Download - Prediction of radar coverage under anomalous propagation condition for a typical coastal site: A case study

Radio Science, Volume 26, Number 4, Pages 909-919, July-August 1991

Prediction of radar coverage under anomalous propagation condition for a typical coastal site' A case study

Samir Hussain Abdul-Jauwad

Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

Pervez Zahir Khan

Metrology, Standards and Materials Division Research Institute, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

Talal Omar Halawani

Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

(Received October 29, 1990; accepted December 12, 1990.)

In this study a model predicting the electromagnetic wave propagation has been developed. The model is initialized with a known refractivity profile and an initial transmitted field. The troposphere is assumed to be inhomogeneous in height and range. The vertical refractivity profile is computed from the meteorological upper air data. Arbitrary as well as experimentally observed profiles could be used as inputs to the model. Inhomogeneity in range is invoked by injecting refractivity profiles at known locations. A transmitter with a Gaussian beam is assumed to generate the initial field based on the transmitter parameters such as vertical beam width, vertical beam elevation, transmitter frequency, antenna height, and type of polarization. A simulation for the typical site on the coast of Arabian gulf has been chosen for the study. The field strength and radar coverage for different conditions of the typical coastal site prevailing a standard and inhomogeneous surface duct demonstrates the sensitivity of the model with the inhomogeneous and homogeneous atmosphere. Radiated fields are predicted on the basis of the initial field, boundary conditions, and atmospheric conditions considering the effect of refractivity in the case of inhomogeneous surface duct and elevated duct. Results reported in this study c•emonstrate the trapping of the energy in ducting atmosphere above the minimum trapping frequency with the predicted field by this model.

1. INTRODUCTION

Most of the time the radar coverage is adversely affected due to presence of nonstandard atmosphere or existence of anomalous propagation (AP) at coastal areas. A slight change in radio refractivity, N, may cause AP to exist. It is broadly categorized in four groups; substandard, super standard, standard, and trapping or ducting. Refractivity is computed from the local parametric values of tem- perature, pressure, and relative humidity. These basic parameters are recorded with the help of an electronic package sent in the upper atmosphere with the help of a balloon. These launches are made at various locations throughout the globe under the

Copyfight 1991 by the American Geophysical Union.

Paper number 91RS00092. 0048-6604/91/91RS-00092508.00

auspices of World Meteorological Organization usually at noon and night times.

The computation of signal strength at microwave frequencies in the troposphere due to a change in refractive indices has been a vital topic of concern. Predictions have been made in the troposphere by geometric optics, physical optics, and normal mode analysis [Kerr, 1951]. Ray tracing derived from geometrical optics model can give concise results, but detailed computations are expensive as well as difficult. The generation of two-dimensional field strength diagrams are also expensive and time consuming. In addition, they are only valid for strong ducting layers. The mode theory uses only one refractivity profile to compute two-dimensional field strength diagrams, i.e., radar coverage, and this is also expensive. The range dependence of refractivity profiles can not be considered in the case of mode theory.

The parabolic approximation is a full wave ap-

909

910 ABDUL-JAUWAD ET AL.: PREDICTION OF RADAR COVERAGE--A CASE STUDY

proach introduced by Leontovich and Folk [1946]. The basic limitation of this forward scatter model is

that it neglects the back scattered field. Computa- tions are limited to low grazing angles. Because of the unavailability of an efficient algorithm and fast computing facilities, it was not frequently used. Hardin and Tappert [1973] used the parabolic approximation, also known as the parabolic equation, of the Helmholtz equation to solve the underwater acoustics problem using the split-step Fourier algorithm. Tappert mentioned the ability of predicting losses in vertical as well as horizontal inhomogeneous acoustic profiles.

The split-step algorithm is applied to obtain the marching solution to the parabolic equation and has been used recently by Jenson and Krol [1975] in underwater acoustics, and by Ko et al. [1983], Ko [1986], Craig [1988], and Dockery [1988] in the prediction of microwave propagation. Effect of the Earth's surface is considered by applying impedance-type boundary conditions [Senior, 1960].

It has been observed that occurrences of the

surface duct are prevalent in the coastal site at 0000 and 1200 hours local time. Two years of upper air data, between May 1986 to April 1988 for 0000 hours and 1200 hours, collected by the Meteorolog- ical Environment and Protection Administration

(MEPA), Saudi Arabia, for a typical coastal site has been studied. It has been observed that a surface

duct is prevalent at the said site with different strengths at 0000 and 1200 hours respectively. The average of surface refractivity has also been computed. It has been mentioned by Skura [1987] that anomalous or nonstandard propagation does exist in the coastal site of Arabian gulf, but a prediction study of electromagnetic wave propagation has never been done for this area. The aim of this work

is to demonstrate the sensitivity of the prediction of the field strengths, and radar coverage to the occurrence of the inhomogeneous surface duct over the standard and homogeneous surface ducts. A case consisting of an elevated duct has also been considered for the study.

2. FUNDAMENTALS OF TROPOSPHERIC

PROPAGATION

The troposphere has been broadly categorized as standard and nonstandard. A standard troposphere is that in which the refractive index decreases

linearly with the increase in height at low altitudes

275 525 375 t•25 0 20 t•O 60 80 100

SURFACE REF. N-UNITS % OCCUR. BY TIt'IE

Fig. I Average cumulative percent occurrence over two years (May 1986-April 1988) for (a) surface refractivity (b) refractivity gradient at I km. altitude from the Earth's surface (dashed lines, for 0000 hours; dot-dashed lines, for 1200 hours; solid line, for 0000 and 1200 hours).

and increases exponentially at high altitudes. No bending of propagating waves occurs in the case of the standard troposphere, but it occurs in the case of the nonstandard troposphere. The latter is fur- ther divided to three different groups depending on the strength of the refractivity gradient. If the refractivity gradient (N/km) exceeds 0 N/km it is called subrefraction. If the gradient is in between -157 N/km and -79 N/km, it is superrefraction and if less than -157 N/km, it is called ducting or trapping. Ducts are again divided to three types, namely, surface duct (SD), elevated duct (ED), and evaporation duct. The evaporation duct appears at the sea surface while SD and ED appear at the Earth's surface. Refractivity at different heights is computed from meteorological data collected by radiosondes.

3. APPEARANCE OF SURFACE DUCT

A study of the occurrence of surface ducts for a typical coastal site of the Arabian gulf over a period of 24 months was made for 0000 and 1200 hours.

The upper air meteorological data is collected by MEPA at various location throughout Saudi Arabia. In this study, data used are collected by a weather station at the coast of the Arabian gulf. A cumulative percent occurrence of the surface refractivity over the same period of time has been computed Figure l a and a cumulative percent occurrence of refractivity gradient (N/km) with respect to time has been computed Figure 1 b. It has been observed that a •urface duct of average thickness of 40 m exists with different strength of the gradient for both times. At 0000 hours the strength of the average gradient is observed to be approximately -175

ABDUL-JAUWAD ET AL.' PREDICTION OF RADAR COVERAGE--A CASE STUDY 911

N/km while at 1200 hours the average gradient is -500 N/km.

4. PARABOLIC EQUATION FOR INHOMOGENEOUS TROPOSPHERE

The parabolic approximation by Folk for a laterally homogeneous atmosphere for a spherical Earth was established in 1946. Recently, vertically and laterally inhomogeneous atmosphere have been considered as an extension to this approach by Jenson et al. [1975], Ko et al. [1983], and Ko [1986]. The vertical and lateral inhomogeneity is adopted by considering the inhomogeneous nature of the permittivity of the atmosphere assuming the relative permeability /at of the atmosphere as unity. Assuming a spherical coordinate system (r, 0, •b), e(r, O) constitutes the inhomogeneity of the permittivity in radial and polar direction. Azimuthal symmetry is assumed about the origin which renders the limitation of the permittivity of the atmosphere into two dimensions, r and 0. The initial field is assumed from a vertical electric dipole (VED) above the Earth's surface. Only Er, Eo, and H• components exist due to the VED source. As E r and E o can be expressed in terms of H• components, Maxwell's wave equation in H• in scaler form will be solved. The field due to VED constitutes the case of vertical

polarization. The horizontally polarized field could be considered using a vertical magnetic dipole (VMD). The field could be predicted in terms of the radial electric field component E r instead of H•. Starting from Maxwell's wave equation and factor- ing out time dependence factor e jøøt and considering a VED, we get

VxVxH--- x X7 x H- w2/xeH = 0 (1)

where e is the permittivity of the space for r > a (effective Earth's radius) and is function of r and 0 and •o is the angular transmitter frequency.

In order to restrict the problem to a two-dimensional one, azimuthal symmetry is assumed. Con- sider a field H4 generated from a VED source expressed in terms of attenuation function u(r, O) as follows:

u(r, O)e ikaO

H 6 (r, 0) = .• (2)

where k = ooV'lxe(a, 0)and e(a, 0) is the permittivity just above the Earth's surface.

Combining (1) and (2) and transforming to rect- angular coordinate system using the transformation z = r - a and x = rO, where z is the altitude and x is the distance along the surface from the initial point, we obtain

• + 2 ik -- + k2(m(x, z) 2 _ 1)x 2 = 0 OZ 2 OX

(3)

where m(x, z) is the modified refractivity which includes the effect of the curvature of the Earth's

surface and is defined as

,(x, z) z m(x, z)- +-

•o a

where e(x, z) is the permittivity of the atmosphere, while e0 is the permittivity of the free space.

For the approximation the following assumptions are made (1)

02U

Ox 2

(2) terms involving the derivatives of e(x, z) has been neglected; and (3) change of Ou/O x over A x is small, i.e., for low grazing angles < 20 ø. These assumptions are very well satisfied in the tropospheric conditions for low grazing angles.

The parabolic equation includes all the diffraction effects due to refractivity of the propagation me- dium and is also valid even for those areas where

ray tracing methods do not work, for example, near caustic and focal points. It also retains the complete coupling effect as in the case of two-dimensional refractivity profile in mode theory.

5. IMPLEMENTATION OF PARABOLIC EQUATION IN NUMERICAL FORM

Solution to the parabolic equation defined as (3) could be found if the applied initial field is known, the value of the field at the boundaries are known, i.e., the boundary conditions at the Earth's surface, and the ionosphere is properly defined. In this case the ionosphere is assumed to be perfectly absorb- ing. With respect to the Earth's surface boundary condition, two cases have been considered' (1) the Earth's surface is assumed to be perfectly conducting corresponding to the case of the sea surface; and

912 ABDUL-JAUWAD ET AL ß PREDICTION OF RADAR COVERAGE--A CASE STUDY

(2) the Earth's surface is assumed to be a finitely conducting corresponding to sand, vegetation, and terrain.

In the case of a finitely conducting Earth's surface, Leontovich's impedance boundary condition is applied [Senior, 1960]. For a vertical polarization, i.e., VED, the boundary condition for u will be satisfied if

Ou -ik --= u z = 0 (4) Oz .1

where ,/is the complex dielectric constant of the Earth and

1

vlxo \eo

but for a horizontal polarization, in case of VMD, boundary condition for u is

• = -ik•qu z = 0 (5) Oz

where

tx 0 permeability of free space (4•rx10 -7 Henry/m); e0 permittivity of free space (8.85 farad/m); e' relative permittivity of the surface of the Earth; o- conductivity of the surface of the Earth (mho -

m/m2); k wave number.

In practice, the boundary conditions (4) and (5) are satisfied if u is considered a vertically symmetric and antisymmetric respectively. On the other hand, if the Earth is assumed to be a perfectly conducting surface, the following conditions satisfy the Earth's boundary condition: (1) u = 0 for vertical polarization, and (2) Ou/Oz = 0 for horizontal polarization.

In such a case, the field is assumed to be symmetric for vertical polarization and antisymmetric for horizontal polarization satisfying the respective boundary conditions. Several numerical tech- niques, such as the finite difference method and the analytical method exist. In the case of the finite difference scheme a system of a large number of simultaneous equation is to be solved with various unknowns and boundary conditions in a closed domain of range. Other analytical methods are too much time consuming. Applying the split-step Fou-

rier algorithm to (3) a marching type solution is obtained in an open domain of range as shown in (6).

U(X + A X, Z) = e -zxx(k/2)(m2- 1)F-l[e-i(p2$x/2k)F[u(x, z)]] (6)

where

U(x, p) = F[u(x, z)]-= u(x, z)e ipz dz (7)

u(x, z) = F-•[U(x, p)]--- U(x, p)e -it'z dp (8)

and F and F-1 stand for forward and backward fast Fourier transform (FFT) respectively. This method exploits the fast speed of the Fourier transform. If the initial field for the current step at the range x kilometer is known and the boundary conditions are properly defined, the field is predicted for the next range (x + A x) km. In each step, one forward and one backward FFT is required. The split-step algorithm is easy to implement and the solution is also stable. This technique has been applied to the problem of underwater acoustics and also, recently, in tropospheric electromagnetic wave propagation.

6. LOWER LIMIT OF TRAPPED FREQUENCY

In the case of a surface duct and an elevated duct, not all the waves will be trapped irrespective of their launch angles. Minimum trapped frequency of the waves does depend upon the duct thickness and the type of polarization being used. The minimum trapped frequency of the wave is defined as, f•, in case of a horizontal polarization and, fv, in case of a vertical polarization.

for horizontal polarization:

3.8 x 10 6

for vertical polarization:

6.3 x 10 6

MHz

MHz

= 1.6579f• MHz


100

• more than -70 dB •, between -70 dB and -80 dB • .... between -80 dB and -90 dB ,•.Y between -90 dB and -100 dB •::::• between -100 dB and -110 dB

open region between -110 dB and -120 dB

...ammmmmmmmmwmmmmmmm•emlmwlm&...

o 50 lOO 150 2O0

RANOE (Krn)

Fig. 2. Field radiated by a vertically polarized 600-MHz Gaussian beam situated at 30 m in a standard atmosphere (10-dB contours).

IO0

.--•mmmmmmmmmammmmmmemwwmm•

.-•mmmmmmmmmmmmmmmmm .•mmwmmmmmmmmmmmmm

.•mmmmmmmmmmm•mmw o•mmmmmmmmmmmmem

.•mmmemmmemmmmm

0 50 100 150 2o0

RANOE (Kin)

Fig. 3. Field radiated by a vertically polarized 600-MHz Gaussian beam at 30 m in a homogeneous surface duct (10-dB contours). See Figure 2 for symbol legend.

914 ABDUL-JAUWAD ET AL ß PREDICTION OF RADAR COVERAGE--A CASE STUDY

2000

1500

"" 1000

-r- 500

30 m

-100

0

km 50kin •' 20 •z t.e

• 60

'::I so

, • i , -100 50 130 80 30

0

-200 0 -1 0 -110 -60 -10 -120 -100 -80 0 100 1 0 20t

FIELD STRENGTH {dB}

Fig. 4. Predicted field strength at various ranges initialised by a vertically polarized 600-MHz Gaussian beam in a standard atmosphere.

where dN/dh is the refractivity gradient in N/km and d is the duct height in case of SD's and duct thickness in case of ED's.

7. CASE STUDY

The model has been applied to the problem of electromagnetic wave propagation in the presence of the surface duct. The effect of the surface duct

over the field strength and the radar coverage are also examined. It also gives a change to study the sensitivity of the field strengths and radar coverages due to minute changes in the refractivity profile observed by radiosondes in the vertical and horizontal direction.

Three profiles have been considered to demonstrate the effect of a standard, homogeneous, and inhomogeneous surface duct. Inhomogeneity is simulated by considering two profiles possessing surface ducts at different heights and of different strength at two selected ranges. The first profile is assumed to be standard with a refractivity gradient of-39.4 N/km. The second profile consists of a surface duct of 40 rn thickness and a refractivity

100 m

i i i i i

0 50 100 150 20'

RANGE (Kms.}

i I I I I

0 50 100 150 200

Fig. 6. Predicted field strength at various ranges in standard atmosphere for selected heights.

gradient of 500 N/km; the profile becomes normal at heights greater than 40 m. The third profile consists of a 150-m surface duct and a refractivity gradient of - 175 N/km. The initial field has been considered for

different transmitter parameter to demonstrate the effect of transmitter height, and different type of polarization and, transmitter frequency over the radar coverage. The vertical beam width (VBW) and the vertical beam elevation (VBELEV) has been considered as 2 ø and 0 ø respectively in all cases. Figure 2 shows the field radiated from a 600-MHz vertically polarized Gaussian beam transmitter at 31.31 rn for a standard troposphere. The field strength (dB) at 10, 50, 100, and 150 km is predicted. The radar coverage is shown in terms of 10-dB contours starting from -110 dB. A normal coverage is obtained in the case of standard atmosphere (Figure 2).

The same transmitter is used in the homogeneous surface duct of 40 m. It is observed that energy is pulled toward the surface duct demonstrating the ducting effect. Trapping of energy does not occur as the minimum trapping frequency in the persisting case is 1250 MHz. The radiated field in this case of

a homogeneous SD is shown in Figure 3. The field strengths in decibels are predicted at 10-, 50-, 100-,

2000 I _ 1500 I • 1000[ ß .r 5oi[ i

-200 -100

50 km km

•-20 30 m 50 m

i i i i i,i o-•,o -.o -,o -•o-'•o •oo •o •,o •o •o •

- - - 100 • • , • •l , • • ' ' ' I I I I I FIELD STRENGTH (dB) 0 50 100 150 200'0 •0 1'00 150 200 • 50 100 150 2000 50 100 150 200

EANGE (Kms.)

Fig. 5. Predicted field strength at various ranges initialised by a vertically polarized 600-MHz Gaussian beam in a homogeneous surface duct.

Fig. 7. Predicted field strength at various ranges in a homogeneous surface ducting atmosphere for selected heights.


L,.I 200- -i-

lOO -

0-

......... ...:;;:.;; ...... .;: .......

..................... ß . :: ............................ ß ß.. • ,-. ß ß ,_ _. • ,•v, ;:• .-'•-• ß ß,. ................

...................

... -'•-'-'.'-"--';;;::::::: :: :: :: :: ::•.'•;-'•;I'-•;-'.'.'•-'::::..;.:;:'G-.:.:.:;:-'; ..............................

RANGE (Kin)

Fig. 8. Field radiated by a vertically polarized 600-MHz Gaussian beam at 200-m height in an inhomogeneous surface duct (10-dB contours). See Figure 2 for symbol legend.

and 150-km range relative to 1 rn when a 600-MHz transmitter with Gaussian beam is kept in a standard atmosphere (Figure 4) and in the surface duct (Figure 5). Predicted field strengths are also illus-

trated for heights 30, 50, 100, and 200 m in the case of the standard atmosphere (Figure 6) and that of a surface duct (Figure 7). To demonstrate the effect of polarization over the predicted field initialised by

lOO

0 5o 100 150 2o0

!;•,NGE (Krn)

Fig. 9. Field radiated by a horizontally polarized 600-MHz Gaussian beam at 200 m height in an inhomogeneous surface duct (10-dB contours). See Figure 2 for symbol legend.

916 ABDUL-JAUWAD ET AL.' PREDICTION OF RADAR COVERAGEmA CASE STUDY

.

oo

...... :ZZ; :_.22:

0 50 100 150 200

RANGE (Kin)

Fig. 10. Field radiated by a vertically polarized Gaussian beam of 600 MHz in -500 N/km duct. Duct height rises from 40 m to 150 m in 100-km range (10-dB contours). See Figure 2 for symbol legend.

the Gaussian beam, the transmitter coverage is illustrated for a horizontally polarized beam (Figure 8), and for a 600-MHz vertically polarized Gaussian beam transmitter, Figure 9, placed at 200 m. The

trapping of energy in the case of an inhomogeneous surface duct constituted by a surface duct of vary- ing thickness from 40 to 150 m, over a range of 50 km with the respective duct strength of -500 and

lOO

,•ememmmmmmmmmmmf,-. .amemmmwmmmmemmmm*

,•mmmmemmmmmmeew•. m•mmmmmm•mmmmm-o.

oam•mmmmmmmmm ß oammmmmmmmmm•m.

.•me•mmm•memm*- o•mmmmmmm•mm•.

,amm•m•memmw-o .amm•mmemmmm.

.ammmmm%mmmfo ,m•mmmmmmmm*

.. m•m•mm•mmmmmm•m•mm•m•m•mm• ß .ammmmmeemmmm•mmmemamm•m•mmmmmemmmmm•...

o.•mmmmmmmm•mmmmmmmmm•mmmmm•m•m•mmwm•m•.oo

oammmmmmmmmmmmmmmmmmmemmmmmmmemmmmmm•ewmmem•-emmm•.e•. emmmmmm•mmmmmmmmmmmm• ................... •m''

..

ß ••r ..... •..

mr .

.... dem, ....

o 5o lOO 15o 2o0

RAN C-,E (Krn)

Fig. 11. Field radiated by a vertically polarized 3000-MHz Gaussian beam in an inhomogeneous surface duct (10-dB contours). See Figure 2 for symbol legend.


.-. -30 I\ Standard • -•,0 Homogeneous SD

•- -50 z

"' -60

• -?0

30 m

D

E I I 0 S0 100 150 2•0

RANGE (Kms.)

-60

-100

-120

-1•,0

v' 50 m ....---. :...-= ...........

/ .... Standard _ • .... Homogeneous SD

------Inhomogeneous SD 'i I I ,I I, 0 50 100 150 200

RANGE (Kms.)

-50

-110

-130

-150

-170

I / .... Standard _ I _[• Homogeneous SD I

7 Inhømøgeneøus SD ! i I i 11 0 5m0 100 150 200

RANGE IKms.)

Fig. 12. Predicted field strengths in range of 200 km for standard atmosphere, homogeneous duct, and inhomogeneous duct of -500 N/km strength and 40 m thickness initialized by a 600-MHz vertically polarized Gaussian beam.

-200 N/km is shown in Figure 10. Trapping occurs after a range of 50 km verifying the minimum trapping frequency criterion. The minimum frequency to be trapped in the presence of- 200 N/km for a duct thickness of 150 m is 242 MHz, which is much less than 600 MHz. Thus causes not to be

trapped until 50 km. The radiated field from a 3-GHz vertically polar-

ized Gaussian beam transmitter of 2 ø VBW and 0 ø

VBELEV placed at a height of 31 m is shown in Figure 11. It is observed that the radar coverage is very poor compared to a 600 MHz transmitter, where most of the energy is trapped within the duct.

Signal fading occurs in the presence of a surface

2000 km

1500

1000

500

0 I -200 -100 0

km 0 km

1•o •o -•o -'1•, o -;o FIELD STRENGTH (dB)

196 k

I

-150 -100 -50

Fig. 13. Predicted field strength at ranges initialized by a 600-MHz vertically polarized Gaussian beam in an inhomogeneous elevated duct atmosphere.

duct. A field comparison has been made in the case of standard homogeneous and inhomogeneous surface ducts. The field strength for the three cases are shown in Figure 12 at altitudes 30, 50, and 100 m when the transmitter is kept at 31.31 m. It is intuitive from Figure 12 that signal fading is more severe for the inhomogeneous duct than homogeneous surface duct.

The behavior of the model in the presence of an elevated duct is also demonstrated. A transmitter

with a vertically polarized 600-MHz Gaussian beam of 1 ø VBW, 0 ø VBELEV radiates the field in an inhomogeneous elevated duct. This duct is 200 m thick and decreases to a 150 m thickness over a

range of 80 km at a height of 800 m and of a refractivity gradient height of-200 N/km. The transmitter is kept at 1014 m. Fields are predicted in 10-, 120-, 180-, and 196-km ranges for all heights (Figure 13) and 802-, 880-, 1014-, and 1104-m heights for all ranges (Figure 14). The radar coverage is shown in Figure 15. It can be seen from the radar coverage (Figure 15) that half of the energy is pulled down because of the ducting effect while some of the energy is reflected from the Earth

0

-20

-Z,O

-60

-•o

-100

802 m •

50 100 150 200 0 0 100 1 0 200 t I i i i i i

'0 510 100 150 ;000 50 100 150 200 RANOE (KMs)

Fig. 14. Predicted field strength at heights initialized by a 600-MHz vertically polarized Gaussian beam in an inhomogeneous elevated duct atmosphere.

918 ABDUL-JAUWAD ET AL.' PREDICTION OF RADAR COVERAGE--A CASE STUDY

1500-

O-

o .•o 1 oo 150 2•

RANGE (Km)

Fig. 15. Field radiated from a vertically polarized 600-MHz Gaussian beam in presence of an inhomogeneous elevated duct (10-dB contours). See Figure 2 for symbol legend.

surface beyond the 100-km range causing an inter- ference pattern. Furthermore, energy starts leaking away from the duct beyond the 150-km range; at altitudes between 500 and 1000 km, a major portion of energy tries to stay within the duct.

Acknowledgments. This research reported herein was funded by the Applied Physics Laboratory of Johns Hopkins University. The authors would like to acknowledge the support of United States Air Force Electronic Systems Division (ESD), the Royal Saudi Air Force, and the Research Institute at King Fahd University of Petroleum and Minerals.

8. CONCLUSIONS

A model which predicts field strength and radar coverage at heights and ranges has been demonstrated for a typical coastal site of the Arabian gulf. It accounts the effect of refractivity, a vital parameter, causing anomalous propagation over the electromagnetic wave propagation in homogeneous and inhomogeneous atmospheres in the presence of surface and elevated ducts. The fields are predicted on the basis of the known initial field with the help of a marching solution, and assumed boundary conditions. Although the Earth's surface is assumed to be perfectly conducting, throughout the study, the effect of a finitely conducting surface is also included. The computation time is very much reduced and the prediction is qualitatively im- proved over ray tracing and mode theory. Radar coverages are predicted for homogeneous and inhomogeneous surface ducts and also for inhomogeneous elevated ducts.

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Craig, K. H., Propagation modelling in the troposphere: Para- bolic equation method, Electron. Lett., 24(18), 1136-1138, 1988.

Dockery, G. D., Modelling electromagnetic wave propagation in the troposphere using the parabolic equation, IEEE Trans. Antenna Propag., 36(10), 1464-1470, 1988.

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variable coefficient wave equations, Am. Rev., 15,423, 1973. Jensen, F. and H. Krol, The use of the parabolic equation

method in sound propagation modelling, Mem. $M-72, Saclantcen La-Spezia, Italy, Aug. 1975.

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ABDUL-JAUWAD ET AL.: PREDICTION OF RADAR COVERAGE--A CASE STUDY 919

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Senior, T. B. A., Impedance boundary conditions for imper- fectly connecting surfaces, Appl. Sci. Res., Sec. B, 8,418-436, 1960.

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S. H. Abdul-Jauwad and T. O. Halawani, Electrical Engineer- ing Department, King Fahd University of Petroleum and Miner- als, Dhahran, Saudi Arabia.

P. Z. Khan, Meteorology, Standards and Materials Division Research Institute, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.