NTIAREPORT 82-100

A Guide to the Use of the

ITS Irregular Terrain Model

in the Area Prediction Mode

G.A. Hufford

A.G. Longley

W.A. Kissick

u.s. DEPARTMENT OF COMMERCE

Malcolm Baldrige, Secretary

Bernard J. Wunder, Jr., Assistant Secretary

for Communications and Information

April 1982

TABLE OF CONTENTS

LIST OF FIGURES ••

LIST OF TABLES

ABSTRACT • • • • •

iv

v

1

1.

2.

3.

4.

5.

INTRODUCTION ••

AREA PREDICTION MODELS. •

THE ITS MODEL FOR THE MID-RANGE FREQUENCIES •

3.1 Input Parameters

3.2 General Description•.

DEVELOPMENT OF THE MODEL.

DETAILED DESCRIPTION OF INPUT PARAMETERS. •

5.1 Atmospheric Parameters.

5.2 Terrain Parameters .•

5.3 Other Input Parameters

1

3

5

6

10

14

17

18

20

22

6. STATISTICS AND VARIABILITY ••• . . . 26

7.

8.

6.1 The Three Dimensions of Variability.

6.2 .A Model of Variability • • .

6.3 Reliability and Confidence

6.4 Second Order Statistics..

SAMPLE PROBLEMS .

7.1 The Operating Range of a Mobile-to-Mobile System .••••

7.2 Optimum Television Station Separation. • • • • •

7.3 Comparison with Data . . . • . . .•.

REFERENCES. . •

28

31

35

37

38

39

42

51

66

APPENDIX A.

APPENDIX B.

LRPROP AND AVAR--AN IMPLEMENTATION OF THE ITS MODEL FOR

MID-RANGE FREQUENCIES • • • • •

QKAREA--AN APPLICATIONS PROGRAM •

iii

69

101
Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

LIST OF FIGURES

A typical plot of reference attenuation versus distance • • •

Minimum monthly mean values of surface refractivity referred

to mean sea level • • • •.•.. • • . . • . . • . • • .

Contours of the terrain irregularity parameter ~h in meters.

The derivation assumed random paths and homogeneous terrain

in 50 km blocks. Allowances should be made for other

conditions. • • • • • . • • . . • • ••• • . • • • • • •

The reference attenuation versus ~h for selected distances.•

Output from a run of QKAREA concerning a mobile-to-mobile

system. • • • •••••••••.•••.•.•...

A triangular grid of cochannel television stations showing

the arrangement of the three offset frequencies . • • • . • •

Fraction of the country receiving an interference-free signal

versus the station separation. We have assumed transmitting

antennas 300 m high and average terrain characteristics

The R3 data at 410 MHz; 44 points . . .

Predicted and observed values of attenuation for the R3 data.

Assumed parameters: f=4l0 MHz, h 1=275 m, hg2=6.6 m,

~h=126 m, N =250 N-units•••..g .••••••••.••.•

s

Predicted and observed curves of observational variability

for the R3 data . • • • • . . . • . • . • . • . . .

11

19

21

23

41

45

50

52

55

57

Figure 11. Predicted and observed medians for the R3

indicate confidence levels for the sample

imately 10% and 90% • . • . . • . • • • •

data. The bars

medians at approx59

Figure 12.

Figure 13.

Figure 14.

Predicted and observed values of attenuation versus distance

for the R3 data. The predictions assumed the transmitters

were sited very carefully • • • • • • • • • • • • • • • •

The sample cumulative distribution of deviations. As indicated in the text, this is a midleading plot. • •

The sample cumulative distribution of deviations assuming

the data are censored when A 0.5 dB •..••••..•

iv

61

62

64
LIST OF TABLES

Design Parameters for a SymmetriG Mobile-to-Mobile System. 40

Design Parameters for a Grid of Channel 10 Television Stations 44

Operational Ranges Under Average Environmental Conditions. 42

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Input Parameters for the ITS ModE~l Together With the Original

Design Limits. . . . . . . . . .

Suggested Values for the Terrain Irregularity Parameter. .

Suggested Values for the Electrical Ground Constants . . . . .

Radio Climates and Suggested Values for N

s

7

8

9

9

v

A GUIDE TO THE USE OF THE ITS IR~~GULAR TERRAIN MODEL

IN THE AREA PREDICTION MODE

George A. Hufford, Anita G. Longley, * and William A. Kissick

The ITS model of radio propagation for frequencies between 20 MHz

and 20 GHz (the Longley-Rice model) is a general purpose model that can

be applied to.a large variety of engineering problems. The model,

which is based on electromagnetic theory and on statistical analyses of

both terrain features and radio measurements, predicts the median

attenuation of a radio signal as a function of distance and the variability of the signal in time and in space.

The model is described in the form used to make "area predictions"

for such applications as preliminary estimates for system design, military tactical situations and surveillance, and land-mobile systems.

This guide describes the basis of the model, its implementation, and

some advantages and limitations of its use. Sample problems are

included to demonstrate applications of the model.

Key words: area prediction; radio propagat.ion model; SHF; statistics;

terrain effects; UHF; VHF

1. INTRODUCTION

Radio propagation in a terrestrial environment is an enigmatic phenomenon

whose properties are difficult to predict. This is particularly true at VHF, UHF,

and SHF where the clutter of hills, trees, and houses and the ever-changing atmosphere provide scattering obstacles with sizes of the same order of magnitude as

the wavelength. The engineer who is called upon to design radio equipment and

radio systems does not have available any precise way of knowing what the characteristics of the propagation channel will be nor, therefore, how it will affect operations. Instead, the engineer must be content with one or more models of radio

propagation--i.e., with techniques or rules of thumb that attempt to describe how

the physical world affects the flow of electromagnetic energy.

Some of these models treat very specialized subjects as, for example, microwave mobile data transfer in high-rise urban areas; others try to be as generally

applicable as Maxwell's equations and to represent, if not all, at least most,

aspects of physical reality. In this report we shall describe one of the latter

models. Called "the ITS irregular terrain model" (or sometimes the Longley-Rice

*The authors are with the Institute for Telecommunication Sciences, National Telecommunications and Information Administration, U.S. Department of Commerce, Boulder,

Colorado 80303.
model; see Longley and Rice, 1968), it is designed for use at frequencies between

20 MHz and 20 GHz, for a wide variety of distances and antenna heights, and for

those problems where terrain plays an important role. It is concerned with the

generally available received power and not with the fine details of channel characterization.

On the other hand the model is avowedly statistical. In the physical world

received signal levels do vary in what appears to be a random fashion. They vary

in time because of changing atmospheric conditions, and they vary in space because

of a change in terrain. It is this variability that the model tries to describe,

thus providing the engineer estimates of not only the general level of expected

received powers but also the magnitude of expected deviations from that general

level.

Being a general purpose model, there are many special circumstances it does

not consider. In what follows we shall try to describe the general nature of the

model, to what uses it may be put, at what points special considerations might

enter, and, if we can, what steps might be taken to allow for them. The number of

possible special circumstances is so great, however, that we have undoubtedly overlooked many important ones. Here, we must depend on the ingenuity of the individual engineer to recognize the circumstance and to determine how to proceed. In

general, we expect the user of this Guide to be somewhat familiar with radio propagation and the effects its sometimes capricious behavior will have on radio systems.

The ITS irregular terrain model is specifically intended for computer use. In

this regard it is perhaps well to introduce here terminology that makes the distinctions computer usage often requires. A model is a technique or algorithm which

describes the calculations required to produce the results. An implementation of

a model is a representation as a subprogram or procedure in some specific computer

language. An applications program is a complete computer program that uses the

model implementation in some way. It usually accepts input data, processes them,

passes them on to the model implementation, processes the results, and produces

output in some form. In some applications programs, radio propagation and the

model play only minor roles; in others they are central, the program being but an

input/output control. For example, the program QKAREA described and listed in

Appendix B is a simple applications program; it calls upon the subprograms LRPROP

and AVAR which are listed in Appendix A and which, in turn, are an implementation of

version 1.2.1 of the ITS irregular terrain model.

2
2. AREA PREDICTION MODELS

Most radio propagation models, especially the general purpose ones, can be

characterized as being either a "point-to-point" model or an "area prediction"

model. The difference is that a point-to-point model demands detailed information

about the particular propagation path involved while an area prediction model

requires little information and, indeed, may not even require that there be a

particular path.

To explain this latter statement, let us consider for what problems a propa- I

gation model should help. There seem to be about five areas of concern:

(1) Equipment design. Given specifications of how new radio equipment is to be used and how reliable communications must be, it

should be possible to predict the values of path loss (and perhaps other characteristics of the channel) for which the equipment must compensate. Conversely, given the properties of

proposed new equipment, one should be able to predict how that

equipment will behave in various situations. In particular, one

should be able to predict a service range--Le., a distance at

which communications are still sufficiently reliable, under the

given conditions.

(2) General system design. This is an ex'tension of the first area.

Here, it is the interaction of radio equipment that is to be

studied. Often, interference, both between elements of the same

system and between elements of one system with another on the

same or adjacent frequencies, is an important part of the study.

Questions such as the proper co-channel spacing of broadcast sta1:ions or the proper spacing of repeaters might be treated.

(3) Specific operational area. In this case one or more radio systems

are to be located in one particular area of the world and, perhaps, operated at one particular season of the year or time of

day. Within this area, however, all terminals are to be located

at random, where "random" may mean not uniformly distributed but

according to some predefined selection scheme. These terminals

may, for example, be mobile so that they will, indeed, occupy

many locations; or they may be "tactical" in that they are to be

set up at fixed locations to be decided upon at a later date,

perhaps only just before they are put. into operation. Questions

3
t6be asked might be similar to those in the previous two areas.

One technique sometimes used is that of a simulation procedure in

which a Monte Carlo approach is taken in the placement of the

terminals or the control of communications traffic.

(4) Specific coverage area. In this case, one of the terminals is at

a specific known location while the other (often many others) is

located at random somewhere in the same vicinity. The obvious

example here is a broadcast station or the base station for a

particular mobile system; but other examples might include radars,

monitoring sites, or telemetry acquisition base stations. The

usual problem is to define a service area within which the reliability of communications is adequate or, sometimes, to find the

strength of interference fields within the service area of a

second station. If calculations are made before the station is

actually set up, one can think of them as part of the decision

process to judge whether the station design is satisfactory.

(5) Specific communications link. In this final case, both terminals

are at specific known locations, and the problem is to estimate

the received signal level. Or more likely, the problem is to

characterize the received signal level as it varies in time.

Again, calculations made here are often used in the design of the

link.

In the last of these areas--the specific communicationslink--one knows, or

presumably can obtain, all the details of the path of propagation. One expects to

obtain very specific answers to propagation questions, and therefore one uses a

point-to-point model.

In the first two areas, however--the design of equipment and of systems--there

is no thought about particular propagation paths. One wants general results, perhaps parametric results, for various types of terrain and types of climate. It is

natural to use an area prediction model.

In the case of a specific operational area or a specific coverage area, one is

confronted with a different problem. Here one has a large multitude of possible

propagation paths each of .which can presumably be described in detail. One might,

therefore, want to consider point-to-point calculations for each of them. But the

sheer magnitude of the required input data makes one hesitant. If a simulation

procedure is used to collect statistics of communications reliability, then the

point-to-point calculations become lost in the confusion to the point where they

4
seem hardly worth the trouble. Ap.?-;J.t~rn9-tive which requires far less input data

is to use an area prediction model, particularly if the model provides by itself

the required statistics.

Even in the case of a specific communications link, the required detailed

information for the propagation path may be unobtainable so that one is forced to

use the less demanding area prediction model. Of course, in doing so one expects

to lose in precision and in the dependability of the results.

In addition to the ITS area prediction model, other widely used models of this

kind include those developed by Epstein and Peterson (1956), Egli (1957), Bullington (1957), the Federal Communications Commission (FCCi Damelin et al., 1966), Okumura

et al. (1968), and the International Radio Consultative Committee (CCIR, 1978a).

By their nature, all these models use empiricism, by which we mean they depend

heavily on measured data of received signal levels. But also, they all depend to a

greater or lesser degree on the theory of electromagnetism. In some cases, theory

is used only qualitatively to help make sense out of what is always a very wide

spread in the measured values. In others of these models theory plays a more

important role, and the empirical data serve to provide benchmarks at which the

model is expected to agree.

3. THE ITS MODEL FOR THE MID-RANGE FREQUENCIES

Originally published by Longley and Rice (1968), the ITS irregular terrain

model is a general purpose model intended to be of use in a very broad range of

problems, but not, it should be noted, in all problems. It is flexible in application and can actually be operated as either an area prediction model or as a pointto-point model. We speak here of two separate "modes" of operation. In the pointto-point mode, part of the input one must supply consists of certain "path parameters"

to be determined from the presumably known terrain profile that separates the two

terminals. In the area prediction mode these Same parameters are simply estimated

from a knowledge of the general kind of terrain involved. The two modes use almost

~dentical algorithms, but their different sets of input data and their different

ranges of application make it inconvenient to discuss them both at once. This

report treats only the area prediction mode.

In the present section we shall provide a t>rief general description of the

model inclUding its design philosophy, a list of its input parameters, and a discussion of some of the physical phenomena involved in radio propagation and whether

they are or are not treated by the model.

5
Before continuing, however, we should first consider the units in which received

signal levels are to be measured. Here we come upon a confusion, for each discipline

of the radio industry se~ms to have chosen its own separate unit. Examples include

electromagnetic power flux, electric field intensity, power available at the terminals of the receiving ancenna, and voltage at the receiver input terminals. If one

wants to divorce the propagation channel from the equipment, one also speaks of

transmission loss, path loss, or basic transmission loss. Most of these quantities

are described by Rice et ale (1967; Section 2), but the important property to note

is that under normal conditions--when straightforward propagation takes place without near field effects or standing waves and when mismatches are kept to a minimum--

all these quantities are easily transformed one to another. Indeed, in this report

we use the term "signal level" so as to be deliberately vague about what precise

unit is intended, because we feel the question is unimportant.

For each of the quantities that might represent a signal level, it is possible

to compute a free space value--a value that would be obtained if the terminals were

out in free space unobstructed by terrain and unaffected by atmospheric refraction.

This free space value is a convenient reference point for radio propagation models

in general and for the ITS model in particular. Our own preference for a measure of

signal level is therefore the attenuation relative to free space which we always

express in decibels. In what follows we shall use the simple term "attenuation,"

hoping that the context will supply the reference point. The quantity is sometimes

also referred to as an "excess path loss." To convert to any other measure of signal level, one simply computes the free space value in decibels relative to some

standard level and then subtracts the attenuation (adds, if one is computing a

loss).

3.1 Input Parameters

In Table 1 we list all the input parameters required by the ITS area prediction

model. Also indicated there are the allowable values or the limits for which the

model was designed. Here we shall try to define the terms involved. As it happens,

however, some of the terms are by nature somewhat ambiguous, and we shall defer more

complete descriptions to Sections 5 and 6.

The system parameters are those that relate directly to the radio system

involved and are independent of the environment. Counting the two antenna heights,

there are five values:

Frequency. The carrier frequency of the transmitted signal. Actually,

the irregular terrain model is relatively insensitive to the frequency,

and one value will often serve for a fairly wide band.

6
Table 1. Input Parameters for the ITS Model Together With the

Original Design Limits

System Parameters

Frequency

Distance

Antenna heights

Polarization

Environmental Parameters

Terrain irregularity parameter, ~h

Electrical ground constants

Surface refractivity

Climate

Deployment Parameters

Siting criteria

Statistical Parameters

Reliability and confidence level

20 r-mz to 20 GHz

1 km to 2000 km

0.5 m to 3000 m

vert:ical or horizontal

250 to 400 N-units

one of seven; see Tabl~ 4

random, careful, or very careful

O.H to 99.9%

Distance. The great circle distance between the two terminals.

Antenna Heights. For each terminal, the height of the center of

radiation above ground. This may sound straightforward, and often it

is; but neither the center of radiation nor the ground '.evel is

always easy to determine. For further discussion see Section 5.

Polarization. The polarization, either vert:ical or horizontal, of

both antennas. It is assumed that the two antennas do llave the same

polarization aspect.

The environmental parameters are those that describe the environment or,more

precisely, the statistics of the environment in which the system is to operate. They

are, however, independent of the system. There are five values:

Terrain Irregularity Parameter ~h. The terrain that separates the two

terminals is treated as a random function of the distance away from

one of the terminals. To characterize this random function, the ITS

model uses but a single value ~h to represent simply the size of the

irregularities. Roughly speaking, ~h is the interdecile range of

7
Table 2. Suggested Values for the Terrain Irregularity Parameter

bh (meters)

Flat (or smooth water)

Plains

Hills

Mountains

Rugged mountains

For an average terrain, use bh=90 m.

o

30

90

200

500

The atmospheric constants, and in particular

terrain elevations--that is, the total range of elevations after the

highest 10% and the lowest 10% have been removed. further discussion

of this important parameter will be found in Section 5. Some suggested

values are in Table 2.

Electrical Ground constants. The relative permittivity (dielectric

constant) and the conductivity of the ground. Suggested values are in

Table 3.

Surface Refractivity N .

s

the atmospheric refractivity, must also be treated as a random function of position and, now, also of time. For most purposes this

random function can be characterized by the single value N repre- s

senting the normal value of refractivity near ground (or surface)

levels. Usually measured in N-units (parts per million), suggested

values are given in Table 4. Further discussion will be found in

Section 5.

Climate. The so-called radio climate, described qualitatively by a

set of discrete labels. The presently recognized climates are listed

in Table 4. Together with N , the climate serves to characterize the

s

atmosphere and its variability in time. Further discussion is given

in Section 5.

The way in which a radio system is positioned within an environment will often

lead to important interactions between the two. DeplOyment parameters try to characterize these interactions. The irregular terrain model has made provision for one

such interaction that is to be applied to each of the two terminals.

Siting Criteria. A qualitative description of the care which one

takes to site each terminal on higher ground. Further discussion is

given in Section 5.

8
Table 3. Suggested Values for the Electrical Ground Constants

Relative Conductivity

Permittivity (Siemens per Meter)

Average ground 15 0.005

Poor ground 4 0.001

Good ground 25 0.020

Fresh water 81 0.010

Sea water 81 5.0

For most purposes, use the constants for an average ground.

Table 4. Radio Climates and Suggested Values for N'

s

N (N-units) s

Equatorial (Congo)

Continental Subtropical (Sudan)

Maritime Subtropical (West Coast of Africa)

Desert (Sahara)

Continental Temperate

Maritime Temperate, over land

(United Kingdom and continental west qoasts)

Maritime Temperate, over sea

360

320

370

280

301

320

350

For average atmospheric conditions, use. a Continental Temperate climate

and N =301 N-units.

s

9
Finally, the statistical parameters are those that describe the kind and variety

of statistics that the user wishes to obtain. Very often such statistics are given

in the form of quantiles of the attenuation. For a discussion of this subject, and

of the meanings we like to give to the terms reliability and confidence, see Section

6.

Aside from the statistical parameters which will vary in number according to the

necessities of the problem, there are some twelve parameter values that one must

define. Although this seems a rather'long list, the user should note that in many

cases several of these parameters have little significance and may be replaced by

simple nominal values. For example, the only use to which the polarization and the

two electrical ground constants are put is to determine in combination the reflectivity of smooth portions of the ground when the incident rays are grazing or nearly

so. At high frequencies this reflectivity is nearly a constant. When both terminals

are more than about 1 wavelength above the ground or more than 4 wavelengths above

the sea, these three parameters have little significance, and one may as well assume,

say, "average" ground constants. At frequencies below about 50 MHz the effect of the

conductivity is dominant; otherwise the relative permittivity is the more important.

Similarly, on short paths less than about 50 km, the atmosphere has little

effect, and one may as well assume average conditions with a Continental Temperate

climate and N =301 N-units. And finally, for the siting criteria one will usually s

assume that both terminals are sited at random. Thus, in a large proportion of

practical problems, one is left with only five parameter values to consider: frequency, distance, the two antenna heights, and the terrain irregularity parameter.

3.2 General Description

Given values for the input parameters, the irregular terrain model first computes several geometric parameters related to the propagation path. Since this is an

area prediction model, the radio horizons, for example, are unknown. The model uses

empirical relations involving the terrain irregularity parameter to estimate their

position.

Next, the model computes a reference attenuation, which is a certain median

attenuation relative to free space. The median is to be taken over a variety of

times and paths, but only while the atmosphere is in its quiet state--well-mixed and

conforming to a standard atmospheric model. In continental interiors such an atmosphere is likely to be found on winter afternoons during fair weather. On oversea or

coastal paths, however, such an atmosphere may occur only rarely.

10
As treated by the model, this reference attenuation is most naturally thought of

as a continuous function of distance such as that portrayed in Figure 1. As shown

there, it is defined piecewise in three regions, called the line-of-sight, diffraction, and forward scatter regions. The "line-of-sight" region is somewhat misnamed;

it is defined to be the region where the general bulge of the earth does not interrupt the direct radio waves, but it still may be that hills and other obstructions do

so. In other words, this region extends to the "smooth-earth" horizon distance,

which is probably farther than the actual horizon distance. In this region the

reference at1:enuation is computed as a combined logarithmic and linear function of

distance; then in the diffraction region there is a rather rapid linear increase; and

this is followed in the scatter region by a much slower linear increase. Parameters

other than distance enter into the calculations by determining where the three

regions fall and what values the several coefficients have. But once the system and

its deployment (in a homogeneous environment) have been fixed, the notion of attenuation as a function of distance should be a convenient one for many problems.

III

.... 20

\ free space

CIl

.Q

-

CIl

t::l

"'

CIl

\,. 40

<:t

01--......

60

line-of-sight diffraction scatter

o 20 40 60 80 100 120

Distance. Kilometers

140 160 180 200

Figure 1. A typical plot of reference attenuation versus distance.

11
The reference attenuation is a good representative value to indicate to a

designer how a proposed system will behave. For some problems, knowing it alone will

be sufficient. For most problems, however, one must also obtain the statistics of

the attenuation. To do this, the model first subtracts a small adjustment for each

climate to convert the reference attenuation to an all-year median attenuation.

Then from this median attenuation further allowances are subtracted to account for

time, location, and situation variability in the manner described in Section 6.

For its calculations, the model utilizes theoretical treatments of reflection

from a rough ground, refraction through a standard atmosphere, diffraction around the

earth and over sharp obstacles, and tropospheric scatter. It combines these using

empirical relations derived as fits to measured data. This combination of elementary

theory with experimental data makes it a semi-empirical model which on the one hand

should agree with physical reality at certain benchmark values of the parameters and

on the other hand should comply with physical laws sufficiently well to allow us to

interpolate between and extrapolate from these benchmark values with a good degree

of confidence. Thus the model is a general purpose one that should be applicable

under a wide variety of "normal" conditions--particularly those conditions that

correspond to the land mobile and broadcast services.

The data used in developing the empirical relations clearly have influenced the

model itself. It should then be noted that these data were obtained from measurements made with fairly clear foregrounds at both terminals. In general, ground cover

was sparse, but some of the measurements were made in areas with moderate forestation. The model, therefore, includes effects of foliage, but only to the fixed

degree that they were present in the data used.

There are several phenomena that the model ignores, chiefly because they occur

only in special circumstances. In cases of urban conditions, dense forests, de1ib.,

erate concealment of the terminals, or concerns,about the time of day or season of

the year, it is possible to make suitable extra allowances or additions to the basic

model. This, of course, requires an engineer who knows the situation involved and

the probable magnitude of the consequent effects.

The possibility of ionospheric propagation is what makes us limit the model to

frequencies above 20 MHz. Still, there will be occasional cases of ionospheric

reflection at frequencies near this lower limit, and scatter from sporadic E will

occur at frequencies below about 100 MHz. Such effects, however, will be apparent

only on very long paths and only for very small fractions of time.

Atmospheric absorption--particu1arly the water vapor line at 22 GHz--is what

limits the model at the higher frequencies. The effects are measurable above about

12
13
terminals are sited in extreme, rather than typical, locations, the calculated attenuation will not represent the median of measurements. An example of such an a~ypical

situation would be propagation along a narrow, steep-sided valley, where the radio

signal may be repeatedly reflected from the walls of the valley.

4. DEVELOPMENT OF THE MODEL

During the years prior to 1960, a good deal of information was obtained regarding

radio propagation through the turbulent atmosphere over irregular terrain. For

paths with fixed terminals a number of prediction models had been developed to

describe the power available at the receiver over known profiles by means of lineof-sight, diffraction, and forward scatter propagation. A good deal of data had

also been accumulated from high-powered broadcast transmitting antennas to rather

well-sited receivers. However, land-mobile types of communication systems were

becoming increasingly important. In such applications some of the terminals are

highly mobile, with randomly changing locations. Little information was available

for such systems, especially where low antenna heights and ready mobility are prime

requirements.

A theoretical and experimental program was undertaken by the National Bureau of

Standards to study propagation characteristics under conditions resembling the

operation of army units in the field. Tactical situations may often require that

antennas be low and placed as inconspicuously as possible, and that receivers be

highly mobile. A report by Barsis and Rice (1963) describes the planned measurement

program and proposed terrain analysis. The measurements were to be carried out in

various types of terrain, including the open plains of eastern Colorado, the foothills and rugged mountains of Colorado, and the rolling, wooded hills of northeastern Ohio. The report describes terrain profile types in terms of a spectral

analysis which depends on a,discrete, finite-interval, harmonic analysis of terrain

height variations over the great circle path between terminals. Characteristics of

terrain profiles of any given length were described relative to a least-squares fit

of a straight line to heights above sea level.

As the study progressed, the harmonic analysis of terrain was replaced by a

single parameter bh, which is used to characterize the statistical aspects of terrain irregularity. Terrain statistics were developed for the areas described above

by reading a large number of terrain profiles. Each profile was represented by

discrete elevations at uniform distances of half a kilometer. Within each region

selected for intensive study, 36 profiles 60 km in length were read in each of six

directions, providing a total of 216 profiles that form a rather closely spaced grid

14
over a 100 km square area. Each profile was considered, in lengthso;f 5, 10, 20 •••

60 km to study the effects of path length on the various terrain parameters.

The interdecile range 6h(d) of terrain heights above and below a straight line,

fitted by least squares to elevations above sea level, was calculated at each of

these distances. Usually the median values of 6h(d) for a specified group of profiles increase with path length to an asymptotic value, ~h, which is used to characterize the terrain. This definition of 6h differs from that used by the CCIR and

by the FCC as noted in Section 5.

An estimate of 6h(d) at any desired distance may be Obtained from the following

empirical relationship:

For homogeneous terrain, values of ~h(d)

6h (d) :::; ~h I1-0. 8 exp (-diDo)]

where the scale distance D equals 50 kID.

o

measured at each distance agree well with those obtained from (1).

(1)

As the terrain

in a desired area becomes less homogeneous, the scatter of measured values of 6h(d)

increases.

For an area prediction where individual path profiles are not available, median

values of terrain parameters to be expected are calculated as empirical functions of

the terrain irregularity parameter ~h, the effec1:ive earth's radius, the antenna

heights, and the siting criteria employed.

Even at first, the model was designed to calculate the reference attenuation

below free space as a continuous function of distance. This could be easily converted to basic transmission loss by adding the free-space loss at each distance.

These reference values of basic transmission loss, with a small adjustment for

climate, represent the median, long-term values of transmission loss predicted for

the area.

To provide a continuous curve as a function of distance, this median attenuation is calculated in three distance ranges as shown in Figure 1, Section 3: a) for

distances less than the smooth-earth horizon distance dLsi b) for distances just

beyond the horizon from d to d i and c) for distances greater than d. The model Ls x x

does not provide predictions for distances less ·than 1 km. For distances from 1 km

to d

Ls , the predicted attenuation is based on two-ray reflection theory and extrapolated diffraction theory. For distances from dLS to d

x ' the predicted attenuation

is a weighted average of knife-edge and smooth-earth diffraction calculations. The

weighting factor in this region is a function of frequency, terrain irregularity,

and antenna heights. For distances greater than d , the point where diffraction and

x

15
scatter losses are equal, the reference attenuation is calculated by means of a

forward scatter formulation.

In developing the original model, comparisons with data were made and empirical

relationships were established. These include expressions for calculating horizon

distances and horizon elevation angles, based on information obtained during the

terrain study. The weighting factor, used to obtain the weighted average between

rounded earth and knife-edge diffraction calculations, is based on radio data taken

from two series of measurements. The first of these provided a large amount of data

at 20, 50, and 100 MHz, obtained with low antennas in Colorado and Ohio. The

results of these measurements are reported by Barsis and Miles (1965) and by Johnson

et al. (1967). The other large body of measurements at VHF and UHF was provided to

the Television Allocations Study Organization (TASO). These measurements were made

in 1958 and 1959 in the vicinity of several cities in the United States, and the

results are summarized by Head and Prestho1dt (1960). Signals from television

stations at frequencies of about 60 and 600 MHz were measured at uniform distances

along radials with 3 and 9 m receiving antenna heights. These measurements were

made with both mobile, and stationary receivers in terrain that ranged from smooth

plains to mountains.

After the model was developed and published (Longley and Rice, 1968), comparisons were made with a large amount of data at frequencies from 20 MHz to 10 GHz.

These comparisons are reported by Longley and Reasoner (1970). Further comparisons,

reported by Longley and Hufford (1975), were made with data at 172 MHz and 410 MHz

taken with very low antennas.

Concerning the question of statistics, recall that the original purpose was to

provide an area prediction model for land-mobile applications. Such systems involve

low antennas and low transmitter powers with consequently short ranges. For such

short paths, over land, the path-to-path variabiiity is considerably greater than

the time variability, and therefore the latter was treated rather casually. A

Continental Temperate climate was assumed and represented by a cumulative distribution with two slopes--two "standard deviations"--to allow for the observed greater

variability of the strong fields than of the weak ones.

As the use of the model was extended to broadcast coverage, with high power

radiated from transmitting antennas on tall towers, the effects of differences in

climate became more important in terms of possible interference between systems.

For such applications, we included sets of mathematical expressions that reproduce

the variability curves for various climates defined by the CCIR (1978b) and listed

in Table 4. Two other climates, Mediterranean and Polar, are described in the CCIR

16
Report, but curves are not presen~~d for them. For land-mobile services in the

United States, the continental Temperate climate is nearly always chosen.

The original "Longley-Rice" model was published in 1968. Shortly afterward a

new version was developed which improved the formulation for the forward scatter

prediction, and later the computer implementation was changed to improve its efficiency and i.ncrease the speed of operation. Since then, minor but hlportant modifications have been made in the line-of-sight calculations.

To keep track of the various versions, most of which are presently being used

at some facility, we have recently begun numbering them in serial fashion. Following the original (which might be called version 0), here is a list of the more

important versions, together with approximate dates when they were first distributed:

1.0

1.1

1.2

1.2.1

2.0

2.1

2.2

January 1969

August 1971

March 1977

April 1979

May 1970

February 1972

September 1972

Version 1.2.1 corrects an error in version 1./.; it is the currently recommended version and is the one whose implementation is listed in Appendix A. The second series,

beginning with version 2.0, used considerably modified diffraction calculations and

tried to incorporate a groundwave at low frequencies. It is not now recommended

and is no longer maintained by its developers.

5. DETAILED DESCRIPTION OF INPUT PARAMETERS

The various parameters required as input to the ITS area prediction model were

described briefly in Section 3 of this guide. Further description and an explanation of their use is provided here.

The primary emphasis of the model is a consideration of the effects of irregular terrain and the atmosphere on radio propagation at frequencies from 20 MHz to

20 GHz. One of the chief parameters used to describe the atmosphere is the surface

ret'ractivity N

s ' while the terrain is characterized by the parameter D.h. A discussion of both atmospheric and terrain parameters is presented here.

17
5.1 Atmospheric Parameters

Atmospheric conditions such as climate and weather affect the refractive index

of air and play important roles in determining the strength and fading properties of

tropospheric signals. The refractive index gradient of the atmosphere near the

earth's surface is the most important atmospheric parameter used to predict a longterm median value of transmission loss. This surface gradient largely determines

the amount a radio ray is bent, or refracted, as it passes through the atmosphere.

In this model we define an "effective" earth's radius as a function of the surface

refractivity gradient or of the mean surface refractivity N. This allows us to

s

consider the radio rays as being straight so long as they lie within the first

kilometer above the earth's surface. At very much higher elevations, the effective

earth's radius assumption over-corrects for the amount the ray is refracted and may

lead to serious errors. In this propagation model we use minimum monthly mean

values of N to characterize reference atmospheric conditions. Since such values are

s

less apt to be influenced by temporary anomalous conditions such as superrefraction

or subrefraction, they provide a rather stable reference which is exactly suited to

computations of the reference attenuation.

The minimum monthly mean value of N , which in the northern hemisphere often

s

corresponds to values measured in February, may be obtained from local measurements

or estimated from maps of a related parameter N. The re~ractivity N is the value

o 0

of surface refractivity that has, for convenience, been reduced to sea level.

Figure 2 from Bean et al. (1960) is a world-wide map of minimum monthly mean values

of N. The corresponding value of surface refractivity is then:

o

N

s

N exp (-h /H ) o s s

N-units (2)

where h is the elevation of the earth's surface and the scale height H equals s s

9.46 km.

The effective earth's radius is directly defined as an empirical function of

N , increasing as N increases. It is common to set N equal to 301 N-unitsi this

s s s

corresponds to an effective earth's radius of 8497 km, which is just 4/3 times the

earth's actual radius. Values of the effective earth's radius are used in computing

the horizon distances, the horizon elevation angles, and the angular distance e for

transhorizon paths.

For short distance ranges the model is not particularly sensitive to changes in

the value of surface refractivity. For this reason, in land-mobile systems we may

often assume that N has the nominal value of 301 N-units. For distances greater s

than 100 km, changes in N

s

have a definite effect on the amount of transmission

loss.

18
IBO W 160 140 120 100 80 60 40 20WOE20 40 60 80 100 120 140 160 E 180

~·~r;,n~1 I 1~~N-L,.J. I j

~~~~J)r-r20

40

30

20

50

70

30

50

60

40

60

10

N

o

S

10

70

80

5

N

80

t~r-TnITi'-i-1 I I I I I I"l--LLi_-1300

1--

300V

31Of.:-U lOS __ _ _ _ _ _ _ _

- - -- - '--ld~Pt=Eh; _-r

-t-

?9Ol:~.-..:~~*---t---j-t--t-f""t---=F=:f='T-I---t-----jII-t-"""I:=t=:jt::=:t=::t==t::=1-I"I'-I'""'t===-:$:=1=== 290 CRPL BASE MAP

MODIFIED CYLINDRICAL. PROJECTION

:t;

N

80

70

60

50

40

30

20

10

N

0

I-'

\D II~

20

30

40

50

60

70

80

5

180 W 160 140 120 100 80 60 40 2OWOE20 40 60 80 100 120 140 160 E 180

Figure 2. Minimum monthly mean values of surface refractivity referred to mean sea level

(from Bean, Horn, and Ozanich, 1960).
Other atmospheric effects, such as changes in the refractive index and changes

in the amount of turbulence or stratification, lead to a variability in time that

may be allowed for by empirical adjustments described in Section 6.

5.2 Terrain Parameters

In VHF and UHF propagation over irregular terrain near the earth's surface, a

number of parameters are important. Early studies by Norton et al. (1955), Egli

(1957), LaGrone (1960), and others indicated that for transhorizon paths the most

important of these parameters appears to be the angular distance e. For within-thehorizon paths the clearance of a radio ray above the terrain between the terminals

is one of the most important factors.

In considering terrain effects, we usually assume that we need allow only for

the terrain along the great circle path between terminals. The angular distance e

is then defined as the acute angle in the great circle plane between the radio

horizon rays from the transmitting and the receiving antennas. The angular distance

e is positive for transhorizon paths, zero at grazing incidence, and negative for

line-of-sight paths.

When detailed profile information is available for a specific path, then the

horizon distances, the horizon elevation angles, and the angular distance e may be

computed directly. In an area prediction, however, specific path profiles are not

available, and these same terrain parameters must be estimated from what we know of

the statistical character of the terrain involved. As described in Section 4,

examination of a large number of terrain profiles of different lengths in a given

area showed that median values of ~h(d) increase with path length to an asymptotic

value ~h. This parameter ~h, defined by (1), is used to characterize terrain.

We should note here that this definition of ~h differs from the one used by the

CCIR (1978a) and by the FCC (Damelin et al., 1966). Their definition is simply the

interdecile range of elevations above sea level in the range 10 to 50 km from the

transmitter. This definition results in smaller values of ~h than our asymptotic

value. We estimate that in most cases the CCIR value will equal approximately 0.64

times our value. For instance, while we would say that a world-wide average value

for ~h is about 90 m, the FCC uses the value of 50 m.

In homogeneous terrain the values of ~h(d) measured over a large number of

paths agree well with those calculated using the relationship in (1). Where the

terrain is not homogeneous, a wider scatter of values occurs, and the estimated

value of ~h(d) may not represent a true median at each distance. In such circumstances we may allow for a greater location variability in the prediction, or at

20
times we may consider different sectors of an area and predict for each sector. An

example of this would be an area that includes plains, foothills, and mountains.

The losses predicted for each sector could be determined for the value of the terrain parameter computed for that sector.

The terrain parameter ~h may be obtained in one of several ways. The method

selected will depend on the purpose for which it is used and on the terrain itself.

In the original work to determine ~h for an area, a large number of profiles were

read at uniform intervals. These profiles criss'-crossed the area in such a way as

to provide a rather fine grid. The interdecile range ~h(d) was obtained for each

profile and plotted as a function of distance. The median value at each distance

was then used to obtain a smooth curve of ~h(d), whose asymptotic value is the

desired parameter ~. This method is quite laborious and may not be necessary for

the desired application. One can now use general maps of the terrain irregularity

parameter as shown in Figure 3, or one may still go directly ~o topographic maps of

the desired area and from them estimate the proper value. To do this, one may

select a random set of paths, compute the value ~h from each path, and use the

median of these calculated values to describe the terrain irregularity. with practice and a few elevations read from the map, one can even estimate ~h by eye.

30

40

25

120 70 65

30

50~--'----'--""----'---'---""----T--"""--""---""_-""-""'-'

Figure 3. Contours of the terrain irregularity parameter ~h in meters.

The derivation assumed random paths and homogeneous terrain

in 50 kIn blocks. Allowances should be made for other conditions.

21
A major problem is that the area of interest is rarely homogeneously irregular.

In such a case one must exercise judgment in selecting paths that will be representative of those that will actually be used in a proposed deployment. For example,

if the desired paths will always be along or across valleys, one should not choose

terrain profiles that cross the highest mountains. When virtually all paths involve

terminals on facing hillsides along the same valley, a highly preferential situation

exists.

Some qualitative descriptions of terrain and suggested values of ~h are listed

in Table 2. Whether or not one needs 'a better estimate, based on computed values,

depends on the sensitivity of the predicted values of transmission loss to changes

in ~h. This sensitivity is quite complicated, depending on the value of ~h itself,

on the antenna heights, distance range, siting criteria, and the radio frequency.

This is probably most readily illustrated by an example. Figure 4 shows plots of

attenuation relative to free space as a function of ~h at various distances. These

curves are for a land-mobile system over irregular terrain at a frequency of 150

MHz. The upper figure represents base-to-mobile communication with antenna heights

of 30 m and 2 m. The lower figure is drawn for mobile-to-mobile units with both

antennas 2 m above ground. For small values of ~h the sensitivity to change is

quite appreciable, especially at distances in the line-of-sight and diffraction

regions. Here the decrease in attenuation (a phenomenon that might be likened to

"obstacle gain") may be as much as 10 dB as ~h increases fl.'om 0 to 25 m. For

larger values of ~h from about 50 to 150 m, there is little change in attenuation

while for still larger values of ~h and for distances in the scatter region the

increases in attenuation are quite regular and less sensitive to change than for

small values of ~h.

The area prediction model depends heavily on the parameter ~h, which characterizes terrain, and on the surface refractivity, N. Median values of all the

s

other terrain parameters are computed from these two values when antenna heights are

specified. Estimates of signal variability in time and space are also dependent on

these two basic parameters. The relationships between the secondary parameters and

the terrain irregularity parameter ~h were developed mainly in rural areas where

antenna sites were always chosen with open foreground and were located on or near

roads. In these areas the ground cover was usually sparse, but some moderate

forestation was present.

5.3 Other Input Parameters

The way a system is deployed--particularly the way the terminal sites are

chosen--can have a marked effect on observed signal levels. Unfortunately, there

22
---

_d==5 km_

10

V 20

---- 1/ -30

50- - 75- - -

L---'"" -100 - 125 - - -200- - - 300

-- ----500--

-- -

---- d==5 km

10

--------:::-- 20 - - - 30 - 50- - V/~__- 75-

- V~- 100-

125 ---

-- - 200

-300_ '---

20

11\ 40 .....

CII

Q

lJ-

..... GO

CII

"'l

80

100

o

20

11\ 40 .....

CII

Q

-

80

100

o

(a)

50

f =

50

100

150 MHz, hgl =

100

150 200

i1n. Me I@rS

30 ro, h g2 = 2 ro

150 200

i1n. MeIers

250

250

300

300

350

350

(b) f = 150 MHz, hgl = 2 ro,

Figure 4. The reference attenuation versus ~h for selected distances.

23
have been very few studies of these effects that could provide us with useful guidance. Nevertheless, the area prediction model does require the siting criteria,

which are qualitative descriptions of the care with which each 0;1; the two terminal

sites is chosen so as to improve communications. The e;l;fect of these criteria on

the model is based on reasonable assumptions, but the validity of the results has

been tested in only a limited number of examples. One should therefore exercise

caution, both in the selection of values for the siting criteria and in the interpretation of results.

Changes in the value of the siting criterion for one of the terminals affect

the assumed effective antenna height of that terminal. This effective height is

defined to be the height of the antenna above the "effective reflecting plane"

which, in turn, is a characterization of the intermediate foreground. It is actually

this height, not the structural height, that the model uses in nearly all of its

calculations. When the effective height increases, the model predicts less transmission loss and a greater communication range.

When the terminals of a system are usually sited on high ground and some

effort is made to locate them where signals appear to be particularly strong, we

say the siting is very good. When most of the terminals are located at elevated

sites but with no attempt to select hilltops or points where signals are strong, we

would classify these as good sites. Finally, when the choice of antenna sites is

dictated by factors other than radio reception, there is an equal chance that the

terminal locations will be good or poor, and we would assume the selection of

antenna sites to be random. But note that even when antennas are sited randomly we

assume they are not deliberately concealed. For concealed antennas an additional

loss should be allowed, the amount probably depending on the nature of the concealment and on the radio frequency and terrain irreg~larity.

With random siting the effective antenna heights are assumed to be simply

equal to the structural heights. with good siting the effective height is obtained

by adding to the structural height an amount that never exceeds 5 m. With very

good siting the additional amount never exceeds 10 m. In both cases the actual

amount added depends on the terrain irregularity parameter, the notion being that

in more irregular terrain there will be greater opportunity to find elevated ground.

In flat areas/the effective heights will always equal the structural heights no

/

matter what the siting criteria .are. The advantages achieved by good and very good

siting are greatest for low antennas with structural heights less than about 10 m.

If the antenna is on a high tower, the assumed change in effective height has

24
little significance--but it is definitely significant for antennas located just

above the ground.

The effective heights estimated from the sit.ing criteria assume that antennas

will be placed on a good site or on the best site within a very limited area. They

do not assume that antennas will be placed on the highest mountain top within a

total deployment area. But in many special problems, one will actually use just

this kind of site selection. One such problem is illustrated in Section 7.3. In

that case the receiver site was deliberately chosen at the edge of a high mesa

overlooking rather smooth terrain. This is a decidedly atypical situation. One

intuitively feels that it should be treated by setting the effective height of the

antenna equal to the height above the terrain that lies below the mesa. But in the

irregular terrain model it is the structural height that must be adjusted.

We usually define the structural height of an antenna to be the height of the

radiation center above ground. But if the antenna looks out over the edge of a

cliff, then it seems entirely natural to say that the cliff is really a part of the

antenna tower and to include its height in the structural height. Another, more

common, example of this same problem occurs in the design and analysis of VHF and

UHF broadcast stations. There, it is the usual practice to site the antenna atop a

hill or well up the side of a mountain in order to gain a very definite height

advantage. While we no longer have an obvious cliff, this height advantage should

still be accounted for by including it in the structural height.

There are several rules used by various people to determine what the ground

elevation should be above which the antenna height is to be found. The FCC uses

the 2 tolD mi (3 to 16 km) average elevation for the radial of interest. Another

rule that has been suggested is that one should not coUfit as "ground" anything that

has a depression angle from the center of radiation of more than 45°. In our own

work we have sometimes said that consideration of terrain elevations should begin

at a point distant about 15 times the tower height.

The choice of a radio climate may be difficult or confusing for the reader.

The several climates described by the CCIR have not been mapped out as various

zones throughout the world, and there are no hard and fast rules to describe each

of the climates. Since our model is intended for use over irregular terrain, our

preference is to use the Continental Temperate climate unless there are clear

indications to choose another. The CCIRcurves showing variability in time are

entirely empirical and depend on the climate chosen. The curves for Continental

and Maritime Temperate climates are based on a considerable amount of data, while

those for the other climates depend On much smaller data samples. The Continental

25
Temperate climate is common to large land masses in the temperate zone. It is

characterized by extremes of temperature and pronounced diurnal and seasonal changes

in propagation. In mid-latitude coastal areas where prevailing winds carry moist

maritime air inland, a Maritime Temperate climate prevails. This situation is

typical of the United Kingdom and of the west coasts of the United States and

Europe. For paths that are less than 100 km long, there is little difference

between the Continental and Maritime Temperate climates, but for longer paths the

greater occurrence of superrefraction and ducting in maritime areas may result in

much higher fields for periods of 10 percent or less of the year.

In considering time variability, it is important to note that we are concerned

only with long-term variability, the changes in signal level that may occur during

an entire year. The data on which such estimates are based were median values

obtained over short periods of time, an hour or less. The yearly signal distributions are then distributions of these medians. This eliminates much of the

short-term variability, which is usually associated with multipath. The rapid,

short-term, mUltipath fading at a mobile receiver depends on many local factors

including the type of receiving equipment, reflections from buildings and trees,

and the speed at which the recording vehicle travels. In smooth, uncluttered terrain there may be little if any multipath fading, whereas the most severe fastfading is Rayleigh distributed. Even simple diversity techniques will greatly

reduce this short-term multipath type of fading.

6. STATISTICS AND VARIABILITY

We come now to a discussion of how the ITS irregular terrain model treats the

statistics of radio propagation. As we have mentioned before it seems undeniable

that received signal levels are subject to a wide variety of random variations and

that proper engineering must'take these variations into account. Unfortunately,

the problem is considerably more complicated than problems of simple random variables one encounters in elementary probability theory.

The principal trouble is that the population of observed signal levels is

greatly stratified--i.e., not only do the results vary from observation to observation (as one would expect) but even the statistics vary. Now it is not surprising

that this should be the case when one varies the fundamental system parameters of

frequency, distance, and antenna heights; nor is it surprising when one varies the

environment from, say, mountains in a continental interior to flat lands in a

maritime climate, or from an urban area to a desert. But even when such obvious

26
parameters and conditions are accounted for, thE~re remain many subtle and important

reasons why different sets of observations have different statistics.

Our problem here is analogous in many ways to that of taking public opinion

polls. There results depend not only on the qUE!stions asked but also on many

subtleties concerning how, where, and when the questions are asked. If one spends

the working day telephoning people at their homes, then one obtains the opinions of

those people who own telephones and answer them and who have remained at home that

day. This procedure might still be a random sampling and might, indeed, provide

acceptable results, if it were not for the fact that public opinion is, again,

greatly stratified--i.e., that the opinions of one segment of the population can

differ greatly from those of another.

In the case of radio propagation, it is the equipment and h9w, where, and when

it is used that provides an added dimension of variability. Perhaps one or both

terminals are vehicle mounted and constrained to streets and roads. Perhaps,

instead, one antenna is likely to be mounted on a rooftop. Perhaps it is most

probable that both antennas are well removed from trees, houses, and other obstacles; or perhaps it is likely that one of the antennas is close to such an obstacle

or even inside a building, whether this be for convenience or because concealment

is desirable. It may be that two regions of the world appear, even to the expert's

eye, to offer the same set of impediments to radio propagation and yet the differences--whose effects we do not understand--may be important.

In any case, the way in which equipment is deployed has an often important

and unpredictable effect on observed signal levels. We propose here to use the

word situation to indicate a particular deployment, whether in actual use or simply

imagined. In technical terms, a situation is a probability measure imposed on the

collection of all possible or conceivable propagation paths and all possible or

conceivable moments of time. (A good introduction to the theory of probability

measures is given by Walpole and Myers, 1972, Ch. 1.) To choose a path and a time

"at random" is therefore to choose them according to this probability measure.

Insofar as ,ve want to get below the level at which stratification is important, we

would want 1:0 restrict a situation (that is, to restrict the set of paths and times

where the imposed probability is non-zero) to include only paths with a common set

of system parameters, lying within a single, homogeneous region of the world. This

is a natural restriction except, perhaps, as it affects the distance between terminals. The distance is a parameter which is difficult to fix while still allowing

a reasonable selection of paths.

27
If we are concerned with a single, well-defined communications link with fixed

terminals, then the situation involved has only a single isolated path which is to

be chosen with probability one. But the deployment of a land-mobile system in one

single area would define a more dispersed situation. Note, moreover, that if the

mobile units pass from an urban area to a suburban or rural area, then we would

suppose they pass from one situation to another. If one sets out to make a set of

measurements of received signal levels, then one will sample from what is, if the

measurement program has been properly designed, a situation pre-defined by the

program objectives. Often the measurements will be in support of what will become

a system deployment. It is then always proper to ask whether the situation from

which the data are taken corresponds accurately enough to the situation in which

the system will operate.

Once again, all this fussiness would be unnecessary--and radio propagation

engineering would long ago have become a finely honed tool--if it were not that the

population of received signal levels is a stratified one. The system parameters,

the environmental parameters, and the situation in which one is to operate are all

important and each of them has some effect on the final statistics. The complexity

of nature often forces us to empirical studies of these statistics; but the large

number of dimensions involved makes this a difficult task.

6.1 The Three Dimensions of Variability

We turn now to a general discussion of the physical phenomenology involved.

First, we should note that there is a very important part of the variability that

we do not wish to include. This is the short-term or small displacement variability

that is usually attributed to multipath propagation. Although it is probably the

most dramatic manifestation of how signal levels vary, we exclude it for several

reasons. For one, a proper description of mult~ath should include the intimate

details of what is usually known as "channel characterization," a subject that is

beyond our present interests. For another, the effects of multipath on a radio

:;;ystem depend very greatly on the system itself and the service it provides. Often

a momentary fadeout will not be of particular concern to the user. When it is, the

system will probably have been constructed to combat such effects. It will use

redundant coding or diversity. Indeed, many measurement processes are designed so

a:;; to imitate a diversity system. On fixed paths, where one is treating the

received signal level as a time series, it is common to record hourly medians--

i.e., the median levels observed during successive hours (or some comparable time

interval). We may liken the process to a time diversity system. If measurements

28
are made with a mobile terminal, one often reports on selected mobile runs about

30 m in length. Then, again, one records the median levels for each run, thus

simulating a space diversity system. Under the "frozen-in-space" hypothesis concerning atmospheric turbulence, one expects hourly medians and 30-m run medians to

be about the same. (But the analogy becomes rather strained for mUltipath in urban

areas.) To the two measurement schemes above, it would seem reasonable to add a

third to correspond to frequency diversity. This would be a "wideband" measurement

in which the average or median power over some segment of the spectrum were recorded.

In any case, it is only the variation of these local medians that concerns us.

If one still finds it necessary to consider instantaneous values of cw signals,

then the usual practice is simply to tack on an additional variability to those we

shall describe here. Often, one assumes either that the signal is locally steady

(in areas where there is no multipath) or that it is Rayleigh distributed (in areas

with extreme multipath). Occasionally one will assume an intermediate case, using

the Nakagami-Rice (see, e.g., Rice et al., 1967, Annex V) distributions or the

Weibull distributions.

If we set out to measure statistics of local medians, the first step that

occurs to us is to choose a particular fixed link and record measurements of hourly

median received signal levels for 2 or 3 years. The resulting statistics will

describe what we call the time variability on that one path. We could characterize

these observations in terms of their mean and standard deviation; but, both because

the distribution is aSYmmetric and not easily classified as belonging to any of the

standard probability distributions, and because the practicing engineer seems to

feel more comfortable with the alternative, we prefer to use the quantiles of the

observations. These are the values not exceeded for given fractions of the time

and are equivalent to a full description of the cumulative distribution function as

described in the elementary texts on statistics. We would use such phrases as "On

this path for 95% of the time the attenuation did not exceed 32.6 dB."

If we now turn our attention to a second path, we find to our dismay that

things have changed. Not only are individual values different, as we would expect

given the random nature of signal levels, but even the statistics have changed. We

have a "path-to-path" variability caused by the fact that we have changed strata in

the population of observable'signal levels. Suppose, now, that we make a series of

these long-term measurements, choosing sample paths from a single situation. In

other words, we keep all system parameters constant, we restrict ourselves to a

single area of the earth and keep environmental parameters as nearly constant as is

reasonable, and we choose path terminals in a single, consistent way. We still

29
find that the long-term time statistics change from path to path and the variation

in these statistics we call location variability. Of course, if the situation we

are concerned with has to do with a single, well-defined link, then it is improper

to speak of different paths and hence improper to speak of location variability.

But in the broadcast or mobile services, it is natural to consider such changes.

The most obvious reason for the observed variability is the accompanying change in

the profile of the terrain lying between the two terminals; although the outward--

statistical, so to speak--aspects of t]le terrain may remain constant, the actual

individual profiles, together with other, less obvious, environmental changes, will

induce large changes in observed signal level statistics.

If we try to quantify location variability, we must talk of how time variability varies with path location. We have no recourse but to speak of the statistics

of statistics. Clinging to the terminology of quantiles, we would speak of quantiles of quantiles and come up with some such phrase as "In this situation there

will be 70% of the path locations where the attenuation does not exceed 32.6 dB for

at least 95% of the time."

Finally, we must ask what effect there is when one changes from situation to

situation. It should be no surprise to be told that the statistics we have so

painfully collected following the outline above have changed. If we use like

appearing situations--that is, if we change operations from one area to another

very similar area or if we merely change the sampling scheme somewhat--then the

observed changes in the location variability we call situation variability.

In other contexts this last variability is sometimes referred to as "prediction error," for we may have used measurements from the first situation to "predict"

the observations from the second. We prefer here to treat the subject as a manifestation of random elements in nature, and hence as something to be described.

To make a quantitative description however, we must renew our discussion of the

character of a "situation." We have defined a situation to be a restricted probability measure on the collection of all paths and times. But if we are to talk of

changing situations--even to the point of choosing one "at random"--then we must

assume that there is an underlying probability measure imposed by nature on the set

of all possible or conceivable situations. And we must assume that at this level we

have specified system parameters, environmental parameters, and deployment parameters in sufficient detail so that the variability that remains is no longer stratified--in other words, so that any sample taken from this restricted population will

honestly represent that population. It is at this point that "hidden variables"

enter--variables whose effects we do not understand or which we simply have not

30
chosen to control. The values of these variables are at the whim of nature and

differ between what would otherwise be identical situations. The effects of these

differences produce the changes in observed statistics.

We are now at the third level of the statistical descr;iption, and evidently we

must speak of quanti1es of quantiles of quantiles.. This produces the phrase, "In

90% of like situations there will be at least 70!~ of the locations where the attenuation will not exceed 32.6 dB for at least 95% of the time."

In general terms such quanti1es would be represented as a function A{'tr,qL,qS)

of three fractions: 'tr' the fraction of time; qL' the fraction of locations; and

qs' the fraction of situations. The interpretation of this function follows the

same pattern as given above: In qs of like situations there wi1~ be at least qL of

the locations where the attenuation does not exceed A(~,qL,qS) for at least qT of

the time. Note that the inequalities implied by the words "at least" and "exceeds"

are important reminders that we are dealing here with cumulative distribution

functions. Note, too, that the order in which the three fractions are considered is

important. First, one chooses the situation, then the location, and finally the

time.

We recall that if a proposition is true with probability q then it is false

with probability 1-q. Working our way through all the inequalities involved, we may

also say: In l-qs of like situations there will be at least l-qL of the locations

where the attenuation does exceed A(qT,qL' qs) f,or at least 1-qT of the time. This

is the kind of phrase one uses when trying to avoid interference.

6.2 A Model of Variability

As complicated as it is, the three-fold description of quantiles does not

completely specify the statistics. At the first level when we are considering time

variability it is sufficient. But at the very next level we have failed to notice

that we are trying to characterize an entire function of quantile versus fraction

of time qT. To do this completely, we would need to consider all finite sequences

qT1' 'tr2' .•• of fractions of time and to examine the resulting observed quantiles

all at once as a multivariate probability distribution. At the third and final

level, matters become even worse.

Obviously this becomes too complicated for practical applications; nor would a

study following such lines be warranted by our present knowledge. But there are

engineering problems that arise which can be aided by a more complete description

of these statistics. Implicit within the ITS irregular terrain model is a second

model which concerns variability and which can tIe used to provide such a description. It is a relatively simple model using a combination of simple random variables

31
each of which depends on only one of the three different dimensions of variability.

While retaining the features described in the p:r;-evious paragraphs, it allows the

engineer to derive formulas for many needed statistics.

Experience shows us that when signal levels are expressed in decibel notation

the observed distributions tend to be normal or at least approximately normal. It

is from this fact that inspiration for the model is largely derived. The broad

statement of normality does, however, suffer from one important flaw which appears

when we discuss signal levels that exceed free space values. Such signal levels

are possible and are, indeed, observed; but their occurrence is rare and becomes

increasingly more rare as one considers ever higher levels. The distributions we

obtain must be truncated or heavily abbreviated at levels above free space.

As it happens, the terminology the ITS irregular terrain model uses to describe

the magnitude of variability differs in a slight way from that used above. As in

Rice et al. (1967, Annex V), the model considers the positive direction of a

deviation as an increase of signal level rather than of attenuation or loss. There

is, of course, no real significance to this convention, but the introduction of an

extra minus sign does tend to confuse our subsequent arguments. For this one

section, therefore, we shall adopt a different posture. Using lower-case letters

to refer to random variables, we suppose that the object of concern is the signal

level w which we measure in a decibel scale. We leave the precise definition of

this signal level deliberately vague, since it is immaterial here whether we speak

of power density, field strength, receiver power, or whatever. It would be related

to the attenuation a by the formula

w = W - a fs (3)

where Wfs' which is not a random variable, is the signal level that would be

obtained in free space.

The above change in convention requires a slight change in our definition of a

quantile. To retain the same relations as are used in practice, we now say it is

the value which is exceeded for the given fraction. For example, if w were a

simple random variable, we would define the quantile W(q) as being the value which

w exceeds with probability q. We should perhaps refer to this as a "complementary"

quantile, but instead we shall merely depend on the context to determine the implied

inequality. The rule to remember he:r;-e is that we assume the attitude of trying to

detect a wanted signal. It must be sufficiently large with a sufficiently high

probability.

32
Our model of variability is a mathematical representation of how one is to

view the received signal level as a random variable. First we assume the system

parameters, the environmental pa~~~ters, and the deployment parameters have been

fixed. From the set of all situations with thesc~ parameters, we choose at random a

particular one s. Then using that situation (which is, remember, a probability

measure) we choose at random a location and a time t. The triple (t,~,s) forms

our elementary event, and the corresponding received signal level w(t,~,s) becomes

a random variable. The model expresses this function of three variables in a more

explicit and manageable way. We first define a tentative value of the signal

level

w' (t,~, s) = Wo + YS (s) + 0L (s) YL Cit) + 0T (5) YT (t) (4)

where Wo is the overall median signal level; yS' YL' YT are thr~e random variables

called deviations; and 0L' 0T are another two random variables called multipliers.

The three deviations are measured in decibels and their median values are a dB.

The two multipliers are dimensionless, always positive, with medians equal to

unity. We now come to the important assumption that the five random variables here

are all mutually independent. This enables us to treat each of them separately and

then to combine them using standard probability theory.

The final step in our model is to write

w(t,~,s) = M(w' (t,~,s» (5)

where M is a modifying function which corrects values greater than the free space

value. For values of w' less than the free space value, we set M(w')=w'; but

otherwise M puts an upper limit on values Qr at any rate reduces them considerably.

As presently constituted, the ITS irregular terrain model cuts back the excess over

free space by approximately a factor of 10. Thus, if Wfs is the free space value

of received signal level, we have M(w') 0.9 Wfs + 0.1 w' when w' > Wfs·

The statistics of the three deviations and the two multipliers depend on the

system parameters, the environmental parameters, and the deployment parameters.

Except that the two multipliers must be positive, the five random variables are

approximately normally distributed. The deviations have standard deviations on the

order of 10 dB, while the multipliers have standard deviations equal to 0.3 or less.

The actual values have been derived from empirical evidence and engineering judgmente

Using this model we can, for example, recov'er the three-dimensional quantiles

discussed previously by following the prescribed procedure step by step. At the

33
first step we would assume there is a fixed situation and a fixed location at which

we observe the received signal level as a function of time. Now one very useful

property of quantiles has to do with the composition of random variables with monotonically increasing functions. If, say, u is a random variable with quantiles U(q)

and if F is a monotonically increasing function, then, as one can easily show, the

random variable F(u) has the quantiles F(U(q». Since 0T(s) is positive, the righthand side of (4) is a monotonically increasing function of YT' and therefore the

time variant quantiles are given by

(6)

where YT(qT) is the quantile of YT. At the next step we would have a fixed

situation and a fixed time variant quantile, and we would look at (6) as a function

of location alone. Again, if YL(qL) is the qL quantile of YL' we quickly find what

is now a twofold quantile

(7)

At the third step we must consider (7) as a random variable since the situation s is

now to be chosen at random. But here we have a new problem. The right-hand side of

(7) is the sum of a fixed number Wand three mutually independent random variables.

o

The statistics of W2must therefore be computed from the convolution of the corresponding three probability distributions. When this has been done, we would pick

off the desired quantile and finally come upon the threefold expression W' (qT,qL,qS).

In the last step, we recall that the modifying function M is monotonically increasing

and so

(8)

The only difficult part in this sequence of computations appears when we must

find the convolution required by (7). To do this the ITS irregular terrain model

uses an approximation sometimes called pseudo-convolution. This is a scheme described

by Rice et aL (1967) to treat several applications problems where the sum of independent random variables is concerned. For completeness and because it is useful in

many applications of the model, we pause here to provide our own description.

In the general case we would have two independent random variables u and v with

corresponding quantiles U(q), V(q). We then seek the quantiles W(q) of the sum

w=u+v. We first form the deviations from the medians which we recognize as having

quantiles

34
Yu(q) = U(q) - U(0.5)

YV(q) = V(q) - V(0.5)

and then we simply use a root-sum-square to derive

W(q) ;:::: U(0.5) + V(0.5)

(9)

+ sign(0.5-q) (10)

If u and v are normally distributed, this expression is exact. For other distributions we can only say that results are reasonable and that in our own experience

using distributions that arise naturally in the applications the expression is surprisingly accurate. It is, however, meant to be only an approximation and must

always be treated as such.

Note that the extension of (10) to more than two summands is straightforward.

The expression even shares with actual convolution the property of being associative

and commutative so that the order in which summands are combined is immaterial.

6.3 Reliability and Confidence

The use of the three-dimensional quantiles is perhaps best illustrated by its

application to the broadcast services. A broadcaster will need to provide an

adequate service to an adequate fraction of the locations at some given range. But

"adequate service" in turn implies an adequate signal level for an adequate fraction

of the time. For television channels 7 to 13, for example, in order to provide

Grade A service the broadcaster must deliver (O'Connor, 1968) a field strength 9 m

above the ground which exceedes 64 dB~ for more than 90% of the time, and that in

at least 70% of the locations. Spectrum managers and also the broadcast industry

will in turn want to assure that a sufficient fraction of the broadcasters can meet

their objectives. If we assume that each broadcaster operates in a separate

"situation," then this last fraction is simply a quantile of the situation variability.

For other services, however, it is often difficult to see how the threedimensional quantiles fit in, and indeed it is probably the case that they do not.

Consider again the broadcast service. A single broadcaster will want to know the

probability with which a given service range will be attained or exceeded. Since

"service range" involves specified quantiles of location and time, the probability

sought concerns situation variability and we return to three-dimensional statistics.

35
On the other hand, consider the same problem from the point of view of an individual

receiver. That individual will want to know only the probability at that one

location of receiving adequate service--that is, of receiving an adequate signal

level for an adequate fraction of the time. The distinction between location variability and situation variability will be of no concern and should not enter into

our considerations.

Using our model as in (4) and (5) we quickly note how we can accommodate a new

kind of analysis. We can suppose that first both the situation and the location

are chosen simultaneously and then, second, the time. The first choice will have

said that all four random variables in (4), excepting only YT' are to be treated at

once and are to be combined into a single deviation yS + c

L YL and a single multiplier cT' What we would have remaining is a twofold description of variability

involving time variability and a combined situation/location variability, and this

is precisely the description that the individual receiver of a broadcast station

would find useful.

To .continue our discussions, we find it convenient here to introduce the term

reliability. This is a quantile of that part of the variability which enters into

the notion of "adequate service." For the individual receiver of a broadcast

station, reliability is concerned with a fraction of time. For a broadcaster,

however, reliability must be expressed as a twofold quantile involving time and

location variability separately. For the remaining variability--always at a higher

level in the hierarchy--we use the term confidence; and we mean this term in the

sense that if one makes a large number of engineering decisions based on calculations that use the same confidence level, then, irrespective of what systems or

even what types of systems are involved, that same fraction of the decisions should

be correct--and, of course, the remainder should be incorrect. Reliability is a

measure of the variability that a radio system will observe during the course of

its deployment. Confidence will be moasureable only in the aggregate of a large

number of radio systems. Clearly, differentiation between the two will depend on

the point of view one takes. To a broadcaster, confidence will be a measure of the

situation variability; to an individual receiver of a broadcast station, it will be

a measure of a combined situation and location variability. But the spectrum

planner of the broadcast service will not speak of confidence at all; from that

point of view all of the variability is observable and is part of the system.

Remembering that we must retain the order in which the three kinds of variability appear, there are four different ways that one can treat them in combination,

all of which have legitimate uses in one kind of service or another. We call these

36
the four modes of variability, although they are really four different ways of

treating the subject of variability.

Two of these four modes we hav~ ~~+eady discussed. In the broadcast mode we

treat all three kinds of variability separately. The typical user of this mode

would be the broadcast~r for whom reliability would measure both location and time

variability and confidence would measure situation variability. In the individual

mode situation and location variability are combined so that there remain this combined variability and time variability. Here, the typical user would be the

individual receiver of a broadcast station for whom reliability means the time

availability, and confidence measures the combined situation/location variability.

It would also be legitimate to combine location and time variability. We call

the result the mobile mode, since to a mobile radio unit changes in location

translate into changes in time. The typical user of this mode would be a mobile

system employing a single base station. Reliability would refer to the combined

location/time variability; it would probably translate into fraction of attempts at

establishing communications. Confidence would be a measure of the situation

variability.

Finally, in the single message mode we combine at once all three of the kinds

of variability, thus obtaining the more usual sort of one-dimensional random variable. The statistics to be used here are much simpler than those we have been

discussing; but, we think, the useful applications are somewhat limited. One

application might be for a communications link that will be used but once. Examples

might include a disaster warning system or a radio link attached to a self-destructing device. The statistics involved would then be couched in terms of confidence

levels. A more important application, however,would be for a mobile-to-mobile

system where the two mobile units are to be deployed worldwide. The statistics

would translate into first-try success probabilities (Hagn, 1980) and thereby

become expressions of reliability.

6.4 Second Order Statistics

Until now we have been discussing only first order statistics--that is, the

statistics of received signal levels for a single path at a single time. But there

are many problems in which more needs to be known. These are problems that depend

on the relative signal levels on separate paths or at separate times. For example,

the problem of interference comes first to mind. Also, there is the question of

37
what happens on~uccessive hops in a chain of communication links, or how to treat

the connectivity of a network of repeaters such as has been suggested for military

use.

The resolution of such problems depends on second or higher order statistics

where one considers the joint probabilities of obtaining given signal levels over

two or more paths. The most common statistic employed here is the correlation

coefficient, but in the general case one might well be forced to use something more

complicated.

Unfortunately, almost nothing is known about the subject. There have been

studies concerning diversity systems in which correlation coefficients have been

found for the two time series obtained when two receiving antennas are separated by

only a few wavelengths or in frequency by only a small fraction of the carrier.

But when it is a matter of the local median levels, studies of their possible

relationships have been rare and inconclusive.

In attacking problems where higher order statistics are required, we seem

forced to devise ad hoc approaches. In our own work on interference, for example,

we have said that time and location variabilities are independent while situation

variabilities are completely dependent. In other words, we have returned to the

model in (4) and (5) and supposed two sets of these equations--one for the desired

link and one for the undesired link. Thus we find a grand total of ten random variables to consider. Now in each set of five we have assumed these to be mutually

independent; but one can still ask about correlations between terms of opposite

equations. Our assumption, based on very meager information, has been that terms

involving time and terms involving location are again mutually independent. On the

, other hand, we have argued that the situation involving the receiver is the same,

or approximately the same, whether one considers the desired or the undesired

transmitter. It would then follow, for example, that the two values of ys are

equal and therefore simply cancel out when one computes the desired-to-undesired

signal ratio. Clearly, these assumptions must be viewed suspiciously; they enjoy

only the benefit that they appear to give reasonable looking results.

7. SAMPLE PROBLEMS

In this section we give a few examples of how the ITS irregular terrain model

can be used to solve engineering problems. They have been selected with an eye

towards variety, and because of the opportunities they provide to illustrate different techniques. In none of them do we claim to have carried the solution to its

38
final completion. Our objective has been to set up the problem, to describe how it

relates to the model, and to provide only a few illustrative results.

In this section all calculations pertaining to radio propagation have been made

by computer using the subprograms LRPROP and AVAR as described and listed in Appendix A. No alterations, however slight, have been permitted. On the other hand, we

have not hesitated to prepare short applications programs which process input data

into a form that can be used by the model and which process the output of the model

in ways that satisfy the requirements of the problem at hand.

We have tried to introduce as little new terminology as possible; what does

appear we hope is standard and recognizable throughout the engineering profession.

7.1 The Operating Range of a Mobile-to-Mobile Syst~m

Our first problem is a simple one which we shall treat with immediately available tools. We study a particular kind of transceiver meant to operate from vehicles, and we ask to what range they should be able to communicate. More precisely,

remembering the many sources of variability within such a system, we must ask to

what range they should be able to communicate reliably. The answer, then, will

depend on what one will accept as adequate reliability.

On the other hand, we can also turn the problem around and ask, for any given

range, what the probability of communications is. Using the terminQ~Q9Y of Hagn

(1980), upon whose report much of the analysis of this section is ba~d, we speak of

"first-trY success probability"--the probability that a communications channel is

established, disregarding repeated attempts.

We assume that the system will operate worldwide and that the "first-tries" are

to be made in many areas. Thus, we assume it is the single message mode of variability that we must use to describe the statistics of propagation. Other parameters of the assumed system are given in Table 5. It is a low band FM system in

which both terminals use the same kind of equipment. The required signal-to-'noise

ratio is the predetection.value. To allow for tlle possibility of multipath fading,

we have introduced a safety margin of 6 dB. Assuming Rayleigh statistics, this

amount below the median will pick up about 85% of the points in any short run.

There are many other sources of variability in the system besides that due to

radio propagation. A particular transceiver will not have exactly the parameters

given in Table 1, neither as regards the transmitter output nor on the receiver side

as regards the sensitivity factors such as noise bandwidth. The antennas, particularly when one thinks of random orientations, will have variable ga~n&. Man-made

noise varies both in location and time.

39
Table 5. Design Parameters for a Symmetric Mobile-to-Mobile System

Frequency

Polarization

Antenna heights

Transmitter power

Antenna gains

Line losses

kTB (B=25 kHz)

Rural noise, above kTB

Required (rf) signal-to-noise ratio

Margin for mUltipa~h (Rayleigh) fading

Margin for uncertainties

45 MHz

Vertical

2 m

16 dBW

-3 dBi

1 dB

-160 dBW

21 dB

6 dB

6 dB

7 dB

In the report cited above, Hagn has carefully estimated the standard deviations

of all these factors and, along with the propagation variability, combined all

sources of variability into one, using the root-sum-square. Here, however, we have

adopted a simpler approach in which we have merely introduced an additional safety

margin. Although this approach is often used by designers, we should note how it

changes the proper interpretation of the results: For a large fraction of the transceivers deployed, the observed performance should exceed the predicted performance.

If we start with the transmitter power, add to it the two antenna gains, and

subtract the two line losses, we find that the power available to the receiver

equals 8 dBW less the losses in the propagation channel. For the other side of the

ledger, if we start with the noise power and add to it the required signal-to-noise

ratio and the two safety margins, we find satisfactory reception if the received

power exceeds -120 dBW. It follows that the system will tolerate a basic transmission loss of as much as 128 dB.

To calculate the propagation losses, we have turned to the applications program

QKAREA listed in Appendix B. This is a simple program which accepts as input the

parameters needed by the ITS irregular terrain model and then lists selected quantiles of basic transmission loss at selected distances. In Figure 5 we have reproduced the output from one such run. The input consisted of the parameters in Table

5 together with values prescribing average terrain characteristics, ground constants, , and climate. The principal feature in the figure is the table of quantiles versus

40
OPERA.TIONAL RANGr:S FOR A MOBILE-TO-MOBILE SVS~rEM

FREQUENCY

A.NTENN,; HEIGHTS

EF~CTlVE HEIGHTS

TERRAIN, DELTA H

45. MHZ

2.0 2.0. r1

2.0 2.0 M (SITING=O,O)

90. M.

POL=1, EPS=15., SGM= .005 81M

CLIM=5, NO=301., NS=101., K= 1.133

SINGLE-MESSAGE SERVTCE

ESTIMATED QUANTILES OF BASIC TRANSMISSION LOSS(DB)

DIST FREE WITH CONFIDENCE

KM SPA.CE 95.0 90.0 80.0 70.0 50.0 20.0 10.0

1.0 65.5 113.4 109.7 105.2 102.1 96.9 88.5 84.1

2.0 71.5 123.0 119.3 114.8 111.6 106.4 97.9 93.5

3.0 7r:...1 -128.8 125.1'\ 120.5 117 .1 112.0 103.6 99.1

4.0 77.6 133.1 -129.2 124.7 121.5 116.2 107.7 103.1

5.0 7C).5 116.5 132.6 -128.0 124.8 119.5 110.9 106.3

6.0 81. 1 139.3 135.4 130.8 127.6 122.2 113.6 109.0

7.0 82.4 141.7 137.<:) 133.2 -130.0 124.6 116.0 111.4

8.0 83.6 141.9 140.0 135.4 132.1 126.7 118.0 113.4

9.0 84.6 145.9 142.0 137.3 134.0 -128.6 119.9 115.3

10.0 85.5 147.7 143.7 139.1 135.8 130.3 121.6 116.9

15.0 89.0 151.4 149.4 144.7 141.4 135.9 127.1 122.3

20.0 91.5 157.2 1'13.2 148.4 145.0 139.5 -130.6 125.9

25.0 93.5 160.4 156.3 151.5 148.1 142.6 1'B.6 -128.8

30.0 95.1 163.2 159.1 154.3 150.9 145.3 136.3 131.4

Figure 5. Output from a run of QKAREA concerning a ~obi1e-to-mobi1e system.

The arrows point to intervals where the quanti1es become equal to

128 dB.

distance. The columns are headed by percentages referred to as "confidence levels."

For the present problem this is, as we have noted before, a misnomer and should be

replaced by "reliability" or "first-try success probability."

Finally, to find the operational ranges we simply read down each column to find

the first distance at which the quantile of basic transmission loss exceeds the

tolerable limit of 128 dB. Doing this and using simple linear interpolation on the

crucial interval, we find the values listed in Table 6. Note that this table

really lists the quantiles of operational range as though1:::.his latter were a random

variable, as indeed it is. If one requires high reliability, one must be content

41
Table 6. Operational Ranges Under Average Environmental Conditions

Reliability

Range

95%

2.9 km

90%

3.7 km

80%

5.0 km

70%

6.2 km

50%

8.7 km

20%

16 km

10%

24 km

with rather short ranges. On the other hand, there will always be a few locations in

a few areas of the world where a first-try succee~s even when the distance is greater

than 20 km.

Note that we have first talked about worldwide operations and next about

average environmental conditions. What we really mean, of course, is anywhere in

the world where conditions are average. For a more complete study of this system,

we could continue on to find operational ranges for other than average conditions--

for plains, hills, and mountains, for example, and for poor, average, and good

ground constants. Then we could display the results as though the operational

ranges were a function of the environmental conditions. Or we could package them

all together into one truly worldwide table of range quantiles. This last step,

however, would require a knowledge of the fraction of attempted communications to be

made under each set of conditions.

7.2 Optimum Television Station Separation

In this next sample problem we treat what might be called a system design. We

determine possible values for some of the parameters related to the placement of a

network of cochannel television stations. Our aim will be to maximize the area that

such a network will serve.

It seems clear that we are speaking here of an interference-limited service.

If we imagine any arrangement of transmitters, we could require that they all

operate at very low powers so that there is no interference. The service is then

noise limited. But we can then increase the area covered simply by increasing the

transmitter powers; and we can continue to do this until the stations begin to

interfere with each other. At this point it is useless to increase powers further

since not only do the desired signals increase but also the undesired signals

increase.

To fix our ideas we imagine a plane surface infinite in extent which we shall

often call "the country" and on which we shall situate the broadcast stations. We

42
assume this plane has homogeneous terrain characteristics and a uniform climate. To

maximize the coverage area, we use the rules of closest packing and assume the stations form a perfect triangular grid and the transmitters all have identical characteristics--that is, that they are all at the same height above the terrain and

that they radiate the same power levels from omnidirectional antennas.

For design purposes the united States is divided into three "zones." Zone 1

consists of the urban northeast extending west to include Illinois; Zone 3 includes

a narrow region surrounding the ·Gulf of Mexico; and Zone 2 includes everywhere else.

Using Zone 2 as a comparison, stations in Zone 1 are presently packed closer together

using lower transmitter heights, while stations i.n Zone 3 are placed further apart

in anticipation of better propagation conditions and higher interference fields.

Also for design purposes one speaks of Grade A service and Grade B service, a distinction that refers not so much to the quality of reception as it does to the

quality and mobility of the assumed receiving systems. Grade A service pictures an

urban environment with relatively inexpensive receiving systems and with relatively

tight constraints on where the receiving antennas are located.

For our sample problem we have chosen to examine a grid of Channel 10 stations

and to require Grade A service. Channel 10 is in the center of the so-called high

VHF band (Channels 7-13), and Grade A service makes the problem a little more interesting. Following O'Connor (1968), we have listed the design parameters in Table 7.

'These parameters are concerned mostly with the assumed receiving system, but they

also include required interference ratios, required reliabilities, and for later

comparison with the values we shall obtain, the transmitter parameters presently

prescribed for Zone 1.

Adequate service to one location is defined to mean a satisfactory signal for

at least 90% of the time. Grade A service then requires adequate service to at

least 70% of the locations. In contrast, Grade B service requires the same adequate

service to only 50% of the locations.

When we examine the required desired-to-undesired ratios, we find that a new

complication is introduced. In television, co-channel stations actually operate on

three different frequencies, since by doing so the required interference ratios can

be drastically reduced. This has to do with synchronization of the horizontal

sweep, which is the first thing to become affectE~d when interference enters. The

three frequencies consist of a nominal frequency and two others, 10 kHz above and

below the nominal. Precision required is 1 kHz. If two stations are separated in

frequency by either 10 or 20 kHz, they are said t:o operate on offset frequencies;

43
Table 7. Design Parameters for a Grid of Channel 10

Television Stations

Frequency 193 MHz

Polarization Horizontal

Receiver Antenna Height

Receiver Antenna Gain*

Receiver Line Losses*

kTB (B=4 MHz)

Urban noise, above kTB*

Required signal-to-noise ratio

Required Diu ratios:

Offset frequencies

Non-offset frequencies

Required reliability*

Transmitter power, EIRP**

Transmitter antenna height**

Service range**

Station separation**

*These entries define Grade A service.

9 m

2 dBi

2 dB

-138 dBW

19 dB

30 dB

28 dB

4'5 dB

70% locations, 90% time

57 dBW

305 m

64 km

274 km

**These entries are the maximum values presently used in Zone 1.

The service range is a calculated value for Grade A service

assuming 50% confidence and no interference.

otherwise they are non-offset. As Table 7 shows, the difference in required desiredto-undesired ratios is a full 17 dB.

To take advantage of this feature, we should arrange to have all neighboring

stations on offset frequencies. We superimpose on our triangular grid what is

essentially a three-channel network in the manner portrayed in Figure 6. In that

figure the zero, plus, and minus signs indicate the positions of stations operating

on the normal, high, and low frequencies, respectively. Let s be the "separation

distance"--the distance between adjacent nodes of the grid. Then we note from

Figure 6 that each station is surrounded by a circle of six offset stations at the

distance s and by a second circle of six non-offset stations at the distance ,/3 'so

We have succeeded in removing the most grievous interference problems to a further

distance.

44
o + o +

+

+

+ +

+

+

o + I] +

Figure 6. A triangular grid of cochannel television stations showing the

arrangement of'thethree'offset frequencies.

At anyone location a receiver will attempt to view a desired station while

being subject at the same time to. interference from all the other co-channel stations. That there are more than One undesired station is called the "multiple

interference problem." Presumably the signals from the several undesired stations

will add together incoherently to fOrm a total interfering signal; the total power

will be the sum of the individual powers. But when two signal levels differ by only

a few decibels the corresponding powers dif.fer by a great deal and the smaller

contributes very little to the sum. We shall therefore assume that of the many

interfering signals only the strongest is of concern. Going a step further, we

shall assume that the desired station is the nearest and that the one undesired

station of concern is the next nearest.

Suppose a receiver location is on the direct line joining the desired station

with an adjacent offset station. If itis·distant x from the desired station, it

will be distant s-x from the undesired station. We can then compute the desired-toundesired ratio R(x); or more precisely we can compute the necessary quantile of

45
this ratio. For small x we expect R to be very large; as x increases, R will

decrease monotonically, reaching very large negative numbers as x approaches s.

Indeed, since we are using quantiles on the high side and thereby favoring the undesired signal, R will reach zero when x is somewhat short of the midpoint at s/2. If

R

o

is the required desired-to-undesired ratio, there will be a distance r at which

R(r) = R

o (11)

Since R is positive, r will be less than s/2. For distances greater than r, the

o

ratio R(x) will be less than the required ratio, and interference will be intolerable. But for distances less than r, the observed ratio exceeds the required ratio

and interference is tolerable. We say that r is the "interference-free range"

although, since we are not really free of interference, the term is somewhat of a

misnomer.

If we consider another radial leading out from our desired station, then the

interference-free range along that radial will be larger than r since the distance

to the undesired station is somewhat larger. However, we must not forget that there

are six undesired stations surrounding the desired station at a distance s. When

the bearing of the receiver location changes sufficiently it will be subject to

interference from another one of these six. In consequence there will be a region

surrounding the desired station which we may call the interference-free service area

of that station. It will have hexagonal symmetry and an inscribed circle of radius

r.

Generally, this region will look very much like a regular hexagon with, howeVer, convex, curvilinear sides. But note that at the vertex of this region there

will be two equidistant undesired stations. One may well imagine, therefore, that

this is a real case of multiple interference, that the true interference-free range

will be somewhat less than indicated by the position of the corner, and that we

should suppose these corners somewhat rounded off. Without entering into too many

details, we may then suppose that the actual interference-free service area is

precisely a circle of radius r.

We must also consider the effects of interference from the six nearest nonoffset stations. Ignoring for the moment the offset stations, we find again a

circle within which service will suffer only tolerable interference. Thus for any

given separation distances we find two service ranges, one concerned with the offset

stations and one with the non-offset stations. We shall assume that the interference-free range r which is to be the primary result of our calculations is equal to

46
the smaller of the two. Thus in our final picture surroundin~ each node of our

trian~ular ~rid there is a circle within which one can obtain satisfactory service.

The area of a single one of the triangles that form the grid is (13/4}s2 while

the area of that triangle which is covered by an interference-free signal from one

of its three vertices is (~/2)r2. Thus the fraction of a single triangle (and hence

of the entire country) that is served is the number

p = (2~/13) (rls) 2 (l2)

It is then our aim to find the optimum value of s--the value of s that maximizes

this fraction.

We must. hasten to note that this is a narrow use of the term- "optimum." While

it does seem to be a natural definition, in actual practice there may well be other

influences that should be considered. For example, the resulting ranges may be too

small to make an individual station economically viable~ or the ranges may be too

large so that overcoming noise requires more transmitter power than is reasonable.

Of course, the most severe criticism of our approach is that we seem to be covering

land whereas we should be trying to reach people. It is one of our implicit assumptions that the two problems are equivalent, or, at least, that the solution of one

provides useful information for the solution of the other.

It is interesting to see what happens to our problem if we use a simple model

of radio propagation. For example, let us assume that free-space calculations suffice. For the offset stations we find

R(x) = 20 log [ (s-x) Ix]

and setting R(r}=R and solving for r we obtain

o

-1

rls = (1 + 10Ro/20)

(l3)

(14)

Similarly, ifRl is the required desired-to-undesired ratio for the non-offset stations, we find

US}

In both cases the ratio rls, and hence also the fraction p, is independent of the

separation distance s. There is no optimum value, and from this point of view it

makes no difference how far apart the stations are placed. Using values from Table

7, we find that the non-offset stations are overpowering and that the final fraction

47
of interference-free service coverage is a paltry 0.03%. Clearly, the bulge of the

earth plays a very important role in allowing television service at all.

Turning to the ITS irregular terrain model, a particular question will be how

to treat properly the statistics involved. Clearly, we require something very like

the broadcast mode of variability modified, however, by the need to find quanti1es

of a ratio of two signals.

As we have noted in Section 6.4, it will be our assumption that location and

time variabilities of the two signals are statistically independent, but that situation variabilities are exactly correlated. Now to each service area there presumably

corresponds a separate situation and therefore, one would suppose, a separate interference-free range--the range is a random variable. But because the situation

variabilities are exactly correlated, they have a strong tendency to cancel against

each other in the ratio; consequehtly the range should exhibit only a very small

variability. In any case, however, since we seek the fraction of the country

covered, it is really the average service area we need. This average we may approximate with the median, and then it will follow that we require the medians of the

situation variabilities.

Going immediately to median values for the terms involving situation variability,

we find from (3) and (4) that at a particular receiver location !/, and a particular

time t the desired-to-undesired ratio becomes

R(X) = 20 log[(s-x)/x] - AOD + AoU

(16)

where the additional subscripts D and U refer to the desired and undesired transmitters, respectively, and where the A 's are overall medians of attenuation.

o

Remembering how the required reliability is stated, we first note that satisfactory

service is achieved at the location !/, provided R exceeds the required desired-toundesired ratio R for at least 90% of the time. In other words, we must first

o

compute the 0.9 quantile of R for each fixed location. As one sees, this may be

reduced to the simple problem of finding the quantile YTR(.9) for the difference

between the two independent random variables, YTD and YTU. At the next step we must

ask whether this 0.9 quantile exceeds R at a sufficient number of locations. We

a

must therefore compute the 0.7 quantile of the time-v~riant quantile. Again, this

reduces to the problem of finding the quantile YLR (·7)

the two independent random variables, YLD and YLU· In

quantile of R(x) required by the problem is given by

48

for the difference between

the end we find that the
R(x) = 20 10gI(s-x)!x] - A + A 00 oU

(17)

To compute the two quantiles of deviations, we use the method of pseudo-convolution to obtain

(18)

where the additional Y's are the indicated quantiles of the corresponding random

variables. Note that both quantiles of the ratio are negative since both refer to

fractions greater than 0.5. Note also that for the undesired station quantiles we

use the complementary quantile. This is because to form the convolution we must add

the two random variables Yo and -yu; and if Yu has the quantiles Yu(q) then -yu has

the quantiles ~Yu(l-q).

There remains the problem of how to obtain the four quantiles of individual

deviations. We suppose we have available to us the threefold quantiles of attenuation A(~,qL,qS) as described in Section 6, and from these we shall obtain the

required deviations. Always putting qs=0.5, we may write

YTO (·9) = (.9, .7, .5) - ~(.5, .7, .5) (19)

YTU(·l) = ~(.l, .3, .5) - ~(.5, .3, .5)

Y

LO (.7) = ~( .5, .7, .5) - ~(.5, .5, .5)

'¥LU(·3) = ~(.5, .3, .5) - ~(.5, .5, .5)

Where, we note, the and are different functions since they refer to different

distances.

At this point we have gathered together the parameters and the formulas needed.

It remains to make the calculations. To do this we have assembled a short applications program that considers a s~quence of proposed station separations s, computes

desired-to-undesired ratios, solves (11) for the interference-free range r, and

prints it out together with other pertinent data.

The results of one run of this program are shown in Figure 7. To emulate conditions in Zone 1, we have assumed transmitter heights of 300 m, hilly terrain with

Ah=90 m, and a continental temperate climate with N =301 N-units. The curve in

s

49
100 150 200 250 300 350

SIslion S~psrslion. Ki/~/~rs

50

./

/'

---

I

/

./

v

o

o

....

c:

10

\"

l.

30

Figure 7. Fraction of the country rece~v~ng an interference-free signal

versus the station separation. We have assumed transmitting

antennas 300 m high and average terrain characteristics.

Figure 7 shows the fraction p as a function of the separation distance s. The maximum appears when s=2l0 km at which separation distance the interference-free range

is 48.6 km, implying that 19.5% of the country is covered with an interference-free

signal. Note that the curve has two corners, one at the optimum distance of 210 km

and another at about 145 km. They appear because for distances between them it is

the non-offset stations that determine the interference-free range whereas for other

distances the range is determined by the closer offset stations. At the optimum

distance, therefore, both offset and non-offset stations are contributing equally to

the interference. It may also be interesting to note that for a separation distance

of 274 km, which is the distance presently invoked for Zone 1, the interference-free

range is 56.5 km and 15.4% of the country is covered.

With the interference-free range determined, we can also find the transmitter

power required to overcome noise. Using the receiver characteristics listed in

Table 7, and now requiring 90% confidence, we find that for the conditions that

obtain at the optimum spacing, we need 53.3 dBW EIRP.

If we were to continue with this problem, we would eValuate these same quantities for varying terrain types, climates, transmitting antenna heights, and

50
frequencies (television channels). The way in which the optimum separation distance

varies might then lead to a second phase of the problem which attempts a broader

look at how best to provide television service to the country.

7.3 comparison with Data

As a final example, we want to show a comparison of predictions from the ITS

area model with a single set of measured data. This will give us the opportunity to

display some of the more obscure aspects of the model and, perhaps, to demonstrate

the terrible intransigence of measured data. It should be emphasized at the outset

that no single set of data can claim to "verify" the model; that requires a large

number of sets of data. Our purpose here can only be to show some of the techniques

that might be used in an extensive study.

The set of measured data we have selected is that reported by McQuate et ale

(1970). It is familiarly known as the R3 data si.nce it_involved the third receiving

site in a sequence of special measurement programs. We have chosen this set on a

whim, by which we hope to mean "at random." It i.s immediately available to us;

there is no previously published account of its analysis; and it serves to illustrate several problems.

In the measurement program six frequencies were used ranging from 200 MHz to

9 GHz. The transmitter was mobile; it would go t:o a preselected point, set up

there, and begin operations. The receiving antennas were mounted on a carriage

which could be continuously moved up and down a 15 m tower; received signal levels

were recorded as a function of height.

For this particular set the receiver tower was erected atop North Table Mountain in central Colorado (to be .precise, at 39"47'30" N., 105"11'59" W.); it was at

the edge of a cliff from where it could look out to the north and east across the

plains. The experiment was specifically designed to simulate a low-flying aircraft

above the plains, and so the transmitter sites were all in this sector. The plan

for choosing these sites involved taking a map of the region, drawing on it circles

of convenient radii (5, 10, 20 km, etc.) with centers at the receiver site, and

finding where these circles crossed convenient roads. Except for the necessity of

keeping to roads, this procedure appears to provide a suitably random selection. We

shall have more to say about this below.

Again at whim--and perhaps because the data might also be construed to simulate

a UHF television transmitter--we have chosen to use the data for 410 MHz. This

leaves us with a specific one of the receiver heights to choose. If we examine the

published curves, we note that for increasing receiver height the signal levels tend

51
to start out with rather low values, increase dramatically in the first several

meters, and then show a moderate, but not excessive, amount of lobing. Now actually,

in order to allow for sufficient guying, the tower was positioned some 8 m back

from the edge of the cliff. It is therefore not surprising that at the lower

receiver heights we observe ground effects which rapidly disappear at higher heights.

The lobing pattern we would attribute to a scattering of energy from the cliff's

edge.

In keeping with the purpose of the experiment, we should therefore choose one

of the higher receiver heights. And in keeping with the spirit of the kind of radio

propagation model we are considering where mUltipath effects are kept to a minimum,

we should choose not a specific height but a smoothed-out average over a range of

heights. What we have done is to use signal levels from heights of 10, 11, 12, 13,

and 14 m; to find their median value; and to ascribe that value to a height of 12 m

above the cliff top.

The resulting data for the 44 different paths on which these measurements were

taken are displayed in Figure 8 where we have plotted the observed attenuation

relative to free space versus distance. A glance at this plot can only bring dismay. There seems to be no structure at all to this set. Over half of the points

are clustered around free space, and if there is any trend it would almost seem

o

.

'..

... 20

...

...

,,8 ~Il 0 0 , (l -' , 0 ..

0

0

( 0

0

nO 0

0

0

, c

(I

Q 0

40

o 20 40 60

Ois/.nc•• Kilam.I.rs

80 100

Figure 8. The R3 data at 410 MHz; 44 points.

52
that attenuation decreases with distance. Our ~irst inclination is to abandon this

set and to look elsewhere. But remembering our aim is merely to provide an illustrative example, we continue. If one is truly interested in sets of data concerning

air-to-ground or UHF television propagation, summaries of others may be found in

Johnson and Gierhart (1979) or in Damelin et al. (1966).

In contrast to the previous examples, here we are concerned with a very particular region of the world. We must therefore determine parameter values for the

model from that particular region, and, we should emphasize, from that particular

situation.

We would assume that the complete set of data represents a sample drawn from

one single situation--although the wide range of distances might argue for a different conclusion. As for the kind of statistics involved, we recall that the

measurements are spot measurements; that is, the transmitter was dispatched to a

random location and at some random time the measurements were r-ecorded. Thus time

variability is inextricably entangled with the more naturally expected location

variability. OUr treatment of statistics should therefore be in terms of the

mobile mode of variability. We use the notation A(qL,qC) to indicate the quantile

of attenuation which, with confidence qc' is not exceeded for at least qL of the

observations. The subscript L serves to remind us that in reality time variability

is small at the short distances we shall consider, and the observed variability will

be dominated by location variability. We refer to confidence rather than "fraction

of like situations" since we have no set of like situations, nor do we ever expect

to obtain one. We can only suppose if we make a large number of (independent)

statements with a given confidence qc' then qc of them will be correct irrespective

of whether the situations involved are like or unlike.

To decide on the proper environmental parameters, our first thought might be to

use Figures 2 and 3 together with the coordinates given above. Using digitized

ve~~tons' of these maps, we find values ~h=444 m, N =236 N-units, and also an average s

terrain elevation of 2120 m. But for the situation here these values are wrong.

Directly west of North Table Mountain and directly west of the sector in which the

measurements were made, there rise abruptly the foothills of the Front Range of the

Rocky Mountains. The mountains of this range are majestic, being among the highest

in the contiguous United States; and the foothills provide spectacular changes in

elevation. Now the values given above assume that it is just as likely that a path

goes west into these ~oothills as that it goes east into the Colorado plains, and

thus they reflect the properties of the foothills to a consi.de:t'able extent. The

;paths in whi.ch we are interested, however, stay entirely clear of the foothills and

exhibit quite different characteristics.

53
In the original data report quoted above, plots are provided of the terrain

profiles for each measurement path. From these plots, or digitized versions of

them, we can derive directly the parameters we need. For example, a short study of

the plots shows an average terrain elevation of about 1700 m. From Figure 2 we find

N =300 N-units and hence from (2) we derive N =250 N-units.

o s

It is clear that the cliff and the mountain below must be included as part of

the antenna structure; this is particularly so when we remember the primary purpose

of the experiment. We must therefore estimate the height of the mountain above its

base as one important parameter. In addition we also need the terrain irregularity

parameter flh.

If we are given a terrain profile leading away from the receiver tower, we can

do two things: We can compute the (asymptotic) flh for that path, and we can find a

linear least squares fit to the profile (indeed, that may have been part of the computation for flh), extend that back beneath the receiver tower, and so find a value

for the height of the cliff top above that profile. In the computations here, we

should be careful not to use the cliff as part of the profile; we are assuming that

it is part of the antenna "tower." And furthermore, at the bottom of the cliff

there is a steep talus slope that, in some directions, may extend outwards nearly 1

km. This, too, should not be considered part of the profile. Our computations

should use only a portion of the profile, and that portion should begin at least

1 km away from the receiver site.

Using the path profiles of the experiment, we have made such calculations.

Restricting ourselves to the 34 paths that are nominally 10 km or more in length, we

have found that the individual values of the asymptotic flh vary from 31 to 212 m

with a median of 126 m. It is this latter that we would propose to use in predictions. Similarly, we have found that the top of the cliff lies between 156 and

390 m above the profiles and,that the median value is 263 m. Adding the 12 m tower

height, we would propose 275 m for the structural height of the receiver.

The calculations on each profile were made by computer using subroutines

related to the ITS model in the point-to-point mode and listed in Appendix A. It

may seem odd that we use point-to-point techniques to treat an area prediction; but

then the determination of the environmental parameters of a specific area is clearly

related to the determination of these parameters on individual paths. Still more

odd, however, is the fact that we have used precisely thoSe paths on which the measurements were made. To justify this we would argue (l) that the profiles are conveniently available, having been determined as part of the measurement program, and

(2) that there seems no better way to assure that we are making our determinations

54
from the same "situation" as that from which the measurements came. An alternative

is possible and would have been forced on us if 'we had been making predictions prior

to the measurement program. We simply proceed as before, choosing, however, our own

set of paths. While this set is arbitrary, it should imitate as closely as possible

the situation of the measurement program. We would suppose a sequence of radials

beginning at the receiving site and extending into the sector of interest. And

finally we would use terrain profiles for that portion of those radials that extend

from 1 km to perhaps 40 or 50 km.

For the final parameters, we note that the transmissions were horizontally

polarized, and we would assume average ground and a continental temperate climate.

At the high frequency and the short distances involved, these assumptions are not

critical.

In Figure 9 we have replotted the data of Figure 8 and superimposed on them

predictions from the prediction model using the parameters given above. To be

precise, we have plotted five quantiles of the expected attenuation as functions of

distance. The central solid curve is the overall median prediction--the "best

estimate." For the other four we can interpret them as meaning that with 90% confidence at most 10% of the observations will lie above the upper dotted curve and at

...... . . . . . .. .. . .. . ---f--- . . . . . nB --- ..... 0 .. -.... +- il 0 0 ~~ " 0 u

0

0

0 ( '~ .00 ,

0

, "- ""b- , 0 '/1 (

, , ·1, ·1).

( ~, ", , ........... ,

0 ~, "''''''',,- '-1( .........

" 'S' 4( , ./) '-1 '$ ,

' .... . ('$ "$)

4(,.9 '(" '''-.... .............

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

I' '.9) ... "

"', .........

c:

C)

"

....

c:

CII

....

....

'o:t

o

20

40

o 20 40 60

OislsnC9. KilQm919rs

80 100

Figure 9. Predicted and observed values of attenuation for the R3 data.

Assumed parameters: f=4l0 MHz" hgl=275 m, hg2=6. g m, t>h=l26 m,

Ns=250 N-units.

55
most 10% will lie below the lower dotted curve; and with the s~e confidence at most

50% of the observations will lie above the upper dashed curve and at most 50% below

the lower dashed curve. Indeed, of the 44 observations available to us, we find

that 2 lie above the upper dotted curve and none below the lower dotted curve; that

14 lie above the upper dashed curve and 5 below the lower dashed curve. Note that

for distances greater than 60 km, the difference between the two dotted curves is

about 45 dB. According to the ITS irregular terrain model, if we want to consider

an interval within which we are fairly confident that a large majority of observed

signal levels will lie, then that interval must be very wide.

One notable aspect of Figure 9 is the tendency of both predictions and observations to level out at free space values where the attenuation vanishes. In the

case of the predictions, this tendency demonstrates both the "free space" region

where the reference attenuation vanishes and the effects of the "modifying" function

described in Section 6. The observations, we note, may take on rather large positive values of attenuation but will form tight clusters about zero. Of course, we

also note that at large distances the observations still show a stubborn tendency

towards free space values that the predictions fail to reproduce.

The wide scatter of attenuation values that one observes in Figure 8 exhibits

the "observational variability"--which we are here interpreting as a combined location and time variability. Although it may not be apparent from Figure 8, in

principle the statistics of this observational variability should depend on the

parameters of the measurements and, in particular, on the distance. Following this

stricture we may look at observational variability at particular distances by

plotting, as we have done in Figure 10, the cumulative distribution functions of

attenuation. In Figure 10(a) are the results for a distance of 20 km. The horizontal axis is a normal probability scale so that straight lines on the graph represent

normal distributions. The solid curve is the "best estimate" distribution function

as obtained from the model, and the two dashed curves give 10% and 90% confidence

ranges. The dotted curve is the sample cumulative distribution function using the

10 measurements made on paths with distances of approximately 20 km. This figure

demonstrates again how both predictions and data level off at free space values.

Note also how the slope of the sample distribution function at the higher quantiles

agrees with that of the predictions. Figure 10(b) gives similar results for a

distance of 50 km. The wide disparity here between predictions and observations

restates the tendency mentioned above for the observed data to attain free space

values even at the longer distances.

56
I-~------- -_._- --- ------ -- -- --- --- - -1-- ..... .. . . ............ ....... '. ....... . t"-.. ....... ,

'. . 9C~O./ ....... , .. ....... . .. r-..... .......

....... .. ... ....... ....... ...... t"-..9C~O ......... .$

.. ..... ' ..............

'-..... . ............... , 9C~O.!} ....... -,

,

o

.

c:

a....

...! 20

c:

...•

...

40

I 10 50

P.r C.nl Obs.rv.' ions

90

J

99

Ca) Distance, about 20 km; 10 points.

I-- ~---- --....... --_

..... :-: ~- .......

....... .. , ........ , .... ....... , ........ , .- "'" .... ....

~~ ........ - . . .

' .....

.. . .. .... .... .... .... , .......

""" , .... , ............. ,

,

, 9C~O./ .... 9C~O .... ....... , '$ .......

~~.!}

I' .... ,

o

c:

a

'..

...

• 20

:;)

c:

CII

...

...

40

I 10 50

P9r C9nl ObS9rvslions

90 99

(b) Distance, about 50 km; 11 points.

Figure 10. Predicted and observed curves of observational variability

for the R3 data.

57
Let us examine this dependence on distance in a little more detail. In developing a propagation model, one technique often employed is to group the data into

successive intervals of values of a parameter one wants to examine--in the present

case, distance. If enough data are available, the use of overlapping intervals is

particularly desirable. Then one finds median values in each group, plots these

medians against the corresponding parameter, and tries to construct a curve that

passes adequately near those medians.

There are too few data in our present set to allow us to make any very profound

discoveries. Indeed, this is a paradox common to most measurement programs. There

are several thousand data exhibited in the original report. By restricting ourselves to a single frequency and a single receiver height, we have suddenly reduced

this number to a mere 44; and if we further restrict ourselves to a small range of

distances, we find very few data remaining.

All of which brings us to the question of sampling error. Presumably a set of

measurements represents a limited sample drawn from the population that comprises a

situation. Any statistics we compute from this sample, such as the cumulative distributions of Figure 10 or the medians we want to consider, are random variables

subject to the laws of probability. Any value we obtain is no more "correct" than

is the face after a single coin toss the correct face. If it is at all possible,

when we report a sample statistic, we should also estimate its probable error; that

is, we should provide some indication as to how far from the corresponding population statistic it might reasonably be, simply because it was estimated from a

limited sample. Of course, as the sample size increases the probable error here

should decrease, tending to zero. Nevertheless, an indication of its magnitude is

quite valuable.

As an example of how such estimates might be made, consider the sample median.

If the sample size n is large enough and if the sample comes from a normally distributed population, then the standard deviation of the sample median is approximately

where cr is the standard deviation of the population. This represents the probable

error of the value we obtain and is the estimate we might provide. The formula

comes from the theory of large samples, but we do not require here very great

accuracy and the formula is probably adequate even for fairly small sized samples.

To particularize, the observational variability predicted by the ITS irregular terrain model for anyone distance has a standard deviation of about 10 dB; so if we

58
have a sample of 10 data then the standard deviation of the sample median will be

approximately 4 dB; the 10% and 90% confidence levels for the population median will

differ from the sample median by about 5 dB. To put this another way, we would then

have only 80% confidence that the population median lies within a 10 dB range.

In Figure 11 we have plotted medians for the few groups of distances available

to us. The vertical bars are drawn at the median distances for each group; the

ticks across them indicate the sample medians of attenuation; and their end points

determine an approximately 10% to 90% confidence interval for the population medians.

In deciding on these confidence intervals, we have not used the large sample theory

described above; instead we have employed thetmore robust scheme given by Walsh

(1962; ch. 6). This scheme depends only on the sample values and is nonparametric

in that it assumes very few properties of the population(s) from which the data are

drawn. It provides a discrete sequence of exact confidence levels from which we

have picked the one lying closest to 10% or 90%. Above each bar in Figure 11 we

have indicated what the exact level is. For example, at 20 km 6% of such samples

will be drawn from populations whose medians lie above the bar and a like number

from populations whose medians lie below the bar. The bar itself thus comprises an

88% confidence interval.

100

.........

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

80

11 % 6%

qO 60

DislsnC9. Kilom919rs

-

20

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

t-- ......_...... ---",O..-- ~---4r.$

4( ~'I)

'$ ...... ...... '$) ................. ........

AO .........----l"-------L----'---....L..__....I-__ __..L.....::~_L__ ___I.:::::::.._ ___J

o

0

11\

....

QI

.Q.,

\J

QI

c:

()

....

QI 20

;:)

c:

QI

....

....

":t

Figure 11. Predicted and observed medians for the R3 data. The bars indicate

confidence levels for the sample medians at approximately 10% and

90%.

59
Note that in Figure 11 the term "confidence ll is being u~ed to describe two

quite different quantities. The vertical bars indicate confidence intervals pertaining to sampling error arising from a limited sample and induced, presumably, by

observational variability at the given distance. On the other hand, the curves

which also form a part of Figure 11 delineate a confidence interval which is a

prediction of the model and pertains to situation variability. Nevertheless, both

confidence intervals refer to the same value--the population median. For example,

let us consider the four measurements taken at almost 80 km. As indicated by the

vertical bar we have 94% confidence that the sample was drawn from a population

whose median attenuation was less than about 14 dB. On the other hand, we predict

from the model that with 90% confidence the same population should have a median

attenuation greater than about 25 dB. While these two statements are not entirely

contradictory, the disparity is certainly considerable~'

Very often a comparison, such as we are making here, between model predictions

and measured data, is as much a critique of the data as it is of the model. Consider again the four measurements taken at nearly 80 km. As one can tell from the

abrupt change in slope of the predicted median attenuation, the smooth earth horizon

distance implied by our parameters is about 76 km. If we account also for an irregular earth, paths at this distance should mostly be well beyond line of sight. And

yet, looking at the corresponding profiles in the original data report; we find that

two of the four are clear line-of-sight paths while the other two are just barely

obstructed. The hills, instead of obstructing these paths, seem to have provided

platforms which elevate the transmitter above the terrain. Two conjectures come to

mind: (1) there is some phenomenon of nature in which hills do indeed tend to

elevate more terminals than they obstruct, and that the ITS irregular terrain model

fails to recognize this; and (2) for some reason we shall never know there was a

tendency in the experiment to pick transmitter sites with favorable positions; in

other words, these sites were not, after all, chosen "at random."

For the second conjecture we can devise a test of sorts to examine whether the

paths do show a bias. We look at the immediate foreground of the transmitter in the

direction of the receiver and ask what the slope of the terrain is. If the sites

are chosen at random, it would seem reasonable to expect that the probability with

which the ground slopes down towards the receiver is the same as the probability

with which it slopes up; we would also expect there will be cases in which the

transmitter is at the top of a hill and some in which it is at the bottom of a

depression. The test, then, consists of comparing the number of paths which slope

down with the number which slope up; of course, there will be sampling error so that

60
we would not expect these numbers to be equal, but we would expect them to be reasonably close. While the test is obviously inspired by the theory of radio propagation, note that it is purely geometric and in itself is entirely independent of

what use is to be made of the paths involved.

To carry out such a test for the R3 data, we have simply examined the published

terrain profiles and tried in that way to categorize each transmitter site. For the

four paths nearly 80 km long we found three of them slope down towards the receiver

and the fourth is at the top of a hill. For the 11 paths at approximately 50 km six

slope down, one slopes up, two are at the top of hills, one is at the bottom of a

depression, and the last is located in a long level stretch that we cannot classify.

For all 34 paths whose lengths are nominally 10 km or more, 12 slope down, 3 slope

up, 3 are at tops of hills, 2 at bottoms of depressions, and the remaining 14 are on

long level stretches. In actual practice we discovered that these classifications

are somewhat subjective, and so we would not give too much credence to the exact

results. Nevertheless, there is a stron9 indication here that the paths are indeed

biased towards high received signal levels.

Perhaps, then, our choice of parameters is wrong in that we have assumed random

siting for the transmitter. Going to the other extreme, we have redrawn Figure 9

using, for the predicted quantiles, all the same parameters except that we have

assumed the transmitter is very carefully sited. The results are shown in Figure 12 •

..... ... ... ...... ------- ...... •..•. b --- ....... ,,8 --.. ill ... .. 0: • "'lI !' - 0 ..... ....... o' . 0 0 .......... 'jj(

0 ....... . I•• I)' .. 11'0 .......

- ....... 0

- ....... 0 ....... 0 ........ ....... 4 ....... ....... c ....... ....... ....... ....... .... r-.... .... 4l 0 ......... ............. jj( •.5' ...... ..... . ....... ....... r-....jj( './) . ....... -. . ....... '.5' ....... jj( •.5' • •.5')........... . • .!J) . to ....... .

jj(.!J ....... .......

• .!J). -....... I •

-

o

.

c:

c:

'..

.... ! 20

c:

....•

....

40

o 20 40 60

Disf.nc•• Kilam.f.rs

80 100

Figure 12. Predicted and observed values of attenuation versus distance for

the R3 data. The predictions assumed the transmitters were sited

very carefully.

61
Note that the discrepancy between predictions and observations has decreased, but

not dramatically so. In any case, this is "post-act" analysis, carried out after we

have discovered a discrepancy. It should be avoided in any serious statistical

study.

For many, the only interesting question to ask in a comparison of measurements

with predictions is the question of how the totality of measurements compares with

the "best estimate" (Le., median) prediction. The idea is to compute the deviations

y = A(.5,.5) - A b (20) , 0 s

and to treat them as samples of a random variable. Note that the sign here is in

keeping with the sign in (4) and that the predicted median must be recomputed for

each observation, but only because the distance changes. Note, too, that the random

variations in these deviations are a consequence of observational variability, and

that, since the locations (and the times) must be assumed to have been independently

selected, the deviations should be all mutually independent.

Returning to the randomly sited predictions of Figure 9, we have plotted in

Figure 13 the cumulative distribution of the deviations for all 44 observations.

But this plot is wrong. It is wrong because the points from which it is made come

....

--ll.

-20

11\

-CII

Q

'-

c::

() '-

40

20

o

i __

""--

\

------- ->

-40

1 10 50

P~r C~nf Obs~rVBfions

90 99

Figure 13. The sample cumulative distribution of deviations. As indicated

in the text, this is a misleading plot.

62
from very different distributions, and they cannot be combined in this way into a

single distribution. Now, while it is true that the standard deviation of observational variability changes with distance, the change is very slight and this is not

what troubles us. The real difficulty involves the same feature we have noted

before--that both data and predictions flatten out at free space values--and at

different distances this flattening appears at different deviations from the predicted medians. Thus the fact that in Figure 13 the median deviation is about 0 dB

and that this is only one point in a long, flat interval of the curve, merely

reflects the fact that a great many of the observations were made at distances less

than about 20 km. If the data set had not so emphasized these shorter distances,

the long flat interval would not have been so pronounced. Similarly, the flat

interval at about 20 dB is due to the accident that several paths had distances

clustering at about 50 km. The consequence of such contamination is that the cumulative distribution function of Figure 13 can Slerve no useful purpose.

There is a way out of the problem. If the high fields represented by free

space values had saturated the receiver or sent measuring devices off-scale, we

would have been in a similar quandary. We could not then have recorded received

signal levels but could only have reported they were too high to measure. Similarly,

in the present case we can imagine that the propagation channel itself has become

"saturated." Indeed, this is entirely in keeping with the model of variability

described in Section 6.3, for the modifying function described there is analogous to

the saturation curve of a receiver. We can suppose that what we really want to

measure is the unmodified attenuation ai, because we expect that the statistics of

the corresponding deviations are nearly invariant with distance. However, we must

also suppose that when the measured attenuation is nearly zero or less the value of

a l cannot be determined.

At this point we have an example of what is known as a sample with censored

data (Walsh, 1962; Efron, 1979) where for some of the data we have properly measured

values while for others we know only that they lie in certain infinite intervals.

The latter are the censored data; they should be neither discarded as useless nor

accepted on an equal footing with the remaining data.

Using the method of Kaplan and Meier (1958), we can construct a sample cumulative distribution function for such censored samples. In Figure 14 we have plotted

the results for the R3 data when we assume that all data are to be censored if the

measured attenuation is less than or equal to 0.5 dB. We feel that this gives a

fairly accurate picture of the true statistics of the deviations. Note that a full

19 of the 44 data were censored.

63
o

40

III 20 ....

..

oQ

...

CJ

.

c:

o...

-.....

:.

-20

"

,

"--.

-

-40

1 10 50

P.r C.n' Obs.r....';ons

90 99

Figure 14. The sample cumulative distribution of deviations assuming the

data are censored when A 0.5 dB.

Another very important example of censored data, which does not appear in the

present data set but does in many others, occurs when signal levels fall below the

sensitivity of the receiver. Here we truly have attenuations that are unmeasurable,

but only because the radio system used is inadequate. Since one important purpose

of a measurement program is to discover when it is that signal levels might be below

sensitivity, the very fact that we have found such locations is of prime importance

to us. Such "data" should never be discarded as being useless; they should simply

be treated as censored data. In other words, although we cannot say what the actual

attenuation is, we can say that it definitely exceeds some known threshold value.

Experimentalists should take note here. It is an important part of the report on

system parameters to provide an estimate of the system sensitivity.

From Figure 14 we can see that the sample median of the deviations is 11 dB and

that the sample 20% to 80% range is 27 dB. From the latter number we find that the

average slope (on the normal probability scale we have used) is 16 dB, a value that

can be likened to a standard deviation. These two numbers can be used to describe

the discrepancy between the measured data and the "best estimate" predictions.

64
The ITS irregular terrain model predicts that, barring the possibility of

signal levels greater than free space, the observational variability is very nearly

normally distributed. Indeed, the curve of Figure 14 appears to the eye to be as

nearly linear as one could expect. If we insist that the deviations involved do

indeed come from a normal distribution then, despite the fact that we have numerous

censored data, we can derive more precise estimates for the underlying mean and

standard deviation by resorting to the maximum likelihood estimate.

In general, we would suppose a sample of size n of which m of the deviations

have observed values Yl••• Ym and r(=n-m) are known only to exceed threshold values

nl••• n

r • We first define the likelihood function

1:. z(Yi -ll) Q(nafll)·

i=l a a j=l

(21)

where Z(x) is the probability density function for the standard normal distribution

and Q(x) its complementary cumulative distribution function. Then the estimates we

seek are the values of II and a which maximize L.

Using the same censored data as they appear in Figure 14, we have made these

calculations and find an estimated mean of 10.6 dB and an estimated standard deviation of 14 dB. It is interesting to note how these numbers relate to our model of

variability. If we rewrite (4) slightly to allow for the assumed mode of variability, we might suppose

(22)

and then what we have said is that for the singlE~ situation of the R3 data the value

of ys(s) is estimated to be 10.6 dB. Furthermore, since the standard deviation of

YL is about 10 dB, we can also say that, again for this one situation, we estimate

0L(s) to be 1.4. Note how, in presenting these sample statistics, we have violated

our own stricture to always include an estimate of the probable error. Unfortunately,

that must remain a problem for future work.

To summarize this section, we have subjected the ITS irregular terrain model to

several tests involving a small subset of data from a single measurement program.

Not surprisingly, the model has passed some of these tests and failed others. But

recall that our aim here has not been concerned with whether the tests were passed

or failed, but rather with illustrating some of the techniques one might use in a

more extensive study.

65
8 • REFERENCES

Barsis, A. P., and M. J. Miles (1965), Cumulative distributions of field strength

over irregular terrain, using low antenna heights, NBS Report 8891.

Barsis, A. P., and P. L. Rice (1963), Prediction and measurement of VHF field

strength over irregular terrain using low antenna heights, NBS Report 7962.

Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data of

the radio refractive index for the United States and the World, NBS Monograph

No. 22 (U.S. Government Printing Office, Washington, D.C.).

Bullington, K. (1950), Radio propagation variations at VHF and UHF, Proc. IRE 38,

pp. 27-32.

Bullington, K. (1957), Radio propagation fundamentals, BSTJ 36, pp. 593-626.

Causebrook, J. H., B. Davis, and R. S. Sandell (1969), The prediction of co-channel

interference to television broadcasting services in the frequency range 470 to

960 MHz, B.B.C. Research Dept Report 1969/33.

CCIR * (1978a) , VHF and UHF propagation curves for the frequency range from 30 MHz

to 1000 MHz, Doc. XIVth Plenary y, Recom. No. 370-3.

CCIR (1978b) *, Propagation data required for trans-horizon radio-relay systems,

Doc. XIVth Plenary y, Report No. 238-3.

Damelin, J., W. Daniel, H. Fine, and G. Waldo (1966), Development of VHF and UHF

propagation curves for TV and FM broadcasting, FCC Report R-6602.

Et'ron, B. (1979), Computers and the theory ot' statistics: Thinking the unthinkable,

SIAM Review ~' pp. 460-480.

Egli, J. J. (1957), Radio propagation above 40 Mc over irregular terrain, Proc.

IRE 45, pp. 1383-1391.

Epstein, J., and D. W. Peterson (1956), A method of predicting the coverage of a

television station, RCA Rev. 17, pp. 571-582.

Hagn, G. H. (1980), VHF radio system performance model for predicting communications

operational ranges in irregular terrain, IEEE Trans. Commun. COM-28, pp. 1637-

1644.

Head, H. T., and O. L. Prestholdt (1960), The measurement of television field

strengths in the VHF and UHF bands, Proc. IRE 48, pp. 1000-1008.

Johnson, M. E., M. J. Miles, P. L. McQuate, and A. P. Barsis (1967), Tabulations of

VHF propagation data obtained over irregular terrain at 20, 50, and 100 MHz,

IER 38-ITSA38, Parts 1, 2, and 3. Available from NTIS, Access. Nos. AD-655-

854, AD-662-7l3, AD 667-530.

*Published by the International Telecommunication Union, Geneva, Switzerland. Also

available from the National Technical Information Service, Springfield, VA 22161,

with Access. No. PB-298-025.

66
Johnson, M. E., and G. D. Gierhart (1979), Comparison of measured data with IF-77

propagation model predictions, DOT Rpt. FAA-RD-79-9. Available from NTIS,

Access. No. AD-A076-508.

Kaplan, E. L., and P. Meier (1958), Nonparametric estimation from incomplete observations, J. Am. Stat. Assoc. ~, pp. 457-481.

LaGrone, A. H. (1960), Forecasting television service fields, Proc. IRE 48, No.6,

pp. 1009-1015.

Longley, A. G., and G. A. Hufford (1975), Sensor path loss measurements analysis

and comparison with propagation models, OT Report 75-74. Available from NTIS,

Access. No. PB-247-638/AS.

Longley, A. G., and R. K. Reasoner (1970), Comparison of propagation measurements

with predicted values in the 20 to 10,000 MHz ,range, ESSA Tech. Report ERL

l48-ITS 97. Available from NTIS, Access. No. AD-703-579.

Longley, A. G., and P. L. Rice (1968), Prediction of tropospheric radio transmission loss over irregular terrain--a computer method 1968, ESSA Tech.

Report ERL 79-ITS 67. Available from NTIS, Access. No. AD-676-874~

McQuate, P. L., J. M. Harman, M. E. McClanahan, and A. P. Barsis (]970), Tabulations of propagation data over irregular tel,rain in the 230 to 9300 MHz

frequency range; Part III, North Table Mountain--Golden, ESSA Tech. Report ERL

65-ITS 58-3. Available from NTIS, Access. No. AD-7l5-753.

Norton, K. A., P. L. Rice, and L. E. Vogler (1955), The use of angular distance in

estimating transmission loss and fading range for propagation through a turbulent atmosphere over irregular terrain, Proc. IRE 43, pp. 1488-1526.

0' Connor, R. A. (1968), Understanding television I s Grade A and Grade B service

contours, IEEE Trans. Broadcasting BC-14, Pl? 137-143.

Okumura, Y., E. Ohmori, T. Kawano, and K. Fukuda (1968), Field strength and its

variability in VHF and UHF land-mobile radio service, Rev. Tokyo Elec. Commun.

Lab. 16, pp. 825-873.

Rice, P. L., A. G. Longley, K. A. Norton, and A. P. Barsis (1967), Transmission

loss predictions for tropospheric communication circuits, NBS Tech. Note 101,

Vols. I and II. Available from NTIS, Access. Nos. AD-687-820 and AD-687-821.

Walpole, R. E., and R. H. Myers (1972), probability and Statistics for Engineers

and Scientists (The Macmillan Company, New York, N.Y.).

Walsh, J. E. (1962), Handbook of Nonparametric Statistics (D. van Nostrand Co.,

Princeton, N.J.).

67

APPENDIX A. LRPROP AND AVAR--AN IMPLEMENTATION OF THE ITS MODEL

FOR MID-RANGE FREQUENCIES

In this appendix we give the explicit source code listing for a computer implementation of version 1.2.1 of the ITS irregular terrain model (the Longley-Rice

model) for radio propagation at frequencies between 20 MHz and 20 GHz. Accompanying

the listing are directions to the programmer for introducing the proper subroutines

into an applications program.

The language used is FORTRAN and conforms to the 1966 ANSI standards. We

believe it is also compatible with the 1977 ANSI standards. On most modern computers the routine should be usable with no modification. For satisfactorily

accurate results we require floating point numbers having at least six significant

decimal figures and a range at least as large as 10±35.

The routines have been constructed so as to be both flexible and efficient.

Redundant or unnecessary computations have been avoided, and there are no iterative

processes involved. The routines may be used for either the "area prediction" mode

or the "point-to-point" mode; and if the desired output consists only of the reference attenuation, one may entirely divorce the calculations from those concerned

with statistics.

The two modes of operation use very similar calling sequences, and they are

treated below in parallel.

Using the Longley-Rice model of radio propagation generally involves three

consecutive steps: the preparation of parameters, the computation of the reference

attenuation, and then the computation of selected statistics. Around these processes the programmer will put others which assemble the required input and which

manipulate the resulting output. These latter we leave largely to the user's ingenuity, and in what follows we try to describe o~y the central three processes.

Parameter preparation is accomplished by one of two subroutines: QLRA for the

area prediction mode and QLRPFL for the point-to-point mode. Also useful is the

subroutine QLRPS. The reference attenuation is computed by LRPROP and the statistics

by the function subprogram AVAR. Internally, most of the input and output is contained in the three common blocks /PROP/, /PROPA/, and /PROPV/. A few of the variables involved there must also be accessed directly by the user.

1. Common blocks.

COMMON/PROP/KWX,AREF,MDP,DIST,HG(2),WN,

DH,ENS,GME,ZGND,HE(2) ,DL(2) ,THE (2)

COMPLEX ZGND

69
This is the collection of the principal system and path parameters. It also

includes the reference attenuation and an error marker. Note that all heights and

distances are measured in meters.

KWX

AREF

MDP

DIST

HG

WN

DH

ENS

GME

ZGND

HE

DL

THE

Error marker. Indicates by its value the severity

of the warning:

o no warning

I caution; parameters are close to limits

2 impossible parameters; default values

have been substituted

3 internal calculations show parameters out

of· range

4 parameters out of range

Reference attenuation. This is computed by the subroutine LRPROP.

Mode of the propagation model. Values:

-1 point-to-point

1 area prediction; to initialize

o area prediction; to continue

For further remarks see note 2 below.

Distance between terminals. See note 3 below.

Heights of the antennas above ground.

Wave number of the radio frequency.

Terrain irregularity parameter.

Surface refractivity.

Effective earth's curvature.

Surface transfer impedance.

Effective antenna heights.

Horizon distances.

Horizon elevation angles.

Note 1. The error marker KWX is meant to serve as a warning to the user that

one or more of the parameters have values that make the results dubious or unusable.

Except when it has the value 2, there is no effect on the computations. The value

is cumulative in that after a series of calculations it will retain its highest

value. Since it is never reset to 0, the user must do this himself.

Note 2. The value of MOP is handled automatically by QLRA and QLRPFL. In the

area prediction mode it must first be set to I whereupon LRPROP will initialize

various constants and set MDP to o. On subsequent calls where it is only the distance that varies, LRPROP need not recompute those constants.

Note 3. The value of DIST is entered in two ways. In the point-to-point mode

it is entered directly into /PROP/. This is done automatically by QRPFL. In the

area prediction mode the distance is an actual parameter in the call to LRPROP.

1 COMMON/PROPV/LVAR,SGC,MDVAR,KLIM

This is the collection of instructions for treating variability in the subroutine AVAR.

70
2

3

1

+20

Climate

1

2

3

4

5

6

7

Level to which coefficients in AVAR must be defined.

Each time the parameter indicated below is changed,

LVAR must be set to at least:

level parameter

o none

1 DIST

2 HE, etc.

3 WN

4 MDVAR

5 KLIM

The subroutine AVAR will compute the necessary coefficients and reset LVAR to O.

The standard deviation of confidence. Output by

AVAR, it may be used to compute a confidence level.

Mode of variability calculations. Values:

o Single message mode: Time, location, and

situation variability are combined

together to give a confidence level.

Individual mode: Reliability is given by

time availability. Confidence is a combination of location and situation variability.

Mobile mode: Reliability is a combination

of time and location variability. Confidence is given by the situation variability.

Broadcast mode: Reliability is given by

the two-fold statement of at least qT of

the time in qL of the locations. Confidence

is given by the situation variability.

In addition, to these values may be added eithe~or

both of the numbers 10 and 20 with the meanings~

+10 Location variability is to be eliminated as

it should when a well-engineered path is

being treated in the point-to-pointmode.

Direct situation variability is to be eliminated as it should when considering interference problems. Note that there may still

be a small residual situation variability.

code. Values:

Equatorial

Continental subtropical

Maritime subtropical

Desert

continental temperate

Maritime temperate over land

Maritime temperate over sea

LVAR

MDVAR

SGC

KLIM

COMMON/PROPA/DLSA,DX,AEL,AKl,AK2,AED,EMD,AES,EMS,

DLS(2) ,DLA,THA

The collection of parameters and coefficients which define the reference attenuation as a function of distance. Ordinarily of no interest to the user.

71
COMMON/SAVE/••.

A collection of miscellaneous constants and coefficients which must remain

defined in certain of the subroutines. Used in place of the SAVE directive and of

no interest to the user.

2. Parameter preparation.

The reference attenuation requires the variables

MDP, DIST, HG, WN, DH, ENS, GME, ZGND, HE, DL, THE

and also an attention to KWX. The statistics require the variables

MDVAR, KLIM, ZT, ZL, ZC

and also an attention to LVAR; note that the value of MDVAR determines the meanings

of ZT, ZL, ZC. The following subroutines should be used to introduce many of these

variables.

CALL QLRPS(FMHZ,ZSYS,ENO,IPOL,EPS,SGM)

This will define WN, ENS, GME, ZGND in /PROP/

FMHZ

ZSYS

ENO

IPOL

EPS,SGM

Fr~quency in MHz.

Average elevation above sea level of the system;

if 0, ENO will be interpreted as ENS.

Minimum monthly mean surface refractivity reduced

to sea level; if it is desired to introduce ENS

instead, then set ZSYs=o.

Polarization code: 0, horizontal; 1, vertical.

Ground constants.

CALL QLRA(KST,KLIM,MDVAR)

Prepares parameters for the area prediction mode. Prior to this call one should

define HG, DH and WN, ENS, GME, ZGND in /PROP/. The present routine will then define

HE, DL, THE, LVAR, and optionally KLIM, MDVAR. It sets MDP=l.

KST

KLIM

MDVAR

Siting criterion code for each terminal; an array of

length 2.

Climate code. If greater than 0, the routine will

put this value in /PROPV/ and set LVAR=5.

Mode of variability. If non-negative, the routine

will put this value in /PROPV/ and set LVAR to at

least 4.

In any case, the routine sets LVAR to at least 3.

72
.__.. _. ---------_.~--_._-

l__C_AL_L_Q_L_RP_F_L_Cp_F_L_,_KL_IM__,MD_V_A_R_) - ....1

Prepares parameters for the point-to-point 'mode and calls LRPROP thus defining

also the reference attenuation AREF. Prior to this call one should define HG and WN,

ENS, GME, ZGND in /PROP/. One should also have prepared a terrain profile in the

array PFL. For this we imagine a sequence of elevations Po' PI"'" Pnp taken at

equal intervals from the point under the first terminal to that under the second.

Note that the path distance is then n ~. p

PFL

KLIM

MDVAR

Terrain profile. An array packed with the values

np ' ~, Po' ••• , Pnp' in that order. Thus PFL(l) is

the floating point representation ofnp ' PFL(2)

equals the interval ~between profile points, and

PFLCi+3) equals Pi' i=O, ••• , np ' Le., the elevation of the point distant i~ from the first terminal•. The total length of the array is np+3.

Climate code. If greater than 0, the routine will

put this value in /PROPV/ and set LVAR=5.

Mode of variability. If non-negative, the routine

will put this value in /PROPV/ and set LVAR to at

least 4.

In any case the routine sets LVAR to at least 3.

It should be noted that the Longley-Rice model is silent on many of the details

for defining some of the path parameters. This is particularly true of the effective

heights h

el,2 and, to some lesser degree, of the terrain irregularity parameter ~h.

The effective height, for example, is defined as the height above the "effective

reflecting plane," and in the past the investigator has been urged to use his own

best judgment as to where that plane should be placed. The subroutine QLRPFL, in

trying to automate the definition of all parameters, has been forced to define explicitly all missing details. It has done this in ,a way that seems reasonable and in

full accord with the intent of the model. These techniques should not, however, be

construed to have any "official" standing.

3. The reference attenuation.

After defining all necessary parameters, the next step is to compute the reference attenuation. This is done by a single call.

l__C_A_L_L_L_RP_R_O_P...:.CD_)~ --__......-.-@__I

This will define the reference attenuation AREF in /PROP/. prior to this call

one should have defined MDP, WN, HG, DH, ENS, GME, ZGND, HE, DL, THE in /PROP/. In

73
the point-to-point mode (when MDP~-l), the distance should also have been defined

as DIST in /PROP/. The formal parameter D will be ignored. In the area prediction

mode (when MDP~l or 0), D represents the distance and LRPROP will replace DIST in

/PROP/ by this value. Also, on the first entry after a set of parameters has been

defined, one should set MDP~l. Then LRPROP will set switches, define certain constants, and reset MDP to O. On subsequent calls, if it is only the distance that

changes, One should not redefine MDP.

In the area prediction mode there is also a special call obtained by setting

D~O. In general, a call to LRPROP will result in the definition of only those coefficients that are necessary to compute the reference attenuation at the indicated

distance. In this special call, however, all coefficients in /PROPA/ will be defined.

If desired, the user can then consider these coefficients to be additional output

from LRPROP.

4. Statistics.

Statistics are available through the function subprogram AVAR in the form of

quantiles--i.e., values of attenuation which are not exceeded for a fraction q of

the samples. Rather than using the fraction q directly, however, we convert our

terminology to an equivalent standard normal deviate z defined by

-1/2 foo -t2/2

q Q(z) (2~) e dt

z

The function Q is the complementary normal probability function as defined in most

texts on statistics. This standard normal deviate is used because the random variables involved are all normally distributed or very nearly normally distributed, and

calculations using them are greatly simplified. We use the complementary function

rather than the direct function because we usually think in terms of a received signal level rather than a loss or an attenuation and would like to say that this level

is at least so large for a fraction q=Q(z) of the samples.

Note that Q is a monotonically decreasing function and that as q goes from 0 to

1, z goes from 00 to _00 For example, Q(0)~0.5, Q(1.28lS5)~0.l, and Q(-1.28l55)~0.9.

Before using AVAR, one should have defined all system and path parameters in

/PROP/ and also the reference attenuation AREF. In addition, one should define

LVAR, MDVAR, KLIM in /PROPV/.

Then the function AVAR can be evaluated. It has three formal parameters whose

meanings are determined by the mode of variability as specified in MDVAR. In what

follows we use freely a notation such as QC, ZC to indicate a pair consisting of a

probability and its corresponding standard normal deviate.

74
Single message mode (MDVAR=O).

A=AVAR(O.,O.,ZC)

Then with confidence QC the attenuation will not exceed A. The first two

parameters are unused.

Individual and mobile modes (MDVAR=l or 2).

A=AVAR(ZR,O.,ZC)

Then with confidence QC the attenuation will not exceed A with a reliability at

least as large as QR. The second of the three parameters is unused.

Broadcast mode (MDVAR=3).

A=AVAR(ZT ,ZL,ZC)

Then with confidence QC there will be at least QL of the locations where the

attenuation will not exceed A for at least QT of the time.

In addition to AVAR there are two small function subprograms which, if desired,

can be used to facilitate the translation between probabilities and standard normal

deviates.

Q=QERF(Z)

Z=QERFI(Q)

These are the Q error function and the inverse of the Q error function respectively.

75
5. Suggested operational flow.

TO USE THE AREA PREDICTION MODE

SET KWX =0

DEFINE HG, DH AND CALL QLRPS

OPTIONALLY, DEFINE MDVAR, KLIM

CALL QLRA

LOOP FOR SELECTED DISTANCES D

t SET LVAR = MAX(LVAR, 1)

t CALL LRPROP(D)

t LOOP FOR SELECTED QUANTILES

t t A=AVAR (... )

t t OUTPUT A

t t--REPEAT

t--REPEAT

CHECK KWX

ENDTO

TO USE THE POINT-TO-POINT MODE

SET K~JX =a

DEFINE PFL, HG AND CALL QLRPS

OPTIONALLY, DEFINE MDVAR, KLIM

CALL QLRPFL

LOOP FOR SELECTED QUANTILES

t A= AVAR'(. .. )

t OUTPUT A

t--REPEAT

CHECK KWX

ENDTO

6. Source code listings.

The subprograms on the following pages are arranged in logical order. First

is LRPROP followed by several ancillary subprograms. Then comes AVAR followed by

the additional routines QERF and QERFI. The last group consists of the preparatory

routines QLRPS, QLRA, and QLRPFL, the latter followed by several ancillary subprograms.

76
SUBROUTINE LRPROP(D)

C COMPUTES AREF, THE REFERENCE VALUE OF RADIO ATTENUATION

C VERSION 1.2.1 (AUG 71/MAR 77/APR 79)

C OF THE LONGLEY-RICE (1968) MODEL

C PRINCIPAL CHANGESC 1.1. A SIMPLIFICATION OF THE LINE-OF-SIGHT AND SCATTER

C ROUTINES

C 1.2. A CHANGE IN THE LINE-OF-SIGHT ROUTINE AND IN THE

C SUBSEQUENT CALCULATIONS. RESULTS ARE IMPROVED WHEN

C ONE OR BOTH ANTENNAS ARE HIGH.

C VALID ONLY FOR••.

C FREQUENCIES BETWEEN 20 MHZ AND 20 GHZ

C ANTENNA HEIGHTS BETWEEN 0.5 M AND 3000 M

C ELEVATION ANGLES LESS- THAN 200 MRAD

C

COMMON/PROP/KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPA/DLSA,DX,AEL,AK1,AK2,AED,EMD,AES,EMS,DLS(2),DLA,THA

C

COMMON/SAVE/SAVA(6),WLOS,WSCAT,DMIN,XAE,SAVB(40)

C

LOGICAL WLOS,WSCAT

C

DATA THIRD/0.3333333/

C

IF(MDP) 10,32,10

C

10 CONTINUE

DO 11 J=1,2

11 DLS(J)=SQRT(2.*HE(J)/GME)

DLSA=DLS(1)+DLS(2)

DLA=DL(1)+DL(2)

THA=AMAX1 (THE (1 )+TlIE (2) ,-DLA*GME)

WLOS=.FALSE.

WSCAT=.FALSE.

C

C CHECK PARAMETER RANGES

IF(ENS .LT. 250 ••OR. ENS .Gr. 400.) TO 154

IF(GME ,.LT. 75E-9 .OR. GME'GT~250E-9) GO TO 154

IF (REAL (ZGND ) •LE. ABS (AlMAG ( ZGND ) » GO TO 154

DO 121 J=1,2

IF(ABS(THE(J» .GT. 200E-3) GO TO 153

IF(DL(J) .LT. O.1*DLS(J) .OR. DL(J) .GT. 3.*DLS(J»

X GO TO 153

121 CONTINUE

IF(WN .LT. 0.838 .OR. WN .GT. 210.) GO TO 151

DO 122 J=1,2

IF(HG(J) .LT. 1••OR. HG(J) .Gr.• 1000.) GO TO 151

122 CONTINUE

GO TO 158

153 KWX=MAXO(KWX,3)

151 KWX=MAXO(KWX,1)

77
IF(WN .LT. 0.419 .OR. WN .GT. 420.) GO TO 154

DO 132 J=1, 2

IF(HG(J) .LT. 0.5 • OR. HG(J) .GT. 3000.) GO TO 154

132 CONTINUE

GO TO 158

154 KWX=4

158 CONTINUE

DMIN=ABS(HE(1)-HE(2»/200E-3

C

C COEFFICIENTS FOR THE DIFFRACTION RANGE

C

Q=ADIFF ( 0 • )

XAE=(WN*GME**2)**(-THIRD)

D3=AMAX1(DLSA, 1. 3787*XAE+DLA)

D4=D3+2.7574*XAE

A3=ADIFF(D3 )

A4=ADIFF(D4)

EMD=(A4-A3)/(D4-D3)

AED=A3-EMD*D3

C

IF(MDP) 33,32,31

31 MDP=O

32 DIST=D

IF(DIST .LE. 0.) GO TO 38

33 CONTINUE

IF(DIST .GT. 1000E3) KWX=MAXO(KWX,1)

IF(DIST .LT. DMIN) KWX=MAXO(KWX,3)

IF(DIST .LT. 1E3 .OR. DIST .GT. 2000E3) KWX=4

38 CONTINUE

C

IF(DIST .GE. DLSA) GO TO 50

C

IF(WLOS) GO TO 48

C

C COEFFICIENTS FOR THE LINE-OF-SIGHT RANGE

C

Q=ALOS(O.)

D2=DLSA

A2=AED+D2*EMD

DO=1.908*WN*HE(1)*HE(2)

IF(AED .LT. 0.) GO TO 41

DO=AMIN 1 (DO,O.S*DLA)

D1=DO+0.25*(DLA-DO)

GO TO 42

41 D1=AMAX1(-AED/EMD,O.25*OLA)

42 A1=ALOS(D1)

IF(DO .GE. D1) GO TO 43

AO=ALOS(DO)

Q=ALOG(D2/DO)

AK2=AMAX1(0.,«D2-DO)*(A1-AO)-(D1-DO)*(A2-AO»/

X «D2-DO)*ALOG(D1/DO)-(D1-DO)*Q»

IF(AK2 .GT. 0.) GO TO 44

IF(AED .GE. 0.) GO TO 44

78
43 AK2==0.

AK1=(A2-A1)/(D2-D1)

IF(AK1 .GT. 0.) GO TO 46

GO TO 45

44 AK1=(A2-AO-AK2*Q)/(D2-DO)

IF(AK1 .GE. 0.) GO TO 46

AK1=0.

AK2=DIM(A2,AO)/Q

IF(AK2 .GT. 0.) GO TO 46

45 AK1=EMD

46 AEL=A2-AK1*D2-AK2*ALOG(D2)

WLOS=. TRUE •

48 IF(DIST .LE. 0.) GO TO 50

AREF=AEL+AK1*DIST+AK2*ALOG(DIST)

GO TO 60

c

50 IF(WSCAT) GO TO 58

C

C COEFFICIENTS FOR THE SCATTER RANGE

C

Q=ASCAT(O.)

D5=DLA+200E3

D6=D5+200E3

A6=ASCAT(D6)

A5=ASCAT(D5)

IF(A5 .LT. 1000.) GO TO 51

EMS=EMD

AES=AED

DX=10E6

GO TO 52

51 EMS=(A6-A5)/200E3

DX=AMAX1(DLSA,DLA+0.3*XAE*ALOG(47.7*WN),

X (A5-AED-EMS*D5)/(EMD-EMS)

AES=(EMD-EMS)*DX+AED

52 WSCAT=.TRUE.

C

58 IF(DIST .GT. DX) GO TO 59

AREF=AED+EMD*DIST

GO TO 60

59 AREF=AES+EMS*DIST .

C

60 AREF=DIM(AREF,O.)

RETURN

END

79
------------------ --.- ---

FUNCTION ADIFF(D)

C THE *DIFFRACTION ATTENUATION* AT DISTANCE D

C A CONVEX COMBINATION OF SMOOTH EARTH DIFFRACTION AND

C DOUBLE KNIFE-EDGE DIFFRACTION

C A CALL WITH D=O SETS UP INITIAL CONSTANTS

C

COMMON/PROP/KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPA/DLSA , DX, AEL, _lU< 1 , AK2 , AED, EMD, AES , EMS, DLS (2 ) , DLA , THA

C

COMMON/SAVE/W01 ,XD1 ,AFO,QK,AHT ,XHT, SAVE (44)

C

DATA THIRD/0.3333333/

C

IF(D .GT. 0.) GO TO 10

C

Q=HG( 1 )*HG(2)

QK=HE(1)*HE(2)-Q

IF(MDP .LT. 0) Q=Q+10.

WD1=SQRT(1.+QK/Q)

XD1=DLA+THA/GME

Q=(1.-0.S*EXP(-DLSA/50E3»*DH

Q=0.7S*Q*EXP(-(Q/16.)**0.25)

AFO=AMIN 1 ( 15. , 2. 171 *ALOG ( 1 • +4 • 77E-4 *HG ( 1) *HG (2 )*WN*Q ) )

QK=1. /CABS (ZGND)

AHT=20.

XHT=O.

DO 1 J=1,2

A=0.5*DL(J)**2/HE(J)

WA=(A*WN)**THIRD

PK=QK/WA

Q=(1.607-PK)*151.0*WA*DL(J)/A

XHT=XHT+Q

AHT=AHT+FHT(Q,PK)

1 CONTINUE

ADIFF=O.

GO TO SO

C

10 CONTINUE

TH=THA+D*GME

DS=D-DLA

Q=0.0795775*WN*DS*TH**2

ADIFF=AKNFE(Q*DL(1)/(DS+DL(1»)+AKNFE(Q*DL(2)/(DS+DL(2»)

A=DS/TH

WA=(A*WN)**THIRD

PK=QK/WA

Q=(1.607-PK)*151.0*WA*TH+XHT

AR=0.05751*Q-4.343*ALOG(Q)-AHT

Q=(W01+XD1/D)*AMIN1«(1.-0.S*EXP(-D/50E3»*DH*WN),62S3.2)

WD=25.1/(25.1+SQRT(Q»

ADIFF=(AR-ADIFF)*WD+ADIFF+AFO

SO RETURN

END

SO
FUNCTION ALOS(D)

C THE *LINE-OF-SIGHT ATTENUATION* AT DISTANCE D

C A CONVEX COMBINATION OF PLANE EARTH FIELDS AND

C DIFFRACTED FIELDS

C A CALL WITH D=O SETS UP INITIAL CONSTANTS

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPA/DLSA,DX,AEL,AK1,AK2,AED,EMD,AES,EMS,DLS(2),DLA,THA

C

COMMON/SAVE/WLS, SAVE (49 )

C

COMPLEX R

C

ABQ(R)=REAL(R)**2+AIMAG(R)**2

C

IF(D .GT. 0.) GO TO 10

C

WLS=0.021/(0.021+WN*DH/AMAX1(10E3,DLSA»

ALOS=O.

00 TO SO

C

10 CONTINUE

Q=(1.-0.S*EXP(-D/50E3»*DH

S=0.7S*Q*EXP(-(Q/16.)**0.25)

Q=HE(1)+HE(2)

SPS=Q/SQRT(D**2+Q**2)

R=(SPS-ZGND)/(SPS+ZGND)*EXP(-WN*S*SPS)

Q=ABQ(R)

IF(Q .LT. 0.25 .OR. Q .LT. SPS) R=R*SQRT(SPS/Q)

ALOS=EMD*D+AED

Q=WN*HE(1)*HE(2)*2./D

ALOS=(-4.343*ALOG(ABQ(CMPLX(COS(Q),-SIN(Q»+R»-ALOS)*WLS+ALOS

C

SO RETURN

END

81
FUNCTION ASCAT(D)

C THE *SCATTER ATTENUATION* AT DISTANCE D

C USES AN APPROXIMATION TO THE METHODS OF NBS TN101 WITH

C CHECKS FOR INADMISSABLE SITUATIONS

C FOR PROPER OPERATION, THE LARGER DISTANCE (D=D6)

C MUST BE THE FIRST CALLED

C A CALL WITH D=O SETS UP INITIAL CONSTANTS

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPA/DLSA,DX,AEL,AK1,AK2,AED,EMD,AES,EMS,DLS(2),DLA,THA

C

COMMON/SAVE/AD, RR, ETQ, HOS, SAVE (46)

C

IF(D .GT. 0.) GO TO 10

C

AD=DL ( 1 )- DL (2 )

RR=HE (2 )/HE ( 1 )

IF(AD) 1,2,2

1 AD=-AD

RR=1./RR

2 ETQ=(5.67E-6*ENS-2.32E-3)*ENS+0.031

HOS=-15.

ASCAT=O.

GO TO 80

C

10 CONTINUE

IF(HOS .GT. 15.) GO TO 12

TH=THE(1)+THE(2)+D*GME

R2=2. *WN*TH

R1=R2*HE(1)

R2=R2*HE(2)

IF(R1 .GT. 0.2 .OR. R2 .GT. 0.2) GO TO 11

ASCAT=1001.

GO TO 80

11 SS=(D-AD)/(D+AD)

Q=RR/SS

SS=AMAX1 (0 • 1 , SS )

Q=AMIN1(AMAX1(0.1,Q),10.)

ZO=(D-AD)*(D+AD)*TH*0.25/D

ET=(ETQ*EXP(-AMIN1C1.7,ZO/8.0E3)**6)+1.)*ZO/1.7556E3

ETT=AMAX1 (ET, 1. )

HO=(HOF(R1,ETT)+HOF(R2,ETT»*0.5

HO=HO+AMIN1(HO,(1.38-ALOG(ETT»*ALOG(SS)*ALOG(Q)*0.49)

HO=DIM(HO,O.)

IF(ET .LT. 1.) HO=ET*HO+(1.-ET)*4.343*ALOG«(1.+1.4142/R1)*

X (1.+1.4142/R2»**2*(R1+R2)/{R1+R2+2.8284»

IF(HO .LE. 15 ••OR. HOS .LT. 0.) GO TO 13

12 HO=HOS

13 H08=HO

TH=THA+D*GME

ASCAT=AHDCTH*O)+4.343*ALOG(47.7*WN*TH**4)-

X 0.1*(ENS-301.)*EXP(-TH*O/40E3)+HO

80 RETURN

END

82
FUNCTION AKNFE(V2)

C KNIFE-EDGE DIFFRACTION

C THE FRESNEL INTEGRAL AS A FUNCTION OF V**2

C

IF(V2 .GT. 5.76) GO TO 2

1 AKNFE=6.02+9.11*SQRT(V2)-1.27*V2

GO TO B

2 AKNFE=12.953+4.343*ALOG(V2)

B RET~N

END

83
FUNCTION FHT(X,PK)

C THE HEIGHT GAIN OVER A SMOOTH SPHERICAL EARTH

C TO BE USED IN THE *THREE RADII* METHOD

C

IF(X .LT. 200.) GO TO 2

FHT=0.05751 *X-4. 343*ALOG(X)

IF(X .GE. 2000.) GO TO 8

W=0.0134*X*EXP(-0.005*X)

FHT=(1.-W)*FHT+W*(17.372*ALOG(X)-117.)

GO TO 8

2 IF(PK .GT. 1.E-5) GO TO 3

IF(X .GT. 1.) GO TO 4

FHT=-117 •

GO TO 8

3 W=-ALOG(PK)

IF(X*~*3 .GT. 5495.) GO TO 4

FHT=2.5E-5*X**2!PK-8.686*W-15.

GO TO 8

4 FHT=17.372*ALOG(X)-117.

8 RETURN

END

84
FUNCTION HOF(R,ET)

C THE HO FUNCTION FOR SCATTER FIELDS

C

DIMENSION A(5),B(5)

C

DATA A( 1 ) ,A( 2) ,A( 3) ,A( 4) ,A( 5)

X / 25., 80.,177.,395.,705./

DATA B(1),B(2),B(3),B(4),B(5)

X / 24., 45., 68., 80.,105./

C

IT=ET

IF ( IT) 2, 2, 1

1 IF(IT-5) 5,4,3

2 IT=1

GO TO 4

3 IT=5

4 Q=O.

00 TO 6

5 Q=ET-FLOAT(IT)

6 X=( 1./R)**2

HOF=4.343*ALOG((A(IT)*X+B(IT»*X+1.)

IF(Q .NE. 0.)

X HOF=(1.-Q)*HOF+Q*4.343*ALOG((A(IT+1)*X+B(IT+1»*X+1.)

RETURN

END

85
FUNCTION AHD(TD)

C THE F(TH*D) FUNCTION FOR SCATTER FIELDS

C

DIMENSION A(3),B(3),C(3)

C

DATA A(1),A(2),A(3)/133.4,104.6,71.8/

DATA B(1),B(2),B(3)/O.332E-3,O.212E-3,O.157E-3/

DATA C(1),C(2),C(3)/-4.343,-1.086,2.171/

C

1=1

IF(TD .LE. 10E3) GO TO 1

1=2

IF(TD .LE. 70E3) GO TO 1

1=3

1 AHD=A(I)+B(I)*TD+C(I)*ALOG(TD)

RETURN

END

86
FUNCTION AVAR(ZZT,ZZL,ZZC)

C QUANTILES OF ATTENUATION RELATIVE TO FREE SPACE

C INCLUDES LONG-TERM TIME VARIABILITY, LOCATION VARIABILITY,

C AND SITE VARIABILITY

C

COMMON/PROPjKWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPV/LVAR,SGC,MDVAR,KLIM

C

COMMON/SAVE/SAVE(13),KDV,WL,WS,DEXA,DE,VMD,VSO,SGL,

X SGTM,SGTP,SGTD,TGTD,GM,GPrCV1,CV2,YV1,YV2,YV3,

X CSM1 ,CSM2, YSM1, YSM2, YSM3,CSP1 ,CSP2, YSP1, YSP2, YSP3,CSD1 ,ZD,

X CFM1,CFM2,CFM3,CFP1,CFP2,CFP3

C

DIMENSION BV1(7),BV2(7),XV1(7),XV2(7);XV3(7)

DIMENSION BSM1 (7) ,BSM2(7) ,XSM1 (7) ,XSM2(7) ,XSM3(7)

DIMENSION BSP1 (7) ,BSP2 (7) ,.XSP1 (7) ,XSP2 (7) ,XSP3 (7)

DIMENSION BSD1(7),BZD1(7)

DIMENSION BFM1 (7) ,BFM2 (7) ,BFM3 (7) ,BFP1 (7) ,BFP2 (7) ,BFP3 (7)

C

LOGICAL WS,WL

C

C

C

EQUATOR, CON SUB, MAR SUB, DESERT, CON TMP, MAR LND, MAR SEA

DATA BV1 ( 1 ), BV1 (2), BV1 (3 ), BV1 (4), BV1 (5 ), BV1 (6), BV1 (7 )

X / -9.67, -0.62, 1.26, -9.21, -0.62, -0.39, 3.15/

DATA BV2(1), BV2(2), BV2(3), BV2(4), BV2(5), BV2(6), BV2(7)

X / 12.7, 9.19, 15.5, 9.05, 9.19, 2.86, 857.9/

DATA XV1(1), XV1(2), XV1(3), XV1(4), XV1(5), XVH6), XV1(7)

X /144.9E3,228.9E3,262.6E3, 84.1E3,228.9E3,141.7E3,2222.E3/

DATA XV2(1), XV2(2),XV2(3), XV2(4), XV2(5), XV2(6), XV2(7)

X /190.3E3,205.2E3,185.2E3,101.1E3,205.2E3,315.9E3,164.8E3/

DATA XV3(1), XV3(2), XV3(3), XV3(4), XV3(5), XV3(6), XV3(7)

X /133.8E3,143.6E3, 99.8E3, 98.6E3,143.6E3,167.4E3,116.3E3/

DATA BSM 1 ( 1 ) , BSM 1 ( 2 ) , BSM 1 ( 3 ) , BSM 1 (4 ) , BSM 1 ( 5 ) , BSM 1 (6 ) , BSM 1 (7 )

X / 2.13, 2.66, 6.11, 1.98, 2.68, 6.86, 8.51/

DATA BSM2(1),BSM2(2),BSM2(3),BSM2(4),BSM2(5),BSM2(6),BSM2(7)

X / 159.5, 7.67, 6.65, 13.11, 7.16, 10.38, 169.8/

DATA XSM 1 ( 1 ) , XSM 1 ( 2) , XSM 1 ( 3 ) , XSM 1 (4 ) , XSM 1 (5 ) , XSM 1 (6 ) , XSM 1 (7 )

X /762.2E3,100.4E3,138.2E3,139.1E3, 93.7E3,187.8E3,609.8E3/

DATA XSM2(1),XSM2(2),XSM2(3),XSM2(4),XSM2(5),XSM2(6),XSM2(7)

X /123.6E3,172.5E3,242.2E3,132.7E3,186.8E3,169.6E3,119.9E3/

DATA XSM3(1),XSM3(2),XSM3(3),XSM3(4),XSM3(5),XSM3(6),XSM3(7)

X / 94.5E3, 136.4E3, 178.6E3, 193.5E3, 133.5E3, 108.9E3, 106.6E3/

DATA BSP1(1),BSP1(2),BSP1(3),BSP1(4),BSP1(5),BSP1(6),BSP1(7)

X / 2.11, 6.87, 10.08, 3.68, 4.75, 8.58, 8.43/

DATA BSP2( 1) ,BSP2 (2) ,BSP2 (3) ,BSP2 (4) ,BSP2 (5) ,BSP2 (6) ,BSP2 (7)

X / 102.3, 15.53, 9.60, 159.3, 8.12, 13.97, 8.19/

DATA XSP1(1),XSP1(2),XSP1(3),XSP1(4),XS~1(5),XSP1(6),XSP1(7)

X /636.9E3,138.7E3,165.3E3,464.4E3, 93.2E3,216.0E3,136.2E3/

DATA XSP2(1),XSP2(2),XSP2(3),XSP2(4),XSP2(5),XSP2(6),XSP2(7)

X /134.8E3,143.7E3,225.7E3, 93.1E3,135.9E3,152.0E3,188.5E3/

87
DATA XSP3 (1) ,XSP3 (2) ,XSP3 (3) ,XSP3(4) ,XSP3 (5) ,XSP3 (6) ,XSP3,(7)

X / 95.6E3, 98.6E3,129.7E3, 94.2E3,113.4E3,122.7E3,122.9E3/

DATA BSD 1( 1 ) , BSD 1 ( 2 ) , BSD 1 ( 3 ) , BSD 1 (4 ) , BSD1( 5 ) , BSD 1 (6 ) , BSD1( 7 )

X / 1.224, 0.801, 1.380, 1.000, 1.224, 1.518, 1.518/

DATA BZD1 (1) ,BZD1 (2) ,BZD1 (3) ,BZD1 (4) ,BZD1 (5) ,BZD1 (6) ,BZD1 (7)

X / 1.282, 2.161, 1.282, 20., 1.282, 1.282, 1.282/

DATA BFM 1 ( 1 ) , BFM 1 (2 ) , BFM 1 (3 ) , BFM 1 (4 ) , BFM 1 (5 ) , BFM 1(6) , BFM 1 (7 )

X / 1., 1., 1., 1., 0.92, 1., 1./

DATA BFM2(1),BFM2(2),BFM2(3),BFM2(4),BFM2(5),BFM2(6),BFM2(7)

X / o. , o. , o. , o. , 0.25, o. , o. /

DATA BFM3(1),BFM3(2),BFM3(3),BFM3(4),BFM3(5),BFM3(6),BFM3(7)

X / o. , o. , o. , o. , 1.77, o. , o. /

DATA BFP1(1),BFP1(2),BFP1(3),BFP1(4),BFP1(5),BFP1(6),BFP1(7)

X / 1 • , 0.93 , 1• , 0 • 93 , 0.93 , 1• , 1./

DATA BFP2(1),BFP2(2),BFP2(3),BFP2(4),BFP2(5),BFP2(6),BFP2(7)

X / 0., 0.31, 0., 0.19, 0.31, 0., 0./

DATA BFP3(1),BFP3(2),BFP3(3),BFP3(4),BFP3(5),BFP3(6),BFP3(7)

X / 0., 2.00, 0., 1.79, 2.00, 0., 0./

C

DATA RT,RL/7.8,24./

C

DATA THIRD/0.3333333/

C

CURV(C1,C2,X1,X2,X3)=(C1+c2/(1.+«DE-X2)/X3)**2»*

X «DE/X1)**2)/(1.+«DE/X1)**2»

C

IF(LVAR .EQ. 0) GO TO 60

IF(LVAR .LT. !?)

X GO TO (10,20,30,40),LVAR

C CLIMATE

IF(KLIM .GT. 0 • AND. KLIM .LE. 7) GO TO 51

KLIM=5

KWX=MAXO (KWX, 2 )

51 CV1=BV1(KLIM)

CV2=BV2(KLIM)

YV1=XV1 (KLIM)

YV2=XV2 (KLIM)

YV3=XV3 (KLIM)

CSM1=BSM1 (KLIM)

CSM2=BSM2(KLIM)

YSM1=XSM1 (KLIM)

YSM2=XSM2(KLIM)

YSM3=XSM3(KLIM)

CSP1=BSP1(KLIM)

CSP2=BSP2(KLIM)

YSP1=XSP1 (KLIM)

YSP2=XSP2(KLIM)

YSP3=XSP3(KLIM)

CSD1=BSD1 (KLIM)

ZD=BZD1 (KLIM)

CFM1=BFM1 (KLIM)

CFM2=BFM2(KLIM)

CFM3=BFM3(KLIM)

88
CFP1=BFP1(KLIM)

CFP2=BFP2(KLIM)

CFP3=BFP3(KLIM)

C MODE OF VARIABILITY

40 KDV=MDVAR

WS=KDV .GE. 20

IF(WS) KDV=KDV-20

WL=KDV .GE. 10

IF(WL) KDV=KDV-10

IF(KDV .GE. 0 .AND. KDV .LE. 3) GO TO 41

KDV=O

KWX=MAXO (KWX, 2 )

41 KDV=KDV+ 1

C FREQUENCY

30 Q=ALOG(0.133*WN)

GM=CFM1+CFM2/«CFM3*Q)**2+1.)

GP=CFP1+cFP2/«CFP3*Q)**2+1.)

C SYSTEM PARAMETE~S

20 DEXA=SQRT (18E6*HE (1 »+SQRT (18E6*HE (2»+ (575. 7E12/WN) **THIRD

C DISTANCE

10 IF(DIST .GE. DEXA) GO TO 11

DE=130E3*DIST/DEXA

GO TO 12

11 DE=130E3+DIST-DEXA

12 VMD=CURV( CV1 , CV2, YV1, YV2, YV3)

SGTM=CURV (CSM 1, CSM2, YSM 1, YSM2, YSM3) *GM

SGTP=CURV(CSP1,CSP2,YSP1,YSP2,YSP3)*GP

SGTD=SGTP*CSD1

TGTD=(SGTP-SGTD)*ZD

SGL=O.

VSO=O.

IF(WL) GO TO 13

Q=(1.-0.8*EXP(-AMIN1(20.,DIST/50E3»)*DH*WN

SGL=10.*Q/(Q+13.)

13 IF(WS) GO TO 14

VSO=(5.+3.*EXP(-AMIN1(20.,DE/100E3»)**2

14 CONTINUE

C

LVAR=O

C

60 CONTINUE

ZT=ZZT

ZL=ZZL

ZC=ZZC

GO TO (600,601,602,603),KDV

600 ZT=ZC

601 ZL=ZC

GO TO 603

602 ZL=ZT

603 CONTINUE

IF ( ABS ( ZT ) •GT • 3. 10) GO TO 605

IF(ABS(ZL) .GT. 3.10) GO TO 605

IF(ABS(ZC) .GT. 3.10) GO TO 605

89
C

GO TO 608

605 KWX=MAXO(KWX,1)

608 CONTINUE

IF(ZT .GT. 0.) GO TO 611

SGT=SGTM

GO TO 618

611 IF(ZT .GT. ZD) GO TO 612

SGT=SGTP

GO TO 618

612 SGT=SGTD+TGTD!ZT

618 CONTINUE

VS=VSO+(SGT*ZT)**2!(RT+ZC**2)+(SGL*ZL)**2/(RL+ZC**2)

GO TO (620,621,622,623) ,KDV

620 YR=O.

SGC=SQRT(SGT**2+SGL**2+VS)

GO TO 628

621 YR=SGT*ZT

SGC=SQRT(SGL**2+VS)

GO TO 628

622 YR=SQRT(SGT**2+SGL**2)*ZT

SGC=SQRT(VS)

GO TO 628

623 YR=SGT*ZT+SGL*ZL

SGC=SQRT(VS)

628 CONTINUE

AVAR=AREF-VMD-YR-SGC*ZC

IF(AVAR .LT. 0.) AVAR=AVAR*(29.-AVAR)!(29.-10.*AVAR)

RETURN

END

90
FUNCTION QERF(Z)

C THE STANDARD NORMAL COMPLEMENTARY PROBABILITY

C APPROXIMATION DUE TO C. HASTINGS, JR.

C MAX ERROR 7.5E-8

C

DATA B1,B2,B3,B4,B5/0.319381530,-0.356563782,1.781477937,

X -1.821255987,1.3302744291

DATA RP,RRT2PI/4.317008,0.398942280/

C

X=Z

T=ABS(X)

IF(T ~LT. 10.) GO TO 1

QERF=O.

00 TO 2

1 T=RP/(T+RP)

QERF=EXP(-O.5*X**2)*RRT2PI*««B5*T+B4)*T+B3)*T+B2)*T+B1)*T

·2 IF(X .LT. 0.) QERF=1.-QERF •

RETURN

END

91
FUNCTION QERFI(Q)

C THE INVERSE OF QERF, GIVES THE STANDARD NORMAL DEVIATE AS A

C FUNCTION OF THE COMPLEMENTARY PROBABILITY

C TRUNCATED AT 0.000001 AND 0.999999

C APPROXIMATION DUE TO C. HASTINGS, JR.

C MAX ERROR 4.5E-4

C

DATA CO,C1,C2/2.515516698,0.802853,0.010328/

DATA D1,D2,D3/1.432788,0.189269,0.001308/

C

X=0.5-Q

T=AMAX1(0.5-ABS(X),0.000001)

T=SQRT(-2.*ALOG(T»

QERFI=T-( (C2*T+C1 )*T+cO)/( ( (D3*T+D2 )*T+D1 )*T+1. )

IF(X .LT. 0.) QERFI=-QERFI

RETURN

END

92
SUBROUTINE QLRPS(FMHZ,ZSYS,ENO,IPOL,EPS,SGM)

C PREPARES PARAMETERS

C SETS--

C WN,ENS,GME,ZGND

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE ( 2 ) , DL( 2 ) , THE ( 2 )

COMPLEX ZGND

C

COMPLEX ZQ

c

DATA GMA/157E-9/

C

WN=FMHZ/47.7

ENS=ENO

IF(ZSYS .NE. 0.) ENS=ENS*EXP(-ZSYS/9460.)

GME=GMA*(1.-0.04665*EXP(ENS/179.3»

ZQ=CMPLX(EPS, 376. 62*SGM/WN)

ZGND=CSQRT ( ZQ-1 • )

IF(IPOL .NE. 0) ZGND=ZGND/ZQ

RETURN

END

93
SUBROUTINE QLRA( KST, KLIMX,MDVARX)

DIMENSION KST(2)

C PREPARES THE LONGLEY-RICE MODEL IN THE AREA PREDICTION MODE

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

CCMMON/pROPV/LVAR, SGC, MDVAR, KLIM

C

DO 10 J=1,2

IF(KST(J)-1) 11,12,13

11 HE(J)=HG(J)

GO TO 15

12 Q=4.

00 TO 14

13 Q=9.

14 IF(HG(J) .LT. 5.) Q=Q*SIN(0.3141593*HG(J»

HE(J)=HG(J)+(1.+Q)*EXP(-AMIN1(20.,2.*HG(J)/AMAX1(1E-3,DH»)

15 Q=SQRT(2.*HE(J)/GME)

DL(J)=Q*EXP(-0.07*SQRT(DH/AMAX1(HE(J),5.»)

THE(J)=(0.6S*DH*(Q/DL(J)-1.)-2.*HE(J»/Q

10 CONTINUE

C

MDP=1

LVAR=MAXO(LVAR,3)

IF(MDVARX .LT. 0) GO TO 21

MDVAR=MDVARX

LVAR=MAXO(LVAR,4)

21 IF(KLIMX .LE. 0) .GO TO 22

KLIM=KLIMX

LVAR=5

22 CONTINUE

RETURN

END

94
SUBROUTINE QLRPFL(PFL, KLIMX, MDVARX )

DIMENSION PFL(5)

C

C SETS UP AND RUNS THE LONGLEY-RICE MODEL IN THE POINT-TO-POINT

CMODE USING THE TERRAIN PROFILE IN PFL.

C PFL(1)=ENP, PFL(2)=XI, PFL(3)=Z(0), •••

C

COMMON/PROPjKWX,AREF ,MDP ,DIST,.HG(2) ,WN,DH, ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPV/LVAR,SGC,MDVAR,KLIM

C

DIMENSION XL(2)

c

DIST=PFL(1)*PFL(2)

NP=PFL( 1)

CALL HZNS(PFL)

C FIND DELTA H

DO 11 J=1,2

11 XL(J)=AMIN1 (15.*HG(J) ,0.1*DL(J).)

XL(2)=DIST-XL(2)

DH=DLTHX(PFL,XL(1),XL(2»

C FIND EFFECTIVE HEIGHTS HE

IF(DL(1)+DL(2) .LT. 1.5*DIST) GO TO 25

C LINE-OF-SIGHT

CALL ZLSQ1(PFL,XL(1),XL(2),ZA,ZB)

HE(1)=HG(1)+DIM(PFL(3),ZA)

HE(2)=HG(2)+DIM(PFL(NP+3),ZB)

DO 21 J=1,2

21 DL(J)=SQRT(2.*HE(J)/GME)*EXP(-0.07*SQRT(DH/AM~X1(HE(J),5.»)

Q=DL(1)+DL(2)

IF(Q .GT. DIST) GO TO 23

Q=(DIST/Q)**2

DO 22 J=1,2

HE(J)=HE(J)*Q

22 DL(J)=SQRT(2.*HE(J)/GME)*EXP(-0.07*SQRT(DH/AMAX1(HE(J),5.»)

23 GO TO 28

C TRANSHORIZON

25 CALL ZLSQ1(PFL,XL(1),0.9*DL(1),ZA,Q)

CALL ZLSQ1(PFL,DIST-0.9*DL(2),XL(2),Q,ZB)

HE(1)=HG(1)+DIM(PFL(3),ZA)

HE(2)=HG(2)+DIM(PFL(NP+3),ZB)

28 CONTINUE

C

MDP=-1

LVAR=MAXO(LVAR,3)

IF(MDVARX .LT. 0) GO TO 31

MDVAR=MDVARX

LVAR=MAXO(LVAR,4)

31 IF(KLIMX .LE. 0) GO TO 32

KLIM=KLIMX

LVAR=5

32 CONTINUE

95
c

c

CALL LRPROP ( 0 • )

RETURN

END

96
FUNCTION DLTHX(PFL,X1,X2)

DIMENSION PFL(5)

C

C COMPUTES THE TERRAIN IRREGULARITY PARAMETER DH FROM THE

C PROFILE PFL BETwEEN POINTS AT X1 .LT. X2.

C

DIMENSION S(247)

C

NP=PFL( 1)

XA=X1!PFL(2 )

XB=X2!PFL(2)

DLTHX=O.

IF(XB-XA .LT. 2.) GO TO SO

KA=0.1*(XB-XA+S.)

KA=MINO (MAXO (4, KA), 25)

N=10*KA-5

KB=N-KA+1

SN=N-1

S( 1 )=SN

S(2)=1.

XB=(XB-XA)!SN

K=XA+1.

XA.=XA-FLOAT (K )

DO 10 J=1,N

11 IF(XA .LE. 0.) GO TO 12

IF(K .GE. NP) GO TO 12

XA=XA-1.

K=K+1

GO TO 11

12 S(J+2)=PFL(K+3)+(PFL(K+3)-PFL(K+2»*XA

10 XA=XA+XB

CALL ZLSQ1(S,0.,SN,XA,XB)

XB=(XB-XA)!SN

DO 15 J=1,N

S(J+2)=S(J+2)-XA

15 XA=XA+XB

C

DLTHX=QTILE(N,S(3),KA)-QTILE(N,S(3),KB)

DLTHX=DLTHX!(1.-0.S*EXP(-AMIN1(20.,(X2-X1)!50E3»)

SO RETURN

END

97
SUBROUTINE HZNS(PFL)

DIMENSION PFL(5)

C

C TO FIND HORIZONS FROM ANTENNAS WITH HEIGHTS HG AT THE TWO

C ENDS OF THE PROFILE PFL.

C PFL(1)=ENP, PFL(2)=XI, PFL(3)=Z(D), •••

C OtJrPUT--DISTANCES DL, TAKE-aFF ANGLES THE.

C DL=DIST IF THE PATH IS LINE OF SIGHT

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

C

LOGICAL WQ

C

NP=PFL( 1 )

XI==PFL(2 )

ZA=PFL( 3 )+HG ( 1 )

ZB=PFL(NP+3)+HG(2)

QC=D.5*GME

Q=QC*DIST

THE(2)=(ZB-ZA)/DIST

THE(1)=THE(2)-Q

THE(2)=-THE(2)-Q

DL( 1 )=DIST

DL(2)=DIST

IF(NP .LT. 2) GO TO 18

SA=D.

SB=DIST

WQ=.TRUE.

DO 1D I=2,NP

SA=SA+XI

SB=SB-XI

Q=PFL(I+2)-(QC*SA+THE(1»*SA-ZA

IF(Q .LE. D.) GO TO 11

THE(1)=THE(1)+Q/SA

DL(1 )=SA

WQ=.FALSE.

11 IF(WQ) GO TO 1D

Q=PFL(I+2)-(QC*SB+THE(2»*SB-ZB

IF(Q .LE. D.) GO TO 1D

THE(2)=THE(2)+Q/SB

DL(2 )=SB

1D CONTINUE

C

18 RETURN

END

98
FUNCTION QTILE(NN,A,IR)

DIMENS ION A(NN )

C

C REORDERS A SO THAT A(J),J=1 ••• IR ARE ALL .GE.

C ALL A(I),I=IR••• NN. IN PARTICULAR, A(IR) WILL HAVE THE· SAME

C VALUE IT WOULD HAVE IF A WERE COMPLETELY SORTED IN

C DESCENDING ORDER.

C RETURNS QTILE=A(IR)

C

M=1

N=NN

K=MINO (MAXO (;1 , IR) , N)

10 CONTINUE

Q=A(K)

IO=M

J1=N

11 CONTINUE

DO 12 I=IO,N

IF(A(I) .LT. Q) GO TO 13

12 CONTINUE

I=N

13 J=J1

DO 14 JJ=M,J1

IF(A(J) .GT. Q) GO TO 15

14 J=J-1

J=M

15 IF(I .GE. J) GO TO 16

R=A(I)

A(I)=A(J)

A(J)=R

10=1+1

J1=J-1

GO TO 11

16 IF(I .GE. K) GO TO 17

A(K)=A(I)

A(I)=Q

M=I+1

GO TO 10

17 IF(J .LE. K) GO TO 20

A(K)=A(J)

A(J)=Q

N=J-1

GO TO 10

20 QTILE=Q

RETURN

END

99
SUBROUTINE ZLSQ1(Z,X1,X2,ZO,ZN)

DIMENS ION Z( S )

c

C LINEAR LEAST SQUARES FIT BE'IWEEN X1, X2 TO THE FUNCTION

C DESCRIBED BY Z--

C Z( 1 )=EN , NUMBER OF INTERVALS, Z ( 2 )=X I, INTERVAL LENGTH,

C Z(J+3), J=O, ••• ,EN, FUNCTION VALUES.

C Ot1rPUT-- VALUES OF THE LINE, ZO AT 0, ZN AT XT.

C

XN=Z( 1 )

XA=AINT(DIM(X1/Z(2) ,0.»

XB=XN-AINT(DIM(XN,X2/Z(2»)

IF(XB .GT. XA) GO TO 1

XA=DIM (XA, 1• )

XB=XN-DIM(XN,XB+1.)

1 JA=XA

JB=XB

N=JB-JA

XA=XB-XA

X=-O.S*XA

XB=XB+X

A=0.S*(Z(JA+3)+Z(JB+3»

B=0.S*(Z(JA+3)-Z(JB+3»*X

IF(N .LT. 2) GO TO 11

DO 10 I=2,N

JA=JA+1

X=X+1.

A=A+Z(JA+3 ).

B=B+Z(JA+3)*X

10 CONTINUE

11 A=A/XA

B=B*12./«XA*XA+2.)*XA)

ZO=A-B*XB

ZN=A+B*(XN-XB)

RETURN

END

100
APPENDIX B. QKAREA--AN APPLICATIONS PROGRAM

The routine QKAREA (Quick Area) is a main program designed to illustrate one

way to use the Longley-Rice model in the area prediction mode. Written under the

constraints of 1966 ANSI Fortran, it is meant to be used in a batch environment. It

reads cards to define parameters and prints out tables showing estimated quantiles

of basic transmission loss versus distance for a set of confidence levels. In

addition to the system and environmental parameters, the user may choose from among

four different modes of variability analysis and the specific quantiles, both of

reliability and of confidence, for which computations will be made.

Input.

Input is through a sequence of cards of ten different "types." Each type

introduces a particular set of parameters or directs the program to perform a particular operation. The cards may appear in any number and in almost any order and as

many output pages for as many different systems as desired may be produced. The

concept is that the information on anyone card will change values for the indicated

parameters while leaving all other values as they were previously defined. When all

desired changes have been made, the user requests an execution run, whereupon computations are made and a page of output produced. At the very beginning all parameters are assigned default values which are therefore the values used unless the

user explicitly changes them. Furthermore, most of the parameters by their nature

must have strictly positive values; if they are given zero or negative values on the

input cards, the program will simply ignore those values and therefore retain those

previously given. In particular, since Fortran interprets a blank field as a zero

value, such a blank field may be used to indicate that the corresponding parameter

is not to be changed.

The cards are read in 10-column fields, the first of which is a sequence of

digits and the remaining seven floating point numbers read with an FlO.O format.

Column 1 is always an "execute" indicator: if it is any non-zero digit, then as

soon as the card is processed the program makes its computations using parameter

values as are then defined. The digit in column 2 indicates the card type and

defines how the remaining digits in the first ten columns and how the floating point

numbers in the remaining fields are to be interpreted. In outline, the ten types

can be described as follows:

101
Col.

123 11, •.•

Stop x0

Title Xl

Distance X2

Reliability X3V

Confidence x4

Environment X5C

System X6NPSS

(Alternate) X7NPSS

Execute X8

Reset X9

(or a blank card)

(a 60 column title on the next card)

do,dl,dsl,d2,ds2

lih,N ,Z ,e,a o s

fMHZ,hgl,hg2

fMHz,hgl,hg2,lih,Ns,e,a

In this table we have begun the representation of a card with the execute digit

X in column 1 and the card type digit in column 2. Following this there may be

single upper case letters indicating particular parameters with single digit values

and then there comes a sequence of variables. These latter are to be replaced by

floating point values in successive 10-column fields beginning at column 11. The

card types and definitions of the variables are described in detail below.

Card Type O. Stop. When this card is read the program has finished the job

run. If the execute digit is set, a page of output is produced first. This card,

which can be entirely blank should be the last card of the input deck. On many

computers, however, it is unnecessary. This is true if, upon reading an end of file

marker, the computer either leaves the input variables unchanged or zeroes them out.

Card Type 1. Title. If desired the user may supply a short title which will

appear at the top of the output pages. This title will consist of the first 60

columns on the card that immediately follows a card of type 1. The default title

reads "Area predictions from the Longley-Rice model, version 1.2.1."

Card Type 2. Distances. This card defines the distances at which quantities

of basic transmission loss will be computed and displayed. There are three intended

formats: (i) d alone with the remaining variables omitted--only the single diso

tance do; (ii) do' dl , dsl alone--distances from do to dl in steps of dsl; and (iii)

all five variables--after stepping through to dl as above, distances continue on to

102
d 2 in steps of ds2 • All distances are measured in kilometers. If do and dl are

nonpositive, the card is ignored; otherwise the entire schema of distances is

changed irrespective of what was previously defined. If some of the variables such

as the step sizes are undefined, the program will use its own algorithms to define

them. In particular, if d is nonpositive, it will be replaced by d 1. In the

o s

original default condition, distances go from 10 to 150 km in steps of 10 km and

then on to 500 km in steps of 50 km.

Card Type 3. Reliability. This card defines the mode of variability and if

needed, the quantiles of required reliability. The mode is given in terms of the

service intended by the system under study and is defined by the digit V with values

V = 0, single-message service

1, individual service

2, mobile service

3, broadcast service.

Since V can have the value 0, it must always be explicitly given whenever this card

type is used. If V=O, there is no meaning to reliability and any quantiles given on

the card are ignored. If V is 1 or 2 the quantile in the first field, measured as a

percentage, will be used as the required reliability; that in the second field will

be ignored. If V is 3, then is the required percent of time and qL the required

percent of locations. If a quantile needed by the variability mode is missing or

nonpositive, it will be replaced by the value of 50 percent; thus, if this card is

used, all indicated data should probably be explicitly defined. The original default

condition assumes the broadcast service (V=3) with both and qL equal to 50 percent.

Card Type 4. Confidence. This card defines a sequence of confidence levels

which will be used as the second independent vari.able (the first being distance) for

which quantiles of basic transmission loss will be computed. There may be from one

to seven of these quantiles, all measured as percentages. Each of the listed quantiles will be used in the order given to head a column in the output table. Each

use of the card redefines the entire list of confidence levels. If none are specified, the program defaults to a single level equal to 50 percent. The original

default condition uses three levels equal to 50, 90, and 10 percent, in that order.

Card Type 5. Environmental parameters. With this card one can specify parameters relating to the terrain, the atmosphere, and the ground. The digit C defines

the radio climate according to the coded values:

103
C = 1,

2,

3,

4,

5,

6,

7,

equatorial

continental subtropical

maritime subtropical

desert

continental temperate

maritime temperate overland

maritime temperate oversea.

If the digit is 0 or blank it is ignored, and the climate code used will be the one

previously defined. The terrain irregularity parameter Ah is measured in meters;

since zero is a legitimate value, when this card is used one must always be sure to

define this parameter explicitly. (If a strictly negative value is used, however,

the program will employ the previously defined value.) The parameter N , measured

o

in N-units, is the surface refractivity of the atmosphere as reduced to sea level,

value of N , then one should merely set Z

s s

The dielectric constant £ of the ground and the

of N

o

wants

while Z , measured in meters, is the average elevation of the ground surface for

s

which the actual surface refractivity N will be computed. If the indicated value

s

is positive, then the value of Z must also be supplied; but note that if one

s

instead to introduce directly the

to zero or leave that field blank.

conductivity 0 measured in siemens per meter are in the last two fields. These are

treated as a pair; if the value of £ is positive, then the value of 0 must also be

supplied; if the value of £ is nonpositive, both fields will be ignored. The

default values use a continental temperate climate (C=5), an average terrain irregularity with Ah=90 m, a four-thirds earth with N =301 N-units, and an average ground s

with £=15 and 0=0.005 S/m.

Card Type 6. System parameters. On this card one can specify the parameters

that define the system under study and how that system is to be deployed. The digit

P defines the polarization: 0 for horizontal and 1 for vertical polarization. The

two digits S give the siting criterion codes, first for terminal 1 and then for

terminal 2; values: 0 for random siting, 1 for careful siting, and 2 for very

careful siting. Since these three digits can have the value 0, they must always be

explicitly defined. Some relief, however, is offered by the "no read" digit N. If

N=l then none of the three digits PSS will be read. In any other case all three

will be read and used. The frequency f

MHz is measured in megahertz and is the value

of the carrier frequency. And the heights hgl , hg2 , measured in meters, are the

heights above ground of the antennas at the two corresponding terminals. If there

is a question, it is the so-called center of radiation of the antenna that should be

used. The default values use vertical polarization (P=l), random siting (S=O) at

both terminals, a frequency of 100 MHz, and antenna heights of 3 m at both terminals.

104
The latter, of course, are not meant to be useful but only to assure that there is

no such thing as an undefined parameter.

Card Type 7. An alternative to card types 5 and 6. This card may be used ~o

introduce most system and environmental parameters at once. The variables have the

same meaning and the same input conventions described above. Note that it is the

actual surface refractivity N that is used here; and recall that ~h should always s

be defined and that if the digit N is not I then all three digits PSS should be

defined.

Card Type 8. Execute. When this card is rE~ad, the program inunediately makes

its computations using the data that have been previously defined. This is an

alternative to setting the execute digit on the preceding card.

Card Type 9. Reset. When this card is read, the program will reset all parameters to their default values. Any following inI)ut cards may proceed on the assumption that they are effectively the first cards in the job run.

Installation.

The program is written in Fortran and conforms to the 1966 ANSI standards. We

think it is also compatible with the 1977 ANSI standards. Thus on most modern

computers it should be operable with very few modifications. The few changes that

might be necessary are listed below.

(1) Some compilers require a PROGRAM card as the first card of the

main program. When this is so, a suitable version of such a

card should replace the first comment card which now reads

"PROGRAM QKAREA."

(2) The input and output files are identified through the two variables KIN and KOT. These in turn are assigned fixed values in a

DATA statement near the beginning of the main program. Presently,

the values are 5 and 6, respectively.' When other values are

necessary or desired, one should replace that DATA statement

with a suitable alternative.

(3) The only use of Hollerith characters, aside from Hollerith

constants within FORMAT statements, is in the reading and

writing of the title cards. Presently, this is done with a l5A4

format which will work adequately on many computers. When it is

necessary or desirable to make adjustments here, one should

change the FORMAT statements 1001 and 2011 and also the length

of the array ITL where the title card is stored.

105
(4) The only non-trivial use of the printer control character is for

positioning the printer page at the beginning of a page of

output. When it is necessary to delete this use, one should

replace FORMAT statement 2001 with a suitable alternative.

(5) With one exception the output lines never exceed 65 characters.

The exception is the actual table of computed quantiles of basic

transmission loss. The lines there can, if completely filled,

occupy as many as 77 characters. When this is too long, one

should merely refrain from requesting the full seven different

quantiles.

We should also mention the common block !SAVE!. A few of the subprograms are

entered several times and expect some of their local variables to have retained the

values previously defined. The use of the common block to store such values will,

since the block is also present in the main program, assure that the values are

retained. Many Fortran processors, however, do always retain the values of local

variables. When this is so, the common block !SAVE! may be deleted wherever it

appears without affecting any results. In 1977 ANSI Fortran a suitablY phrased SAVE

directive would be preferable.

The Longley-Rice model has been successfully installed on fairly small minicomputers. For such computers that do not use the full Standard Fortran, however,

the present implementation will probably require considerable modification. The areas

where it violates the Standard Basic Fortran, and therefore where one might encounter

difficulties, include the following: (i) symbolic names of six characters, (ii)

logical variables and logical IF statements, (iii) complex variables and some complex arithmetic, (iv) FUNCTION subprograms that redefine entities in common, (v)

array dec1arators in COMMON statements, (yi) labeled common blocks, (vii) DATA

statements, and (viii) the "A" format descriptor.

In general, the program should be operable and results satisfactorily accurate

on any computer with floating point numbers having at least six significant decimal

±35 figures and a range at least as large as 10 •

106
C PROGRAM QKAREA

C *QUICK AREA*

C TO ILLUSTRATE THE USE OF THE LONGLEY-RICE MODEL

C IN THE AREA PREDICTION MODE

C

C INPUT IS IN 10-COL FIELDS, THE FIRST OF WHICH IS

C A SEQUENCE OF DIGITS

C IN PARTICULAR,

C COL 1 IS THE *EXECUTE* COLUMN--A NON-ZERO DIGIT

C WILL FORCE OUTPUT

C COL 2 INDICATES THE CARD TYPE--

C C~

C 12 11, •••

C STOP- XO (OR A BLANK CARD)

C TITLE- X1 (NEXT CARD HAS 60-COL TITLE)

C DISTANCES- X2 DO,D1,DS1,D2,DS2

C RELIABILITY- X3V QT,QL

C CONFIDENCE- X4 QC1,QC2, •••

C ENVIRONMENT- X5C DH,NO,ZS,EPS,SGM

C SYSTEM- X6NPSS FMHZ,HG1,HG2

C ( ALTERNATE) X7NPSS FMHZ, HG 1, HG2, DH, NS, EPS, SGM

C EXECUTE- X8

C RESET- X9

C

COMMON/PROP!KWX,AREF,MDP,DIST,HG(2),WN,DH,ENS,GME,ZGND,

X HE(2),DL(2),THE(2)

COMPLEX ZGND

COMMON/PROPV/LVAR,SGC,MDVAR,KLIM

C

COMMON/PROPA/DLSA,DX,AEL,AK1,AK2,AED,EMD,AES,EMS,nLS(2),DLA,THA

COMMONI SAVElSAVE (50)

C

DIMENSION JIN(6),XIN(7)

DIMENSION ITL(15)

DIMENSION KST(2)

DIMENSION QC(7),ZC(7),XLB(7)

C

LOGICAL WQIT,WCON,WTL

C

C THE 1/0 UNITS ARE.DEFINED HERE

DATA KIN,KOT/5,61

C

DATA GMA/157E-9/

C

DATA DB/8.6858901

DATA AKM/1000.j

C

WQIT=.FALSE.

WCON=.TRUE.

GO TO 190

C

10 CONTINUE

C READ INPUT SEQUENCE

107
C

1000

1001

C

C

x

FORMAT(6I1,4X,7F10.0)

FORMAT ( 1SA4 )

JIN (1 )=0

JIN(2)=0

READ(KIN,1000) JIN,XIN

WCON=JIN(1) .EQ. 0

JQ=JIN(2)

IF(JQ .NE. 0)

GO TO (110,120,130,14D,1S0,160,170,180,190),JQ

WQIT=.TRUE.

GO TO 20

110 CONTINUE

READ(KIN,1001) ITL

WTL=.TRUE.

GO TO 20

120 CONTINUE

XIN(1)=DIM(XIN(1),D.)

Q=XIN(2)-XIN(1)

IF(Q .GT. 0.) GO TO 121

IF(XIN(1) .EQ. 0.) GO TO 128

DO=XIN (1 )

DS=O.

DSC=D.

NO=1

NDC=O

GO TO 128

121 IF(XIN(3) .LE. 0.) XIN(3)=AMAX1(1.,AINT(Q/20.+0.5»

IF(XIN(1) .LE. D.) XIN(1')=XIN(3)

DO=XIN (1 )

DS=XIN(3 )

DSC=DS

NO=DIM(XIN(2),XIN(1»/DS+1.75

NDC=O

IF(XIN(4) .LE. XIN(2» GO TO 128

IF(XIN(S) .LE. D.) XIN(S)=S.*XIN(3)

DSC=XIN(S)

JQ=(XIN(4)-XIN(2»/DSC+D.75

NDC=ND

ND=ND+JQ

128 GO TO 20

130 CONTINUE

MDVAR=MINO(JIN(3),3)

LVAR=MAXO(LVAR,4)

QT=SD.

QL=SO.

ZT=O.

ZL=D.

IF(XIN(1) .LE. 0.) GO TO 131

QT=XIN (1 )

ZT=QERFI(QT/10D. )

108
131 IF(XIN(2) .LE. 0.) GO TO 138

QL=XIN(2)

ZL=QERFI (QL/1 00. )

138 GO TO 20

140 CONTINUE

NC=O

DO 141 JC=1,7

IF(XIN(JC) .LE. 0.) GO TO 141

NC=NC+1

QC(NC)=XIN(JC)

ZC(NC)=QERFI(QC(NC)/100.)

141 CONTINUE

IF(NC .GT. 0) GO TO 148

NC=::=1

QC( 1 )=50.

ZC(1)=0.

148 GO TO 20

150 CONTINUE

IF (JIN (3) •LE • 0) GO TO 151

KLIM=JIN(3)

LVAR=5

151 IF(XIN(1) .GE. 0.) DH=XIN(1)

IF(XIN(2) .LE. 0.) GO TO 152

ENO=XIN(2)

ZSYS=XIN ( 3 )

152 IF(XIN(4) .LE. 0.) GO TO 158

EPS=XIN(4 )

SGM=XIN(5)

158 GO TO 20

160 CONTINUE

IF(JIN(3) .EQ. 1) GO TO 161

I POL=MIN0 (JIN (4 ) , 1 )

KST(1)=MINO(JIN(5),2)

KST(2)=MINO(JIN(6),2)

161 IF(XIN( 1) .GT. 0.) FMHZ=XIN( 1)

IF(XIN(2) .GT. 0.) HG(1)=XIN(2)

IF(XIN(3) .GT. 0.) HG(2)=XIN(3)

GO TO 20

170 CONTINUE

IF(JIN(3) .EQ. 1) GO TO 171

IPOL=MINO(JIN(4),1)

KST(1)=MINO(JIN(5),2)

KST(2)=MINO(JIN(6),2)

171 IF(XIN(1) .GT. 0.) FMHZ=XIN(1)

IF(XIN(2) .GT. 0.) HG(1)=XIN(2)

IF(XIN(3) .GT. 0.) HG(2)=XIN(3)

IF(XIN(4) .GE. 0.) DH=XIN(4)

IF(XIN (5) .LE. 0.) GO TO 172

ENO=XIN(5)

ZSYS=O.

172 IF(XIN(6) .LE. 0.) GO TO 178

EPS=XIN(6)

SGM=XIN(7)

109
178 GO TO 20

180 CONTINUE

WCON=.FALSE.

00 TO 20

190 CONTINUE

FMHZ=100.

HG(1)=3.

HG(2)=3.

DH=90.

ENO=301.

ZSYS=O.

EP5=15.

SGM=0.005

IPOL=1

KST( 1)=0

KST(2)=0

KLIM=5

MDVAR=3

LVAR=5

NC=3

QC(1)=50.

QC( 2 )=90.

QC(3)=10.

QT=50.

QL=50.

ZC(1)=0.

ZC(2)=-1.28155

ZC(3)= 1.28155

ZT=O.

ZL=O.

DO=10.

D5=10.

DSC=50.

ND=22

NDC=15

WTL=.FALSE.

C

20 CONTINUE

IF(WCON) GO TO 30

C

C EXECUTION

C

KWX=O

CALL QLRPS(FMHZ,ZSYS,ENO,IPOL,EPS,SGM)

CALL QLRA(KST,-1,-1)

C

C WRITE HEADING

2001 FORMAT(1H1/1HO)

2002 FORMAT(1H)

2010 FORMAT(3X,

.62HAREA PREDICTIONS FROM THE LONGLEY-RICE MODEL, VERSION 1.2.1 . )

2011 FORMAT(3X,15A4)

2015 FORMAT(12X,9HFREQUENCY,F12.0,4H MHZ)

no
2016

2017

2018

C

211

212

C

2021

2022

C

FORMAT(6X,15HANTENNA HEIGHTS,2F8.1,2H M)

FORMAT(4X,17HEFFECTIVE HEIGHTS,2F8.1,

12H M (SITING=,I1,1H"I1,1H»

FORMAT(5X,16HTERRAIN, DELTA H,F12.0,2H M)

WRITE(KOT,2001)

IF(WTL) GO TO 211

WRITE(KOT,2010)

GO TO 212

WRITE(KOT,2011) ITL

WRITE(KOT,2002)

WRITE(KOT,2002)

WRITE (KOT, 2015) FMHZ

WRITE(KOT,2016) HG

WRITE(KOT,2017) HE,KST

WRITE(KOT,2018) DH

WRITE (KOT, 2002)

FORMAT(3X,4HPOL=,I1,6H, EPS=,F3.0,6H, SGM=,F6.3,4H S!M)

FORMAT(3X,5HCLIM=,I1,5H, NO=,F4.0,5H, NS=,F4.0,4H, K=,F6.3)

Q=GMA!GME

WRITE(KOT,2021) IPOL,EPS,SGM

WRITE(KOT,2022) KLIM,ENO,ENS,Q

WRITE (KOT, 2002)

C

2030 FORMAT(3X,22HSINGLE-MESSAGE SERVICE)

2031 FORMAT(3X,18HACCIDENTAL SERVICE!

8X,F5.1,27H PER CENT TIME AVAILABILITY)

2032 FORMAT(3X,14HMOBILE SERVICE!

8X,21HREQUIRED RELIABILITY-,F5.1,9H PER CENT)

2033 FORMAT(3X,17HBROADCAST SERVICE!

8X,21HREQUIRED RELIABILITY-,F5.1,14H PER CENT TIME!

29X,F5.1,19H PER CENT LOCATIONS)

C

IF(MDVAR .NE. 0)

X GO TO (231,232,233),MDVAR

C

WRITE(KOT,2030)

GO TO 238

231 WRITE(KOT,2031) QT

GO TO 238

232 WRITE(KOT,2032) QT

GO TO 238

233 WRITE(KOT,2033) QT,QL

238 WRITE(KOT,2002)

C

2040 FORMAT (3X ,

.62HESTIMATED QUANTILES OF BASIC TRANSMISSION LOSS(DB)

2041 FORMAT(7X, 4HDIST, 5X , 4HFREE,4X, 15HWITH CONFIDENCE!

8X,2HKM,5X,6HSPACE ,7F8.1)

2045 FORMAT(2X,3F9.1,6F8.1)

C

111
C COMPUTE AND PRINT VALUES

WRITE (KOT, 2040 )

WRITE(KOT,2002)

WRITE(KOT,2041) (QC(JC),JC=1,NC)

WRITE(KOT,2002)

DT=DS

D=DO

DO 240 JD=1,ND

LVAR=MAXO(1,LVAR)

CALL LRPROP(D*AKM)

FS=DB*ALOG(2.*WN*DIST)

DO 241 JC=1,NC

241 XLB(JC)=FS+AVAR(ZT,ZL,ZC(JC»

WRITE (KOT, 2045 ) D,FS,(XLB(JC) ,JC=1 ,NC)

IF(JD .EQ. NDC) DT=DSC

D=D+DT

240 CONTINUE

C

2081

2082

2083

2084

FORMAT(3X,

.62H**WARNING- SOME PARAMETERS ARE NEARLY OUT OF RANGE.

3X,

.62H RESULTS SHOULD BE USED WITH CAUTION.

FORMAT(3X,

.62H**NOTE- DEFAULT PARAMETERS HAVE BEEN SUBSTITUTED

3X,

.62H FOR IMPOSSIBLE ONES.

FORMAT(3X,

.62H**WARNING- A COMBINATION OF PARAMETERS IS OUT OF RANGE.

3X,

.62H RESULTS ARE PROBABLY INVALID.

FORMAT(3X,

.62H**WARNING- SOME PARAMETERS ARE OUT OF RANGE.

3X,

.62H RESULTS ARE PROBABLY INVALID.

I

I

I

I

C

IF(KWX .EQ. 0) GO TO 28

C

C PRINT ERROR MESSAGES

WRITE (KOT, 2002 )

GO TO (281,282,283,284),KWX

281 WRITE (KOT , 2081 )

GO TO 28

282 WRITE(KOT,2082)

GO TO 28

283 WRITE(KOT,2083)

GO TO 28

284 WRITE (KOT, 2084)

28 CONTINUE

C

30 CONTINUE

IF( .NOT. WQIT) GO TO 10

C

STOP

END

112
QKAREA, a Sample Run

Input Deck:

· 1

• TEST PROBLEM 1 , QKAREA

· 2 O. 100. 10. 400. 25.

· 32 70.

· 4 10. 50. 90. 95.

.870110 400. 10. 1 • 200.

· 1

• TEST PROBLEM 2, QKAREA

.861 25 •

• 99

· 1

• TEST PROBLEM 4, QKAREA

· 31 10.

· 57 30. 350. 300. 25. 0.02

· 60000 1200. 5. 1000.

.88

This deck gives rise to four pages of output which are reproduced in the following.

113
TEST PROBLEM 1, QKAREA

FREQUENCY 400. MHZ

ANTENNA HEIGHTS 10.0 1.0 M

EFFECTIVE HEIGHTS 14.5 1.0 M (SITING=1,0)

TERRAIN, DELTA H 200. M

POL=1, EPS=15. , SGM= .005 SIM

CLIM=5, NO=301., NS=301., K= 1.333

MOBILE SERVICE

REQUIRED RELIABILITY- 70.0 PER CENT

ESTIMATED QUANTILES OF BASIC TRANSMISSION LOSS(DB)

DIST FREE WITH CONFIDENCE

KM SPACE 10.0 50.0 90.0 95.0

10.0 104.5 136.1 145.7 155.3 158.1

20.0 110.5 149.3 158.4 167.4 169.9

30.0 114.0 156.5 165.1 173.7 176.1

40.0 116.5 162.6 170.8 179.0 181. 3

50.0 118.5 168.0 175.9 183.8 186.1

60.0 120.1 172.8 180.6 188.3 190.5

70.0 121.4 177 .4 185.0 192.7 194.8

80.0 122.6 181.7 189.3 196.9 199.0

90.0 123.6 185.9 193.4 201.0 203.1

100.0 124.5 190.0 197.4 204.9 207.0

125.0 126.4 197.4 204.8 212.1 214.2

150.0 128.0 200.0 207.2 214.5 216.5

175.0 129.4 202.5 209.6 216.8 218.8

200.0 130.5 205.0 212.1 219.1 221.1

225.0 131.5 207.6 214.6 221.5 223.4

250.0 132.5 210.2 217.1 223.9 225.9

275.0 133.3 212.8 219.6 226.4 228.3

300.0 134.0 215.2 222.0 228.7 230.6

325.0 134.7 217.6 224.3 231.1 232.9

350.0 135.4 219.9 226.6 233.3 235.2

375.0 136.0 222.1 228.8 235.5 237.3

400.0 136.5 224.3 230.9 237.6 239.5

114
TEST PROBLEM 2, QUREA

FREQUENCY

ANTENNA HEIGHTS

EFFECTIVE HEIGHTS

TERRAIN, DELTA H

25.

10.0

14.5

200.

MHZ

1.0 M

1.0 M

M

(SITING=1,0)

POL=1, EPS=15., SGM= .005 S/M

CLIM=5, NO=301., NS=301., K= 1.333

MOBILE SERVICE

REQUIRED RELIABILITY- 70.0 PER CENT

ESTIMATED QUANTILES OF BASIC TRANSMISSION LOSS(DB)

DIST

KM

FREE

SPACE

WITH CONFIDENCE

10.0 50.0 90.0 95.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

125.0

150.0

175.0

200.0

225.0

250.0

275.0

300.0

325.0

350.0

375.0

400.0

80.4

86.4

90.0

92.5

94.4

96.0

97.3

98.5

99.5

100.4

102.3

103.9

105.3

106.4

107.5

108.4

109.2

110.0

110.6

111.3

111.9

112.5

110.2

125.3

131.3

136.1

140.4

144.2

147.7

151.0

154.1

157.1

164.0

170.3

176.3

182.1

187.8

193.7

199.7

205.8

211.9

218.0

220.3

222.5

120.1

134.9

140.6

145.1

149.1

152.7

156.0

159.2

162.1

165.0

171.6

177.8

183.6

189.3

194.9

200.7

206.6

212.6

218.7

224.7

227.0

229.1

130.1

144.51

149.8

154.1

157.9

161.2

164.4

167.3

170.2

172.9

179.2

185.2

190.9

196.4

202.0

207.7

213.5-

219.4

225.4

231.4

233.7

235.7

132.9

147.2

152.5

156.7

160.4

163.7

166.7

169.6

172.4

175.1

181.4

187.3

193.0

198.4

204.0

209.6

215.4

221.4

227.3

233.3

235.5

237.6

**WARNING- SOME PARAMETERS ARE NEARLY OUT OF RANGE.

RESULTS SHOULD BE USED WITH CAUTION.

115
AREA PREDICTIONS FROM THE LONGLEY-RICE MODEL, VERSION 1.2.1

FREQUENCY

ANTENNA HEIGHTS

EFFECTIVE HEIGHTS

TERRAIN, DELTA H

100.

3.0

3.0

90.

MHZ

3.0 M

3.0 M

M

(SITING=O,O)

POL=1, EPS=15., SGM= .005 SiN

CLIM=5, NO=301., NS=301., K= 1.333

BROADCAST SERVICE

REQUIRED RELIABILITY- 50.0 PER CENT TIME

50.0 PER CENT LOCATIONS

ESTIMATED QUANTILES OF BASIC "TRANSMISSION LOSS(DB)

DIST

KM

FREE

SPACE

WITH CONFIDENCE

50.0 90.0 10.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

110.0

120.0

130.0

140.0

150.0

200.0

250.0

300.0

350.0

400.0

450.0

500.0

92.5

98.5

102.0

104.5

106.4

108.0

109.4

11 0.5

111.5

112.5

113.3

114.0

114.7

115.4

116.0

118.5

120.4

122.0

123.3

124.5

125.5

126.4

132.8

143.8

150.0

155.1

159.6

163.6

167.3

170.8

174.2

177 .6

180.8

184.0

187.2

190.3

193.4

198.9

204.2

209.4

214.3

218.9

223.1

227.1

142.5

153.0

158.8

163.5

167.7

171.4

174.9

178.2

181.6

184.9

188.0

191. 1

194.2

197.2

200.3

205.6

210.8

215.9

220.8

225.3

229.5

233.5

123.2

134.6

141.2

146.7

151.4

155.7

159.7

163.3

166.9

170.3

173.7

176.9

180.1

183.3

186.4

192.1

197.6

202.9

207.8

212.4

216.7

220.7

116
TEST PROBLEM 4, QKAREA

FREQUENCY

ANTENNA HEIGHTS

EFFECTIVE HEIGHTS

TERRAIN, DELTA H

1200. MHZ

5.0 1000.0 M

5.0 1000.0 M (SITING=O,O)

30. M

POL=O, EPS=25. , SGM= .020 S!M

CLIM=7, NO=350., NS=339., K= 1.447

ACCIDENTAL SERVICE

10.0 PER CENT TIME AVAILABILITY

ESTIMATED QUANTILES OF BASIC TRANSMISSION LOSS(DB)

DIST FREE WITH CONFIDENCE

KM SPACE 50.0 90.0 10.0

10.0 114.0 114.0 130.0 110.2

20.0 120.1 119.9 135.9 116.2

30.0 123.6 123.2 139.1 119.7

40.0 126.1 125.5 141.2 122.2

50.0 128.0 127.1 142.6 124.1

60.0 129.6 128.5 143.6 125.6

70.0 130.9 129.6 144.2 126.9

80.0 132.1 130.5 144.7 128.0

90.0 133.1 131.3 144.9 128.9

100.0 134.0 132.3 146.2 129.9

110.0 134.9 136.9 ·152.3 131.4

120.0 135.6 142.9 158.3 132.9

130.0 136.3 148.8 164.2 134.7

140.0 137.0 154.6 170.0 139.2

150.0 137.6 160.2 175.6 144.8

200.0 140.1 175.1 190.8 159.5

250.0 142.0 175.3 191.2 159.3

300.0 143.6 179.8 195.6 164.0. 350.0 144.9 186.1 201.7 170.5

400.0 146.1 192.2 207.6 176.7

450.0 147.1 197.7 213.0 182.3

500.0 148.0 202.7 218.0 187.4

117

FORM NTIA-29

(4-80)

U.S. DEPARTMENT OF COMMERCE NAT"L. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION NO.

NTIA Report 82-10C

2. Gov't Accession No. 3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Guide to the Use of the ITS Irregular Terrain Model

in the Area Prediction Mode

7. AUTHOR(S)

George A. Hufford, Anita G. Longley, &< William A.• Kissick

8. PERFORMING ORGANIZATION NAME AND ADDRESS

U.S. Department of Commerce

National Telecommunications & Information Administration

Institute for Telecommunication Sciences

Boulder, CO 80303

11. Sponsoring Organization Name and Address

5. Publication Date

April 1982

6. Performing Organization Code

910.01

9. Project/Task/Work Unit No.

9108107

10. Contract/Grant No.

12. Type of Report and Period Covered

U.S. Department of Commerce

National Telecommunications & Information Administration

Institute for Telecommunication Sciences ~1~3-.-------------------------1

Boulder, CO 80303

14. SUPPLEMENTARY NOTES

15. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature

survey, mention it here.)

The ITS model of radio propagation for frequencies between 20 MHz and 20 GHz

(the Longley-Rice model) is a general purpose model that can be applied to a large

variety of engineering problems. The model, which is based on electromagnetic

theory and on statistical analyses of both terrain features and radio measurements,

predicts the median attenuation of a radio signal as function of distance and the

variability of the signal in time and in space.

The model is described in the form used to make "area predictions" for such

applications as preliminary estimates for system design, military tactical situations and surveillance, and land-mobile systems. This guide describes the basis of

the model, its implementation, and so~e advantages and limitations of its use.

Sample problems are included to demonstrate applications of the model.

16. Key Words (Alphabetical order, separated by semicolons)

Area prediction; radio propagation model; SHF; statistics; terrain effects; UHF;

VHF

17. AVAILABILITY STATEMENT 18.. Securily Class. (This report) 20. Number of pages

UNLIMITED.

o FOR OFFICIAL DISTRIBUTION.

Unclass ~i-=f-=i,-,=e,:,:d:c-:-__---, -f--::-:---7-1=2-=-0 ---t

19. Securily Class. (This page) 21. Price:

Unclassified

U. S. GOVERNMENT PRINTING OFFICE 1982 - 578-541/224 Reg. 8