the compleik subroutine of the ionort-isp system for...

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Advances in Space Research 53 (2014) 201–218

The COMPLEIK subroutine of the IONORT-ISPsystem for calculating the non-deviative absorption:

A comparison with the ICEPAC formula

Alessandro Settimi ⇑, Marco Pietrella, Michael Pezzopane, Bruno Zolesi,Cesidio Bianchi, Carlo Scotto

Istituto Nazionale di Geofisica e Vulcanologia (INGV), Sezione di Geomagnetismo, Aeronomia e Geofisica Ambientale (ROMA 2),

Via di Vigna Murata 605, I-00143 Rome, Italy

Received 13 May 2013; received in revised form 16 October 2013; accepted 31 October 2013Available online 13 November 2013

Abstract

The present paper proposes to discuss the ionospheric absorption, assuming a quasi-flat layered ionospheric medium, with small hor-izontal gradients. A recent complex eikonal model (Settimi et al., 2013b) is applied, useful to calculate the absorption due to the iono-spheric D-layer, which can be approximately characterized by a linearized analytical profile of complex refractive index, covering a shortrange of heights between h1 = 50 km and h2 = 90 km. Moreover, Settimi et al. (2013c) have already compared the complex eikonal modelfor the D-layer with the analytical Chapman’s profile of ionospheric electron density; the corresponding absorption coefficient is moreaccurate than Rawer’s theory (1976) in the range of middle critical frequencies. Finally, in this paper, the simple complex eikonal equa-tions, in quasi-longitudinal (QL) approximation, for calculating the non-deviative absorption coefficient due to the propagation acrossthe D-layer are encoded into a so called COMPLEIK (COMPLex EIKonal) subroutine of the IONORT (IONOspheric Ray-Tracing)program (Azzarone et al., 2012). The IONORT program, which simulates the three-dimensional (3-D) ray-tracing for high frequencies(HF) waves in the ionosphere, runs on the assimilative ISP (IRI-SIRMUP-P) discrete model over the Mediterranean area (Pezzopaneet al., 2011). As main outcome of the paper, the simple COMPLEIK algorithm is compared to the more elaborate semi-empirical ICE-PAC formula (Stewart, undated), which refers to various phenomenological parameters such as the critical frequency of E-layer. COMP-LEIK is reliable just like the ICEPAC, with the advantage of being implemented more directly. Indeed, the complex eikonal modeldepends just on some parameters of the electron density profile, which are numerically calculable, such as the maximum height.� 2013 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Ionospheric D-layer; Quasi-longitudinal propagation; Non-deviative absorption; ICEPAC formula; ISP-IONORT; Complex eikonal theory

1. Introductive review

Absorption is the process by which ordered energy ofthe radio wave is transformed into heat and electromag-netic (e.m.) noise by electron collisions with neutral mole-cules and ionized particles.

0273-1177/$36.00 � 2013 COSPAR. Published by Elsevier Ltd. All rights resehttp://dx.doi.org/10.1016/j.asr.2013.10.035

⇑ Corresponding author. Tel.: +39 06 51860719; fax: +39 06 51860397.E-mail addresses: [email protected] (A. Settimi), marco.pie-

[email protected] (M. Pietrella), [email protected] (M. Pezzopane),[email protected] (B. Zolesi), [email protected] (C. Bianchi),[email protected] (C. Scotto).

1.1. Collisions

Collisions of free electrons with neutrals, heavy ions orother electrons are important for various macroscopic phe-nomena. At higher altitudes (E-region), they determine thethermal and electrical conductivity of the plasma and thusthe current systems which give rise to geomagnetic varia-tions. Another important role of electron collisions is theabsorption of radio waves which occurs at lower altitudes(D-region). Conversely, the collision frequency can bededuced from radio wave propagation data with an appro-priate theory (Davies, 1990).

rved.

http://dx.doi.org/10.1016/j.asr.2013.10.035mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.asr.2013.10.035http://crossmark.crossref.org/dialog/?doi=10.1016/j.asr.2013.10.035&domain=pdf

202 A. Settimi et al. / Advances in Space Research 53 (2014) 201–218

The collision cross-section r for electrons in the nitrogenN2 — the most abundant atmospheric constituent up to100 km — was measured by Phelps and Pack (1959) andfound to be proportional to the root-mean-square electronvelocity vrms. In consequence, Sen and Wyller (1960) gener-alized the Appleton–Hartree magneto-ionic theory to: (a)include a Maxwellian velocity distribution of the electrons,and (b) extend the findings of Phelps and Pack (1959) to allconstituents of air, i.e. r � vrms. A further consequence isthat the momentum transfer collision frequency m of elec-trons with thermal energy kBT becomes proportional topressure p (being kB the Boltzmann’s constant and T theabsolute temperature).

The atmosphere in the E and D layers mainly consists ofnitrogen N2 (about 80%), with atomic and molecular oxy-gen O2 as the next most important constituents. The rela-tively large cross section for N2 makes it likely that, asfirst-order approximation, the height variation of collisionfrequency m is proportional to the partial pressure of the N2present. Experiments show that the cross section for O2also varies with the square root of T so that the two contri-butions can be combined (Thrane and Piggott, 1966).

When there is complete mixing of the atmosphere gases(Budden, 1988):

dpp¼ dqN

qNþ dT

T¼ � dH

H; ð1:aÞ

where p is the total pressure, qN the number density, T theabsolute temperature of molecules and H = kBT/mg theatmospheric scale height, being g the gravity accelerationand m the mean molecular mass.

Just for this reason, the collision frequency m varies withthe height h above ground as:

mðhÞ ¼ m0pðhÞp0¼ m0 exp �

h� h0H

� �: ð1:bÞ

According to Budden (1965), the generalized theory(Sen and Wyller, 1960) is important in the detailed quanti-tative interpretation of certain experiments, but, for mostpractical radio propagation problems, the classical theory(Appleton and Chapman, 1932) suffices, especially whenappropriate values of the effective collision frequency areused.

1.2. Measurement of absorption

The three principal techniques for measuring absorptionare called the A1, A2, and A3 methods. These methods aredescribed in detail in Rawer (1976) and Hunsucker (1991).

The A1 or pulse reflection method consists of the mea-surement of the amplitude E1, E2, E3, . . . , etc., of echoesreflected once, twice, etc., from the ionosphere. Themethod has been discussed by Piggott et al. (1957). Theeffective reflection coefficient q is given with just one iono-spheric reflection by

q ¼ aE1h0; ð2:aÞ

with two ionospheric reflections and one ground reflectionby

q2qg ¼ 2aE2h0 ð2:bÞ

and so on; i.e.

qnqn�1g ¼ naEnh0: ð2:cÞ

Here qg is the reflection coefficient of the ground, a is a cal-ibration that depends on factors such as transmitter powerand antenna gain, and h0 is the group height that accountsfor spatial diffraction of the wave. Hence

qqg ¼2E2E1¼ 3E3

2E2¼ � � � ðnþ 1ÞEnþ1

nEn; ð2:dÞ

which eliminates the need to know a. The ground reflectioncoefficient can be estimated from the ground characteris-tics. Alternatively, qqg can be measured late at night whenthe absorption is small and q � 1. When q is known fromEq. (2.d), substitution in Eq. (2.a) gives a, which makespossible the measurement of absorption during the day,when only one echo may be present. The effects of distanceattenuation on high frequency (HF) absorption measure-ments have been discussed by Whitehead and Jones(1972). The measurement of absorption is complicated byecho fading, with amplitude variations on order of 10:1over fading periods varying from a few seconds to tens ofminutes. Amplitudes can be measured visually, photo-graphically or, preferable, by digital sampling of the echopeak. Some additional problems arise because of the super-position of ordinary and extraordinary pulses, by partialreflections, by the dispersion of echoes, and by the presenceof interfering signals.

The A2 or riometer (relative ionospheric opacity meter)technique (Little and Leinbach, 1959) measures the inten-sity of the wideband noise that impinges on the Earth fromdeep space. The equipment consists of a stable receiver, asuitable antenna, a recorder, and a noise-diode calibrator.The calibration is carried out periodically by switchingfrom the antenna to the calibrator. The received noisepower is a function of sidereal (star) time, since the antennabeam will sample the same part of sky each day, as theEarth rotates. The absorption is, therefore, the differencebetween the signal actually received and that which wouldhave been received at the same sidereal time in the absenceof absorption (Krishnaswamy et al., 1985). In using thistechnique a compromise frequency has to be selected. Onthe one hand, one would prefer to use a frequency as highas possible to avoid deviative effects but on higher frequen-cies the galactic noise level decreases and the ionosphericabsorption decreases. A typical frequency used with thistechnique is 30 MHz, with which absorption changes ofabout 0.1 dB can be measured. The problem of interferencecan be partially removed by recording the minimum signalreceived while the receiver frequency is swept over a smallrange (but many bandwidths). Another difficulty arisesduring solar radio bursts. Thus a polar-pointing antenna

A. Settimi et al. / Advances in Space Research 53 (2014) 201–218 203

is desirable. For discussion of the limitations of the tech-nique see Little (1957).

In the A3 method, the output of a continuous-wave(CW) transmitter is recorded. The method has the advan-tage of simplicity and sensitivity but it is incapable of dis-criminating between various echoes. This difficulty can bepartially removed by a suitable choice of frequency; forexample, near the gyro-frequency the contributions fromthe extraordinary ray wave and the higher order echoesare small. In the HF bandwidth, a CW system can bestbe calibrated by assuming the night-time absorption to bezero. At low (LF) and middle (MF) frequencies, whichare used for studying the night-time absorption, the systemis calibrated using the ground wave. This technique isattractive since broadcast transmitters are available. Also,we are usually interested in changes of absorption so thataccurate knowledge of the zero absorption level is not crit-ical. For further discussion of this technique see Schwentek(1966). This technique is subdivided according to frequencybandwidth: A3(a) on frequencies higher than 2 MHz andA3(b) on frequencies lower than 2 MHz (see Rawer(1976) and Hunsucker (1991) for further details).

2. The ICEPAC formula

This section describes the equations used to calculatelosses for one ionospheric reflection (1 hop) mode. Themodel is intended to cover the frequency bandwidth from3 to 30 MHz. All modes which result from a complete elec-tron density profile are covered, as well as sporadic-Emodes. The theoretical background and method of mea-surements are given in the manual on absorption measure-ments (Rawer, 1976). The equations described below areintended to be used on a worldwide basis, and are limitedby the available worldwide prediction data base. The equa-tions are based on the CCIR 252-2 (1970) loss equationusing a philosophy that modifications are made only whenmeasured values demand a change. The CCIR loss equa-tion was primarily derived from F2 low-angle modes withoperating frequencies not greater than the frequency ofoptimum traffic (FOT). For these conditions, there is noneed to modify the equation.

Free-space losses result from the geometrical dispersionof energy as the radio wave propagates away from thetransmitter. In ionospheric propagation, the incrementalcross section of the ray bundle at the receiver depends uponthe physical properties of the ionosphere medium and thegeometry of the propagation path. Simplifying assump-tions are made in the program so that transmission lossescan be calculated in a practical manner. In the simplestmodel of sky-wave propagation, it is assumed that theEarth and the ionosphere are both flat and that the reflec-tion is specular (mirror-like). For this type of propagation,the energy density diminishes as the inverse square of theray-path distance (Piggott, 1953).

The absorption in the D–E regions of the ionosphericmedium is usually the major loss (after free space) in radio

wave propagation via the ionosphere. To describe thiseffect, Martyn’s (1935) theorem relating vertical and obli-que path and the quasi-longitudinal approximation to theabsorption loss is used (Davies, 1990):

LðfobÞ ¼ LðfvÞ cos u0; ð3Þ

LðfvÞ ¼ CZ hðfvÞ

h0

Nml dh

ðfv þ fHÞ2 þ m2p� �2 ; ð4Þ

where:

C = constant,h0 = height at bottom of ionospheric medium,h(fv) = height of reflection for frequency fv,N = N(h) electron density profile,m = m(h) collision frequency,l = refractive index,fob = oblique sounding frequency,fv = vertical sounding frequency,fH = gryofrequency, (fL, the longitudinal component ofgyrofrequency, is usually put into equation)u0 = angle of the Earth’s normal to ray path at heighth0.

Eq. (4) can be put into the form (Davies, 1990)

LðfvÞ ¼ ðm=cÞðh0 � hpÞ; ð5Þ

where:

m = effective collision frequency,h0 = virtual height of reflection,hp = phase height of reflection,c = velocity of light.

Since hp is bounded and h0 is not, there is a strong depen-

dence on frequency which cannot be explained simply byan inverse-frequency-squared law. The usual method ofanalysis has been to write equation (4) in the form (defini-tion of the global absorption parameter A):

LðfvÞ ¼AðfvÞ

ðfv þ fHÞ2 þ ðm=2pÞ2; ð6Þ

Most analyses of absorption are based on estimated A(fv).This has sometimes been done by assuming non-deviativeabsorption only, and thereby trying to ignore the frequencydependence. When this is done with a small data base andno adjustment is made for the frequency dependence, theresults can be misleading (and possibly inaccurate).

The CCIR absorption equation is based upon the USArmy Signal Radio Propagation Agency study (Laitinenand Haydon, 1962), with a modification for lower frequen-cies by Lucas and Haydon (1966). The exponent of the fre-quency term (fv + fH) and values for the terms (m/2p)

2 andA(fv) were determined by least square curve fitting. Theoblique loss measurements were normalized to virtual lossmeasurements to give a standard comparison method, aswell as to expand the data base, by the equation:


LðfobÞ ¼LðfvÞ½ðfv þ fLÞ2 þ ðm=2pÞ2�ðfob þ fHÞ2 þ ðm=2pÞ2

sec u0: ð7Þ

The data used were for F-layer modes only. The fittedequation is the ICEPAC formula (Stewart, undated):

LðfobÞ ¼677:2 � I � sec u0ðfob þ fLÞ1:98 þ 10:2

: ð8Þ

That is, the averaged value of A(fv) is 677.2�I, where(Stewart, undated):

I ¼ �0:04þ expð�2:937þ 0:8445 � foEÞ: ð9Þ

The formula for absorption index I is in terms of E plasmafrequency foE which includes the variation in zenith angleand solar activity. There have been attempts to modifyand replace Eq. (8) by independent researchers. Barghausenet al. (1969), as Schultz and Gallet (1970), used the meth-ods described by Piggott (1953) on a smaller data base thanused to derive Eq. (8), but did not add the frequency depen-dence (non-deviative absorption) suggested by Piggott(1953): essentially the same as Eq. (5) is obtained, but fora parabolic E-layer. George (1971) has developed anabsorption equation using an A(fv) which has an implicit(h0 � hp) curve. The method of supplementing Eq. (8) de-scribed by Stewart (undated) was based upon area cover-age and radar backscatter data and was developed in twosteps, first to correct Eq. (8) for E-layer modes and thenfor frequency dependence (deviative absorption). It isessentially based upon the suggestion in Piggott (1953)and in George (1971), but with the advantage of havingthe electron density profiles available on a worldwide basis(see also Rawer (1976), Bibl et al. (1961), and Fejer (1961)for further discussion).

3. The IONORT-ISP system

Pezzopane et al. (2011) described how the joint utiliza-tion of autoscaled data such as the F2 peak critical fre-quency foF2, the propagation factor M(3000)F2, and theelectron density profile coming from two reference stations(Rome and Gibilmanna), and the regional SIRMUP (Sim-plified Ionospheric Regional Model Updated) and globalIRI (International Reference Ionosphere) models can pro-vide a valid tool for obtaining a real-time three-dimen-sional (3-D) electron density mapping of the ionosphericmedium. Preliminary results of the proposed 3-D modelare shown by comparing the electron density profiles givenby the model with the ones measured at three testingionospheric stations (Athens, Roquetes, and San Vito).

Azzarone et al. (2012) described an useful software tool,called IONORT (IONOspheric Ray Tracing), for calculat-ing a 3-D ray tracing of high frequency (HF) waves in theionospheric medium. This tool runs under Windows oper-ating systems and its user-friendly graphical interface facil-itates both the numerical data input/output and the two/three-dimensional visualization of the ray path. In orderto calculate the coordinates of the ray and the three

components of the wave vector along the path as depen-dent variables, the core of the program solves a system ofsix first order differential equations, the group path beingthe independent variable of integration. IONORT uses a3-D electron density specification of the ionosphere, as wellas geomagnetic field and neutral particles–electrons colli-sion frequency models having validity in the area ofinterest.

The 3-D electron density representation of the iono-sphere medium computed by the assimilative IRI-SIR-MUP-P (ISP) model was tested using IONORT, thesoftware application for calculating a 3-D ray-tracing forHF waves in the ionosphere (Settimi et al., 2013a). A radiolink was established between Rome (41.89�N, 12.48�E) inItaly, and Chania (35.51�N, 24.02�E) in Greece, withinthe ISP validity area, and for which oblique soundingsare conducted. The ionospheric reference stations, fromwhich the autoscaled foF2 and M(3000)F2 data and real-time vertical electron density profiles were assimilated bythe ISP model, were Rome and Gibilmanna (37.99�N,14.02�E) in Italy, and Athens (37.98�N, 23.73�E) in Greece.IONORT was used, in conjunction with the ISP and theIRI 3-D electron density grids, to synthesize oblique iono-grams. The comparison between synthesized and measuredoblique ionograms, both in terms of the ionogram shapeand the maximum usable frequency characterizing theradio path, demonstrates both that the ISP model can moreaccurately represent real conditions in the ionosphere thanthe IRI, and that the ray-tracing results computed byIONORT are reasonably reliable.

Many collision frequency models are available. Forinstance, the constant collision frequency, the exponentialor tabular profiles (Jones and Stephenson, 1975). To addother collision frequency models, the user must write a sub-routine that will calculate the normalized frequency Z andits gradients (oZ/or, oZ/oh, oZ/ou) as a function of posi-tion in spherical polar coordinates (r,h,u). The normalizedfrequency is defined as Z = m/2pf, where m is the collisionfrequency between electrons and neutral air moleculesand f is the wave frequency. The coordinates (r,h,u) referto a computational coordinates system, which is not neces-sarily the same as geographic coordinates. In particular,the ionospheric dipole model of the Earth’s magnetic fielduses the axis of computational coordinate system as theaxis for dipole field. When using this dipole model, thecomputational coordinate system is a geomagnetic coordi-nate system, and the Earth’s magnetic field is defined ingeomagnetic coordinates.

The IONORT-ISP system applies a frequency modelconsisting of a double exponential profile (Jones andStephenson, 1975) (see Fig. 1)

mðhÞ ¼ m1e�a1ðh�h01Þ þ m2e�a2ðh�h

02Þ; ð10Þ

where h is the height above the ground.Specifying for the first exponential:

50

100

150

200

250

100 102 104 106 108

νν νν(h) [s-1] (log)

h [k

m]

Fig. 1. Semi-logarithmic plot of the collision frequency m, in units of s�1,as function of the height above ground h, in the range (50.0 km, 250.0 km)(see Eq. (10)).


collision frequency at height h01, m1 = 3.65 � 104 colli-sions per second;Reference height, h01 = 100 km;exponential decrease of m with height, a1 = 0.148 km

�1.

Specifying for the second exponential:

collision frequency at height h02, m2 = 30 collisions persecond;reference height, h02 = 140 km;exponential decrease of m with height, a2 = 0.0183 km

�1.

3.1. The oblique sounding subroutine

Figs. 2a and 2b show a flowchart of the IONORT-ISP(or IRI) system, calling in cascade the OBLIQUESOUNDING and COMPLEIK subroutines (based onthe COMPLex EIKonal model) that produce respectively“SYNTHESIZED IONOGRAM.txt” and “L_COMP-LEIK(dB) vs FREQUENCY(MHz).txt” as output files.The IONORT program runs on a file, named “GRIDPRO-FILES.txt”, which is the output of the ISP (or IRI) model,reproducing the 3-D electron density grid over the Mediter-ranean area on a chosen date (in the format dd/mmm/yyyy). IONORT reads an input file, named “RAY TRAC-ING INPUT.ini”, to initialize the default inputs related toall the computational parameters needed by the ray-tracingalgorithm, including the geographical position of the trans-mitter (TX), represented in a geocentric spherical coordi-nate system (RTX, THTX, PHTX) (RTX in km, THTX andPHTX in radians). The coordinates of the transmitter canbe expressed in terms of its height HTX = RTX � RT (inkm) from ground level, where RT = 6371 km is theaveraged radius of the Earth, and in terms of latitudeLATTX = (180�/p)(p/2 � THTX) and longitude LONTX =

(180�/p)PHTX angles (in degrees). The program iterativelycalculates a ray path for each nested loop cycle K relativeto the frequency FK, the elevation ELK, and the azimuthAZK angles, specified by a frequency-step FK = FK + -FSTEP, elevation-step ELK = ELK + ELSTEP, and azimuth-step AZK = AZK + AZSTEP procedure, withK = START, . . . ,END (see Fig. 2a). The computing timefootprint of a complete IONORT simulation can be esti-mated as Dt � NTOT, if Dt is assumed as the computing timeof every loop cycle, where NTOT = NF � NEL � NAZ is thetotal number of cycles, with NF = 1 + [(FEND � FSTART)/FSTEP] the number of frequencies, NEL = 1 + [(ELEND -� ELSTART)/ELSTEP] and NAZ = 1 + [(AZEND -� AZSTART)/AZSTEP] the number of elevations andazimuths respectively.

At the end of each Kth nested loop cycle, the OBLIQUESOUNDING subroutine lists the parameters of ray-tracingoutput, i.e. the geographical position of the arrival point,still represented in the geocentric spherical coordinate sys-tem (RK, THK, PHK), and also by the corresponding groupdelay time TK (in ms). Moreover, the OBLIQUE SOUND-ING subroutine reads an input file, named“RECEIVER.txt”, to read the geographical position ofthe receiver (RX), again represented in the geocentricspherical coordinate system, and the tolerated accuracyof the RX position, defined by the triplet of relative errors(ERROR_R, ERROR_TH, ERROR_PH). These relativeerrors should not exceed the following reasonable upperlimits: ERROR_R 6 emax, ERROR_TH 6 0.1%,ERROR_PH 6 0.1%. In fact, emax, i.e. the component 42of vector W represented in the input file “RAY TRACINGINPUT.ini” used by IONORT is defined as the maximumallowable relative error in any single step length for any ofthe equations being integrated. If ERROR_TH andERROR_PH are less than 0.1%, then the arrival point isdisplaced from the receiver by not more than a few kilome-ters. Finally, the OBLIQUE SOUNDING subroutine com-pares the geographical positions of the arrival point andthe receiver, assessing how close the arrival point is tothe receiver. More precisely, OBLIQUE SOUNDING sub-routine assumes that the target has been reached, or ratherthat the arrival point may be approximated with the recei-ver, when it falls within a circle centred on the receiver, ofradius defined by the tolerated accuracies (ERROR_R,ERROR_TH, ERROR_PH) of the RX position. Formally,the subroutine evaluates the following logical condition(see Fig. 2b):

ðABSððRK � RRXÞ=RRXÞ � ERROR RÞ: AND:ABSððTH K � TH RXÞ=TH RXÞ� ERROR THÞ:AND:ABSððPH K � PH RXÞ=PH RXÞ� ERRORP H :

If this expression is false, then it means that the trans-mitter–receiver radio link has not been established at fre-quency FK, elevation ELK, and azimuth AZK. In thiscase, the OBLIQUE SOUNDING subroutine returns

Fig. 2a. Flowchart of the IONORT-ISP (or IRI) system, calling in cascade the OBLIQUE SOUNDING and COMPLEIK (COMPLex EIKonal)subroutines that produce respectively “SYNTHESIZED IONOGRAM.txt” and “L_COMPLEIK(dB) vs FREQUENCY(MHz).txt” as output files. Theiterative procedure consists of a nested loop cycle with frequency, elevation and azimuth steps.


control to the IONORT system, so the Kth loop cycle isconcluded while the next cycle starts. By contrast, the trans-mitter and receiver are radio linked and in this case, OBLI-QUE SOUNDING updates the MUF variable(MUF = FK) which, at the end of whole IONORT simula-tion, will coincide with the maximum usable frequency(MUF) of the radio link. Simultaneously, the subroutinestores in the final output file named “SYNTHESIZED

IONOGRAM.txt”, the values of frequency FK and groupdelay TK that represent the Kth ionogram point PK = (FK, -TK) relative to the Kth cycle giving rise to a transmitter–receiver radio link (see Fig. 2b). At the end of the completesimulation, the final output file will include all the N points,PK1 = (FK1,TK1), PK2 = (FK2, TK2), . . . ,PKN = (FKN,TKN),composing the ionogram trace of the radio link. It is worthnoting that the nested loop cycle in the azimuth transmission

Fig. 2b. Exploded details of the logical box shown at the centre of Fig. 2a containing the IONORT program and both the OBLIQUE SOUNDING andCOMPLEIK subroutines.


angle is optional. If the loop cycle in azimuth angle is notinitiated, then the azimuth between the transmitter and thereceiver becomes a fixed constant in the ray-tracing, i.e.AZK = AZSTART = AZEND, which can be calculated simplyby applying spherical trigonometry:

tgðAZSTARTÞ ¼ sinðTH RXÞ sinðTH TXÞ sinðPH RX � PH TXÞ=½cosðTH RXÞ � cosðTH TXÞ cosðDTX�RXÞ�;

where

cosðDTX�RXÞ ¼ cosðTH RXÞ cosðTH TXÞþ sinðTH RXÞ sinðTH TXÞ cosðPH RX � PH TXÞ:

Omitting the azimuth cycle is convenient when possible,as the computing time footprint of the complete IONORTsimulation is reduced to Dt � NF � NEL, with Dt the comput-ing time for every cycle.

Fig. 3 plots the ordinary and extraordinary trace of theoblique ionograms, synthesized by the IONORT-ISP sys-tem, over the Rome–Chania radio link on 3 July 2011 at17:00 UT, 4 July 2011 at 20:00 UT, 6 July 2011 at12:00 UT, 7 July 2011 at 17:00 UT, 8 October 2011 at06:15 UT, 9 October 2011 at 02:15 UT, 21 October 2010at 12:00 UT, 29 May 2010 at 12:00 UT. WF (with field) indi-cates that the oblique ionograms were computed by takingthe geomagnetic field into account. In the synthesized iono-grams, with one ionospheric reflection (1 hop), the azimuth

cycle is applied, with the elevation angle step set to 0.2� andthe RX accuracy to 0.1%.

Each synthesized ionogram was computed without apply-ing the collision frequency model (10), in order to plot the ion-ogram regardless of the absorption. Excluding the days 4 Julyand 9 October of the year 2011, respectively at the solar termi-nator on 20:00 UT and in the night time on 2:15 UT, for all theother cases, the oblique ionogram is composed by: (1) thetrace corresponding to the ionospheric F1–F2 layers at highaltitudes (h > 150 km) and characterized by a low absorptioncoefficient (L 6 20 dB); (2) the trace corresponding to the E-layer at lower altitude (90 km < h 6 150 km) and with anhigher absorption coefficient (L 20 dB). Generally, the ion-ogram computed applying the collision frequency model (10)does not show the trace corresponding to the E-layer for thehigh absorption.

By the way, the E-layer must not be confused with thesporadic E-layer (Davis and Johnson, 2005). IONORTruns on ISP, modelling just the normal and cyclic ioniza-tion properties of the ionospheric (non sporadic) E-layer,which is absent in the night time and occurs during the day-light hours, especially at the noon. Indeed, no conclusivetheory has yet been formulated as to the origin of spo-radic-E layer. As its name suggests, sporadic-E layer is anabnormal event, not the usual condition, but can happenat almost any time. Sporadic-E activity peaks predictablyin the summertime in both hemispheres.

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2 6 10 14 18 22 26 30


FREQUENCY [MHz]

GR

OU

P D

ELA

Y [m

s]


1

3

5

7

9

11

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

GR

OU

P D

ELA

Y [m

s]


1

3

5

7

9

11

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

GR

OU

P D

ELA

Y [m

s]

ROME - CHANIA 29 MAY 2010 12:00 UTIONORT - ISP 1 hop

Fig. 3. Ordinary and extraordinary trace of the oblique ionograms, synthesized by the IONORT-ISP system, over the Rome–Chania radio link on 3 July2011 at 17:00 UT, 4 July 2011 at 20:00 UT, 6 July 2011 at 12:00 UT, 7 July 2011 at 17:00 UT, 8 October 2011 at 06:15 UT, 9 October 2011 at 02:15 UT, 21October 2010 at 12:00 UT, 29 May 2010 at 12:00 UT. WF (with field) indicates that the oblique ionograms were computed by taking the geomagnetic fieldinto account. In the synthesized ionograms, with one ionospheric reflection (1 hop), the azimuth cycle is applied, with the elevation angle step set to 0.2�and the receiver (RX) accuracy to 0.1%.



4. The complex eikonal model for the ionospheric D-layer

Settimi et al. (2013b) conducted a scientific review on thecomplex eikonal, extrapolating the research perspectives onthe ionospheric ray-tracing and absorption.

As regards the scientific review, the eikonal equation isexpressed, and some complex-valued solutions are definedcorresponding to complex rays and caustics.

As regards the research perspectives, the authors con-tinue the topics discussed by Bianchi et al. (2009), propos-ing a novelty with respect to the other referencedbibliography: indeed, the medium is assumed as dissipative.Under these conditions, suitable generalized complex eiko-nal and transport equations are derived.

Settimi et al. (2013c) conducted a scientific review onionospheric absorption, extrapolating the research perspec-tives of a complex eikonal model for one-layer ionosphere.

As regards the scientific review, there is deduced a quasi-longitudinal (QL) approximation for non-deviative absorp-tion which is more refined than the corresponding equationreported by Davies (1990).

As regards the research perspectives, there is analyzed indepth a complex eikonal model for one layer ionosphere,already discussed by Settimi et al. (2013b). A simple for-mula is deduced for a simplified problem. A flat layeringionospheric medium is considered, without any horizontalgradient. The authors demonstrate that the QL non-devia-tive absorption coefficient according to the complex eiko-nal model is more accurate than Rawer’s theory (1976) inthe range of middle critical frequencies.

4.1. A simple formula for a simplified problem

The cited paper (Settimi et al., 2013b) proposed a newformula, useful to calculate the absorption due to the prop-agation across the ionospheric D-layer, which can beapproximately modelled by a linearized complex refractiveindex, covering a short range of heights betweenh1 = 50 km and h2 = 90 km.

According to Budden (1988), rocket techniques haveevidenced that, in the daytime, the D-layer shows, almostas a rule, its maximum (and minimum) of electron densityin the vicinity of 80 km (and 85 km). In authors opinion,this evidence is not so strong, and even if the refractiveindex nR(h) is not a monotonically decreasing function ofheight h along the D-E layers valley, anyway this is not asubstantial correction: nR(h) can be linearized up toh2 = 90 km about.

Settimi et al. (2013c), assuming the analytical continuityof complex eikonal model with the quasi-longitudinal (QL)approximation for non-deviative absorption, demonstratesthe necessary and sufficient condition to equate the colli-sion frequency deriving from the altitude profile adoptedfor complex refractive index (Settimi et al., 2013b),

nðhÞ ¼ 1; h < h0; ð11:aÞ

nðhÞ ¼ nRðhÞ þ inIðhÞ; h P h0nRðhÞ ¼ n0 þ aRðh� h0ÞnIðhÞ ¼ aI

8><>: ;where h0 is the height of the ionospheric bottom andn0 = n(h0) the refractive index across the ionosphere–neu-tral atmosphere boundary (in a first-order approximation,the boundary refractive index n0 could be assumed as a realnumber slightly different from 1, i.e. in any case n0 – 1, sothat the refractive index n(h) is a discontinuous function ofheight h, i.e. crossing h = h0), to the variation of collisionfrequency with the altitude (Budden, 1988),

mðhÞ ¼ mmax exp �h� hmax

H

� �

ffi mmax 1�h� hmax

H

� �; jh� hmaxj � H ; ð12Þ

where H is the atmospheric scale height, mmax is a constant,i.e. mmax = m(hmax), and hmax is the height corresponding tothe maximum electron density Nmax, i.e. Nmax = N(hmax)(the constant mmax is not the maximum collision frequencybut the collision frequency at the “maximum height” hmax).

The QL approximation for non-deviative absorption, asdeduced by Settimi et al. (2013c), is more refined than thecorresponding equation reported by Davies (1990); and,linearizing the involved equations, there are obtained thecoefficients aR and aI as functions of the angular frequencyx, the collision frequency mmax, the scale height H, themean magnetic gyro-frequency hxH i, and the height ofthe ionospheric bottom h0, the refractive index across theionosphere–neutral atmosphere boundary n0 = n(h0):

aR ¼ �1

2

1�n20

H

1þ hmax�h0H; ð13:aÞ

aI ¼ �ð1� n0Þ2

Hmmax

x� hxH i1þ mmax

x� hxH i

� �2" #: ð13:bÞ

A reasonable hypothesis should be assumed for Eq. (13):the boundary refractive index n0 is a real number slightlyless than 1, i.e. n0 < 1, so that both the coefficients aRand aI are negative, i.e. aR < 0 and aI < 0.

Therefore, the linearized analytical profile for complexrefractive index (13.b) can be re-arranged as:

nRðhÞ ffi n0 �1� n20

2

ðh� h0Þ=H1þ ðhmax � h0Þ=H

; h >> h0: ð14:aÞ

nI ¼ �ð1� n0Þ2

Hmmax

x� hxH i1þ mmax

x� hxH i

� �2" #: ð14:bÞ

Just the reasonable hypothesis assumed below Eq. (13)could imply the expected conclusions for Eq. (14): the realrefractive index nR(h) is a decreasing function of height h,while the imaginary refractive index nI is negative, as sub-stantially correct for any ionospheric profile of the D-layer(Budden, 1988).


Moreover, considering a vertical radio sounding withone ionospheric reflection (1 hop), once calculated the opti-cal path c : h1 ! h2 (Settimi et al., 2013c),

DlðvÞ12 ¼Z h2

h1

nRðhÞdh

ffi ðh2 � h1Þ n0 �1� n20

4

ðh1 þ h2 � 2h0Þ=H1þ ðhmax � h0Þ=H

� �;

jh2 � h1j � h0 < hmax;

ð15Þ

it results proportional to the integral absorption coefficient(Settimi et al., 2013c):

bðvÞ12 ffi 21� n01þ n0

DlðvÞ12 1þhmax � h0

H

� �mmax

c

� xxþ hxHi

1þ mmaxxþ hxH i

� �2" #: ð16Þ

Note that the refractive index n0 = n(h0) can be compu-tationally assumed as n0 = 1 � emax for any ray-tracingprogram, where emax is defined as the maximum allowablerelative error in single step length for any of the equationsbeing integrated. To get a very accurate (but expensive) raytrace, it is possible use a small emax (about 10

�5 or 10�6).For a cheap, approximate ray trace, a large emax (10

�3 oreven 10�2) should be used. For cases in which all the vari-ables being integrated increase monotonically, the total rel-ative error can be guaranteed to be less than emax (Jonesand Stephenson, 1975).

Instead, considering an oblique radio sounding with oneionospheric reflection (1 hop), the Martyn’s (1935) absorp-tion theorem assures that the integral absorption coefficientbðobÞ12 of a wave at angular frequency x incident on a flationospheric medium with angle u0 is further dependenton the secant of u0. It results a simple formula for a simpli-fied problem (Settimi et al., 2013c):

bðobÞ12 jxob¼x ¼ bðvÞ12 jxv¼x cos u0 cos u0

¼ 2 1� n01þ n0

DlðvÞ12 1þhmax � h0

H

� �mmax

c

� xxþ hxH i

1

sec u0þ mmax

xþ hxH i

� �2sec u0

" #:

ð17Þ

4.3. The COMPLEIK subroutine of the IONORT-ISPsystem

Implying the reader to be familiar with the mathemati-cal symbols and their physical meaning, as introducedand discussed by Jones and Stephenson (1975), Azzaroneet al. (2012), and Settimi et al. (2013a,b,c), let us reportin Appendix A the FORTRAN code for the COMPLEIK(COMPLex EIKonal) subroutine of the IONORT-ISP sys-tem (see Figs. 2a,b). The FORTRAN code is enriched bysome comments. All variables are computed in the Interna-

tional System (SI) of units. Few auxiliary variables areemployed, referring to the D-layer, which is modelled asone ionospheric layer between the heights h1 = 50 kmand h2 = 90 km, being characterized by suitable mean val-ues for the gravity acceleration g_MEAN, the absolutetemperature T_MEAN, and the atmospheric scale heightH. The auxiliary variables refer even to the whole iono-spheric model, involving the refractive index n_h0 acrossthe ionosphere–neutral atmosphere boundary h0, the meanvalue of magnetic gyro-frequency FH_MEAN, the heightrelative to the maximum electron density hMAX and thecorresponding collision frequency NU_hMAX. TheCOMPLEIK subroutine implements the complex eikonalmodel equations. The Martyn’s (1935) absorption theoremis applied to transform a vertical to an oblique radiosounding with one ionospheric reflection (1 hop), calculat-ing the secant SEC of incident angle PH0, the vertical opti-cal path OPT_PATH, and the oblique absorptioncoefficient L_COMPLEIK, then reported in decibel unitsL_COMPLEIK_dB.

The COMPLEIK subroutine stores, in the final outputfile so called “L_COMPLEIK(dB) vs FRE-QUENCY(MHz).txt”, the values for frequency FK andoblique absorption coefficient LK which corresponds tothe Kth ionogram point PK = (FK,TK), relative to the Kthcycle giving rise to a transmitter–receiver radio link (seeFig. 2b). At the end of the complete simulation, the finaloutput file will include all the N points, PK1 = (FK1,LK1),PK2 = (FK2,LK2), . . . ,PKN = (FKN,LKN), composing theabsorption profile of radio link.

4.4. Analysis and discussion

Let us consider a radio link, with one ionospheric reflec-tion (1 hop), between the transmitter (TX), Rome, Italy(lat1 = 41.89�N, lon1 = 12.48�E) and the receiver (RX),Chania, Crete (lat2 = 35.51�N, lon2 = 24.02�E) stations.The IRI (2007) model allows computing the E plasma crit-ical frequency foE on any date and time, relative to the mid-point M between TX and RX, M = [latM = (lat1 + lat2)/2 = 38.70�N, lonM = (lon1 + lon2)/2 = 18.25�E]. The IRIparameter foE is an input of the ICEPAC formula (Stewart,undated), useful to calculate in a phenomenological waythe non-deviative absorption coefficient, in quasi-longitudi-nal (QL) approximation (Eqs. (8), (9)). Let use theIONORT program, simulating a three-dimensional (3-D)ray tracing of high frequency (HF) waves in the iono-spheric medium, in conjunction with the ISP model, whichgenerates a 3-D real-time electron density grid of the iono-sphere over the Mediterranean area, in order to synthesizeoblique ionograms of the radio link between Rome andChania. The IONORT-ISP system allows computing theheight hmax relative to the maximum electron density Nmax,which is an input of the COMPLEIK subroutine (based inthe COMPLex EIKonal model), useful to calculate theo-retically the QL non-deviative absorption coefficient (Eqs.(15)–(17)).

10-2

10-1

100

101

102

2 6 10 14 18 22 26 30

03 JUL 2011 17:00 UT04 JUL 2011 20:00 UT06 JUL 2011 12:00 UT07 JUL 2011 17:00 UT08 OCT 2011 06:15 UT09 OCT 2011 02:15 UT21 OCT 2010 12:00 UT29 MAY 2010 12:00 UT

FREQUENCY [MHz]

AB

SOR

PTIO

N R

ELA

TIVE

DEV

IATI

ON

[%] (

log)

WF, Ordinary ray

10-2

10-1

100

101

102

2 6 10 14 18 22 26 30

03 JUL 2011 17:00 UT04 JUL 2011 20:00 UT06 JUL 2011 12:00 UT07 JUL 2011 17:00 UT08 OCT 2011 06:15 UT09 OCT 2011 02:15 UT21 OCT 2010 12:00 UT29 MAY 2010 12:00 UT

FREQUENCY [MHz]

AB

SOR

PTIO

N R

ELA

TIVE

DEV

IATI

ON

[%] (

log)

WF, Extraordinary ray

Fig. 4. Refer to the caption of Fig. 3. Semi-logarithmic plots of thepercentage relative deviation (%), in the quasi-longitudinal (QL) approx-imation, between the non-deviative integral absorption coefficientsaccording to the complex eikonal model (COMPLEIK subroutine) (seeEqs. (15)–(17)) and the ICEPAC formula (see Eqs. (8) and (9)), as afunction of the frequency f (MHz), for both the ordinary and extraor-dinary rays.


Note that suitable operative conditions must hold toapply the theorems of Breit and Tuve and Martyn (Mar-tyn, 1935; Davies, 1990). Accordingly, the real obliquesounding between the stations of Rome and Chania canbe transformed into a virtual oblique sounding, which obli-que sounding, in turn, is reduced to a virtual verticalsounding along the vertical line of the midpoint M. Thus,the ICEPAC formula can be referred to the radio linkRome–Chania, once the IRI (2007) model has computedthe critical frequency foE relative to the midpoint M.Indeed, the Breit and Tuve’s and Martyn’s theoremsshould be applied during hours of the day in which the ion-ospheric medium is characterized by small horizontal gra-dients, when the azimuth angle of transmission isassumed to be a constant along the great circle path (Set-timi et al., 2013a). By the way, the COMPLEIK subroutinecan be ideally used for a flat layering ionosphere, withoutany horizontal gradient, so characterized by an electrondensity profile dependent only on the altitude (Settimiet al., 2013b,c). The complex eikonal model can be imple-mented more directly than the ICEPAC formula, sincethe height hmax is computed numerically within theIONORT-ISP system, instead the critical frequency foEonly in a phenomenological way, and even applying theIRI (2007) model.

Fig. 4 shows the semi-logarithmic plots of the percent-age relative deviation (%), in the QL approximation,between the non-deviative integral absorption coefficientsaccording to the complex eikonal model (COMPLEIKsubroutine) (see Eqs. (15)–(17)) and the ICEPAC formula(see Eqs. (8) and (9)), as a function of the frequency f(MHz), for both the ordinary and extraordinary rays.

All the radio links Rome–Chania tested at frequencieswithin the HF bandwidth 6 MHz 6 f 6 18 MHz show anabsorption relative deviation that is minimum for theordinary ray, assuming small values in the range (10�1%,10%), and it is maximum for the extraordinary ray, assum-ing larger values in the range (1%, 102%), especially at low-est frequencies (6 MHz 6 f 6 10 MHz). Indeed, accordingto the complex eikonal model (Settimi et al., 2013b,c),the ionospheric D-layer can be approximately character-ized by a linearized analytical profile of complex refractiveindex, covering a short range of heights betweenh1 = 50 km and h2 = 90 km. Note that the complex eikonalmodel is not numerically reliable for the low critical fre-quencies, as the integral absorption coefficient implies tac-itly that not so low altitudes are involved (Settimi et al.,2013c). At any rate, in the HF bandwidth, the non-devia-tive absorption is mostly limited to the D-layer (Rawer,1976). The ordinary ray is partially absorbed just duringthe propagation across the D-layer, thus its QL non-devia-tive absorption coefficient is rightly calculated by theCOMPLEIK subroutine. Instead, according to the ISPmodel (Pezzopane et al., 2011), the ionospheric mediumis characterized by an electron density profile defined onthe heights range 81 km � h � 378 km de facto excludingthe D-layer (50 km 6 h 6 90 km), though including the

E-layer (90 km < h 6 150 km). Note that if the extraordi-nary ray propagates at lowest radio frequencies, then theAppleton–Hartree dispersion formula may not be adequatein the D–E layers (Davies, 1990). Within the QL approxi-mation, the lower the radio frequencies, the more theextraordinary ray is reflected at bottom altitudes comparedto the ordinary ray. At any rate, for HF sounding, there aretwo distinct height ranges from which appreciable contri-butions are to be expected: in the D-layer, where thenon-deviative formulae may be used, and in the vicinityof E-layer reflection height, where the deviative term mustbe taken into account. The situation is similar for higherfrequencies in the HF bandwidth which penetrate theE-layer and are reflected from the F-layer. In this case,some non-deviative absorption on the E-layer may alsobe important (Rawer, 1976). The extraordinary ray spends

0

20

40

60

80

100

2 6 10 14 18 22 26 30

ICEPAC - WF, Ordinary rayICEPAC - WF, Extraordinary rayCOMPLEIK - WF, Ordinary rayCOMPLEIK - WF, Extraordinary ray

FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]


0

20

40

60

80

100

2 6 10 14 18 22 26 30


FREQUENCY [MHz]

AB

SOR

PTIO

N [d

B]

ROME - CHANIA 29 MAY 2010 12:00 UTIONORT - ISP 1 hop

Fig. 5. Refer to the caption of Fig. 3. Semi-logarithmic plots of the QL approximation for non-deviative absorption coefficients (dB), according to thecomplex eikonal model (COMPLEIK subroutine) (see Eqs. (15)–(17)) and the ICEPAC formula (see Eqs. (8) and (9)), as a function of the frequency f(MHz), for both the ordinary and extraordinary rays.



a longer time in the E-layer than in the D-layer, feelingboth the deviative and non-deviative absorptions due tothe E-layer as well as the non-deviative absorption due tothe D-layer. Thus, its QL non-deviative absorption coeffi-cient is more correctly calculated by the ICEPAC formula,modelling even the E-layer, than the COMPLEIK subrou-tine, which models only the D-layer.

Fig. 5 shows the semi-logarithmic plots of the QL approx-imation for non-deviative absorption coefficients (dB),according to the complex eikonal model (COMPLEIK sub-routine) (see Eqs. (15)–(17)) and the ICEPAC formula (seeEqs. (8) and (9)), as a function of the frequency f (MHz),for both the ordinary and extraordinary rays.

Excluding the days 4 July and 9 October of the year2011, respectively at the solar terminator on 20:00 UTand in the night time on 2:15 UT, for all the other cases,the QL non-deviative absorption profile is composed by:(1) a pair of ordinary and extraordinary traces correspond-ing to the ionospheric F1–F2 layers at high altitudes(h > 150 km) and characterized by a low absorption coeffi-cient (L 6 20 dB); (2) another pair of ordinary and extraor-dinary traces corresponding to the E-layer at bottomaltitude, 90 km < h 6 150 km, and with an higher absorp-tion coefficient (L 20 dB) (see Fig. 3). Compared to theICEPAC formula, the COMPLEIK subroutine providesthe best results for the low absorption traces; indeed, theICEPAC–COMPLEIK fitting is optimal for both theordinary ray, in the whole HF bandwidth, and the extraor-dinary ray, in a wide neighbourhood of the maximumusable frequency (MUF). Instead, the COMPLEIK sub-routine provides the worst results for the high absorptiontraces: indeed, the ICEPAC–COMPLEIK fitting tends toworsen, not only for the ordinary ray but much more forthe extraordinary ray, especially in the lowest frequenciesof the HF bandwidth. Note that if the extraordinary raypropagates at lowest radio frequencies, then the ICEPACformula may not be adequate in the D–E layers (Stewart,undated). The complex eikonal model is not numericallyreliable for the highest critical frequencies, when the linear-ized complex refractive index fails, having to be replaced bya parabolic or even cubic profile (Settimi et al., 2013c). Atany rate, these results should be improved by extending thecomplex eikonal model to higher altitudes in order toinclude even the E-layer besides the D-layer. It could notbe worth the effort, since other equations would result forthe non-deviative absorption coefficient, much less handythan Eqs. (15)–(17), and possibly involving numericalproblems due to spurious oscillations.

In this paper, the complex eikonal model (Settimi et al.,2013b,c) has been used in conjunction with the discreteassimilative ISP grid profiles of the ionospheric electrondensity over the Mediterranean area (Pezzopane et al.,2011). As main results, the COMPLEIK subroutine, char-acterizing only the D-layer, has been joined up to theIONORT-ISP system (Settimi et al., 2013a), which simu-lates the 3-D ray-tracing for HF waves in the ionosphericmedium, de facto excluding the D-layer, though including

the E-layer, in order to allow calculating the QL non-devi-ative absorption coefficient due to the radio propagation inthe D-layer (and E-layer). Then, compared to the ICEPACformula, the non-deviative absorption profiles correspond-ing to the COMPLEIK subroutine are reliable for all theordinary and extraordinary rays effectively radio linkingRome and Chania. Indeed, the transmitting frequencies,generally tuned in the neighborhood of the MUF, are par-tially absorbed in the D–E layers and are propagating inthe F1–F2 layers. Instead, the absorption profiles corre-sponding to the COMPLEIK subroutine lose their reliabil-ity just for all the rays that cannot establish the radio linkRome–Chania. Indeed, these non-transmitting frequencies,especially if tuned away from the MUF, are fully absorbedby the D–E layers and so cannot propagate in the F1–F2layers.

5. Conclusions

The IONORT (IONOspheric Ray-Tracing) program(Azzarone et al., 2012), which simulates the three-dimen-sional (3-D) ray-tracing for high frequencies (HF) wavesin the ionospheric medium, runs on the assimilative ISP(IRI-SIRMUP-P) discrete model over the Mediterraneanarea (Pezzopane et al., 2011). Even if the IONORT-ISPsystem (Settimi et al., 2013a) involves a model of collisionfrequency, anyway it is not enabled to compute the absorp-tion coefficient, neither non-deviative nor deviative. Afterall, according to the ISP model, the ionosphere is charac-terized by an electron density profile defined on the heightsrange 81 km � h � 378 km, de facto excluding the D-layer(50 km 6 h 6 90 km), though including the E-layer(90 km < h 6 150 km).

As main outcome of the paper, now the IONORT pro-gram includes a so called COMPLEIK (COMPLex EIKo-nal) subroutine. Thus, the IONORT-ISP-COMPLEIKsystem is enabled to compute, in quasi-longitudinal (QL)approximation, just the non-deviative absorption coeffi-cient, due to the propagation across the ionosphericD-layer. Indeed, the COMPLEIK subroutine encodes thecomplex eikonal equations (Settimi et al., 2013b,c), whichmodel the D-layer approximately by a linearized analyticalprofile of complex refractive index, covering the shortrange of heights between h1 = 50 km and h2 = 90 km.

In this paper, the simple COMPLEIK algorithm hasbeen compared to the more elaborate semi-empirical ICE-PAC formula (Stewart, undated), which refers to variousphenomenological parameters such as the critical fre-quency of E-layer. COMPLEIK is reliable just like ICE-PAC, with the advantage of being implemented moredirectly. Indeed, the compex eikonal model depends onlyon some parameters of the electron density profile, whichare numerically calculable, such as the maximum height.

In a forthcoming paper, the IONORT-ISP-COMPLEIKsystem will be enabled to compute, in quasi-longitudinal(QL) (or quasi-transverse (QT)) approximation, the devia-tive (or non-deviative) absorption coefficient, due to the


propagation (or reflection) within the E-layer. Indeed, thecomplex eikonal theory may be extended to higher alti-tudes in order to join up the discrete model (ISP) to theanalytical model (COMPLEIK) of E-layer, replacing the

C******************************************************************

C COMPLEIK: The algorithm of COMPLEIK subroutineC simulates the non-deviative absorption by ionospheric collisions,C based on the following simplifying hypotheses:C quasi-longitudinal (QL) propagation for the ray-tracing;C a linear profile of electron density for the D-layer;C a dipole model for the geomagnetic field;C an exponential model for the collision frequency.C******************************************************************

SUBROUTINE COMPLEIKC******************************************************************

C******************************************************************

C SEE: Jones and Stephenson (1975), Azzarone et al. (2012).C******************************************************************

COMMON R(20),T,STP,DRDT(20),N2COMMON/CONST/PI,PIT2,PID2,DEGS,RAD,K,C,LOGTENCOMMON/WW/ID,W0,W(400)COMMON/RK/N,STEP,MODE,E1MAX,E1MIN,E2MAX,E2MIN,FACT,R

COMMON/XX/MODX(2),X,PXPR,PXPTH,PXPPH,PXPT,hMAXC******************************************************************

C******************************************************************

C Optical ray incident on a flat ionospheric mediumC forming the secant angle PH0C******************************************************************

COMMON/PH0/PH0C******************************************************************

C******************************************************************

C COMPLEIK OUTPUT: Absorption coefficient (dB).C******************************************************************

COMMON/L_COMPLEIK_dB/L_COMPLEIK_dBC******************************************************************

C******************************************************************

C SEE: Jones and Stephenson (1975), Azzarone et al. (2012).C******************************************************************

COMPLEX N2REAL LOGTENREAL NU1, NU2EQUIVALENCE (RAY,W(1)), (EARTHR,W(2)), (F,W(6)),>(FH,W(201)),>(NU1,W(251)), (z1,W(252)), (A1,W(253)),>(NU2,W(254)), (z2,W(255)), (A2,W(256))C******************************************************************

C******************************************************************

C SEE: Settimi et al. (2013b), Settimi et al. (2013c).C******************************************************************

REAL K1, K2, K3, K4, K5REAL kB,>m, mH, mp, me,>h1, T1,>h2, T2,>g0, g_MEAN,>T_MEAN, H,>ONE, h0, n_h0,>FH_MEAN, NU_hMAX,>ANGLE, SEC,>OPT_PATH, L_COMPLEIK, L_COMPLEIK_dBDATA K1/0.0053024/, K2/0.0000058/,

linearized complex refractive index with a parabolic or evencubic profile.

Appendix A.

************

************

COMPLEIK01************************

************

COMPLEIK02COMPLEIK03COMPLEIK04

STARTCOMPLEIK05COMPLEIK06

************************

************

COMPLEIK07************************

************

COMPLEIK08************************

************

COMPLEIK09COMPLEIK10COMPLEIK11COMPLEIK12COMPLEIK13COMPLEIK14COMPLEIK15

************************

************

COMPLEIK16COMPLEIK17COMPLEIK18COMPLEIK19COMPLEIK20COMPLEIK21COMPLEIK22COMPLEIK23COMPLEIK24COMPLEIK25COMPLEIK26COMPLEIK27

>K3/0.000003086/, K4/0.2857/, K5/0.770982/ COMPLEIK28DATA h1/50./, T1/273./, COMPLEIK29>h2/90./, T2/187./ COMPLEIK30C******************************************************************************

C******************************************************************************

C INTERNATIONAL SYSTEM (SI) OF UNITSC******************************************************************************

C******************************************************************************

C Boltzmann’s constantC******************************************************************************

kB = 1.3806488*1E-23 COMPLEIK31C******************************************************************************

C******************************************************************************

C Electron, proton, hydrogen atom mass and atmosphere mean molecular massC******************************************************************************

me = 9.10938291 1E-31 COMPLEIK32mp = 1.672621777 1E-27 COMPLEIK33mH = me + mp COMPLEIK34m=29.0 mH COMPLEIK35C******************************************************************************

C******************************************************************************

C IONOSPHERIC D-LAYER BETWEEN THE HEIGHTS h1 = 50 km AND h2 = 90 kmC******************************************************************************

C******************************************************************************

C International Gravity Formula (IGF) 1967:C Mean value of gravity acceleration.C******************************************************************************

g0 = 9.780327 COMPLEIK36g_MEAN = g0 (1. + (K1–K2)/2)–K3 1.E3 (h1 + h2)/2 COMPLEIK37C******************************************************************************

C******************************************************************************

C Mean value of absolute temperatureC******************************************************************************

T_MEAN = T2 (1.–K4 ((m g_MEAN)/(2. kB T2)) 1.E3 (h1–h2)) COMPLEIK38C******************************************************************************

C******************************************************************************

C Atmospheric scale heightC******************************************************************************

H = 1.E-3 ((kB T_MEAN)/(m g_MEAN)) COMPLEIK39C******************************************************************************

C******************************************************************************

C In order to smooth out numerical problems due to spurious oscillations,C the constant ONE can be computationally assumed asC ONE = 1.-E1MAX for any ray-tracing program,C where E1MAX is defined asC the maximum allowable relative error in single step lengthC for any of the equations being integrated.C******************************************************************************

ONE = 1. COMPLEIK40ONE = ONE-E1MAX COMPLEIK41C******************************************************************************

C******************************************************************************

C SEE: Pezzopane et al. (2011), Settimi et al. (2013b,c).C******************************************************************************

C******************************************************************************

C h0:C the height of the ionospheric bottom.C According to the ISP (IRI-SIRMUP-P) model,C the ionosphere is characterized by an electron density profileC defined on the heights range 81 km 6 h 6 378 km,C de facto excluding the D-layer (50 km 6 h 6 90 km),C though including the E-layer (90 km < h 6 150 km).C According to the COMPLEIK (COMPLex EIKonal) model,C the D-layer can be approximately characterizedC by a linearized analytical profile of complex refractive index,C covering a short range of heights between h1 = 50 km and h2 = 90 km.


C The COMPLEIK subroutineC has been joined to the IONORT-ISP systemC for including the whole D-layer:C thus, the height of ionospheric bottom h0 must be below h1 = 50 km.C******************************************************************************

h0 = h1 COMPLEIK42C******************************************************************************

C******************************************************************************

C n_h0:C the refractive index across the ionosphere–neutral atmosphere boundary.C In a first-order approximation,C the boundary refractive index n0 could be assumedC as a real number slightly less than 1, i.e. n0 < 1,C so that the real refractive index nR(h)C is a decreasing function of height h,C while the imaginary refractive index nI is negative,C as substantially correct for any ionospheric profile of the D-layerC (Budden, 1988).C******************************************************************************

n_h0 = ONE COMPLEIK43C******************************************************************************

C******************************************************************************

C Mean value of magnetic gyro-frequencyC******************************************************************************

FH_MEAN = K5 FH* COMPLEIK44>(((EARTHR**3) (h1 + h2 + 2 EARTHR))/ COMPLEIK45>(((h1+EARTHR)**2) ((h2 + EARTHR)**2))) COMPLEIK46C******************************************************************************

C******************************************************************************

C Collision frequency modelC consisting of a double exponential profile (Jones and Stephenson, 1975);C hMAX is the height relative to the maximum electron density,C which can be numerically calculated.C******************************************************************************

NU_hMAX = NU1 EXP(-A1 (hMAX-z1)) + NU2 EXP(-A2 (hMAX-z2)) COMPLEIK47C******************************************************************************

C******************************************************************************

C APPLYING MARTYN’S ABSORPTION THEOREM (MARTYN, 1935):C******************************************************************************

SEC = 1/COS(PH0) COMPLEIK48C******************************************************************************

C******************************************************************************

C FROM A VERTICAL RADIO SOUNDING WITH ONE IONOSPHERIC REFLECTIONC******************************************************************************

C******************************************************************************

C The optical path c: h1! h2C results proportional to the vertical absorption coefficientC******************************************************************************

OPT_PATH = (h2–h1)* COMPLEIK49>(n_h0-((1.-n_h0**2)/4)* COMPLEIK50>(((h2 + h1-2. h0)/H)/(1. + ((hMAX-h0)/H)))) COMPLEIK51C******************************************************************************

C******************************************************************************

C TO AN OBLIQUE RADIO SOUNDING WITH ONE IONOSPHERIC REFLECTIONC******************************************************************************

C******************************************************************************

C The oblique absorption coefficientC of a wave incident on a flat ionosphere forming the angle PH0C is further dependent on the secant of PH0C******************************************************************************

L_COMPLEIK = 2. ((1.-n_h0)/(1. + n_h0))* COMPLEIK52>OPT_PATH (1. + ((hMAX-h0)/H))* COMPLEIK53>(NU_hMAX/C) (F/(F + RAY FH_MEAN))* COMPLEIK54>((1./SEC) + SEC ((1.E6 NU_hMAX)/(PIT2 (F + RAY FH_MEAN)))**2) COMPLEIK55C******************************************************************************

C******************************************************************************


C Reporting the absorption coefficient L_COMPLEIK in decibel units (dB)C******************************************************************************

L_COMPLEIK_dB = �(20./LOGTEN) L_COMPLEIK COMPLEIK56C******************************************************************************

C******************************************************************************

C COMPLEIK OUTPUT: Absorption coefficient (dB).C******************************************************************************

OPEN(6,FILE = ’L_COMPLEIK(dB) vs FREQUENCY(MHz).txt’, COMPLEIK57>STATUS = ’unknown’) COMPLEIK58WRITE (6,1) F,ABS(L_COMPLEIK_dB) COMPLEIK591 FORMAT (2(2X,F20.10)) COMPLEIK60C CLOSE (6) COMPLEIK61C******************************************************************************

RETURN COMPLEIK62END COMPLEIK63


Appendix B. Supplementary data

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.asr.2013.10.035.

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The COMPLEIK subroutine of the IONORT-ISPsystem for calculating the non-deviative absorption:A comparison with the ICEPAC formula1. Introductive review2. The ICEPAC formula3. The IONORT-ISP system4. The complex eikonal model for the ionospheric D-layer5. ConclusionsReferences

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