by pink arrows. 

Figure 5. Radar return showing two severe weather cells (red) that obscure returns from areas 

further in range (pink arrows).

Note from (2) that when the atmosphere has no attenuation (i.e., in (2) or in (5)) then . 

That is, without interference, the radiometer sees the cold background ( ~ 2.7 K) of outer space. However, 

when the atmosphere has high attenuation (i.e., in (2) or in (4)), then . As 

long as lies between and , then the radiometer measurement is said to be “not saturated” and a 

value of can, in principle, be derived from using (5). As shown in the sections to follow, an X-band 

radiometer measurement is seldom saturated in the proposed configuration. 

Radiometers cannot replace radars as a tool for severe weather detection, since radiometers cannot directly 

provide weather-distance (i.e., range) information. The following discussion elaborates on this point. Note 

from (4) that the value of the total attenuation ( ) derived from the radiometer measurement is the total or 

integrated value along the entire atmospheric path defined by ̅ in Figure 3. Information in addition to the 

radiometer measurement is needed to determine whether the total attenuation ( ) results from a very deep 

weather cell having low attenuation rate (small ) or a thin weather cell having high . Severe weather 

(e.g., heavy rain) is more strongly correlated with high than high . Hence, for severe weather detection, 

it is more useful to know how varies along ̅ rather than simply knowing . The range information 

provided by radar measurements can be used to transform radiometer measurements of into values of . 

A novel method of deriving rough estimates of using only the radiometer data is discussed in section 0. 
9

However, should the radar fail, a novel all-radiometer severe weather avoidance strategy is possible. This 

strategy is implied by Figure 2 and consists of scanning the area forward of the aircraft to find the region of 

minimum observed brightness temperature. The aircraft then flies towards this region since minimum 

brightness temperature implies minimum path attenuation as well as minimum liquid path (integrated rain rate) 

and thus (hopefully) the safest direction. 

Atmospheric Attenuation Details 

It appears most cost effective to upgrade existing X-band AWRs with either or both a radiometer mode and a 

dual-linear polarization (V and H) antenna. Hence, the focus here is on atmospheric attenuation at 9.3GHz for 

both V and H polarization. Before starting, we make clear the distinction between total atmospheric 

attenuation, in units of Nepers (Np) or decibel (dB), and atmospheric extinction ( or dB/km). As 

shown in (3), the integral over distance of extinction produces attenuation. The units of extinction can be 

either or , where the conversion is = 

Plotted in Figure 6. is the calculated value of extinction caused by water vapor and oxygen in the US Standard 

Atmosphere (i.e., in (2)) [3]. This calculation depends on altitude above the Earth because the density 

of both water vapor and oxygen thins with altitude and, to a lesser extent, because physical temperature at 

altitude has an effect. 

At the frequencies of interest in this report, dry-atmosphere extinction (i.e., ) is typically much 

smaller than the extinction caused by rain and clouds. Rain and clouds can both be characterized as collections 

of liquid water droplets. The basis for extinction, , caused by liquid water comes from Mie theory [3][6], 

which is considered exact when the droplets have spherical shape. The spherical shape approximation is 

accurate for small droplets and the smallest of these remain suspended in the air as clouds. However, as shown 

in Figure 7, the larger droplets will fall through the air at significant velocity as rain. The air resistance at these 

fall velocities can change the droplet shape from spherical to oblate-spheroid. When the drop diameter nears 

 , the fall velocity can cause sufficient air resistance/turbulence to break the droplet apart [7]. The 

oblate-spheroid shape of the droplets causes the extinction of rain to be polarization dependent. Convenient 

formulas that take this polarization dependence into account are given by [8]. These formulas were used to 

calculate the rain extinction at both V-pol and H-pol experienced by a 9.3 GHz wave traveling in the horizontal 

direction thru rain. Results of these calculations are plotted in Figure 6.. 
10

Figure 6. Extinction at 9.3 GHz for rain and for the rain-free US Standard Atmosphere 

Figure 7. Raindrop shape depends on size and fall velocity. Drops larger than 5mm 

in diameter often break into smaller drops [7]. 

Simulated Radiometer Measurements in Severe Weather 

Software was developed to compute values of that are seen by an airborne radiometer. The software uses 

formulas from sections 0 and 0 and can account for the following factors: 

• radiometer operating frequency

• antenna beamwidth and gain pattern

• antenna pointing (elevation angle)

• aircraft flight altitude

• round Earth effects 
11

• Earth surface brightness temperature

• atmospheric physical temperature, pressure, water vapor (all functions of altitude) 

• rain rate

• rain height 

• rain storm radius

• aircraft location within the rain storm

Here, calculations from the software are used to determine what conditions (if any) cause the radiometer 

measurement to saturate at as discussed in section 0. The following notes apply to these calculations. 

1. The radiometer operating frequency is 9.3 GHz, which is the operating frequency of many airborne 

weather radar (AWR). 

2. Antenna beamwidth of is used and is typical for a medium sized AWR antenna. 

3. The rain height is , which is typical for many mid-latitude storms. Rain height is also 

referred to as the height of the melting/freezing layer. Above this height, the water is in its frozen 

state. Frozen water causes much less attenuation than the liquid water located below the rain height 

[3].

Results of our first calculations are plotted in Figure 8 and are for a small severe weather cell, similar in size to 

a small thunderstorm. In particular, this weather cell has a rain height of and the rain-filled region of the 

cell extends to a range of in front of the radiometer/aircraft. The Figure 8 (a) data results when the 

radiometer antenna is pointed straight ahead (i.e., ). The three curves in this plot correspond to three 

separate flight altitudes, with the lower ( ) and upper ( ) altitudes being below and above the rain 

height respectively. The Figure 8 (b) data results when the antenna is pointing up by about one-half of the 

antenna beamwidth (i.e., ). For these calculations, and represents the saturation 

value. Note that saturation is observed when the radiometer is flying below the rain height and only when 

rain rates are extreme ( ). Also note the differences between Figure 8 (a) and Figure 8 (b) at low 

rain rates, which are due to differences in . That is, for the slight up-tilt of the antenna in Figure 8 (b), 

becomes less biased (corrupted) by the Earth brightness temperature (here assumed to be ).

Results of our second calculations are plotted in Figure 9 and are for a large thunderstorm. For this storm, rain 

height is and the rain-filled region of the cell extends to a range of in front of the 

radiometer/aircraft. As in Figure 8, and Earth brightness temperature is . Unlike Figure 

8, the flight altitudes of and are now both below the rain height. Also note that, due to the larger 

extent of the storm, saturation occurs when rain rates are very-heavy ( ) instead of extreme. 
12

Figure 8. Computed brightness temperature of a SMALL thunderstorm for antenna elevation 

angles of (a) 0deg and (b) 4deg and for flight altitudes of 1km, 3km, and 5km. The rain height is 3 

km. 
13

Figure 9. Computed brightness temperature of a LARGE thunderstorm for antenna elevation 

angles of (a) 0deg and (b) 4deg and for flight altitudes of 1km, 3km, and 5km. The rain height is 4 

km. 

Retrieving Total Atmospheric Attenuation from Radiometer Data 

In (5), the substitution, , is used to give (9) where is interpreted as total loss of the atmosphere. 

(8) [ 

] 

Straightforward manipulation of (8) gives the following solution for . 

(9) 

 

 
14

The precision, , associated with a retrieval of from using (9) depends on the measurement precision 

(NEDT), , for . See Section 0, for further discussion on . The relationship between and can be 

derived from (8) as, 

(10) 

|

 

 

|

 

 

 

.

Using , , and two values of (NEDT), can was calculated from (10) and plotted 

in Figure 10. 

Figure 10. Atmospheric attenuation retrieval precision caused by radiometer NEDT. 

However, can also be affected by errors in . For example, using a value of that is either too high 

or too low by ( ) will bias the retrieved value of as shown in Figure 11. When reviewing Figure 

11, note that a error in is also a error in . Thus, for example, when actual , a 

 bias in implies either or . In principle, when the radiometer antenna is pointing at small 

elevation angles, the physical temperature of air near the radiometer/aircraft should serve as an accurate 

estimate for .
15

Figure 11. Atmospheric attenuation retrieval bias caused by error in Teff. 

Retrieving Rain-Rate and Rain-Depth from Radiometer Data 

As discussed in section 0, radiometer measurements are typically used to estimate total path attenuation, 

 ( ). However, estimates of attenuation rate, are more useful for severe weather detection. 

Knowing is equivalent to knowing both rain rate, ( ), and rain depth, ( ). Here, a novel 

method is presented for estimating and from only measurements of the radiometer. 

Using the relationship in (5), the V-pol and H-pol measurements of a radiometer that is viewing rain of depth, 

 , can be approximately calculated as, 

(11) ( ) [ ( ) ] 

Where p is the polarization, commonly V (vertical) or H (horizontal). 

From [8], one finds that at 9.3GHz, 

(12) ( ) ( ) , 

where, ( ) ( )

and ( ) ( ).

Hence, ( ) and ( ) represent two equations of the form shown by (11) that contain two unknowns, and 

 . The closed-form solution for is given by, 

(13) [

 

 

 

 (

 

 ( ))

 (

 

 ( ))

]

 

 

 
16

Once a value of is computed from (13), it can be substituted into (11) to solve for . Note that the 

condition, , makes the solution for both and possible. When , then there is 

effectively only the one unknown, , and separate solutions for and are not possible. 

Further analysis shows that the solutions for and from 9.3 GHz radiometer data described above are quite 

sensitive to . Retrieval error in and for in both ( ) and ( ) was computed and is 

shown in Figure 12. As seen in these plots, retrieval error depends on the actual values for and . 

Figure 12. Retrieval error for (a) rain rate and (b) rain-depth when raw radiometer measurement 

error is 1.0 K. 
17

Brassboard Radiometer Description 

For experimental purposes, a simple radiometer (i.e., brassboard) was constructed. The radiometer is described by 

the block diagram in Figure 13 and by the photo in Figure 14. The radiometer antenna was made by Cobham (part# 

460386) and is commonly used by aircraft weather radar. Receiver components were housed in a Styrofoam 

enclosure to help stabilize receiver physical temperature and reduce radiometer calibration drift. Data acquisition 

from the radiometer utilized an Agilent Technologies spectrum analyzer, part# E4440A. The radiometer was 

calibrated using commercially-available calibrated noise sources and microwave absorbing material. Calibration 

data is shown in Figure 15. 

Error! Reference source not found.Brassboard radiometer receiver. 

Figure 13. Block diagram of the brassboard radiometer. 

G1= 24dB

NF= 1.2dB

L1= 2dB F1 9.1 - 9.7 GHz G2= 24dB

NF= 1.2dB

L3= 2dB L4= 3dB G3= 23dB

NF= 1.5dB

L2= 2dB TREC = 120K, GTOT ~ 70dB WR90, L= 5", fC = 6.6GHz

Antenna, D= 12" 

Slotted Waveguide

Cobham #460386

9.25 – 9.50 GHz Beamwidth (3dB) = 7.5 deg Spectrum Analyzer

(data collection)
18

Figure 14. Brassboard radiometer. 

Figure 15. Calibration curve for the brassboard radiometer. 

Brassboard Radiometer Experiment 

The brassboard radiometer described in the previous section was used to validate our theoretical description of 

the radiometer measurement ( ) given in sections 0 and 0. More specifically, data was collected from the 

ground-based brassboard radiometer as it was tilted to scan from Earth to the sky across the horizon. The 

effective horizon during this experiment was the top of the wide building shown in Figure 14. Measured and 

calculated values of for this experiment are shown in Figure 16. First, note that the horizon for this 

experiment was measured to be at an elevation angle of . Thus, when the radiometer antenna was scanned 

to then half of its antenna beam observes the Earth and half observes clear sky. Hence, the measured 

 at is seen to be about half-way between the clear sky value of and the Earth value of 

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2000 4000 6000 8000 10000 12000 14000 16000

Brightness Temperature (K)

Raw Measurement (microW)

Sky

BlackBody

NoiseSrc#1

NoiseSrc#2

Calibration Line
19

 . Calculations plotted as a solid line in Figure 16 used the actual radiometer antenna beamwidth of 

 . These calculated values of agree well with the measured data, giving us confidence 

with our modeling. The dashed-line calculations shown in Figure 16 used and can be interpreted as 

the -profile of the scene. As discussed in Section 0, the brassboard radiometer measurements represent 

an antenna-beam weighted average of this -profile. These results indicate that an AWR antenna only needs 

to be pointed up in elevation by less than the beamwidth to prevent loss of sensitivity from earth surface 

radiation. 

 

~ 5 deg.

Figure 16. Measured brightness temperature as a function of elevation angle. This shows that a 7.5 

deg beamwidth (1’ diameter) antenna has to be pointed in elevation a minimum of about 5 deg 

above the horizon to prevent significant earth surface radiation interference. A larger antenna with 

narrower beamwidth could be pointed even closer to the horizon. 

Combined Radar/Radiometer Retrieval 

In some cases co-located radar and radiometer measurements can be made. Here “co-located” emphasizes that 

a similar range interval is sampled and detected by the radar and that the same range interval contributes most 

of the path attenuation and thus the radiometer measured brightness temperature. The next analysis shows how 

such co-located radar and radiometer measurements might be combined to better characterize liquid 

precipitation. 

The general radiometric transfer equation for the radiometer observed brightness temperature is given by [24], 

Vol. 3, Ch. 17, pp. 1331, Eq. 17.80 as 

(14) 

         

  

 

  

0 0

0

P d P

BG

PdP

TB AT

T T

ePTPT dP eT

 


20

where 

TBG

is the background radiation , 



is the Zenith angle (



= 90 –> pointing towards the horizon), 

 T

is the total atmospheric absorption coefficient in units of nepers per unit distance at distance P from the 

radar, and 

TAT

is the temperature of the atmosphere at P. 

In a stratified cloud/precipitation, where uniform physical temperature and drop size distribution can be 

assumed, the observed brightness temperature can be approximated with the following much simpler 

expression [24] Vol. 3, Ch. 17, pp. 1331, Eq. 17.83 : 

(15) 

  P

AT

P

B BG eTeTT

 



 



 

1

,

where 

TAT



is the effective (or mean) radiating atmospheric temperature, and 

P

is the extent of the cloud/precipitation layer along the antenna beam direction . 

The exponential attenuation factor, 

P

e



 through the cloud-precipitation along the antenna beam can be 

defined as the total loss factor,

P

T

eL







, and since the comic background radiation is a negligible 2.7 K, 

the observed brightness temperature can be simply approximated as 

(16) 

  B

 T TLT AT 1

.

So if the mean cloud/precipitation physical temperature can be estimated from in situ measurements and 

extrapolated into the cloud/precipitation layer, then from the observed brightness temperature, the total loss 

factor can be calculated. LT can be also expressed as a function of the more commonly used attenuation 

coefficient,  in units of dB/km, where a (nepers/km) = 0.23 (db/km) 

(17) 

P

T

eL





.0 23

.

From the radar measurements, the cloud/precipitation path length 

  P

 can be directly measured and then the 

path attenuation rate  can be calculated. 

The attenuation rate is directly proportional to the liquid water content for hydrometeors that are much smaller 

than the radar wavelength, Rayleigh scatterers (when

5.0

2 



rn

, where n is the index of refraction, r is the 

cloud/precipitation drop radius and  is the radar wavelength) [3]:

(18) 

fm   T v

.1 9 5 3

.6exp 866 5.41 10  

,

where 

m = liquid water content (g/m3

),
21

f is the radar frequency and T is the cloud/precipitation temperature. 

So from the radiometer observed brightness temperature and radar measured cloud/precipitation path length, 

the attenuation rate and then mean liquid water content can be estimated. 

(19) 



 













N

i

VV

i

v DN

D

m

1

3

3

623

4 

,

where 

Diis the ith cloud/precipitation scatterer diameter, DV is an equivalent volume diameter and NV, the equivalent 

volume number density. 

The radar received power for Rayleigh scatterers is proportional to the sixth moment of the drop size 

distribution (see section 4.1.2, equation (68 ):

(20) 





N

i

At i

r

DK

r

PAn c

P

1

4

6

2 5

2

2

32 

 

And similarly, defining an equivalent volume diameter, DV, and number density, NV, the radar measured power 

can be written simply as

(21) 

6

42

2 52

32 VV

At

r DN

r

P An Kc

P







By combining the radiometer measured liquid water content with the radar received backscattered power, the 

equivalent volume diameter and number density can be estimated: 

(22) 

3

42

2 42

16

3

V

At

V

r D

r

P An Kc

m

P







,

so

(23) 

3

2 42

42

3

16

Pm An Kc

rP

D

AtV

r

V







and

(24) 

v

V

V

V m

D

m

N 3

6





.

A measure of drop size and liquid water content are important parameters for characterizing the hazard 

potential of super-cooled precipitation. Higher liquid water content clearly presents an increased hazard 

potential, because there is more liquid available to freeze on the aircraft wing. But drop size is also important, 

because larger drops shatter on impact and can cause enhanced surface roughness and have larger areas of 

impingement, sometimes beyond the ice protection system. 
22

Adding Radiometer Mode to a Weather Radar System 

New-model, digital-output, AWR (e.g. Rockwell RTA-4100) can cost $150,000. Old-model, digital-output 

AWR (e.g., Honeywell PRIMUS-440) cost $80,000 for a new system and used systems cost about $30,000. 

Digital-output AWR have mostly replaced the older analog-output AWR. However, some analog-output 

AWR are still utilized and have a used price of roughly $5,000-$15,000. Major manufacturers of aircraft 

weather radar (AWR) are currently: 

 Rockwell-Collins 

 Honeywell 

 Bendix-King 

 TelePhonics 

 Garmin 

To permit planning for the addition of a radiometer mode to an AWR, technical information was requested 

from some of these AWR manufacturers. However, the contacted manufacturers required a non-disclosure 

agreement from us before providing any technical information. This disclosure agreement seemed 

inappropriate at this early planning stage. So, detailed AWR technical information was not obtained. 

However, as shown below, many aspects of joint AWR/radiometer operation can still be investigated. 

The method of using a radiometer to assist the AWR with severe weather avoidance was described in sections 

0 and 0. For this application, moderate measurement accuracy ( ) is required from the radiometer. In 

principle, existing radar circuitry could be used to make the radiometer measurements and achieve this 

accuracy after some simple hardware modifications. These hardware modifications consist of adding a switch 

and calibrated noise source to the AWR front-end electronics as shown in Figure 17. Note that noise source is 

used as a warm load calibration unpowered, and as a hot load when the power to the noise source is turned ON. 

The radar receiver is then reprogrammed to occasionally toggle both the new switch, S2, and the noise source 

power so that data can be collected from the noise source in both ON and OFF state. This noise source data is 

then used for radiometer calibration in the manner described by Figure 15. As shown in Figure 18, operation 

of the AWR is also modified so that the transmit (TX) pulse is absent or blanked immediately before 

radiometer data collection. Following the blanked-pulse, all the range-gate data samples collected over the 

pulse-repetition-interval (PRI) can be averaged and used as radiometer data. 
23

Figure 17. Aircraft weather radar (a) before and (b) after radiometer mode modification. 

Time PRI

1 2 ... ave Total Number of samples in the Averaging Interval: N = ave*nrg

Blanked TX Pulses

nrg=n range gates

1/Fs 1

2

3

4

.

.

.

.

. nrg 1

2

3

4

.

.

.

.

. nrg 1

2

3

4

.

.

.

.

. nrg

Figure 18. Blanking the transmit (TX) pulses for the radar permits radiometer data collection. 

With this overview in hand, details of radiometer operation and performance can now be discussed. Precision, 

 ( ), of the radiometer measured can be estimated from [3], 

(25) 

√ 

 

where ( ) is the receiver noise temperature, ( ) is the bandwidth of the receiver, and ( ) is the 

time interval over which the data is averaged. As shown in Table 1, modern AWR typically have 

and . When is near , calculation (25) shows that achieving will require 

 . Further, if the radar scans a ( ) ( ) sector forward of the aircraft with a 

 wide antenna beam, then about ( ) ( ) separate radiometer measurements are needed. 

Since each measurement takes , a total of will be required to perform a complete 

T/R ANT Radar Receiver Display

T/R ANT Display NN Noise Source Control

( a )

( b )

S2 Radar Receiver

Switch Control
24

scan. Thus, the aircraft pilot would have severe weather updates from the radiometer at a rate of about once a 

second. 

However, surely the radar receiver bandwidth could be opened up to at least 5 MHz, and modern data 

acquisition system can easily process data at that sample rate. So repeating the above calculation, the 

approximate update rate with this more realistic receiver bandwidth would decrease to 0.4 sec, yielding two 

scan updates per second. 

Table 1 Specifications of a modern aircraft weather radar. 

Design Parameters Derived Parameters

Frequency (MHz) 9500 Wavelength (cm) 3.16

Pulse Width (μs) 2 Receiver Bandwidth (MHz) 0.52

Pulse Repeat Freq (kHz) 6 Maximum Range (km) 25

Rcvr Noise Figure (dB) 3 Rcvr Noise Temperature (K) 290

Radiometer Summary and Conclusions 

Adding a radiometer mode to conventional aircraft weather radar (AWR) will improve hazardous weather 

detection/avoidance by overcoming some radar deficiencies (e.g., insufficient sensitivity in liquid cloud and 

drizzle conditions, severe weather blind-spots) and assisting the radar to determine atmospheric attenuation. 

Radiometer measurements can be processed into estimates of total atmospheric attenuation ( ). However, as 

discussed in section 0, atmospheric extinction ( ) is a more useful indicator of severe weather than 

atmospheric total attenuation. Radiometer measurements can be combined with radar measurements to derive 

atmospheric extinction. A novel method for roughly estimating extinction using only radiometer data was 

presented in Section 0. However, it is generally more accurate to use the combined measurements from radar 

and radiometer to estimate extinction. 

A novel radiometer-only strategy for severe weather avoidance was presented in section 0. This all-radiometer 

strategy consists of flying towards the area of least atmospheric attenuation and should help overcome radar 

blind-spots. It was calculated that radiometer blind-spots (i.e., saturation) occur only under the unusual 

conditions of extreme rain rates that extend more than in front of the aircraft. Tilting the antenna further 

up in elevation reduces the chance of saturation even in severe conditions.. 

Radiometer measurements can be contaminated by interference from the sun, Earth, other aircraft, and manmade transmissions (e.g., radar). However, most of this interference can be avoided or flagged by monitoring 

and controlling the antenna pointing. Much of the remaining interference can be self-detected by the 

radiometer. 

A brassboard radiometer was constructed and used to collect data during a horizon-scanning experiment. 

Radiometeric measurements and calculations from this experiment show excellent agreement and give 

confidence in our modeling. Furthemore, the measurements demonstrated that an AWR antenna only has to be 

pointed above the horizon by less than the antenna beamwidth to prevent loss of sensitivity due to earth surface 

radiation. 

Although we have not been able to obtain detailed technical information from vendors of AWR, planning the 

addition of a radiometer mode to a typical X-band AWR was investigated. Calculations show that severe 

weather updates from this radiometer can be obtained at a minimum rate of about 2 Hz for 48 separate cells in 

 ( ) ( ) sector forward of the aircraft.
25

3 Polarimetry 

Review of Weather Radar Polarimetry 

Current aircraft weather radars collect radar reflectivity (power) and Doppler velocity mean and standard 

deviation (changes in signal phase) at horizontal linear polarization. Hazardous weather conditions are 

identified by high reflectivity and/or by elevated velocity standard deviation (high turbulence or sheer). 

Reflectivity however is unreliable in severe weather conditions, because rain attenuation can mask severe 

weather cores and icing hazard (freezing rain) for example is not necessarily accompanied by elevated 

spectrum width. Dual-polarization radar can measure additional radar parameters that can be used to 

correct for signal attenuation in rain and can identify mixed phase precipitation such as hail (and the 

melting layer). 

The polarization of a radar system refers to the orientation of the electric field oscillations, perpendicular 

to the transverse wave’s direction of propagation. Current weather radars operate using horizontally 

polarized antennas, so when the radar is pointed towards the horizon, the electric field oscillates in the 

horizontal plane as shown in the first (left) graphics of Figure 19. The transmitted field of dualpolarization radar, shown in the second (right) graphics of Figure 19, uses both horizontal and vertical 

polarized fields. The horizontally polarized filed scattering most influenced by the horizontal cross 

section of particles, while the vertical polarization “sees” the vertical dimension. Small raindrops are 

close to perfect spheres due to the surface tension, but larger deform and often modeled as oblate 

spheroid. This deformation is illustrated in Figure 20. When only small hydrometeors are present in a 

range cell of polarimetric radar, the backscattering, attenuation and phase propagation is approximately 

same at H and V polarization. But when larger raindrops are present, the H-pol radar pulse will reflect 

stronger, attenuate more and phase propagation will be slower than V-pol. Consequently, the received 

power ratio (Zdr) and the relative phase (Φpd) between H and V-polarized received signal contain 

information on hydrometeor size. The phase center of mixed phase and non-water particles also different 

at V and H-polarization, so mixed phase precipitation (hail and melting band) and ground debris can be 

effectively detected and differentiated from rain and drizzle using H and V signal correlation (ρHV). The 

following sections describe the polarimetric radar parameters commonly used for characterizing 

precipitation. 

Figure 19. Dual polarization technology gives additional information on hydrometeor size, shape 

and composition to allow the estimation of rain rate, signal attenuation and the detection of hail. 

Source: NOAA 
26

Figure 20. Raindrop shape primarily depends on size. Larger raindrops also have significant shape 

oscillations and often break apart. 

3.1.1 DIFFERENTIAL PHASE SHIFT (ΦDP) 

ΦDP is the difference in the returned phase of the vertical and horizontal returns. The difference in return 

phase is greater from areas of large oblate spheroid drops because the horizontal pulses are slowed 

slightly more than the vertical pulses. This slight difference is enhanced by range from the radar. It is 

generally regarded that ΦDP is best used in its range derivative form: KDP.

3.1.2 SPECIFIC DIFFERENTIAL PHASE SHIFT (KDP) 

KDP is the range derivative of ΦDP in units of deg/km. It is generally immune to large hail and thus is 

considered a useful indicator of actual rain rates and thus attenuation. Larger values indicate higher rain 

rates and higher attenuation. Light rain containing smaller, non-oblate spheroids results in very little 

return phase differences. 
27

Figure 21. Dependence of Rain Rate - Kdp relations on radar frequency, (from Park, S.-G., Maki, 

M., Iwanami, K., Bringi, V.N., Chandrasekar, V., 2005. Correction of radar reflectivity and 

differential reflectivity for rain attenuation at X band. Part I: theoretical and empirical basis. J. 

Atmos. Ocean. Tech.. 22, 1621–1632.). 

3.1.3 DIFFERENTIAL REFLECTIVITY (ZDR) 

ZDR is ten times the log of the ratio of the horizontal and vertical reflectivity return from a sample 

volume. Large drops are oblate spheroids and have increasingly larger returns from the horizontal pulses 

compared with the vertical pulses. For ZDR over 6 dB, the drop diameters are exceeding 8 mm. Drops with 

diameters under about 0.5 mm have ZDR about zero. Unfortunately, hail also has a ZDR of around zero. 

Generally, snow has a low ZDR independent of shape due to its small dielectric constant. ZDR is sensitive 

to drop size, but inversely to propagation through rain with larger raindrops. So the observed ZDR first has 

to be corrected for differential attenuation and then usually combined with Kdp (specific differential phase 

shift and/or Z (reflectivity) to estimate rain rate/attenuation. The linear reflectivity ratio is usually 

differentiated from the LOG ratio by using lower case “dr”: 

(26) 

  ZDR 10 Zdr 10log

3.1.4 CORRELATION COEFFICIENT (ρHV) 

ρHV is a measure of the correlation of the returned power from the horizontal and vertical pulses. Typical 

meteorological values are near unity with a very small variation. In regions of non-uniform scatterers, 

such as mixed phase areas (i.e. – melting layers), the ρHV is lower. Ground clutter and airborne debris also 

have lower values as shown in middle image in the top row of Figure 22. ρHV is rather used for scatterer 

identification than quantitative retrieval. 
28

Figure 22. Example of X-band polarimetric radar data collected at low elevation angle in a 

tornado: Reflectivity (top-left), ρhv debris signature (top-middle), spectrum width (top-right), 

Doppler velocity folded (bottom-left and ) and un-folded (bottom-middle), ZDR (bottom-right). 
29

Rain Rate Estimation from X-band Polarimetric Radar Parameters 

Rain rate is highly correlated with attenuation, so existing models for estimating rain rate from X-band 

polarimetric radar parameters can be used to correct for attenuation. 

Figure 23. examples of k–R relationships established using Mie theory (spherical raindrops) for the 

K- (0.86 cm), Ka- (1.15 cm), X- (3.2 cm), C- (5.6 cm), and S-band (10 cm) wavelengths and the for 

raindrop temperatures T=0 ̊C (dotted line), T=10 ̊C (continuous line), and T=20 ̊C (dashed line). 

From [12]. 

3.1.5 Rain Rate – Z Estimator 

(27) 

.1 76

100ZR eh

from [13] , 

where R is rain rate in mm/hr, and 

Zeh is equivalent reflectivity factor at horizontal polarization, in units of mm6/m3. 
30

3.1.6 Rain Rate – Kdp Estimator 

(28) 

b

R  aKDP ,

where 

Kdp is the specific differential phase shift in units of deg/km, and 

a = ~16.5 and b = ~ 0.75 but slightly dependent on drop size distribution. 

Figure 24. R – Kdp power-law fit relations using size distributions collected in various field 

campaigns. From [14].

3.1.7 Rain Rate – Z, Zdr Estimator 

(29) 

.1 07 .5 97

 .0 0039 eh ZZR dr

,

(from [18]) 

where 

Zdr is differential reflectivity in linear units; ZDR = 10log10(Zdr).
31

3.1.8 Rain Rate – Z, Zdr, Kdp Estimator 

(30) 

b eh ZdZc

DR KaR d p



 10

,

(from [19])

where the a, b, c, d coefficients are given in Table 2. 

Table 2. Coefficients of Equation (30) rain rate estimator (KDP, Z, ZDR). 

Rain type a B C D

Stratiform 0.009 −0.173 0.103 −0.653

Convective 0.198 0.4405 0.035 −0.036

3.1.9 Comparison of various rain rate estimators 

The various rain rate estimators are compared in Figure 25 for sensitivity to variability in drop size 

distribution. As expected, the reflectivity only estimator is very poor in conditions anything other than the 

lowest rain rates. The estimation improves, as more parameters are included with reflectivity. But only 

error free parameters are considered in these simulated results, while Z and Zdr are susceptible to error due 

to unaccounted for attenuation and differential attenuation. Kdp on the other hand is a much easier to 

measure accurately, so the upper right rain rate estimator plot is closer to what can be realistically 

expected than the two lower row plots of the more complex models. 
32

Figure 25. Scatter plots of rain rain calculated from measured drop size distribution and estimated 

from four types of rain rate estimators a) R(Z), b) R(Z, Zdr), c) R(Kdp), d) R(Kdp, Z, Zdr). From [19].

Dual Polarized Aircraft Weather Radar Antenna. 

The cost of upgrading current AWRs to dual-polarized capability is relatively inexpensive. The 

transmitted pulse has to be equally divided to the Vertical and Horizontally polarized antenna ports and a 

second receiver has to be added to the radar. The signal processing is well-documented and straight 

forwards and polarimetric radar parameters, such as differential reflectivity (Zdr) and specific differential 

phase shift (Kdp) can be effectively used to correct for attenuation in moderate to heavy rain conditions. 

The only missing (enabling) technology is a dual-pol antenna, similar in quality to the current standard 

flat plate waveguide array. A dual-pol slotted waveguide array antenna has been patented by Rockwell 

Collins: http://www.google.com/patents/US8098189 but performance data was not available to assess 

potential polarimetric measurement accuracy, at the time of this report. 

Past aircraft weather radars used magnetron transmit sources that required waveguide antenna array 

technology to be able to transmit high peak power. Modern weather radars use lower peak power solid 

state transmitters, which opens up the possibility of using microstrip antenna arrays. Some attempts have 

been made to develop lightweight dual-pol X-band microstrip antennas. In 2001 a design study was 

conducted by Antenna Design Associates to develop a Multi-Frequency (X, Ka and W-band) and dual-pol 

at X-band aircraft weather radar antenna. It concluded that a two-layer aperture coupled microstrip array 

technology (shown in Figure 26) could be used to develop a 1’ to 2’ diameter, medium power (<10 kW 

peak power) flat, lightweight X-band aircraft weather radar antenna. Larger antennas with narrower 

beam-width would be also possible, but without much improvement in gain due to increasing losses, with 
33

increasing antenna size, in the microstrip substrate. More recently, a small experimental dual-pol 

microstrip array (~5” diameter, 25 deg beam width, 19 dB gain) was developed by [26]. 

Figure 26. Dual-polarized X-band microstrip antenna array with interleaved coplanar feed (coax 

feed points are shown as solid circles; offset = 6.96 cm = 0.202D). This design also incorporates a 

0.2D V-H aperture offset for spaced antenna transverse wind measurement. A waveguide power 

divided network is also required in the rear of the antenna. 

3.1.10 Antenna requirements 

Current airborne weather radars use flat plate (or box-horn) array antennas, like shown in the left pane of 

Figure 27. These antennas are ideal for airborne radar applications due to their light weight, low profile 

and excellent electrical properties (low loss and low sidelobes), but are not available in dual-polarization 

version (no information is available on the Rockwell Collins patent of a dual-pol slotted waveguide array, 

http://www.google.com/patents/US8098189). Microstrip antennas are also relatively light weight and 

have low profile, but can be developed with dual polarization capability, so they might be a potential 

candidate for future dual-polarized aircraft weather radars. The following discussion is on the required 

specifications of a dual-pol AWR antenna for hydrometeor classification and implementation of 

polarimetric rain retrieval algorithms. 
34

Figure 27. Various size flat plate (box-horn) waveguide array antennas. 

An ideal weather radar antenna has low loss, low sidelobes, close to the ratio of wavelength (



) and 

antenna diameter (D), 

D



 (rad) half power beamwidth (high gain for a given diameter) and if 

polarimetric, then the orthogonally polarized ports have high isolation, but co-located beams. The high 

gain is important for sensitivity. The low sidelobes (>25 dB one way) are important to ensure confidence 

in the location of the source of the measurements. Sidelobe return from a strong reflectivity core can 

severely bias an observation. Fortunately most well designed antennas have at least 20-25 dB first 

sidelobe levels and sidelobes further from the main beam quickly drop. But more importantly, with single 

antenna radar, the antenna is used for transmission and reception, so interference from a precipitation core 

is attenuated by twice the sidelobe suppression level in dB. Consequently an antenna with a 25 dB 

sidelobe will suppress return from the direction of that sidelobe by 50 dB. Since most airborne weather 

radars rarely measure above 50 dB signal to noise ratio, a 25 dB first sidelobe level is usually sufficient. 

Co-located V and H-pol beams are essential for useful 

 HV

 measurements, with correlations varying 

from 0.99 in light rain and drizzle to 0.90 in hail and as low as 0.80 in mixed phased precipitation 

(melting layer / bright band). Consequently the vertical and horizontally polarized beam patterns of a 

dual-pol antenna should be 99% co-located (overlapped). 

Quantifying the effect of polarization isolation on the measurement accuracy of polarimetric radar 

parameters and consequently on the quality of the retrieved rain rate or attenuation is more complicated. 

Wand and Chandrasekar investigated this problem [21], but only plots of the results were published, not 

the complete expressions. Consequently the effect of finite antenna isolation on rain and attenuation 

estimation required us to re-examine the problem: 

The received field of simultaneous V and H radar transmission can be described as 

(31) 

         













T

j

T

e

APSPAR 

1

,
35

where 

 T

is transmitter phase difference between the V and H channels, and [A] is the antenna distortion matrix, 

accounting for antenna polarization isolation, I, V-channel phase shift and coupling phase shifts 

 HV

 and 

VH

: 

(32) 

  













VH VV

HV

j j

j

eeI

eI

A  



1

, 

[S] is the target scattering matrix: 

(33) 

  













VH VV

HH HV

SS

SS

S

,

and [P] is the propagation distortion matrix, accounting for differential phase shift (

dp

): 

(34) 

  











 



2 0

01

dp j

e

P

.

The resulting received voltages from the H and V antenna ports are: 

(35)

   

   

H V H V T d p H V VV T VH

d p

VV T VH

d p

H V T

jj j j

j

VV

j j j

j

VH

j j j

j

HV

j j

HH

H

eIeeeIeSeeIeIeS

eIeeeSeeIS

R

     

    

  

  













1

1

2

2

and 

(36)

   

   

H V T VV d p VV T VH

d p

VV

VV T VH

d p

VH H V T VH

j j j

j j

VV

j j

j

j

VH

j j j

j

j

HV

j j j

HH

V

eeS eIeeeeSeeI

eIeeeeISeeIeIS

R

      

      

  

  













1

1

2

2

These expressions can be simplified by eliminating the second order terms with 

 2

I

, 

VH IS

and 

HV IS

: 

(37) 

 

VV T d p VV T H V

d p

H V T

j j j

j

VV

j j

j

HV

j j

H HH eeeIeSeeeSeeISR

       





  

2 1

(38) 

 

VV H V T

d p

d p VV VV T VH j j

HH

j

j

VH

jj j j

j

V VV eeISeeSeIeeeeSR

     

  



 

2
36

The resulting becomes a simplified expression for the observed radar reflectivity factor 

Z

ˆ

 by assuming 

that co-pol scattering is uncorrelated with cross-pol backscatter: 

(39) 

 

VV T d p VV T H V

d p

H V T

j j j

j

VV

j j

j

HV

j j

H HH eeeIeSeeeSeeISR

       





  

2 1

(40) 

 

VV H V T

d p

d p VV VV T VH j j

HH

j

j

VH

jj j j

j

V VV eeISeeSeIeeeeSR

     

  



 

2

(41) 

 

 



























  

     

 

d p VV T VH d p VV T VH

H V T H V T

j j j

j j j jj

d r

HV

j j

H H H

eeeeeeee

z

I

eeI

ZRZ      

 



)()(

2

1

ˆ

, 

and the complex exponentials combined to simplify: 

(42) 

   











  d p VV T VH

d r

HH HV T HV

z

I

ˆ

IZZ cos21 2 cos 

,

where zdr is the differential reflectivity in linear units 

V

H

dr Z

Z

z 

; 

  DR dr Z z 10 10log

.

So the error factor bound of the measured reflectivity, 

ZH

E

, is 

(43) 















zdr

E I

HV

ZH



121

, 

and similarly for the vertically polarized reflectivity: 

(44)

     VV d p VV T VH HV d r d p VV HV

ˆ

IZZ 2cos21 2 zI 2cos  

(45) 

  Z HV dr E zI

V

121  

.
37

Figure 28. Estimated reflectivity error bounds as a function of antenna polarization isolation. Solid 

lines are for rain (

 HV

= 0.98) and dashed lines for mixed phase (

 HV

= 0.9). 

The differential reflectivity, zdr, is the ratio of the H and V reflectivity factor (assumed that 

HV

 VH

to simplify): 

(46)

   

    d p VV T VH HV d r d p VV HV

d p VV T VH

d r

HV T HV

d rd r

I zI

z

I

I

zz

 

 



  



2cos21 2 2cos

cos21 2 cos



and so the zdr error factor range without phase control over the transmitted field, 

(47) 

    HV d r

HV

z

HV d r

HV

zI

zdr

I

E

zI

zdr

I

d r 









































121

121

121

121

.
38

Figure 29. Estimated differential reflectivity error bounds as a function of antenna polarization 

isolation for rain (solid; 

 HV

= 0.98) and mixed phase precipitation (dashed; 

 HV

= 0.8). 

Note that if the antenna leakage and propagation phase shifts are the same (

VH HV  VV

 0

) and 

if the differential phase shift is small, then the error can be dramatically reduced by transmitting the V and 

H field components 90 deg out of phase: 

(48) 

    H

d r

HH HV Z

z

I

IZZ  











 

2

2 cos 2cos21

ˆ   

(49) 

   

    d r

HV d r

d r

HV

d rd r

z

I zI

z

I

I

zz 

 

 



2

2 cos 2cos21

2

2 cos 2cos21

  

  



Differential 

  dp

and Specific Differential Phase 

  Kdp

are defined as: 

(50) 

  *

dp  arg RR VH
39

and 

(51) 

dr

d

K

dp

dp





, 

where r is range. 

The measured Differential Phase 

dp



 can be expressed as a sum of the measurement error

  dp

 and 

dp

: 

(52) 

dp dp  dp



.

Similarly, the measured 

Kdp



can be written as: 

(53) 















































d p

d p

d p

d p

d p

d p d p d p d p d p

d p d

d

K

dr

d

d

d

dr

d

dr

d

dr

d

dr

d

K 1





.

(54) 

  *

dp  arg RR VH



(55)

 

 





















 



















   



 







*

2

2 1arg

VV T H V T

d p

d p VV VV T VH

VV T d p VV T H V

d p

H V T

j j

HH

j j

j

VH

jj j j

j

VV

j j j

j

VV

j j

j

HV

j j

d p HH

eeISeeeSeIeeeeS

eeeIeSeeeSeeIS

      

      



By eliminating the second order terms and terms without

dp

, and replacing

N

LDR

HH VH  ZSS H



, where N is the number of independent samples averaged and LDR is the 

linear depolarization ratio, the differential phase expression can be simplified as: 

(56) 

  







































































 

  



     

d r

j j

j

jj j j j j j

j

HV

d p H

Z

eee

N

LDR

eeeeeIeeee

Z

VV T

d p

d p VV VV T H V T VV T VH

1

1

arg

2

 

      





If we can assume that the antenna cross-pol and co-pol phase shifts are the same, such that 

VH HV  VV

 0

,  HV 1and 

Zdr 1

then 
40

(57) 

































 



  2

arg 221

d p

d p T T

j

j j

j

d p e

N

LDR

ee eI

 



(58) 

   

    





































 

 











 

 



2

cos cos2cos2

2

sin sin2sin2

arctan

d p

d p T T

d p

d p T T

d p

N

LDR I

N

LDR I

 

 



.

Assuming that 

I 12

and 

2 1

N

LDR

, we can further simplify as 

(59) 

  









 

 

2

sin2sin2

d p

d p d p T T

N

LDR  I 



(60) 

  









 

  

2

sin2sin2

d p

d p d p d p T T

N

LDR  I 



(61) 





















































2

1 cos21

d p

d p

d p

d p

d p d p N

LDR K

d

d

KK



This result indicates that the Specific Differential Phase 

  Kdp

 error bounds are primarily affected by the 

target linear depolarization ratio (LDR) and averaging (N) as long as the antenna isolation is fairly good 

  I 12 . 

In most conditions 

dp



will be small such that 

1

2

cos 













 dp

, so the error is most likely positive: 

(62) 

N

LDR EKd p

 21
41

Figure 30. Estimated Kdp error bounds as a function of antenna isolation, using Equations (53 and 

(58. An LDR or 25 dB was used in the calculations. 

Figure 31. Estimated attenuation error bounds as a function of antenna isolation, based on the 

results of Figure 30, using Kdp (Equation (28 - solid lines) and Z-Zdr based estimators (Equation 

(29 – dashed lines). 
42

The conclusion of the above analysis is that polarimetric antennas for airborne weather radars should have 

at least 25 dB (30 dB preferred) polarization isolation to be useful for estimating attenuation using the 

various retrieval methods. Kdp-based estimators can be effective with less isolation. The V and H 

components of the transmitted field should be phased close to 90 ̊ for best measurement quality. 

4 Multi-Frequency Radar (MFR) 

The task of adding another frequency to the current X-band AWR channel is significantly more complex 

and potentially more expensive than adding radiometric or polarimetric measurement capability. A 

second radar channel will likely require a separate antenna and radar electronics, although a dual 

frequency antenna, specific for airborne use, might be easier to develop than a lightweight, flat-plate dualpolarized X-band antenna. But most likely, only the data system and power supply sections could be 

shared. So why consider a second frequency? 

The two most important frequency dependent factors influencing weather radar measurements are that 

backscattering efficiency of small scatterers increases as f

4

 (Equation (80)) and attenuation coefficient as 

f

2

 (Figure 48). Attenuation becomes more important with range so long-range weather radars are 

predominately lower frequency (relative to X-band) S-band (10 cm wavelength) in the US and C-band (5 

cm) in Europe. On the other end of the weather radar spectrum, cloud research radars are usually Ka (1 

cm) or W-band (3 mm) to maximize sensitivity at close range. A multi-frequency weather radar has two 

primary benefits: 

1) A greater variety of weather conditions could be effectively measured. When the total path 

attenuation is high (due to long range and/or high attenuation rate), the lower frequency radar channel will 

be more effective, while higher frequency radar should have better sensitivity to small scatterers, when 

attenuation is less of a factor. Of course the availability of quality radar components and transmitter 

power are also important factors determining sensitivity at various frequencies. 

2) The difference between radar parameters measured at two (or more) frequencies can be used to 

estimate cloud and precipitation properties. Particularly differential attenuation, derived from differential 

reflectivity gradient, can be highly correlated with attenuation and liquid water content, and can be used 

to better characterize weather hazards. 

This section investigates the potential use of multiple frequencies with AWRs. First, sensitivity, back 

scattering efficiency and attenuation is investigated at a function of frequency. Then practical system 

parameters are considered along with a model of various cloud, drizzle and rain conditions, to simulate 

the frequency dependence of radar sensitivity, attenuation and measurement range in various weather 

conditions. 

MFR Measurement Concept 

4.1.1 Point Scattering 

The transmitted power density, S, of a radar can be expressed as 

(63) , ( ), 

where is the transmitted power in (W), G is the antenna gain and is the distance to the point target 

2

4 r

GP

S

t

t



 W/m2

Pt r
43

reflector in meters. 

The power density captured by a point target and re-radiated towards the radar is 

(64) , ( ).

The received power at the antenna feed is the product of the antenna effective aperture (Aeff) and the 

incident power density of the scattered field according to 

(65) , 

and since 

(66) , 

the received power can be written in the standard form of the radar range equation for a point target as 

(67) , (W). 

The actual antenna aperture area, A, can be expressed as where is the aperture efficiency 

so the radar range equation can be rewritten as: 

(68) .

4.1.2 Volume Scattering 

The radar range equation for a point target can be extended to a volume target by replacing with the 

product of volume reflectivity, ( ) and the volume of a range cell, V. The range cell volume of 

a pulsed radar, with an approximately Gaussian shaped antenna pattern, may be approximated as 

(69) , ( ),

where, 

r is the range to the center of the volume cell in meters, and are azimuth and elevation angles 

2 42

)4(4 r

GP

r

S

S

t t

r











W/m2

42

)4( r

GP

ASAP

t

effreffr







4/ 

2

eff  GA

43

22

)4( r

GP

P

t

r







 nAA Aeff A n

42

22

4 r

nAP

P

At

r









 32

/ mm

   

 

 



0

2

0

2

3

2 2 2

8/ ln 2

2

)sin(),(

2

d B

c

f rdd

c

rV

3 m

 
44

respectively, and are the half power antenna beamwidths in azimuth and elevation, 

(70) 

m/s, speed of light in free space,

transmit pulse length (sec.). 

For most weather radars, the antenna half power beamwidths in elevation and azimuth may be 

approximated to be equal so (radians). 

The resulting expression for the received power from a volume target is: 

(71) 

(72) 

(73) 

since 

(74) , 

(75) 

(76) 

This result shows that the radar received power from a volume target is a function of the power-aperture 

product, range squared, range resolution ( ) and volume reflectivity of the target. Volume reflectivity 

is dependent on frequency, polarization and beam pointing direction. This backscattering efficiency of 

small scatterers is dramatically higher (f

4

) at higher frequency, as shown next. Note that the Signal to 

3dB 3dB )

)2ln(4

exp()

)2ln(4

exp( 2

3

2

2

3

2

d B d B

f









 

8

c 3 10  

db   33 dB 



V

r

nAP

P

At

r 42

22

4







4 16ln 2

2

3

2

42

22

At dB

r

cr

r

nAP

P 





 64ln 2

2

3

22

22

At dB

r

c

r

nAP

P 

D A

db



3  22





 A

c

r

nAP

P

At

r

64ln 2

2

2

22

22



2

2

32r

P An c

P

At

r





2

c
45

Noise Ratio, SNR, usually increases as range resolution squared, because the noise power decreases with 

increasing pulse length (decreasing bandwidth) and thus range resolution. 

The Volume reflectivity can be written as a sum of all the individual point scatterers in a unit volume: 

(77) .

If each scatterers can be approximated as a sphere, then the radar cross section can be estimated as the 

physical cross section and an efficiency coefficient: 

(78) 

where is the back scattering efficiency of a single scatterer and D is the diameter. 

Figure 32. Radar backscattering efficiency ( ) of a metal sphere as a function of radius (a) and 

radar wavelength ( ) (from: [3]). 

i

N

i

v 



1



4

2

D

4

2

D

 








46

In the Rayleigh region, 

(79) , 

so

(80) , 

where , and n is the complex index of refraction. 

at 9.4 GHz ; at 35 GHz and at 95 GHz . 

In the optical region, , so , when |n| >>1. 

In the Rayleigh region: 

(81) 

In the Optical region: 

(82) 

2

4

4 K

D





















2

4

652

4

K

DD



 

 

1

1

2

2







n

n

K

.0 97 2K  .0 88 2K  .0 72 2K 

 1 4

2

D

 





Nv

i

i

t

r

D

r

AP

P

1

6

42









Nv

i

i

t

r

D

r

AP

P

1

2

2


47

4.1.3 Attenuation 

The attenuation ( ) of liquid clouds and precipitation (liquid Rayleigh scatterers) is inversely 

proportional to wavelength squared and proportional to liquid volume [24]: 

Liquid in the Rayleigh region: 

(83) 

Attenuation of ice clouds is much lower and weaker related to wavelength: 

Ice in the Rayleigh region: 

(84) 

4.1.4 MRF Measurement Parameters 

Ideally we would like to know the particle size distribution , but radar can only measure integrated 

parameters: 

Reflectivity in the Rayleigh region: -> severely biased by the largest particles 

Reflectivity in the Optical region: -> areal cross section 

Attenuation in the Rayleigh region: -> liquid volume 



.1 95

1

3









Nv

i

Di









Nv

i

Di

1

3

  Df

   

DfD

6

   

DfD

2

   

DfD

3
48

Number density (NV) is also an important physical parameter. 

Let’s assume a narrow size distribution such that 

(85) 

for Rayleigh scatterers, 

and 

(86) 

for Optical scatterers. 

where De is an equivalent particle diameter. 

4.1.5 MFR Information Content 

Two frequencies – both in Rayleigh and no differential attenuation (high altitude ice and ash with two 

radars): 

(87) 

(88) 

Two linearly dependent equations, so there is no additional information by having two frequencies: 

(89) → No New Information! 

Two frequencies – one Rayleigh, one Optical and no differential attenuation (high altitude ice and ash 

with radar and lidar sensor): 

6

2

4

5

1

6

2

4

5

1

eV

N

i

i

i

N

i

DNKDK

v v









    

 

2

1

2

1 4 4

eV

N

i

i

i

N

i

DND

v  v 

   

 

6

4

1

1 2 eV

f

t

f DN

r

AP

P







6

4

2

2 2 eV

f

t

f DN

r

AP

P







4

1

4

2

2

1

f

f

f

f

P

P






49

(90) 

(91) 

Two independent equations, two unknowns (De and NV).

(92) → Size Estimate 

From either power measurements, NV can be also estimated. The above two cases (Rayleigh-Rayleigh 

without attenuation and Rayleigh-Optical) is well illustrated in Figure 41. The small, at both frequency 

(Rayleigh), cloud droplets form a flat line at 0 dB F-Zdr for the full range of LWC indicating that those 

radar measurements have no information content regarding LWC. On the other hand, the drizzle and rain 

points form an arcing line with high degree of correlation with liquid. Only scatterers in the Rayleigh to 

Optical transition, called Mie region would be difficult to measure. 

Two frequencies – both in Rayleigh with differential attenuation (super-cooled liquid clouds and drizzle): 

(93) 

(94) 

(95) 

(96) → Liquid Water Content Information 

From either power measurement, NV and De can also be estimated. 

In practice, retrievals are more complicated because the drop size distributions can be broad, so 

6

4

1

1 2 eV

f

t

f DN

r

AP

P







2

2 2 eV

t

f DN

r

AP

P





4

1

4

2

1

f

e

f

f D

P

P





DN  r

r

AP

P eV f

f

t

f  1



6

4

1

1 2

exp 





DN  r

r

AP

P eV f

f

t

f  2



6

4

2

2 2

exp 





  r P

P

ff

f

f

f

f  12  4

1

4

2

2

1

exp 





  















  2

1

2

2

3

12

11

ff

ff DN eV




50

reflectivity measurements are dominated by larger particles than attenuation in the Rayleigh region. Also 

higher frequency radars will often operate in the Mie region (Rayleigh to Optical transition), where 

resonant scattering varies the backscattering efficiency rapidly as a function of . Nevertheless, even 

with wider drop size distribution, an approximate measure of drop size and number density can be 

estimated and used to improve hazard detection and characterization. Clearly, there is potential for 

additional information from dual frequency radar when the differential attenuation rate is sufficient to 

decouple the measurements. The logical next questions are: how much frequency separation is needed; 

and what is the overlapping detection range (sensitivity) of various frequency combinations, based on 

currently available radar components? These questions are addressed next. 

MFR Components – Current Technology 

4.1.6 Antennas 

Airborne weather radar antennas are almost exclusively flat plate slotted arrays as shown in Figure 33. 

These antennas combine exceptional mechanical and electrical properties. Their light weight and narrow 

profile make them ideal for airborne use, the waveguide construction minimizes loss and thus improves 

efficiency and the absence of feed blockage leads to very low sidelobes and high beam efficiency. In size 

they range from 12” on the smaller commuter aircrafts to 20” and 30” in diameter on the mid- to larger 

aircrafts. This antenna type has been manufactured up to Ka-band frequency, but only available in single 

polarization. Dual-polarized microstrip arrays have been designed but their limited power handling 

capability makes them less attractive with weather radars. 

Figure 33. X-band flat plate slotted antenna array. 

For the following multi-frequency radar simulations we will consider the 20” diameter antenna at the 

frequencies shown in Table 1. 



D
51

Table 3. Approximate antenna half-power beamwidth and gain as a function of antenna diameter 

and frequency. Further analysis of this report will consider the optimistic 20” diameter antenna 

size for all frequencies. 

Diameter X Ku Ka W

12” 7.5○(27 dB) 4.7○(31 dB) 2.0○(38 dB) 0.75○(47 dB)

20” 4.5○(31 dB) 2.8○(35 dB) 1.2○(43 dB) 0.44○(51 dB)

30” 3.0○(35 dB) 1.9○(38 dB) 0.8○(46 dB) 0.30○(55 dB)

4.1.7 Transmitters 

Airborne weather radars primarily use magnetron power sources for their small size, high peak power. Xband magnetrons in the 10-25 kW peak power range have been manufactured in high volume for marine 

ship detection and airborne weather radar applications, resulting in per unit cost below $1000 – by far the 

least expensive high power source per unit output power. Figure 34 illustrates the peak power of various 

power sources as a function of frequency. Magnetrons are vacuum devices, available up to Ka-band at 

peak power ranging from about a 1kW to hundreds of kW, and 0.1% maximum duty. Other, usually 

higher average power vacuum tubes include Klystrons, Gyrotrons and Traveling Wave Tube amplifiers, 

but these typically require much larger modulators and power supplies, making them less attractive for 

airborne use. High duty solid state devices and microwave modules are sufficiently compact for airborne 

use and over the past decade have become available with average power comparable or exceeding that of 

magnetrons, but their peak power is low, and it is difficult to utilize more than 5% duty cycle with a 

single antenna radar system. Table 4 through Table 7 lists the various currently available transmit sources 

at various frequencies. For the sensitivity calculation 10 kW peak power was assumed at X, Ku and Ka 

bands and the optimistic (due to cost, size and weight) 1 kW Klystron peak power at W-band. 

Figure 34. Transmitter power versus frequency (from the Litton-L3 MPM web site). 
52

Table 4. X-band transmitters (8 – 12 GHz). 

Type Peak Duty Practical Cost 

($K)

Peak Average

Magnetron 10-1000 kW 0.1% 10-20 kW 10-20 W 1

TWTA 0.1-150 kW CW – 6% - - 100+

Klystron 0.5 – 3 kW CW – 5% - - 100+

MPM 10-100 W 5% - CW 100 W 10 W 10

Solid State PA 20 W 5%-CW 20 W 2 W 10

Solid State Array (20x20) 0.1 W/element 5%-CW 40 W 4 W 10

Table 5. Ku-band transmitters (12 – 18 GHz). 

Type Peak Duty Practical Cost 

($K)

Peak Average

Magnetron 10-1000 kW 0.1% 10-20 kW 10-20 W 1

TWTA 0.1-60 kW 0.1 – 30% - - 100+

Klystron 0.5 – 3 kW 5% - CW - - 100+

MPM 10-100 W 5% - CW 100 W 10 W 10

Solid State PA 20 W 5% - CW 20 W 2 W 10

Solid State Array (20x20) 0.1 W/element 5% - CW 40 W 4 W 10
53

Table 6. Ka-band transmitters (27 – 40 GHz). 

Type Peak Duty Practical Cost 

($K)

Peak Average

Magnetron 10-100 kW 0.1% 20 kW 20 W 50

TWTA 0.1-60 kW 0.1 – 30% - - 100+

Klystron 2 kW 5% 2 kW 100 W 100+

MPM 10-100 W 5% - CW 100 W 10 W 10

Solid State PA 20 W 5% - CW 20 W 2 W 10

Solid State Array (20x20) - - - - -

Table 7. W-band transmitters (75 – 110 GHz). 

Type Peak Duty Practical Cost 

($K)

Peak Average

Magnetron - - - - -

TWTA 0.05 - 3 kW 10 % – CW - - 100+

Klystron 1 kW 5% - CW 1 kW 50 100+

MPM 10-100 W 5% - CW 100 W 10 W 10

Solid State PA 20 W 5% - CW 20 W 2 W 10

Solid State Array (20x20) 0.1 W/element 5% - CW 40 W 4 W 10
54

MFR Detection 

A cloud and precipitation model was used to predict detection (unity single pulse signal to noise ratio) 

using the following assumed radar specifications: 

Table 8. List of radar parameters used in the sensitivity calculations. 

X Ku Ka W

Peak Power (kW) 10 10 10 1

TX Loss (dB) 0.25 0.3 0.5 1

Antenna Gain (dB) 31 35 43 51

RX NF (dB) 1 1.5 3 6

RX Bandwidth (MHz) 2 2 2 1

RF Parts Cost (2013) 10k ? 100k 300k

Transmitted peak power: 10 kW at X, Ku, Ka and 1 kW at W-band. 

Transmitter Loss: 0.25, 0.3, 0.5 dB and 1 dB at X, Ku, Ka and W-band respectively 

Receiver Noise Figure: 1.0, 1.5, 3.0 and 6 dB at X, Ku, Ka and W-band respectively 

Range Resolution: 150 m (1 MHz pulse bandwidth) 

Receiver bandwidth: 2 MHz with X, Ku and Ka (twice the pulse bandwidth with magnetron radars); 1 

MHz with W-band Klystron transmitter. 
55

Figure 35. Single pulse, unity Signal to Noise Ratio (SNR) in clear air with 20” diameter antennas.

4.1.8 Cloud and Precipitation Model 

In this section a cloud and precipitation model is presented for the evaluation the measurement range and 

sensitivity of various radars considered in 4.1.7. The model consists of groups of three types of size 

distribution: 1) Clouds using the Modified Gamma distribution model, 2) Drizzle using a modified 

Marshal-Palmer distribution model and 3) Rain using the Marshall-Palmer distribution [pp. 306 – 317, 

Ulaby et. al, 1984: Microwave Remote Sensing]. 

Cloud model - Modified Gamma drop size distribution: 

The number of drop per cubic meter per drop radius unit interval 

(97) 

     

rp exp  brar
56

where 

r = drop radius, m, 

and are shape parameters, 

b is a constant related to the mode radius, in units of m, 

(98) , 

and a is a function of the liquid water content, , according to 

(99) ; 

(100) 

and is the Gamma function. 

For this simulation the parameters were varied: 

, , and gm-3.

Drizzle and Rain model: 

The number of drop per cubic meter per unit radius interval 

(101) 

where 

d is drop diameter in units of meters, 

m

-4

, for drizzle, 

m

-4

, for rain, 

and rain rate, was varied from 0.1 to 5 mm/hr for drizzle and 0.1 to 150 mm/hr for rain. The resulting 

range in Liquid Water Content as a function of rain rate is illustrated in Figure 36. 

 

c

r







c

r

b 

mv

  2

6

4 10

3

2











bm

a

v







4

2





    75   23.0 mv  21.0

    Ndp 

exp  bd 6 N

 1030

.0 21 5700 

 Rb r

6 N 8 10 .0 21 4100 

 Rb r

Rr
57

Figure 36. Drizzle and Rain data distribution of Rain-Rate vs. Liquid water Content. 

A data base of drop size distributions was generated using the three models for cloud, drizzle and rain. 

This simulated data set was then used to estimate the measurement range and sensitivity of various radar 

configurations. The distribution of mean diameter and the sixth moment (mean Z diameter) of the 

distributions is shown in Figure 37. The deviation of the scatter plot from the straight x=y line is an 

indication of the spread of the drop size distribution and how the radar reflectivity is dominated by the 

largest drops in a volume cell. The data set distribution of radar reflectivity vs. attenuation at C, X and 

Ka-band is shown in Figure 38, Figure 39 and Figure 40. 
58

Figure 37. Combined Cloud, Drizzle and Rain data distribution of Mean Z diameter vs. Mean 

Diamater – an indication of the dominant drop diameter of radar reflectivity factor. 
59

Figure 38. Combined Cloud, Drizzle and Rain data distribution of radar reflectivity factor, Z, vs. 

Attenuation at C-band (5 GHz). 
60

Figure 39. Combined Cloud, Drizzle and Rain data distribution of radar reflectivity factor, Z, vs. 

Attenuation at X-band (9.4 GHz). 
61

Figure 40. Combined Cloud, Drizzle and Rain data distribution of radar reflectivity factor, Z, vs. 

Attenuation at Ka-band (35 GHz). 
62

The potential information content of differential frequency reflectivity measurements is illustrated in Figure 41. The 

cloud particles are Rayleigh scatterers at both frequencies so their differential reflectivity factor is 0 dB, irregardless 

of LWC – forming a line at 0 dB and indicating that LWC cannot be estimated; as was predicted in Equation (89. In 

the Mie (Rayleigh to Optical transition) region at Ka-band, resonnance can actually increase the radar reflectivity 

factor at Ka band compared to that at C-band (negative F-Zdr), but LWC (and/or drop size) estimation is still 

difficult because the same differential reflectivity can indicate multiple solutions to LWC. But when the higher 

frequency is in the optical region (the arc of points from [0 g/m3, -0.75 dB] to [3 g/m3, 8 dB]), LWC can be easily 

estimated, confirming (92. 

Figure 41. Combined Cloud, Drizzle and Rain data distribution of frequency differential 

reflectivity factor, F-Zdr, vs. Liquid Water Content (g/m3). 
63

4.1.9 Detection to 10 and 20 km Range 

In this section, detection of the various simulated cloud, drizzle and rain cases are determined for the 

radar configurations (frequencies and system parameters) tabulated in Table 8 and assuming a 150 m 

range resolution. The attenuation rate and reflectivity was calculated for each cloud, drizzle and raindrop 

size distribution. Then the radar received signal to noise ratio was calculated for each simulated weather 

condition, considering reflectivity and the two-way attenuated propagation to 10 km and 20 km (the 

weather condition, and the corresponding attenuation rate, was assumed to be uniform over the two-way 

propagation distance). Detection was assumed for unity or greater signal to noise ratio. This detection 

threshold is reasonable because averaging was not accounted for, which reduces the false alarm rate. 

Undetected weather events (simulated data base points) are indicated by small black points through 

Figure 42 to Figure 44. 

Multi-frequency detection to 10 km range is shown in Figure 42 and Figure 43, and to 20 km in Figure 

44. The scatter plots clearly show that the lower frequency radars are best suited for the heavier 

precipitation, while the higher frequency millimeter wave systems have better sensitivity at short ranges. 

But even from a range of 10 km, attenuation severely diminishes detection at Ka and W-bands. As 

expected, X-band is the best compromise between sensitivity to small scatterers and attenuation, but Cband is the only of the tested frequencies that can penetrate the heaviest precipitation cases. 

The results indicate that detection of the most hazardous conditions like high LWC cloud and severe rain 

is not reliable at multiple frequencies. The combination of C and X bands (red and orange points in Figure 

42 and Figure 44) is effective in most drizzle and weak to moderate rain events, but C-band lacks 

sensitivity in most clouds and small particle drizzle and X-band attenuation is too strong in the heavier 

rain cases. The X and Ka combination (red and purple points in Figure 42 and Figure 44) can detect more 

cloud cases and the weaker drizzle, but ineffective in anything stronger than moderate rain. Also, from a 

range of 20 km, the attenuation at Ka-band severely reduces the number of detected cases. 

A better reason for multi-frequency is the option to switch to a lower frequency channel when attenuation 

at X-band prevents detection. This improvement in sensitivity at lower frequency in highly attenuation 

conditions is well illustrated in Figure 45, Figure 46 and Figure 47. C-band can detect all but the weakest 

drizzle and rain conditions to 20 km but very few of the clouds. X-band detects more of the clouds and 

just about all the drizzle but is unable to penetrate heavier rain. Attenuation at Ka-band severely limits 

detection to just a few low LWC cases. 
64

Figure 42. C, X, and Ka-band simultaneous, multi-frequency detection to 10 km range. 
65

Figure 43. Simultaneous MFR detection to 20 km range at X, Ka, and W-bands. 
66

Figure 44. Simultaneous MFR detection at C, X and Ka-bands, to 20 km. 
67

Figure 45. C-band detection to 20 km range. 
68

Figure 46. X-band detection to 20 km range. 
69

Figure 47. Ka-band detection to 20 km range. 

MFR Differential Attenuation 

In severe weather conditions X-band attenuation can be quite high, resulting in underestimating 

dangerous weather cores, but worse: can lead to the received signal diminishing below the detectable 

threshold. So by shifting to a lower frequency, attenuation quickly decreases. The attenuation rate (dB 

km-1 mm-1 hr) in rain as a function of radar frequency is illustrated in Figure 48. Attenuation is 

approximately proportional to

2

f

, so a small change in operating frequency, that is available within a 
70

waveguide band, yields very small change in attenuation rate and thus is not worth the added cost and 

system complexity. However, if a C-band channel was available in heavy (50 mm/hr) precipitation, the 

attenuation rate decreases from 1.5 dB/km at X-band to 0.3 dB/km. Over a 10 km wide precipitation (20 

km two way propagation) this adds up to a 24 dB difference in total attenuation, and approximately a 12 

dB improvement in sensitivity; assuming similar radar power-aperture product. So when X-band 

attenuation is so severe that attenuation correction techniques, such as polarimetric or radiometric, 

cannot be employed, C-band radar might still penetrate heavy precipitation and detect and identify 

hazardous weather cores.

Figure 48. Attenuation rate per rain rate as a function of radar frequency (from [3]). 

There is a potential benefit in adding a second, higher frequency. The backscattering efficiency of small 

(Rayleigh) hydrometeors increases as

4

f

 , so for example a Ka-band (33 GHz –> 3.5 times that of a 9.4 

GHz X-band) radar, given the same radar power-aperture product, receiver sensitivity and attenuation, 
71

will have almost 22 dB more sensitivity. The problem however is attenuation. Attenuation can easily 

overcome this gain. The attenuation rate at X-band is approximately 0.006 dB km-1 mm-1 hr while at Kaband is 0.3 dB km-1 mm-1 hr. So even a 10 mm/hr rain will cancel the backscattering sensitivity gain over 

a short, two-way propagation distance of 7.5 km (3.75 km range)! Shifting from X-band to Ka-band only 

makes sense when attenuation is low; i.e. for the detection and mapping of freezing drizzle/light-rain or 

high altitude ice clouds. 

For the same reason, dual-frequency X-Ka band measurements also have limited use. Although the 

differential attenuation rate is sufficiently high for easy detection and estimation of rain rate and 

attenuation, as shown in Figure 52, in those highly attenuating conditions, dual-frequency measurement 

range is severely limited as confirmed by the results shown in Figure 42, Figure 46 and Figure 47. 

The following calculation finds the measurement accuracy that is necessary for useful dual-frequency 

measurements: 

(102) 

 PPP nsm

(103) 

N

PP ns

m



 

; 

 and

(104) 

n

n

n

N

P

 

(105) 

s n

s

N

SNR

N

SNR

P

2 2

11

1 































The standard error deviation 

E

of a reflectivity measurement in units of dB can be expressed as

(106) 

























































  













 



s n

ss

N

SNR

N

SNR

P

P

dBE

2 2

1 0 1 0

11

1

10log 10 1log





.

In high SNR this expression simplifies to 

(107) 













 

N

dBE 1



10 10 1log

.
72

The estimation of differential attenuation (

A

) requires four reflectivity measurements: 

(108) 

           

12

2 1 2 1

21

2 2 1 1

,

rr

rdBZrdBZrdBZrdBZ

ffA

f f f f



  



Consequently, the expected error in the measured differential attenuation, 

dBEA , in high signal to noise 

ratio conditions can be expressed as 

(109) 

r

dBE dBE A

A











4

.

If we are satisfied with a 1 km range resolution and a 1 dB/km differential attenuation as a practical goal, 

then the measurement standard deviation should be no more than 0.5 dB. To achieve this accuracy, the 

required number of independent samples is 

(110) 

67 15.0 10 1 0 1log  











  N

N

Note that the resulting measurement quality is still poor since the 0.5 dB error corresponds to a single 



deviation, yielding only a 68% of the measurements within 0.5 dB of the actual differential attenuation, as 

shown in Figure 49. To increase the confidence interval to 95%, 287 samples are required. Another way 

to look at this is that if the differential attenuation is 1 dB, then our measurements should have much 

smaller than 1 dB standard deviation. If 287 samples are averaged, then the resulting differential 

attenuation measurement standard error (deviation) will be approximately ±0.25 dB. But if ~300 

independent samples are required for a single differential attenuation measurement, then a complete 

volume scan ahead of an aircraft would take a long time. 

If an AWR has a PRF of 5 kHz (30 km maximum range), then approximately 17 (5000/300) 

measurements can be made in a second, so a 100 point volume scan would take about 6 seconds. But this 

calculation assumes that all the samples are independent at all the frequencies. At X-band the approximate 

decorrelation time of precipitation due to differential motion of the particles, is ~10 ms. A rate 50 times 

slower than the 5 kHz PRF assumed above. So either much longer integration time is required, or each 

radar has to be equipped with frequency agility to force decorrelation by shifting the operating frequency 

by at least the transmitted pulse bandwidth [25]. Even if all these obstacles are overcome, what frequency 

combinations and what weather conditions would have 1 dB/km or greater differential attenuation? 

 
73

Figure 49. Gaussian distribution as a function of standard deviation, . The ±σ interval 

corresponds to about 68% of the samples, while ±2σcorresponds to 95% and ±3σto 99.7%. 

The cloud, drizzle and rain model established in Section 4.1.8 is used again to investigate differential 

attenuation of various frequency combinations. The results are shown in Figure 50 through Figure 52. 

Only the heavier rain produces a differential attenuation rate more than 1 dB/km at X-C and X-Ku 

frequency combinations. The X-Ka frequency combination is sufficient in most rain and higher LWC 

cloud conditions, but attenuation also diminishes the sensitivity of a Ka-band radar at longer ranges as 

shown in Figure 50 and Figure 51. 

We can conclude that useful differential reflectivity measurements are quite difficult. The differential 

attenuation needs to be quite high, often resulting in impractically short maximum detection range at the 

attenuating frequency, a large number of independent samples are required, and the resulting retrievals 

have coarse range resolution. 
74

Figure 50. X vs. C-band differential attenuation for a range of cloud, drizzle and rain conditions. 
75

Figure 51. Ku-X differential attenuation estimates are less than 0.5 dB/km for clouds and drizzle, 

and only reach 2 dB/km in the heavier rain conditions. More frequency separation is needed for 

MFR retrieval. 
76

Figure 52. Ka-X Differential attenuation. 

5 Sensitivity 

A limitation of X-band weather radar is often the lack of sensitivity due to either low scatterer cross 

section (e.g.: high altitude ice crystals or volcanic ash), high attenuation (severe thunderstorms) or a 

combination of both (freezing drizzle and rain). The only remedy to attenuation is to shift to a lower 

frequency, but improving sensitivity through adjusting the radar operating parameters and improved 

signal processing will always improve the probability of detection. A few of these methods are presented 

next. 

Following is the explanation why the easiest and quickest way to boost the sensitivity of a weather radar 

is to degrade its range resolution. Low radar cross section is primarily due to the small size of scatterers, 

but low index of refraction can also be an important factor (glass composite of volcanic ash). From (76), 

the received power from a volume scatterer can be expressed as: 
77

(111) 





2

32r

cAP

P

efft

r



.

The corresponding thermal noise floor, Pn, is: 

(112) 



kTF

n

kTBFP 

, 

where k is Boltzman’s constant (1.38E-20 mW/K*Hz), T is temperature is deg. K, F is the receiver noise 

figure and B is the receiver bandwidth (in units of Hz) matched to the transmit pulse length 

 .

Signal to Noise Ratio (SNR) is a key factor of detection, in addition to averaging and setting a threshold 

level to an acceptable probability of false alarm rate. SNR is a ratio of the measured signal and receiver 

thermal noise floor: 

(113) 

kTFr

cAP

kTF

r

cAP

P

P

SNR efft

efft

n

r

2

2

2

32

32 







 

.

Note the square of the transmit pulse length in the numerator of the expression for SNR. This indicates 

that for a given weather radar with fixed peak transmit power, antenna aperture and noise figure, 

degrading range resolution, i.e. transmitting longer pulses, is the most effective way to improve 

sensitivity. With a receiver bandwidth matched to the transmitted pulse, the measurement signal to noise 

ratio improves as square of the range resolution. 

Another method of improving sensitivity is through signal processing. In low turbulence regions, a 

combination coherent (FFT) and non-coherent (power) integration can boost the probability of detection 

without lowering the false alarm rate. The comparison of conventional non-coherent averaging and a 

combination of coherent (FFT) and non-coherent averaging is illustrated in Figure 53. A W-band groundbased radar was used to collect data from a non-precipitating cloud on February 9, 2012 in Amherst MA. 

The radar was configured to transmit at constant PRF and data blocks from 10240 transmit pulses were 

processed three ways: 1) the top figure shows the combination of FFT and non-coherent averaging, where 

the 10240 data points where broken up into groups of 128 for FFT processing and the 80 power 

spectrums where averaged (non-coherently); 2) the middle figure illustrates the conventional power 

averaging of all 10240 data points and the 3) bottom figure is commonly called Coherent Power, 

generated by forming 620 single lag pulse-pairs and averaging those complex correlations and displaying 

the magnitude of the resulting average. 

The false alarm rates are quite similar in all three methods, but the FFT shows the best detection, followed 

by the power average and the coherent-power close third. The Coherent Power image could have been 

improved by forming 10239 pulse pairs from the 10240 pulse train, which likely would have resulted in 

an image quality similar to that of the power average, but with more uniform false alarm background. 

An even better method of detecting weak scatterers is using the Doppler velocity image. Figure 54 shows 

the same data but processed to velocity using the FFT processing (top) and pulse-pair (bottom). The 

coherent integration of the FFT processing still has the best detection, but even the pulse-pair velocity 

image surpasses all three power image detection methods of Figure 53. 
78

Figure 53. Top: Feb 9 09:00 UTC W-band co-pol FFT-based dBZ (top); non-coherent dBZ 

(middle); pulse-pair lag 1 dBZ (bottom). Conditions: =0.5; PRF: 4.960 kHz; FFT: N=128; 

M=80; non-coherent and pulse-pair lag 1: NM=10240. Data gathered at an elevation angle of 145 

degrees. 
79

Figure 54. Top: Feb. 9 09:00 UTC W-band co-pol data FFT-based velocity (top); PP velocity 

(bottom). 

6 References 

[1] Doviak, R. J., and D. S. Zrnic, WSR-88D Radar for Research and Enhancement of Operations: 

Polarimetric Upgrades to Improve Rainfall Measurements (.pdf, 7.01 MB), NOAA/NSSL Report, 

1998, 110 pp., [Supplements and errata (.pdf, 156 kB) — updated 7 Aug 2013]. 

[2] Ryzhkov, A., Rainfall Measurements with the Polarimetric WSR-88D Radar (.pdf, 12.1 MB), 

NOAA/NSSL Report, 2003. 

[3] F. Ulaby, R. Moore, A. Fung, Microwave remote sensing: Active and Passive, Vol I, Microwave 

Remote Sensing Fundamentals and Radiometry: Artech House, Dedham MA, 1986. 
80

[4] C. S. Ruf, S. M. Gross, S. Misra, “RFI Detection and Mitigation for Microwave Radiometry With an 

Agile Digital Detector, IEEE Trans. Geosci. Remote Sens., vol. 44, no.3, pp.694-706, Mar. 2006. 

[5] Goodberlet, M.A. and I. Popstefanija, “RFI Mitigation Using Two-Scale Estimators for Statistical 

Variance,” IEEE Trans. Geosci. Remote Sens. Letters, Vol. 10(4), Jul. 2013. 

[6] G. Mie, Ann. Physik., 25, pp. 377-445, 1908. 

[7] “Rain” at http://en.wikipedia.org/.

[8] Specific attenuation model for rain for use in prediction methods, Recommendation ITU-R P.838-3, 

International Telecommunications Union (ITU), 2005. 

[9] C. Balanis, Antenna Theory: Analysis and Design, Chap 1: Harper & Row Publishers, New York, 

1982. 

[10] Anagnostou, M.N., et al., “Performance evaluation of high-resolution rainfall estimation by X-band 

dual-polarization radar for flash flood applications in mountainous basins,” J. Hydrol., 2010, 

doi:10.1016/j.jhydrol.2010.06.026. 

[11] Park, S.-G., Maki, M., Iwanami, K., Bringi, V.N., Chandrasekar, V., “Correction of radar reflectivity 

and differential reflectivity for rain attenuation at X band. Part I: theoretical and empirical basis,” J. 

Atmos. Ocean. Technol., 22, 1621–1632, 2005. 

[12] Delrieu, G., H. Andrieu and J. Creutin, “Quantification of Path-Integrated Attenuation for X- and CBand Weather Radar Systems Operating in Mediterranean Heavy Rainfall,” J. of App. Meteor, Vol. 39, 

pp. 840-850, 1999. 

[13] Matrosov, S., C. A. Clark, and D. A. Kingsmill,, “A polarimetric radar approach to identify rain, 

melting-layer, and snow regions for applying corrections to vertical profiles of reflectivity,” J. Appl. 

Meteor. Climatol. , 46, 154–166 2007. 

[14] Matrosov, S. “Evaluating Polarimetric X-Band Radar Rainfall Estimators during HMT,” J. Tech., 

Vol. 27, pp 122-134, 2010. 

[15] Matrosov, S.Y., Kingsmill, D.E., Martner, B.E., Ralph, F.M., “The utility of X-band polarimetric 

radar for quantitative estimates of rainfall parameters,” J. Hydrometeorol., 6, 248–262, 2005. 

[16] Matrosov, S.Y., K. A. Clark, B.E., Martner, A. Tokay, F, “X-Band Polarimetric Radar 

Measurements of Rainfall,” J. Applied Meteor., 6, 941–952, 2002. 

[17] Matrosov, S.Y., R. A. Kropfli, R. F. Reinking, B. E. Martner, “Prospects for Measuring Rainfall 

Using Propagation Differential Phase in X- and Ka-Radar Bands,” J. Atmos. Ocean. Technol., 38, 766–

776, 1999. 

[18] Bringi, V. N., and V. Chandrasekar, Polarimetric Doppler Weather Radar: Cambridge University 

Press, 636 pp, 2001. 

[19] Zhao et al., “Improving the rainfall rate estimation in the midstream of the Heihe River Basin using 

raindrop size distribution,” Hydrol. Earth Syst. Sci., 15, 943–951, 2011. 
81

[20] Tobias Otto: “Advances in polarimetric X-band weather radar,” Power Point Presentation, 2012. 

Available http://ieeexplore.ieee.org/. 

[21] Wang, Yanting and V. Chandrasekar, “Polarization Isolation Requirements for Linear DualPolarization Weather Radar in Simultaneous Transmission Mode of Operation,” IEEE Trans. Geosci. 

Remote Sensing, 44, 2019–2028, 2006. 

[22] Anagnostou, M.N., Anagnostou, E.N., Vivekananda, J., “Correction for rainpath specific and 

differential attenuation of X-band dual-polarization observations,” IEEE Trans. Geosci. Remote Sensing, 

44, 2470–2480, 2006. 

[23] Gorgucci, E., Scarchilli, G., Chandrasekar, V., Bringi, V.N., “Rainfall estimation from polarimetric 

radar measurements: composite algorithms immune to variability in raindrop shape–size relation,” J. 

Atmos. Ocean. Technol., 18, 1773–1786, 2001. 

[24] Ulaby, More and Fung, Microwave Remote Sensing, Artech House, 1989. 

[25] Doviak and Zrnic, Doppler Radar and Weather Observations, Academic Press, 1984. 

[26] Yu-Jiun Ren and Yan Zhang, “Ultra-lightweight dual-polarized X-band array antenna for airborne 

weather radar applications,” Microwave and Optical Technology Letters, May 2009, Vol. 51, Issue 5, pp. 

1324–1326, May 2009. 
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