24 IRE TRANSACTIONS ON AERONAUTICAL AND NAVIGATIONAL ELECTRONICS March

Effect of Precipitation on the Designof Radio Altimeters*

RICHARD K. MOOREt

Summary-Radio altimeters operating in the microwave regionmust distinguish between desired signals returned from the groundand undesired signals returned from precipitation. Calculation ofthe relative ground and precipitation returns for a 0.1 microsecondpulse-duration altimeter requiring 10 to 1 desired-undesired signalratio indicates a minimum wavelength of about 2 cm may be usedfor reliable operation in heavy precipitation.

Curves have been computed for minimum wavelength at a givenaltitude for fixed range to rain and for equal rain and ground ranges,for various beam widths. Minimum wavelengths as long as 20 cm areindicated for some conditions. Use of circular polarization may permitaltimeter operation at wavelengths less than 2 cm, even with intenseprecipitation.

INTRODUCTION

m HE RADIO altimeter is a device which measures

the distance from an aircraft to the ground by de-termining the time for an electromagnetic wave to

travel from the aircraft to the ground and back to theaircraft. Usually altitude is shown by some automaticindicator which should be actuated by the ground signalbut which may also be actuated, in error, by returnsfrom other sources or by noise. At extremely high fre-quencies, the detecting device associated with the indi-cator may confuse a signal echo received from precipi-tation, in the form of rain, snow, or clouds, with that re-turned from the ground. In this paper, the ratio of re-

turn from ground to return from rain is calculated forvarious system parameters, and conclusions are drawnfrom calculated values as to minimum usable wave-lengths.

Calculations are made for a sample set of rain andground parameters. Examples are presented where theminimum range for which the receiver will permit a sig-nal to reach the detector is fixed, and also for the situa-tion where the minimum range is a fixed amount lessthan the range to the ground.Although some radio altimeters utilize a frequency

modulated wave, a pulse radar is assumed. It can beshown that most of the effects present with pulse altime-ters are also present with fm altimeters, with difference-frequency in the fm altimeter taking the place of timedelay in the pulse altimeter.

Certain assumptions have been made which facilitateeasy calculation of the necessary parameters and whichdo not differ too widely from the actual situation likelyto be encountered. These assumptions are as follows:

* Manuscript received by PGANE, June 18, 1956; first revisionreceived, October 1, 1956; and second revision received, December 27,1956. Most of this work was done while the author was a consultantfor Sandia Corp.

t Elec. Eng. Dept., University of New Mexico, Albuquerque,N. M.

1) The transmitted pulse is square and the returnpulse is not distorted by the receiving system.

2) The antenna beam is conical. That is, the gain isconstant out to some critical angle, and is zero fromthere on.

3) Return from the ground is made up completelyof area-extensive scatter and this scatter is in-dependent of the angle of incidence.

4) The scattering cross section per unit volume of therain is uniform throughout the illuminated vol-ume.

DERIVATION OF THE EFFECT OF PRECIPITATION

Precipitation Return Calculation

It is well known that radar return from precipitationis made up of the sum of the returns from the individualparticles, added on a power basis. It is possible to ex-press the scattering properties of rain in terms of a radarcross section per unit volume of illuminated particle-containing region.i

Using this assumption, we have for the power returnfrom a differential volume the following:

WtGtGrX2cTpe-2aRdVdWr. = ~~ (41r) 3R4 (1)

whereW,=transmitted powerGr=receiving antenna gainGt = transmitting antenna gainR = distance from the radar to the differential volume

in questionOrp=scattering cross section per unit volumedV=differential volume elementX = wavelengtha =atmospheric attenuation coefficient.

The terms of (1) may be combined and integrationperformed over the illuminated volume, as indicatedby (for Gr= Gt= G)

Wtx2 r G2o-ve-2aRdVW4r= R4

(4 7) 3 IlluminateclVolume R4(2)

The significant factor in radar system design is not theactual power received, but the ratio of the power re-

ceived to that transmitted. This represents the spacegain, which is always less than unity. That gain cor-

I See D. E. Kerr, "Propagation of Short Radio Waves," McGraw-Hill Book Co., New York, N. Y., Ch. 7; 1951.

Moore: Effect of Precipitation on the Design of Radio Altimeters

responding to signal return from precipitation is de-fined here as Lp.

X2 1Ro r ope 2aRLp = --SJ J J sin 6dq5dOdR, (3)

(4r) 3 R ecTI2 R

where Ro is the "radar range" to the limit of the volumeelement involved, c is the velocity of electromagneticwaves, and T is the pulse duration. All other quantitiesremain the same as before, except that spherical coordi-nates (r, 0, q5), centered on the altimeter, have been in-troduced and the volume element is expressed in termsof them.To simplify calculations, a conical beam antenna will

be assumed; its properties are given by

G=Go 0<60<6

G=0 Oo<0<7r (4)2

Go= -1-cos

Here the angle On is the half-width of the antenna beam.The resulting value for the signal attenuation is

X2G 020r RO e- 2aR p G0L 2(4 r)2 0- cT12 R2 dR J sin OdO. (5)

Exact integration of this yields an expression involvingthe difference of exponential integrals. In most cases ofinterest for altimeters the difference in attenuation cor-responding to a single pulse length in space is negligible,and the factor e-2aR may be taken out of the integral.When this is done and (5) is integrated,

X2Go,2 cTope- 2aRo(l - cosS 6)Lp = J

I 1 - Ro2\ 2Roo

(6)

that the return was calculated from a differential ele-ment of volume in the precipitation. Here, instead of aradar cross section per unit volume, the radar crosssection per unit area of the earth O-e has been used. Sum-ming up the returns from all illuminated areas, we ob-tain for external gain for the signal from the earth

Xi2Le =

4 I3I4)'lluminated Area

G2o e-2aRdA

R4 (7)

By use of polar coordinates on the ground, and assump-tion of a plane surface, the indicated integration be-comes

eRO =/ or fo2 G2 edtdRe-2aRe (4i7)(R) cT12) Or h R3 (8)

This equation is based on the assumption that the illumi-nated area on the ground is determined by the pulselength. At lower altitudes the limitation of illuminatedarea may be due to the beam width of the antennas.The gain for the altitude signal (usually that signal

corresponding to the range to the leading edge at thetime when the trailing edge of the pulse has just hit theground) may be calculated from (8) using the assump-tions of (4) and taking e-2aR outside the integral as

La = X2Go2o.ee-2a(h+cI2)cT (1 +

64r2h13 (+cT)

(9)

Pulse-length and beam-width limitations on illumina-tion apply, respectively, when

cT cos 00h > -pulse-length limited

2 (1 - cos 00)

Ground Return and System RequirementsThe over-all internal system gain must be such as to

overcome the over-all external loss or attenuation. Thisis determined by a minimum signal expected to be re-turned from the ground. If the system gain is sufficientlygreat, the return signal from the precipitation will be de-tected as well as that from the ground. Hence, to deter-mine the effect of rain on system performance, it is nec-essary to establish the required system gain or its recip-rocal the maximum external loss which will occur.

It has been established by experiment that most radioaltimeters are actuated by signals which are scatteredfrom large numbers of small targets on the ground. Thespecular or mirror-type reflection is very rare indeedand will not be considered here.2The return from a differential element of surface on

the ground may be calculated in much the same manner

2 R. K. Moore and C. S. XVilliams, Jr. "Radar return at near-vertical incidence," IPROC. IRE, vol. 45, pp. 228-238; February, 1957.Later unpubished work indicates this may not always be neglected.

cT cos 06h < - beam-angle limited.

2 (1- cos 00)(10)

Where the limitation is due to beam angle, the expres-sion for the altitude signal gain is

X2Go2o ee- 2a (h+cT/2)

La= 327r2h2 (1 - cos 6t).

EARTH-SIGNAL To PRECIPITATION-SIGNAL RATIOWe may establish an earth-signal to precipitation-

signal ratio which is in essence the signal-to-noise ratioof the system, where only noise due to return from pre-cipitation is considered. Where the illuminated area onthe ground is limited by pulse length, this ratio may beobtained from (6) and (9), resulting in

La o eRo2e-2a(h+(cT/2)-Ro)

Lp aph3(1 - Cos 06) 1 (12)

1957 25

(11)

26 IRE TRANSACTIONS ON AERONAUTICAL AND NAVIGATIONAL ELECTRONICS March

Below the altitude given by (10) the limitation ofilluminated area on the ground is due to the beam-width. In this case we must take the ratio of (11) to(6), obtaining

La OeR02(l -cos 200) e-2a(h+[cT/2)-Ro] / cT= 1~~~~~~~~~~--I (13)Lp 2oph2cT(l- cos Go) 2RoJ

These ratios may be considered as signal-to-noise ratiosin terms of the signal received at any altitude. Theymay also be thought of as minimum signal-to-noiseratios in which the altitude used is the maximum alti-tude to which the altimeter is required to perform. Inthis case the value of La is the gain for which the systemmust be designed.

INTERPRETATION AND EXAMPLES

This section discusses available precipitation echodata and attenuation data and considers their applica-tions. Next is treated, for fixed minimum range, the casewhere the antenna beam is specified independently offrequency. Finally, the situation is considered wherethe minimum range varies with the indicated altitude.In this connection, an absolute minimum wavelengthis also calculated.

Precipitation DataThe radar cross section of an ensemble of spherical

moisture particles has been calculated theoretically3

64Xr5fp = 4 E di6. (14)

In this expression a summation is carried out over alldrops contained in the volume of rain whose return isbeing calculated. The resulting value is divided by theilluminated volume to obtain a value for the averageradar cross section per unit volume. The expression fora single drop is simply that of (14), where only one termis used in the summation. It is customary to termEdil per unit volume as Z; hence (14) is

1.96 X 104UVp Z (15)

where

Z- Ed. (16)

Because of the extreme sensitivity of the radar cross-section to the diameter of the particles, the largest radarcross sections are normally encountered in precipita-tion and cloud forms having very large particles. Itshould also be noted that the cross section varies in-versely as the fourth power of the wavelength so that,as one goes to short wavelengths, the radar cross sec-tion per unit volume increases very rapidly for a givendrop size distribution.

3See Kerr, op. cit., p. 596.

Rains that have been observed give a wide range ofcross sections per unit volume as indicated by the fol-lowing. (Wavelength is expressed in meters.)

Light rain4 5.40 X 10-13/X4"Violent rain like a cloudburst"4 5.70X10-1/X4Thunderstorm5 4.90 X 10-12/X4Heavy orographic rain5 6.88 X 10-11/X4

Because two of the examples list radar cross sectionalmost as great as 10-10/X4, this value has been chosenin the examples which follow as representative of anextreme case of precipitation echo. In point of fact, it isexpected that echoes at least two powers of 10 greaterthan this may be encountered occasionally.6 No attempthas been made to establish statistically the prevalenceof echoes of this magnitude.

Attenuation is not as easy to calculate as back-scatter, since the variation with drop size is rather com-plex for the size of drops found in heavy rain.7 Sinceaccurate experimental data are hard to find over therange of frequencies considered here, the theoreticalcalculations from Burrows and Attwood have beenused.8 While these may not be completely representa-tive, the values for a precipitation rate of 50 mm/hourshouLld correspond roughly with the reflectivity assumed.Attenuations used for the first following example are:

Wavelength Attenuation(cm) (db/km)1.25 8.03.0 2.255.0 0.258.0 0.0510.0 0.0315.0 0.015

SPECIFIED BEAM-FIXED MINIMUM RANGE

In many cases the beam width of an altimeter antennais determined by the motions through which the air-craft on which it is mounted may go while the altimeteris required to work satisfactorily. Thus, if the aircraft isexpected to bank as much as 30°, while reasonablealtimeter performance is maintained, the antenna musthave a beam width of at least ± 300 in the transversedirection. It is necessary, therefore, to consider altimeteroperation using a relatively broad-beam antenna whosebeamwidth is specified by the performance of the vehiclecarrying the altimeter.

Since for a given system the signal-to-noise ratio re-quired for successful operation is usually determinedby the parameters of the altimeter itself, it may bedesirable to consider this ratio as a specified parameterand rewrite (12) and (13) in terms of the maximum alti-

4Quoted from A. C. Best, in C. R. Burrows and S. S. Attwood,"Radio Wave Propagation," Academic Press, New York, N. Y., pp.279-280; 1949.

5 D. C. Blanchard, "Radar reflectivity and drop-size distribu-tion in the Pacific northeast trades," Proceedings, Third RadarWeather Conference, McGill University, Montreal, p. D-29; 1952. Amore comprehensive report on Blanchard's work is in D. C. Blanchard,"Raindrop size distribution in Hawaiian rains," J. Met., vol. 10, p.457; 1953.

6 Private communication with A. E. Mueller, Illinois State WaterSurvey.

7See Kerr, op. cit., pp. 671-685.8 Burrows and Attwood, op. cit., p. 281.

Moore: Effect of Precipitation on the Design of Radio Altimeters

tude which may be utilized. In these terms, when at-mospheric attenuation is negligible, the maximum alti-tude is given by

1 ( afi eRo2X4ho =

47r} (a) --cos a)

!1/3h, < h.

ROX2 o1e(l - cos 200) (ho )(1 - cosL\ 1

1rZ - (I - cos Oo)2cT

cT II / 2

Ro < h. < hc

where (15) has been used to substitute for o- anthe altitude separating beam-width and pulse-llimitation.The expression for ho may be simplified conside

if the half-pulse-length in space is much smaller themaximum altitude and the minimum range, andbeam angle also is small. In this case we obtain

1 20eRo2X4 1/347r l.................. LaJ

7r2Z o2

R0X2 a. 1/2

ho = Ro Ire(a) l/2

lr 2cTi

The results of these expressions may be seen best byconsidering a specific example. For this example, let usassume the following:

Ro = 500 ft. = 153m

Se = 0.1

-p= 10-10/X4

cT = 30m = 98.6 ft.

La- = 10.

The assumption for Ro is that the receiver is able to re-cover within 1 microsecond after the start of the trans-mitted pulse. The assumption of 0.1 for scattering bythe earth (o-,) appears, on the basis of the Sandia Corp.terrain return measurement program, to be a reasonable

(17) minimum over a fair range of frequencies. A pulse lengthof 1/10 microsecond has been chosen because this is a

I h. is value which is easy to obtain, whereas shorter pulses are

ength quite difficult to attain. A signal-to-noise ratio of 10 hasbeen assumed as reasonable for a simple system. The

zrably results are shown in Fig. 1. The critical altitude for 20in theif the

hc< h

Ro< h <hc. (18)

It should be noted that the maximum permissible alti-tude varies with wavelength, minimum range, and beamangle in a different fashion if this altitude is above thebeam-limited critical altitude than it does if it is below.When attenuation may not be neglected, maximum

altitude is

3 o10h In [e2/1'(Ro-cT/2) 1]

2a R cT

.3~

h, < h

k--ln 1+ [ea(Ro-cTJ2) 1]cT

a l Ro--2

Ro < h < h, (19)

which simplifies for small attenuation to

hoo[ho[1- -a(ho+--Ro)] hK< h

h - ho [1-- (ho +--Ro)]. Ro < h < h1. (20)

1.0.9.8.7.6

.4

;Ev. 3

C:N11

01

a .07; .06> .05

'r .092: .033.07N

.02

__CONDITIONS OF EXAMPLE:,~~~~-l IIMMRNE-50

.0117

2) ANTENNA - CONICAL BEAM OF 1/2 WIDTH esq m

3) EARTH SCATTERING COEFFICIENT - 0.1 sq m10 sq m

4) PRECIPITATION SCATTERING COEFFICIENT 10 cu m5) REQUIRED SI N - 10 db

AIz 7 A.I ,1.

. ., .I,910 0 0I . .5 . .

Fig. 1-Minimum uisable wavelength fixed minimum range.

beam widths is about 80,000 feet. The essential lack ofdependence of maximum altitude on beam width forthe beamwidth-limited case, results in a single line, forall beamwidths, below the critical altitudes associatedwith those beams. Above these critical altitudes, in thepulse length limited region, the maximum usable alti-tude is a function of beamwidth, and this is indicated bythe separate curves for the different beam angles. Be-cause of the assumed minimum altitude capability of500 feet, no minimum altitudes are indicated below thislevel.

It should be noted that, for this example, the radaraltimeter must operate at a wavelength of almost 10 cmto work at 3000 meters. This is because the return fromrain occurs at a range of only 500 feet, whereas that from

1957 27

0L- 3 4 5 6 7 89 1,000 3 4 5 6 7 8 9 10,000 2 3 4 5 6 7 8 9

i

28 IRE TRANSACTIONS ON AERONAUTICAL AND NAVIGATIONAL ELECTRONICS March

the ground occurs at a range 20 times as great. Hencefor the signal from the rain at 500 feet to be 1/10 thesignal from the ground at 10,000 feet it need only be avery small fraction of the signal from the ground at 500feet. If we assume beamwidth limitation, so that thealtitude variation is inversely the square of height, therain return at 500 feet may be 36 db below the earthreturn at 500 feet and still cause trouble for groundreturn of 10,000 feet range.

Although it might be expected that attenuationthrough the rain would have a serious effect on opera-tion altitude, it only causes a 17 per cent reduction at3 cm for 20 beamwidth and lesser amounts for widerbeams and longer wavelengths.

Use of sensitivity-time-control to adjust the receivergain as a function of range would tend to reduce theeffect of the difference in distance between rain echoand ground echo. If this compensation could be perfect,the fixed minimum range example would reduce to thevariable minimum range example which follows. Be-cause altitude variation of the ground return powervaries between inverse square and inverse cube with theexponent a function of terrain type and altitude, com-pensation cannot be perfect. Furthermore, compensa-tion for rain attenuation would be difficult because ofits variability. Hence, the preceding and following ex-amples represent limits on performance of a fixed mini-mum range system.

VARIABLE MINIMUM RANGE-ABSOLUTEMINIMUM WAVELENGTH

In some cases it may be possible to adjust the mini-mum range in such a way that it is a fixed amount lessthan the measured altitude. Of course, if the measuredaltitude is not known, this cannot be fixed; but, if thealtitude signal is picked up at a lower altitude where it isstronger and the aircraft then ascends, a servosystemmay move a gate within which the altitude signal re-mains. Similarly, even while flying continuously at highaltitudes, the altitude signal may be picked up outsidethe area of precipitation and the gate set at that time.In that case, use of such a gate would be quite feasible.Some fm altimeters utilize a relatively narrow-band

filter and a servosystem to adjust some parameter of thesystem so that the difference frequency lies within thisfilter. This may be represented by a gate which is afixed percentage of the altitude. In either case it is pos-sible to obtain results superior to those possible with afixed minimum range because the rain and earth returnscome from more nearly the same distance, and the ratiobetween them is therefore not as sensitive to the alti-tude of operation.

Let us assume a fixed gate width. Here we have forthe minimum range

Ro= h-g=h(=l- ). (21)

Substituting this result into (17) and rewriting so tha:taltitude is the independent parameter and wavelengththe dependent parameter, one obtains

4 = 64ir5Z (-a/ )a,

h(1 - cos 00)(1 - g/j)2

, h,< h

-X4 - 6475Z (La/Lp)oJe

I

L 2(h_g)i

2(h -g)-

2cT(l- cos Oo)(1 - g/h)2(1 - cos 2G0)

0 < t < hC. (22)

For gate widths small enough that attenuation acrossthe gate may be neglected, (22) is valid. Fortunately,one would ordinarily design a gate such that this is so,for if attenuation were to be considered, its variationwith wavelength would complicate the result a greatdeal.On substitution of a value for (1 -cos Oo) from (10),

it can be seen that, provided the ratio of half-pulse-length in space to minimum range and that of gatewidth to altitude are small, the minimum possible wave-length is independent of altitude for the beam limitedcase and varies only as the fourth root of altitude forpulse-length limitation.

It should be noted that the absolute minimum wave-length which may be used under any circumstance isthat where the minimum range and the actual range arethe same; that is, where g= 0. Thus, if the signal fromthe rain and from the ground do not have a suitableratio when both are at the same distance, nothing canbe done to improve this ratio. However, modification-of the system may be possible, which will permit a lowersignal-to-noise ratio to be used for successful operation.The example given has been modified to give this ab-solute minimum wavelength, and results appear in Fig.2. It can be seen that the minimum wavelength for a0.1 microsecond pulse and a 10 to 1 signal-to-noise ratio,with the rain and earth scattering cross sections chosen,is just under 2 centimeters for the narrower beamwidth,becoming higher for wider beamwidths. Thus it appearsthat a successful radio altimeter using a pulse length of0.1 microsecond or more and requiring a 10 to 1 signal-to-noise ratio or better cannot be built for a wavelengthof less than 2 cm.

Moore: Effect of Precipitation on the Design of Radio Altimeters

WF .3 1 _ _ _ _____

CONDITtVNS OF EXAMPLE:2 14 MINIMUM RANGE = OPERATION RANGE

X 2) ANTENNA - CONICAL BEAM OF 1!2 WIDTH

U 3) EARTH SCATTERING COEFFICIENT - 0. Isq

4) PRECIPITATION SCATTERING COEFFICIENT 10Im

> 09 5) REQUIRED S/N 10 db c

0I 2 3 4 5 7 9 1,000 2 3 4 5 0 000 2 4 6 8

Fig. 2-Minimum usable wavelength.

CONCLUSION

It has been shown that, for the chosen example,wavelengths of less than 2 cm are not practical for radioaltimeters which must perform in the presence of heavyprecipitation echo. Wavelengths even longer than thismust be employed if a fixed minimum range (due to

receiver recovery) is imposed, rather than a gate whichmoves with the altitude signal. Use of stc makes thefixed minimum range system approach the performanceof the moving gate system. With the moving gate someprocedure must be established for acquiring the groundif the altimeter is turned on during the time it is in theprecipitation area.There is possibility that this situation may be some-

what improved and that shorter wavelengths may in-deed be used for radio altimeters, provided circularlypolarized antennas are used for transmitting and re-ceiving. It is well known that there is a large reductionin the back scattering from a spherical particle if onecircularly polarized sense is transmitted and the sameantenna is used for receiving, since polarization senseis reversed by a spherical scatterer. Unfortunately it hasnot been established that all grounds are such thatthey will return appreciable signals with the incidentsense. Some experimental evidence has been gatheredby various agencies to indicate that a great many groundswill return such a polarization that the use of circularpolarization would permit a considerable improvementin ground signal-to-rain signal ratio.

c7(..A..5D

291957