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Full wave determination of oblique incidence reflection coefficients for model Es-layers
Full wave determination of oblique incidence reflection coefficients for model &layers ROD 1. BARNES High Frequency Radar Division, Defence Science and Technology Organisation, PO Box 1500,Salisbury. SA 5108, Australia (Received
inj?natform 7 April 1993 ; accepted 15April 1993)
Abstract-Reflection and transmission coefficients for horizontally stratified &layers are numerically calculated using a full wave analysis. The analysis is general in terms of the permitted angles of incidence of the radio wave and the orientation of the geomagnetic field with respect to the wave. Con~ntration profiles of synthetic ES-layers used in the analysis are generated using wind shear theory. The analysis shows that partial reflections and transmissions can occur over a wide frequency range but only become of significant amplitude when the &-layer’s vertical width is less than about 400 m or when the concentration gradient is very steep due to the introduction of non-linear wind shears. Coupling between o and x modes can be very strong for oblique paths and is largely inde~nden~ of the layer width. Under certain circumstances almost complete mode reversal upon reflection is predicted. This effect is very sensitive to the orientation of the geomagnetic field, 8, with respect to the wave normal, n, and approaches a maximum when I! x 41 approaches unity and becomes negligible when In_xEJ approaches 0.
1. INTRODUCX’ION Sporadic-E (I?,) is a term that is used to describe anomalous layers that appear in the E-region of the world’s ionosphere. It generally appears at between 90 and 150 km altitude and can occur at all orientations of the geomagnetic field, although the production mechanism is thought to vary with magnetic latitude. The orientation of the magnetic field in effect divides Es into three fundamentally different forms: auroral, mid-latitude and equatorial. Recent reviews on the different types of E, include those by WHITEHEAD(1989) and HUNSUCKER(1982). This paper is concerned with radio propagation through thin horizontally stratified layers and is most applicable to some forms of E, found at mid-latitudes ; however, the analysis could be. applied to thin layers wherever they occur. Rocket measurements of the vertical width of E,layers have produced values from about 0.5 to 3 km (SEDDON, 1962; BEYNONet al., 1972; SMKH and MECHTLY, 1972). The peak electron concentration within layers is often several times larger than the surrounding quiescent E-layer. Because of these characteristics, vertical gradients in electron concentration corresponding to a scale height of the order of 1 km are usually present in &-layers. Radio waves travelling into these gradients will encounter regions
where the refractive index of the medium is changing signi~cantiy within a wavelength of travel. This situation will accentuate partial transmissions, reflections and mode coupling of the wave. Full wave analysis is often used in studying these cases because simpler techniques that use the WKB approximations, such as ray tracing, become invalid. Partial reflection of LF and MF radio waves is thought to occur from sharp gradients in the D-region (e.g. BUDDEN, 1955a,b; GARDENERand PAWSEY, 1953). The possibility that it occurs with &layers at HF has been investigated at vertical incidence by CHE~~ELL(1971a,b), MILLERand SMITH(1976, 1977) and by JALONENet uf. (1982). Chessell used full wave analysis on several mathematical functions designed to represent thin layers, but neglected to smoothly situate the layers in a background E-region thereby producing discontinuities at the layer boundaries. These discontinuities will produce additional partial reffections which may be confused with the contribution due to the layer gradients (MILLER and SMITH, 1977). MILLER and SMITH (1977) used continuous concentration profiles produced from measurements made by rockets which were flown through &layers. They compared their results with vertical incidence ionograms and concluded that the very weak partial reflections and transmissions expected at vertical inci-
378
R. 1. BARNES
dence could not explain the measured difference between the highest frequency reflected by the E,, f,E,, and the lowest frequency which penetrated the Es, ,fbEs. To explain this result they adopted the cloud model of non-blanketing Es (WHITEHEAD, 1972). Many subsequent measurements have detected this cloudy horizontal structure (e.g. MILLER and SMITH, 1978; FROM, 1983; BARNES, 1991). The purpose of this work is to extend the work of Miller and Smith to the general case where the radio wave can be incident on the ionosphere at any angle. Oblique ionograms, produced by the FMCW oblique ionosonde used in the frequency management system of the Jindalee Over-the-Horizon radar (EARL and WARD, 1987) show Es to be nearly always nonblanketing and it is important to know what role partial transparency can play in this. This paper develops the theory of oblique propagation through horizontally stratified ES-layers and produces expected values of the reflection and transmission coefficients given the layer characteristics and the angle of incidence of the wave with respect to the layer and the Earth’s magnetic field.
H,. H,., H: magnetic field components Zl (POeO)‘iZ T,, FT,. Z,H, and Z,H,,, respectively (E,, -4. I,, SV). e
of the radio wave
Starting with Maxwell’s equations and by eliminating the z components of the electric and magnetic fields it can be shown (e.g. BUDDEN, 1985; YEH and LIU, 1972) that the equation involving the field components of a radio wave of arbitrary angle of incidence on the ionosphere is de = -ikTg
(1)
where the matrix T is given by,
2. THEORY We start by giving a list of the symbols that are used in the following theory. Where possible the same symbols as used by BUDDEN (1985) in describing radio propagation are used here. m e c
w EO
H z
v(z) N(z) 0,
s
C
x 0
n
mass of an electron electronic charge speed of light in a vacuum angular frequency of the propagating radio wave permittivity of a vacuum Earth’s magnetic field intensity altitude effective electron collision frequency electron concentration zenith angle of the radio wave normal below the ionosphere sin (0,) cos (0,) denotes the high frequency branch of the extraordinary mode denotes the ordinary mode is the wave normal vector
X
N:
Y
&@lW2 eB 2 anti-parallel
L,, L,,, Lz “;
direction
to g cosines of r
w&
1-Z V E,, E,., I$ electric field components
of the radio wave
SM,.:
0,
“42
-
0.
(2)
--:
SM,:
-,K
and where A = I+ M,,, and the components Mu, are components of the susceptibility matrix for the ionosphere M, which is taken to be, -X M = C@J’ - y*) v-L,;Y*, iL,YU-
X
i
-iL,YU-L,L,Y*,
L,L,, Y2,
-iL,,YU-L,LzY2,
lJ2- r_: Y’, iL, YU-
L,.L, Y’,
iU.,YU-L,LzY2 -iL.,YU-L,.LzY’ u2-L;Y2
(3) i
Once an initial value for e is determined for a particular height in the ionosphere, equation (1) can be used to find the components at any other height. In our case we wish to start initially above the ES-layer in the background E-region where the gradients are very small, and then to integrate through the ES-layer until we again reach the background E-region where the gradients are again very small. The size of the
Reflection
coefficients
for model &-layers
379
deduced e components below the layer can then be compared with the original components above the layer. As is shown below, if two independent solutions to the equation can be acquired then reflection and transmission coefficient matrices R and T, denoted by
R=
and
T=
can be obtained.
For the matrix components in equation (4) the subscripts indicate the mode of reflection or transmission that the coefficient denotes, e.g. R,,, denotes the expected amplitude of the reflected wave which has ordinary polarization, o-mode, in the background E-region below the ES-layer, when a wave of unit amplitude which has extraordinary polarization, x-mode (no z-mode is expected above or below the layer) in the background E-layer, impinges from below. It is emphasized that the polarization is determined in the background E-region where concentration gradients are assumed to be negligible. It is instructive to study regimes where the plasma frequency of the back.&round E-region is small compared to the transmitted frequency, i.e. Xin the background region is very small. This ensures that the characteristic polarizations of the waves in the background E-region are approaching the free space value and eliminates, as much as possible, effects on the reflection coefficient that the background E-region would have on a signal transmitted from the ground. This gives a good indication of the E, reflection characteristics while still ensuring that the boundary condition of dN/dz = 0 is met. To determine the initial conditions for equation (1) a description of the wave is required in the background E-region above the Es-layer. At sufficient height above the ES-layer the background layer is assumed to be slowly varying and the WKB solution is expected to hold. The field components at oblique incidence can then be described by the following equation
x exp (III~ 1,4
BUDDEN (1985) p. 190.1 The Booker
quartic which needs to be solved to establish the initial e can be simplified by judicious orientation of axes and for this work the wave normal is assumed to be in the s-~J plane (Fig. 1). As stated earlier, two independent solutions are required and so two independent initial conditions will be needed. Well above the ES-layer no downwardly propagating wave is expected (neglecting reflections from higher layers which we are not interested in here). Hence two of the Booker quartic solutions representing upwardly propagating o and x mode waves can be separated out and used with equation (5) to establish two independent initial values of g. Once the initial values of
e = A,F,(a,y,+a,,Ai,Y,A,,u,q,+A,)
j=
Fig. 1. The geometry used in the analysis. The axes are set up so that the wave normal is always in the x-z plane. The orientation of the magnetic field is described by the three unit cosines.
q,dz) (5)
where the four values of j, representing the four characteristic modes, involve, where X is small, upward and downward travelling x-mode and o-mode waves. The q, are the solutions to the Booker quartic (BOOKER. 1938) and the components a, hr A, and F, depend on the ionospheric conditions at the height z. [The actual expressions for these can be found in
Sf
(6)
whereJ’ is a matrix containing the amplitude of the four characteristic waves at the height of interest. This stems directly from the application of Maxwell’s equations to plane wave radio wave propagation in the ionosphere (BUDDEN, 1985). Provided S is nonsingular then this equation can be inverted so that the components offcan be found, i.e. f’= s- ‘e.
(7)
Under the WKB approximation, which should be valid in the background E-layer, the components of S- ’ can be deduced from the ionospheric conditions
R. I. BARNES
380 at that height. The two values of e obtained &-layer will, through equation (7) provide of J The components of the f matrix are tudes of the four characteristic modes and &-layer let them be denoted by
below the two values the amplibelow the
.f“ = (ua’, UX’, do’, dx’) f” = (uo’, u.?, do’, d.x*)
(8)
while above the layer let f” = (L;O’,U.X’,eo’,ex’) f’ = (w*, cx’, eo*, a’).
(9)
For the components in these equations the symbols o and x indicate the modes, u and u indicate ‘upwards’ and d and e indicate ‘downwards’. Superscripts are being used here to discriminate between the two independent solutions. The two initial conditions above the layer of an upwardly propagating o mode and an upwardly propagating x mode both of unit amplitude can be described byfmatrices with component values of .f’ = (l,O,O,O) and .f 2 = (0, 1>0,0),
(10)
respectively. By using the definition of the reflection and transmission coefficients, it can then be readily shown that the initial values for,f; along with the two values extracted from the p vectors below the layer, reflection and transmission yield the desired coefficient through the following matrix equations,
and
where mod = uo’u.x2-uo*ux’
3.
THE INTEGRATION
The integration method is similar to that used by MILLER and SMITH (1976) and by CHE~~ELL(1971a). In our case, a fifth order Runge-Kutta-Fehlberg adaptive method was applied to the solution of equa-
tion (1). The method starts with the initial solutions and steps downwards through the ionosphere, i.e. small negative steps in z, and accumulates the corresponding changes in e. The adaptive numerical method is very efficient because it has the ability to automatically change the step size through the ionosphere according to a requirement that the ‘error’, which is determined by the method, must remain under a predetermined error tolerance for each step. The tolerance value was set to IO- ’ which appeared to be well into the region where the method converged to a stable answer (see Fig. 12). This technique becomes especially useful around points where the components of T become very large, which can happen at so-called resonances. The step size in these regions needs to be reduced to avoid too large a numerical error accumulating. Observation of the step size throughout the integration indicated that the adaptive method was working with the step size being reduced by several orders of magnitude around resonances. A problem that is well known for the solution of equation (1) by this method is that of numerical swamping. This occurs because in several regions one or both of the solutions can be evanescent, that is, the real part of the square of the refractive index is negative and so the amplitude grows exponentially as the integration proceeds downwards. This has two effects that need to be countered. First, the storage capacity of the computer doing the integration can be quickly exceeded. Second, small amounts of the evanescent solution inadvertently introduced into the second solution due to the finite precision of the computer can quickly cause swamping of the wanted solution due to the exponential growth. In this sense swamping means that the finite precision of the computer and the exponential growth of the evanescent waves causes the complete loss of information on the propagating wave. PITTEWAY(1965) noted this problem in similar work in the D-region and showed that the introduction of the constraint that the two solutions remained hermitian orthogonal would ensure that the two remained independent. Pitteway’s constraint was implemented here along with regular scaling of the f to keep the evanescent wave within range of the computer’s capacity. A record of the scaling was kept so that it could be removed after the integration because it affects the calculation of the transmission coefficients (MILLER and SMITH, 1976, 1978).
4.
THE CONCENTRATION
The actual &-layer concentration the calculations are quite important
PROFILES
profiles used in to this work. The
381
Reflection coefficients for model &-layers components of the matrix T vary chiefly through the variation of N,. Some variations due to a height dependence of the magnetic field strength and the effective collision frequency are expected but these are secondary effects to the electron concentration variations (note : in the analysis, the effective collision frequency is included as a function of altitude). It was decided that first-order wind shear theory would be used to generate the concentration profiles, i.e. where the compression of metallic ions due to a wind shear is just balanced by diffusion. As is shown below a linear wind shear generates a Gaussian or normal distribution of electrons around the wind shear node and since several of the measured profiles in MILLER and SMITH (1977) appear to be approximately Gaussian, the profiles generated using wind shear theory are expected to usefully approximate that of real &-layers. The concentration profile of metallic-ions, N,,,(z), for the case of a &-layer produced by the wind shear mechanism, where a shear of strength, S, is limited by diffusion, is described approximately by the following equation, (14)
where D = the diffusion constant, z, is the distance from the wind shear node, p is an odd number representing the non-linearity of the shear, and F, the efficiency factor is approximately given by F
=
w, cos u ~-~v,-.
(15)
Here u is the geomagnetic dip angle, wi is the ion gyrofrequency, and vi is the effective ion collision frequency (WHITEHEAD, 1961). For a linear wind shear, p = 1, which was assumed in Whitehead’s work, but SMITH and MILLER (1980) have produced evidence that non-linear wind shears can occur and in this case a higher power of p would be more appropriate. Other non-linear functions could be used here with equal validity and no attempt is made here to explain why the wind shear might take this form. It is simply convenient to use it in producing a squarish profile like the one shown in SMITH and MILLER (1980). The theory is extended here with p = 1 but is readily developed for higher values. Solving equation (14) yields a Gaussian distribution of ions thus,
Nm= &ex
2
p 0
2
(16)
where z0 = J2D/FS, and A = the total metallic-ion content of the &-layer.
columnar
We are interested in the total electron concentration that is generated by the presence of these ions. SMITH and MILLER (1980) show that under steady-state conditions with the inclusion of recombination the electron concentration N, is given by,
For the calculations to proceed A, z,, and N, need to be specified in equation (16) so that N,,, and subsequently N, can be determined for every value of z,. It is assumed, for simplicity, that the molecular ion concentration, N,, is unaffected by the wind shear due to the short life-time of these ions. It is therefore taken that N, is constant over the region of interest and can be determined from a given value of fE, the peak plasma frequency of the surrounding E-region. z0 is the vertical distance in which the concentration of metallic ions decrease to l/e of the initial value and is called the half-width of the &-layer. It is left as a variable. For a symmetric &-layer (the only type of layer considered here) the full vertical width is simply twice the half-width. At z, = 0 the peak electron concentration of the layer is expected. Now if a value of.f,E, is supplied it can be related to N,(peak) which can subsequently be used to find the peak value of N,,, which with z, - 0 in equation (16) can be used to determine A. In summary, once a vertical half-width for the E,layer, the maximum plasma frequency of the layer and the plasma frequency of the background E-region are specified, the value of NC is determined for all values of 2,. The values of several other parameters are required before the integration can proceed, including the orientation of the transmitted wave, the orientation of the geomagnetic field and its strength and the effective collision frequency as a function of height. The orientation of the field and the wave were left as variables so that the propagation characteristics could be investigated more generally. The other variables were supplied in look up tables to a computer program designed to implement the Runge-Kutta-Fehlberg integration and subsequent determination of the reflection coefficients.
5. RESULTS As a test of the procedure, parameters were set to that of a radio wave vertically incident on an ionosphere identical to that which MILLER and SMITH
382
R.I.BARNES
:: ii bdox
/;
INCIDENT
FREQUENCY
,’ CRITICAL
07 FREWENCY
I‘”
Fig. 2. Comparison of the reflection and transmission coefficient results derived from this analysis (left) with those of CHESSELL (1971b) (right) for a wave vertically incident on a parabolic layer with a half-width of471 m.
(1977) used in a comparison of their method with that used by CHESSELL (1971b). Figure 2 illustrates the result of the test, showing Chessell’s result alongside the result from this analysis. Visually there appears to be good agreement. When the wave normal is slowly tilted off-vertical, the departure from the vertical incidence result is quite smooth giving confidence that there are no gross errors in the analysis suddenly manifesting themselves at off-vertical incidence (Fig. 3). The concentration profile was then switched from Chessell’s to a wind shear Es-layer with a 500 m halfwidth. A vertically incident wave was attempted first for comparison with Miller and Smith’s results. ftEs was set at 7 MHz and fE to 3 MHz in an attempt to get a rough comparison with the example shown in fig. 4 of MILLER and SMITH (1977). which visually appears to approximate a Gaussian distribution. The result, shown in Fig. 4 (of this paper) illustrates quali-
tative agreement with their conclusion, i.e. that for vertical incidence, mode coupling and partial reflections from this scale of layer are very weak when boundary conditions are properly treated. The small differences in detail in the comparison are expected since no attempt has been made to use a precise replica of their concentration profile. The analysis was then performed for obliquely incident waves. Figure 5 shows how the reflection coefficients vary with frequency for wave zenith angles extending from 0 to 85”. Parameters are set so that the waves are travefling in the southern hemisphere from magnetic north towards magnetic south through an &-layer withfbE, = 5 MHz, a half-width of 500 m and where the dip angle is 45”. The .&layer is situated in an E-region of plasma frequency equal to 2 MHz. As the obliquity of the wave is increased, Fig. 5 shows that the frequency which the layer is able to strongly reflect (the Maximum Useable Frequency or MUF in
383
Reflection coefficients for model &-layers
2.5
3.5
3
2.5
TRANSMllTED
FREQUENCY
3
3.5
(MHz)
Fig. 3. Oblique reflection coefficient results for the same layer as used for the result in Fig. 2. The panels show the variation as the zenith angle (Thetz) of the wave is moved from 0 up to 15”. prediction parlance) increases in a manner approximated by the secant rule which is often used when treating rays (DAVIES, 1990). In panel 1 of Fig. 5, the R,,trace crosses the 5 MHz point with a reflection coefficient value of about 0.6. This same reflection power can be compared with other angles of incidence on the same &-layer (other panels in Fig. 5). A graph of the frequency, which here we will call the MUF, corresponding to this power point for all the angles of incidence on the layer is shown in Fig. 6, along with a curve following the secant rule. The comparison is very good for this case and further results have suggested the secant rule is a good approximation for all horizontally stratified layers unless they are very thin. The width used here of 1 km (i.e. 500 m half-width) is typical of experimental measurements (WHITEHEAD, 1989). The validity of the secant rule is an indication that partial reflections do not become more of a problem as obliquity is increased. If this was so the secant rule would be expected to diverge from the measured result at large zenith angles. The downward turn of the reflection coefficient R,, at low frequencies seen in some of the panels in this and later plots, is caused by the transmitted frequency ionospheric
/
5
6
TRANSMITTED
7
FREQUENCY
8
9
(MHZ)
Fig. 4. Vertical incidence reflection coefficient results for a wind shear layer with a 1 km vertical width showing very weak partial reflections and mode coupling in qualitative agreement with MILLERand SMITH(1977). The logarithm (base 10) of the moduli are plotted.
R. I.
4
5
75
20
6
4
25
15
5
20
6
25
4
30
BARNES
5
6
7
20 25 30 35
TRANSMITTED
FREQUENCY
6
8
30
10
40
8
50 30
10 12 14 16
40
50
60
(MHZ)
Fig. 5. Oblique incidence reflection coethcients for north to south propagation, through an &layer 1 km wide and at a dip angle of 45” in the southern hemisphere. The zenith angle of the wave is varied from 0 to 85”. Mode coupling reaches a peak near where the wave normal is perpendicular to the magnetic field. Partial reflections are very weak.
closely approaching the background E-region plasma frequency. This effect is independent of the &-layer and is of no special interest. Observation of Figs 3 and 5 show that as the zenith angle is increased from zero, the components R,, and Rx, depart from the near equality they exhibit at vertical incidence. In Fig. 5 the coupled components increase in magnitude as the obliquity is increased, until around 45” when they begin to decrease again. At 45” angle of incidence the wave normal is perpendicular to the magnetic field and this appears to be critical in the coupling process. This result is consistent with the predictions of COHEN (1960, 1964) who stated that strong coupling would be expected at this orientation because spatial changes of the wave polarizations are large for transverse propagation. The increase in coupling as the wave normal and magnetic field vector become closer to perpendicular
was tested at a different latitude. Figure 7 is a plot of the result for the same &-layer used to produce Fig. 5 but now situated at a low latitude where the magnetic dip is 20”. The plot shows that the coupling is maximized when the orientation of the wave normal is between 10 and 20” from zenith. Inspection of an intermediate value suggested that it occurs at around 15” from zenith, which is 5” from the perpendicular to the field. The reason for the maximum occurring slightly off-~~endicular like this is not known. COHEN(1964) pointed out that the degree of coupling depends on the concentration gradients in the cloud and the above result suggests that to predict the precise orientation for maximum coupling would require a more analytical treatment of the coupling phenomenon than is given here. The coupling occurring at 30” zenith, where R, shows a dip in the reSection coefficient at 4 MNz, is probably caused by the
Reflection coefficients for model .&-layers CALCULATED
AND
SECANT
RULE
1
0
PREDICTED
I
20
40 Zenith
onqle
MUFS
vs ZENITH
385 ANGLE
I
I
60
80
[degrees]
Fig. 6. Comparison of the MUF vs frequency as predicted by this analysis (dotted line) and the secant rule (solid line).
o mode -+ z mode + o mode coupling described by CHESSELL (197la,b) and reported to be observed by JALONENet al. (1982). This coupling provides a means for the incident o mode to penetrate the layer at subcritical frequencies producing the decrease in reflected signal observed in this panel. When the propagation is from magnetic east to west, the plane of the wave normal and the plane of the magnetic field are orthogonal and so the wave normal will be more ‘naturally’ perpendicular to the field. The effect will reach a maximum at the magnetic equator where all wave normals, irrespective of zenith angle, will be perpendicular to the magnetic field. Figure 8 is the result for east to west propagation through a layer with the same parameters and dip angle as the example in Fig. 5. Once again, at oblique incidence, the maximum frequency supported by the ionosphere has increased well above the value of 5 MHz forf,E, and in a way predicted by the secant rule. At zenith angles greater than about 50” the reflection coefficients R,, and R,_, are, for many frequencies, larger than R,, and R,,.Near the MUF this effect is so strong that the term mode reversal on reflection is almost appro-
priate. As the zenith angle is being increased, In_x HI is approaching unity indicating that the increased coupling in these cases is consistent with the previous examples and the predictions of Cohen. Layer width is another parameter than can be varied in the analysis. Figure 9 shows the results of varying the width of the &-layer. The other parameters were set the same as in the last example where the propagation was east to west in the southern hemisphere. As is expected partial transmissions become quite severe when the width of the layer is decreased sufficiently. The smallest value of halfwidth, 50 m (i.e. 100 m full-width) is less than the generally accepted values for &-layers (WHITEHEAD, 1989) and so at oblique incidence partial reflections for most &-layers will occur over a limited frequency range. The width of the layer is also important in the mode coupling effect. As the width increases the gradients in the layer should become closer in magnitude to those found in the normal E-layer. Figure 9 suggests that the mode coupling effect lessens as the width increases. For orientations where In_x I31 are
R. I. BARNFJ
386
_._
4
ix Lxx
II
Thetr
=lOdegs;
4
-1
-1.
-2
-2
-
-
.‘.’
5
6
“p
\ Rxx
ROOi
4
5
6
7
r---.‘---- ..‘. .
4
I_
Thetr
i I
.._ -8
ROO :
=50degs I
nt+
6
7
5
,
,
_
\
wo
!_
8
TRANSMITTED
--
6 ,,
’ K
ROO ;
Thetz
=6Odegs /
6
FREQUENCY
8
i j
1 Thetr
=70degs
I
/
I
,
10
8
70
72
i Ii
14
16
(MHZ)
Fig. 7. Oblique incidence reflection coefficients for north to south propagation, through an &-layer similar to that in Fig. 5 but at a region in the southern hemisphere where the dip angle is 20”. Once again mode coupling reaches a peak near where the wave normal is perpendicular to the magnetic field.
approaching unity it may be possible to use mode discrimination to distinguish between normal E and Es traces on oblique ionograms (provided the correct polarizations for transmissions and receptions can be made). An experiment using portable ionosondes is being planned by the author to test these results. A further calculation involved the use of non-linear wind shears in the coefficient predictions. SMITH and MILLER (1980) showed that non-Gaussian profiles measured by rockets implied non-linear wind shears in the generation mechanism. In the most extreme example shown in their paper, one &-layer exhibited an almost square profile. Very large gradient at the edges of the layer would be expected in this case and it is instructive to see how large the partial reflections produced by these edges can become. The use of non-linear shears, involves using an odd value ofp, greater than 1, in equation (14). The peak
concentration No is kept equal to the value it has with p = 1. This implies that different values of p have different metallic-ion contents, A. A layer with f& = 5 MHz like the one used to produce Fig. 5, but with p = 25, gives the layer a squarish looking profile (Fig. IO). For north to south propagation through this profile, in the southern hemisphere, the reflection coefficient analysis produced the results shown in Fig. 1I. Partial reflections were enhanced substantially to the point where an ionosonde would see a significant band of frequencies exhibiting partial reflections. Very sharp concentration profiles like the one in Fig. 10 are believed to exist in the D-region and to produce most of the partial reflection observed from that region. The sharp gradient used in this simulation produced quasi-periodic variations in the coefficient profiles which may be due to interference of partial reflections from different heights, but they could be due to
387
Reflection coefficients for model &-layers .-.__..L_.IR; Rex-Rxo
letz =40d 4
5
6
hetz
1
I
5
6
7
‘P I
=60degr
8
8
-‘.
Roe-RXX
10
8
10 12 14 16
I -
ROX-RXO ‘... ~~~-...._ Roo-Rx?---
.xtz =82degs 1
15
20
25
15
20
25
30
I
wtz =84degs I
20 25 30 35
TRANSMITTED
FREQUENCY
I
I
30
40
50 30
40
50
60
(MHZ)
Fig. 8. Oblique incidence reflection coefficients for east to west propagation through an &-layer 1 km wide and at a dip angle of 45” in the southern hemisphere. The zenith angle of the wave is varied from 0 to 85’. Mode coupling reaches near mode reversal conditions at zenith angles greater than 60”.
numerical instability, although the Runge-KuttaFehlberg adaptive method is a precaution against this. Interference of o mode reflections from different heights has been described by CHBELL (197la) and has been reported to be observed by JALONEN (198 1). In this case, some o mode is converted to x mode at the coupling point near where X = I, and then travels up to where X = 1+ Y and reflects. Then, on the way down a portion is recoupled to o mode as it passes the coupling point and finally at the receiver, it interferes with the o mode reflected from where X = 1. As a test of the stability of the integration process in this particular case, the error tolerance in the Rung+ Kutta-Fehlberg algorithm was varied and the convergence to the result shown in Fig. 11 was examined. Figure 12 shows that when the error tolerance is set to 10. ’ or less, changes in the reflection coefficients are negligible for our purposes. For safety an error
tolerance examples
of lo-’ was used in the calculation shown in this paper.
6.
of all the
CONCLUSIONS
Predicted reflection and transmission coefficients have been numerically calculated for radio waves obliquely incident on &-layers. The profiles of the layers were derived using wind shear theory and care was taken in placing the layers in a background Elayer. The highest frequency at which these simulated ESlayers strongly reflect increases as the elevation angle of the wave decreases, closely obeying the secant rule as does the region of frequencies that produce partial reflections. The departure from the secant rule increases as the width of the layer is decreased, but this only becomes significant for layers of 400 m width
0
-1
2
w-2 0 LL LL E o-3
0 Rex-Rxo
I
F
RowRxo
Row-Rxx
b/2
20
30
40
width
=200
20
, 30
40
TRANSMITTED
20
30
FREQUENCY
40
20
30
(MHZ)
Fig. 9. Oblique incidence reflection coefficients for east to west propagation with a zenith angle of 80” and at a dip angle of 45’ in the southern hemisphere. The half-width of the layer is varied from 50 to 5000 m. Partial reflections are quite significant for very thin layers.
I
-
I
,
I
10"
I
3x10”
profile of a synthetic
I
1
I
2x10" Electron
Fig. 10. Concentration
I
I
Concentration
&layer
4x10”
I
5x10”
I
I
6X10”
Lm-‘1
produced
by a non-linear
wind shear.
40
Too
1RANSMl
I IEU
or less. These very thin model layers produce detectable partial reflections over a significant frequency range as do sharp gradients produced by non-linear wind shears. &-layers which have been observed to have squarish profiles are likely to produce significant amounts of partial reflection over a relatively wide band of frequencies. Mode coupling, upon reflection, between o and .Y modes is predicted in several examples. The effect reaches a maximum near where the magnetic field and the wave normal are perpendicular but becomes negligible when the wave normal and the field are parallel or anti-parallel. The degree of coupling also appears to depend on the nature of the concentration gradient. Decreasing the concentration gradients in the layer reduces the amount of coupling seen. and it may be that since the E-region is a much thicker layer than the &-layer. measurement of polarization could be used to distinguish between the two on oblique ionograms.
;
rf
(Ml
Ir)
Fig. 1I. Reflection and transmission coelhcients for the E,layer shown in Fig. 10. Although the layer is relatively wide the sharp gradient produced by the non-linear shear has produced significant partial reflections.
20
25
30
20
20
25
20 30 TRANSMITiEt>
Ackno,~lecrlqc,ments-I wish to thank all members of the Ionospheric Effects group at DSTO for helpful discussion, criticism and support.
25
30
25 30 t REOUCNCY
20
25
30
70
25
30
(Mt tz)
Fig. 12. Reflection coefficients for the &layer in Fig, IO, calculated with a varying value of error tolerance in the Runge-Kutta-Fehlberg algorithm. Once the value of the tolerance is below lo- ’ the changes in reflection coefficients are negligible. For safety a value of 10eh was used for all the examples in this paper.
390
R. I. BARNEY REFERENCES
BARNESR. I. BEYNONW. J. G.. MAUDEA. 0. and RATCLIFFG. BOOKERH. G. BUDDENK. G. BUDDENK. G. BUDDENK. G.
1991 1972 1938 1955a 1955b 198.5
CHE~SELL C. 1. CHESSELL C. I. COHENM. H. COHENM. H. DAVIESK. EARL G. F. and WARD B. D. FROMW. R. GARDENERF. F. and PAWSEYW. L. HUNSUCKER R. D. JAL~NENL. JALONEN L., NYGRENT. and TURENEN T MILLERK. L. and SMITHL. G.
1971a 1971b 1960 1964 1990 1987 1983 1953 1982 1981 1982 1976
MILLERK. L. and SMITHL. G. MILLERK. L. and SMITHL. G. PITTEWAYM. L. V. SEDDONJ. C.
1977 1978 1965 1962
SMITHL. G. and MECHTLYL. A. SMITHL. G. and MILLERK. L. WHITEHEAD J. D. WHITEHEAD J. D. WHITEHEAD J. D. YEH K. C. and LIU C. H.
1972 1980 1961 1972 1989 1972
J. attnos. terr. Phys. 53,99. Planet. Space Sci. 20, 89. Phil. Trans. R. Sot. London A231,41 I. Proc. R. Sot. A221, 516. Phil. Trans. R. Sot. A24,45. The Propagation of Radio Waves : The Theory of Radio Waves of Low Power in the Ionosphere and Magnetosphere. Cambridge University Press. J. atmos. terr. Phys. 33, 1515. J. atmos. terr. Phys. 33, 1803. Astrophys. J. 131, 664. J. geophys. Res. 69, 1935. Ionospheric Radio. Peter Peregrinus Ltd. London. Radio Sci. 22, 275. Planel. Space Sri. 31, 1397. J. atmos. terr. Phys. 15, 321. Rev. Geophys. Space Phys. 20,293. J. almos. terr. Phys. 43, 1285. J. atmos. [err. Phys. 44, 731. Aeronomy Report No. 75, University of Illinois, Urbana. J. atmos. (err. Phys. 39, 899. J. geophys. Res. 83, 3761. Phil. Trans. R. Sot. A257,219. Sporadic-E as observed with Rockets. Extract from Ionospheric Sporadic-E, E. K. SMITH and S. MATSUSHITA (eds). Pergamon Press, New York. Radio Sci. I, 367. J. atmos. terr. Phys. 42, 45. J. atmos. terr. Phys. 23,49. Radio Sci. 7, 355. J. almos. terr. Phys. 51,401. Theory of Ionospheric Waves. Academic Press, New York.
inj?natform 7 April 1993 ; accepted 15April 1993)
Abstract-Reflection and transmission coefficients for horizontally stratified &layers are numerically calculated using a full wave analysis. The analysis is general in terms of the permitted angles of incidence of the radio wave and the orientation of the geomagnetic field with respect to the wave. Con~ntration profiles of synthetic ES-layers used in the analysis are generated using wind shear theory. The analysis shows that partial reflections and transmissions can occur over a wide frequency range but only become of significant amplitude when the &-layer’s vertical width is less than about 400 m or when the concentration gradient is very steep due to the introduction of non-linear wind shears. Coupling between o and x modes can be very strong for oblique paths and is largely inde~nden~ of the layer width. Under certain circumstances almost complete mode reversal upon reflection is predicted. This effect is very sensitive to the orientation of the geomagnetic field, 8, with respect to the wave normal, n, and approaches a maximum when I! x 41 approaches unity and becomes negligible when In_xEJ approaches 0.
1. INTRODUCX’ION Sporadic-E (I?,) is a term that is used to describe anomalous layers that appear in the E-region of the world’s ionosphere. It generally appears at between 90 and 150 km altitude and can occur at all orientations of the geomagnetic field, although the production mechanism is thought to vary with magnetic latitude. The orientation of the magnetic field in effect divides Es into three fundamentally different forms: auroral, mid-latitude and equatorial. Recent reviews on the different types of E, include those by WHITEHEAD(1989) and HUNSUCKER(1982). This paper is concerned with radio propagation through thin horizontally stratified layers and is most applicable to some forms of E, found at mid-latitudes ; however, the analysis could be. applied to thin layers wherever they occur. Rocket measurements of the vertical width of E,layers have produced values from about 0.5 to 3 km (SEDDON, 1962; BEYNONet al., 1972; SMKH and MECHTLY, 1972). The peak electron concentration within layers is often several times larger than the surrounding quiescent E-layer. Because of these characteristics, vertical gradients in electron concentration corresponding to a scale height of the order of 1 km are usually present in &-layers. Radio waves travelling into these gradients will encounter regions
where the refractive index of the medium is changing signi~cantiy within a wavelength of travel. This situation will accentuate partial transmissions, reflections and mode coupling of the wave. Full wave analysis is often used in studying these cases because simpler techniques that use the WKB approximations, such as ray tracing, become invalid. Partial reflection of LF and MF radio waves is thought to occur from sharp gradients in the D-region (e.g. BUDDEN, 1955a,b; GARDENERand PAWSEY, 1953). The possibility that it occurs with &layers at HF has been investigated at vertical incidence by CHE~~ELL(1971a,b), MILLERand SMITH(1976, 1977) and by JALONENet uf. (1982). Chessell used full wave analysis on several mathematical functions designed to represent thin layers, but neglected to smoothly situate the layers in a background E-region thereby producing discontinuities at the layer boundaries. These discontinuities will produce additional partial reffections which may be confused with the contribution due to the layer gradients (MILLER and SMITH, 1977). MILLER and SMITH (1977) used continuous concentration profiles produced from measurements made by rockets which were flown through &layers. They compared their results with vertical incidence ionograms and concluded that the very weak partial reflections and transmissions expected at vertical inci-
378
R. 1. BARNES
dence could not explain the measured difference between the highest frequency reflected by the E,, f,E,, and the lowest frequency which penetrated the Es, ,fbEs. To explain this result they adopted the cloud model of non-blanketing Es (WHITEHEAD, 1972). Many subsequent measurements have detected this cloudy horizontal structure (e.g. MILLER and SMITH, 1978; FROM, 1983; BARNES, 1991). The purpose of this work is to extend the work of Miller and Smith to the general case where the radio wave can be incident on the ionosphere at any angle. Oblique ionograms, produced by the FMCW oblique ionosonde used in the frequency management system of the Jindalee Over-the-Horizon radar (EARL and WARD, 1987) show Es to be nearly always nonblanketing and it is important to know what role partial transparency can play in this. This paper develops the theory of oblique propagation through horizontally stratified ES-layers and produces expected values of the reflection and transmission coefficients given the layer characteristics and the angle of incidence of the wave with respect to the layer and the Earth’s magnetic field.
H,. H,., H: magnetic field components Zl (POeO)‘iZ T,, FT,. Z,H, and Z,H,,, respectively (E,, -4. I,, SV). e
of the radio wave
Starting with Maxwell’s equations and by eliminating the z components of the electric and magnetic fields it can be shown (e.g. BUDDEN, 1985; YEH and LIU, 1972) that the equation involving the field components of a radio wave of arbitrary angle of incidence on the ionosphere is de = -ikTg
(1)
where the matrix T is given by,
2. THEORY We start by giving a list of the symbols that are used in the following theory. Where possible the same symbols as used by BUDDEN (1985) in describing radio propagation are used here. m e c
w EO
H z
v(z) N(z) 0,
s
C
x 0
n
mass of an electron electronic charge speed of light in a vacuum angular frequency of the propagating radio wave permittivity of a vacuum Earth’s magnetic field intensity altitude effective electron collision frequency electron concentration zenith angle of the radio wave normal below the ionosphere sin (0,) cos (0,) denotes the high frequency branch of the extraordinary mode denotes the ordinary mode is the wave normal vector
X
N:
Y
&@lW2 eB 2 anti-parallel
L,, L,,, Lz “;
direction
to g cosines of r
w&
1-Z V E,, E,., I$ electric field components
of the radio wave
SM,.:
0,
“42
-
0.
(2)
--:
SM,:
-,K
and where A = I+ M,,, and the components Mu, are components of the susceptibility matrix for the ionosphere M, which is taken to be, -X M = C@J’ - y*) v-L,;Y*, iL,YU-
X
i
-iL,YU-L,L,Y*,
L,L,, Y2,
-iL,,YU-L,LzY2,
lJ2- r_: Y’, iL, YU-
L,.L, Y’,
iU.,YU-L,LzY2 -iL.,YU-L,.LzY’ u2-L;Y2
(3) i
Once an initial value for e is determined for a particular height in the ionosphere, equation (1) can be used to find the components at any other height. In our case we wish to start initially above the ES-layer in the background E-region where the gradients are very small, and then to integrate through the ES-layer until we again reach the background E-region where the gradients are again very small. The size of the
Reflection
coefficients
for model &-layers
379
deduced e components below the layer can then be compared with the original components above the layer. As is shown below, if two independent solutions to the equation can be acquired then reflection and transmission coefficient matrices R and T, denoted by
R=
and
T=
can be obtained.
For the matrix components in equation (4) the subscripts indicate the mode of reflection or transmission that the coefficient denotes, e.g. R,,, denotes the expected amplitude of the reflected wave which has ordinary polarization, o-mode, in the background E-region below the ES-layer, when a wave of unit amplitude which has extraordinary polarization, x-mode (no z-mode is expected above or below the layer) in the background E-layer, impinges from below. It is emphasized that the polarization is determined in the background E-region where concentration gradients are assumed to be negligible. It is instructive to study regimes where the plasma frequency of the back.&round E-region is small compared to the transmitted frequency, i.e. Xin the background region is very small. This ensures that the characteristic polarizations of the waves in the background E-region are approaching the free space value and eliminates, as much as possible, effects on the reflection coefficient that the background E-region would have on a signal transmitted from the ground. This gives a good indication of the E, reflection characteristics while still ensuring that the boundary condition of dN/dz = 0 is met. To determine the initial conditions for equation (1) a description of the wave is required in the background E-region above the Es-layer. At sufficient height above the ES-layer the background layer is assumed to be slowly varying and the WKB solution is expected to hold. The field components at oblique incidence can then be described by the following equation
x exp (III~ 1,4
BUDDEN (1985) p. 190.1 The Booker
quartic which needs to be solved to establish the initial e can be simplified by judicious orientation of axes and for this work the wave normal is assumed to be in the s-~J plane (Fig. 1). As stated earlier, two independent solutions are required and so two independent initial conditions will be needed. Well above the ES-layer no downwardly propagating wave is expected (neglecting reflections from higher layers which we are not interested in here). Hence two of the Booker quartic solutions representing upwardly propagating o and x mode waves can be separated out and used with equation (5) to establish two independent initial values of g. Once the initial values of
e = A,F,(a,y,+a,,Ai,Y,A,,u,q,+A,)
j=
Fig. 1. The geometry used in the analysis. The axes are set up so that the wave normal is always in the x-z plane. The orientation of the magnetic field is described by the three unit cosines.
q,dz) (5)
where the four values of j, representing the four characteristic modes, involve, where X is small, upward and downward travelling x-mode and o-mode waves. The q, are the solutions to the Booker quartic (BOOKER. 1938) and the components a, hr A, and F, depend on the ionospheric conditions at the height z. [The actual expressions for these can be found in
Sf
(6)
whereJ’ is a matrix containing the amplitude of the four characteristic waves at the height of interest. This stems directly from the application of Maxwell’s equations to plane wave radio wave propagation in the ionosphere (BUDDEN, 1985). Provided S is nonsingular then this equation can be inverted so that the components offcan be found, i.e. f’= s- ‘e.
(7)
Under the WKB approximation, which should be valid in the background E-layer, the components of S- ’ can be deduced from the ionospheric conditions
R. I. BARNES
380 at that height. The two values of e obtained &-layer will, through equation (7) provide of J The components of the f matrix are tudes of the four characteristic modes and &-layer let them be denoted by
below the two values the amplibelow the
.f“ = (ua’, UX’, do’, dx’) f” = (uo’, u.?, do’, d.x*)
(8)
while above the layer let f” = (L;O’,U.X’,eo’,ex’) f’ = (w*, cx’, eo*, a’).
(9)
For the components in these equations the symbols o and x indicate the modes, u and u indicate ‘upwards’ and d and e indicate ‘downwards’. Superscripts are being used here to discriminate between the two independent solutions. The two initial conditions above the layer of an upwardly propagating o mode and an upwardly propagating x mode both of unit amplitude can be described byfmatrices with component values of .f’ = (l,O,O,O) and .f 2 = (0, 1>0,0),
(10)
respectively. By using the definition of the reflection and transmission coefficients, it can then be readily shown that the initial values for,f; along with the two values extracted from the p vectors below the layer, reflection and transmission yield the desired coefficient through the following matrix equations,
and
where mod = uo’u.x2-uo*ux’
3.
THE INTEGRATION
The integration method is similar to that used by MILLER and SMITH (1976) and by CHE~~ELL(1971a). In our case, a fifth order Runge-Kutta-Fehlberg adaptive method was applied to the solution of equa-
tion (1). The method starts with the initial solutions and steps downwards through the ionosphere, i.e. small negative steps in z, and accumulates the corresponding changes in e. The adaptive numerical method is very efficient because it has the ability to automatically change the step size through the ionosphere according to a requirement that the ‘error’, which is determined by the method, must remain under a predetermined error tolerance for each step. The tolerance value was set to IO- ’ which appeared to be well into the region where the method converged to a stable answer (see Fig. 12). This technique becomes especially useful around points where the components of T become very large, which can happen at so-called resonances. The step size in these regions needs to be reduced to avoid too large a numerical error accumulating. Observation of the step size throughout the integration indicated that the adaptive method was working with the step size being reduced by several orders of magnitude around resonances. A problem that is well known for the solution of equation (1) by this method is that of numerical swamping. This occurs because in several regions one or both of the solutions can be evanescent, that is, the real part of the square of the refractive index is negative and so the amplitude grows exponentially as the integration proceeds downwards. This has two effects that need to be countered. First, the storage capacity of the computer doing the integration can be quickly exceeded. Second, small amounts of the evanescent solution inadvertently introduced into the second solution due to the finite precision of the computer can quickly cause swamping of the wanted solution due to the exponential growth. In this sense swamping means that the finite precision of the computer and the exponential growth of the evanescent waves causes the complete loss of information on the propagating wave. PITTEWAY(1965) noted this problem in similar work in the D-region and showed that the introduction of the constraint that the two solutions remained hermitian orthogonal would ensure that the two remained independent. Pitteway’s constraint was implemented here along with regular scaling of the f to keep the evanescent wave within range of the computer’s capacity. A record of the scaling was kept so that it could be removed after the integration because it affects the calculation of the transmission coefficients (MILLER and SMITH, 1976, 1978).
4.
THE CONCENTRATION
The actual &-layer concentration the calculations are quite important
PROFILES
profiles used in to this work. The
381
Reflection coefficients for model &-layers components of the matrix T vary chiefly through the variation of N,. Some variations due to a height dependence of the magnetic field strength and the effective collision frequency are expected but these are secondary effects to the electron concentration variations (note : in the analysis, the effective collision frequency is included as a function of altitude). It was decided that first-order wind shear theory would be used to generate the concentration profiles, i.e. where the compression of metallic ions due to a wind shear is just balanced by diffusion. As is shown below a linear wind shear generates a Gaussian or normal distribution of electrons around the wind shear node and since several of the measured profiles in MILLER and SMITH (1977) appear to be approximately Gaussian, the profiles generated using wind shear theory are expected to usefully approximate that of real &-layers. The concentration profile of metallic-ions, N,,,(z), for the case of a &-layer produced by the wind shear mechanism, where a shear of strength, S, is limited by diffusion, is described approximately by the following equation, (14)
where D = the diffusion constant, z, is the distance from the wind shear node, p is an odd number representing the non-linearity of the shear, and F, the efficiency factor is approximately given by F
=
w, cos u ~-~v,-.
(15)
Here u is the geomagnetic dip angle, wi is the ion gyrofrequency, and vi is the effective ion collision frequency (WHITEHEAD, 1961). For a linear wind shear, p = 1, which was assumed in Whitehead’s work, but SMITH and MILLER (1980) have produced evidence that non-linear wind shears can occur and in this case a higher power of p would be more appropriate. Other non-linear functions could be used here with equal validity and no attempt is made here to explain why the wind shear might take this form. It is simply convenient to use it in producing a squarish profile like the one shown in SMITH and MILLER (1980). The theory is extended here with p = 1 but is readily developed for higher values. Solving equation (14) yields a Gaussian distribution of ions thus,
Nm= &ex
2
p 0
2
(16)
where z0 = J2D/FS, and A = the total metallic-ion content of the &-layer.
columnar
We are interested in the total electron concentration that is generated by the presence of these ions. SMITH and MILLER (1980) show that under steady-state conditions with the inclusion of recombination the electron concentration N, is given by,
For the calculations to proceed A, z,, and N, need to be specified in equation (16) so that N,,, and subsequently N, can be determined for every value of z,. It is assumed, for simplicity, that the molecular ion concentration, N,, is unaffected by the wind shear due to the short life-time of these ions. It is therefore taken that N, is constant over the region of interest and can be determined from a given value of fE, the peak plasma frequency of the surrounding E-region. z0 is the vertical distance in which the concentration of metallic ions decrease to l/e of the initial value and is called the half-width of the &-layer. It is left as a variable. For a symmetric &-layer (the only type of layer considered here) the full vertical width is simply twice the half-width. At z, = 0 the peak electron concentration of the layer is expected. Now if a value of.f,E, is supplied it can be related to N,(peak) which can subsequently be used to find the peak value of N,,, which with z, - 0 in equation (16) can be used to determine A. In summary, once a vertical half-width for the E,layer, the maximum plasma frequency of the layer and the plasma frequency of the background E-region are specified, the value of NC is determined for all values of 2,. The values of several other parameters are required before the integration can proceed, including the orientation of the transmitted wave, the orientation of the geomagnetic field and its strength and the effective collision frequency as a function of height. The orientation of the field and the wave were left as variables so that the propagation characteristics could be investigated more generally. The other variables were supplied in look up tables to a computer program designed to implement the Runge-Kutta-Fehlberg integration and subsequent determination of the reflection coefficients.
5. RESULTS As a test of the procedure, parameters were set to that of a radio wave vertically incident on an ionosphere identical to that which MILLER and SMITH
382
R.I.BARNES
:: ii bdox
/;
INCIDENT
FREQUENCY
,’ CRITICAL
07 FREWENCY
I‘”
Fig. 2. Comparison of the reflection and transmission coefficient results derived from this analysis (left) with those of CHESSELL (1971b) (right) for a wave vertically incident on a parabolic layer with a half-width of471 m.
(1977) used in a comparison of their method with that used by CHESSELL (1971b). Figure 2 illustrates the result of the test, showing Chessell’s result alongside the result from this analysis. Visually there appears to be good agreement. When the wave normal is slowly tilted off-vertical, the departure from the vertical incidence result is quite smooth giving confidence that there are no gross errors in the analysis suddenly manifesting themselves at off-vertical incidence (Fig. 3). The concentration profile was then switched from Chessell’s to a wind shear Es-layer with a 500 m halfwidth. A vertically incident wave was attempted first for comparison with Miller and Smith’s results. ftEs was set at 7 MHz and fE to 3 MHz in an attempt to get a rough comparison with the example shown in fig. 4 of MILLER and SMITH (1977). which visually appears to approximate a Gaussian distribution. The result, shown in Fig. 4 (of this paper) illustrates quali-
tative agreement with their conclusion, i.e. that for vertical incidence, mode coupling and partial reflections from this scale of layer are very weak when boundary conditions are properly treated. The small differences in detail in the comparison are expected since no attempt has been made to use a precise replica of their concentration profile. The analysis was then performed for obliquely incident waves. Figure 5 shows how the reflection coefficients vary with frequency for wave zenith angles extending from 0 to 85”. Parameters are set so that the waves are travefling in the southern hemisphere from magnetic north towards magnetic south through an &-layer withfbE, = 5 MHz, a half-width of 500 m and where the dip angle is 45”. The .&layer is situated in an E-region of plasma frequency equal to 2 MHz. As the obliquity of the wave is increased, Fig. 5 shows that the frequency which the layer is able to strongly reflect (the Maximum Useable Frequency or MUF in
383
Reflection coefficients for model &-layers
2.5
3.5
3
2.5
TRANSMllTED
FREQUENCY
3
3.5
(MHz)
Fig. 3. Oblique reflection coefficient results for the same layer as used for the result in Fig. 2. The panels show the variation as the zenith angle (Thetz) of the wave is moved from 0 up to 15”. prediction parlance) increases in a manner approximated by the secant rule which is often used when treating rays (DAVIES, 1990). In panel 1 of Fig. 5, the R,,trace crosses the 5 MHz point with a reflection coefficient value of about 0.6. This same reflection power can be compared with other angles of incidence on the same &-layer (other panels in Fig. 5). A graph of the frequency, which here we will call the MUF, corresponding to this power point for all the angles of incidence on the layer is shown in Fig. 6, along with a curve following the secant rule. The comparison is very good for this case and further results have suggested the secant rule is a good approximation for all horizontally stratified layers unless they are very thin. The width used here of 1 km (i.e. 500 m half-width) is typical of experimental measurements (WHITEHEAD, 1989). The validity of the secant rule is an indication that partial reflections do not become more of a problem as obliquity is increased. If this was so the secant rule would be expected to diverge from the measured result at large zenith angles. The downward turn of the reflection coefficient R,, at low frequencies seen in some of the panels in this and later plots, is caused by the transmitted frequency ionospheric
/
5
6
TRANSMITTED
7
FREQUENCY
8
9
(MHZ)
Fig. 4. Vertical incidence reflection coefficient results for a wind shear layer with a 1 km vertical width showing very weak partial reflections and mode coupling in qualitative agreement with MILLERand SMITH(1977). The logarithm (base 10) of the moduli are plotted.
R. I.
4
5
75
20
6
4
25
15
5
20
6
25
4
30
BARNES
5
6
7
20 25 30 35
TRANSMITTED
FREQUENCY
6
8
30
10
40
8
50 30
10 12 14 16
40
50
60
(MHZ)
Fig. 5. Oblique incidence reflection coethcients for north to south propagation, through an &layer 1 km wide and at a dip angle of 45” in the southern hemisphere. The zenith angle of the wave is varied from 0 to 85”. Mode coupling reaches a peak near where the wave normal is perpendicular to the magnetic field. Partial reflections are very weak.
closely approaching the background E-region plasma frequency. This effect is independent of the &-layer and is of no special interest. Observation of Figs 3 and 5 show that as the zenith angle is increased from zero, the components R,, and Rx, depart from the near equality they exhibit at vertical incidence. In Fig. 5 the coupled components increase in magnitude as the obliquity is increased, until around 45” when they begin to decrease again. At 45” angle of incidence the wave normal is perpendicular to the magnetic field and this appears to be critical in the coupling process. This result is consistent with the predictions of COHEN (1960, 1964) who stated that strong coupling would be expected at this orientation because spatial changes of the wave polarizations are large for transverse propagation. The increase in coupling as the wave normal and magnetic field vector become closer to perpendicular
was tested at a different latitude. Figure 7 is a plot of the result for the same &-layer used to produce Fig. 5 but now situated at a low latitude where the magnetic dip is 20”. The plot shows that the coupling is maximized when the orientation of the wave normal is between 10 and 20” from zenith. Inspection of an intermediate value suggested that it occurs at around 15” from zenith, which is 5” from the perpendicular to the field. The reason for the maximum occurring slightly off-~~endicular like this is not known. COHEN(1964) pointed out that the degree of coupling depends on the concentration gradients in the cloud and the above result suggests that to predict the precise orientation for maximum coupling would require a more analytical treatment of the coupling phenomenon than is given here. The coupling occurring at 30” zenith, where R, shows a dip in the reSection coefficient at 4 MNz, is probably caused by the
Reflection coefficients for model .&-layers CALCULATED
AND
SECANT
RULE
1
0
PREDICTED
I
20
40 Zenith
onqle
MUFS
vs ZENITH
385 ANGLE
I
I
60
80
[degrees]
Fig. 6. Comparison of the MUF vs frequency as predicted by this analysis (dotted line) and the secant rule (solid line).
o mode -+ z mode + o mode coupling described by CHESSELL (197la,b) and reported to be observed by JALONENet al. (1982). This coupling provides a means for the incident o mode to penetrate the layer at subcritical frequencies producing the decrease in reflected signal observed in this panel. When the propagation is from magnetic east to west, the plane of the wave normal and the plane of the magnetic field are orthogonal and so the wave normal will be more ‘naturally’ perpendicular to the field. The effect will reach a maximum at the magnetic equator where all wave normals, irrespective of zenith angle, will be perpendicular to the magnetic field. Figure 8 is the result for east to west propagation through a layer with the same parameters and dip angle as the example in Fig. 5. Once again, at oblique incidence, the maximum frequency supported by the ionosphere has increased well above the value of 5 MHz forf,E, and in a way predicted by the secant rule. At zenith angles greater than about 50” the reflection coefficients R,, and R,_, are, for many frequencies, larger than R,, and R,,.Near the MUF this effect is so strong that the term mode reversal on reflection is almost appro-
priate. As the zenith angle is being increased, In_x HI is approaching unity indicating that the increased coupling in these cases is consistent with the previous examples and the predictions of Cohen. Layer width is another parameter than can be varied in the analysis. Figure 9 shows the results of varying the width of the &-layer. The other parameters were set the same as in the last example where the propagation was east to west in the southern hemisphere. As is expected partial transmissions become quite severe when the width of the layer is decreased sufficiently. The smallest value of halfwidth, 50 m (i.e. 100 m full-width) is less than the generally accepted values for &-layers (WHITEHEAD, 1989) and so at oblique incidence partial reflections for most &-layers will occur over a limited frequency range. The width of the layer is also important in the mode coupling effect. As the width increases the gradients in the layer should become closer in magnitude to those found in the normal E-layer. Figure 9 suggests that the mode coupling effect lessens as the width increases. For orientations where In_x I31 are
R. I. BARNFJ
386
_._
4
ix Lxx
II
Thetr
=lOdegs;
4
-1
-1.
-2
-2
-
-
.‘.’
5
6
“p
\ Rxx
ROOi
4
5
6
7
r---.‘---- ..‘. .
4
I_
Thetr
i I
.._ -8
ROO :
=50degs I
nt+
6
7
5
,
,
_
\
wo
!_
8
TRANSMITTED
--
6 ,,
’ K
ROO ;
Thetz
=6Odegs /
6
FREQUENCY
8
i j
1 Thetr
=70degs
I
/
I
,
10
8
70
72
i Ii
14
16
(MHZ)
Fig. 7. Oblique incidence reflection coefficients for north to south propagation, through an &-layer similar to that in Fig. 5 but at a region in the southern hemisphere where the dip angle is 20”. Once again mode coupling reaches a peak near where the wave normal is perpendicular to the magnetic field.
approaching unity it may be possible to use mode discrimination to distinguish between normal E and Es traces on oblique ionograms (provided the correct polarizations for transmissions and receptions can be made). An experiment using portable ionosondes is being planned by the author to test these results. A further calculation involved the use of non-linear wind shears in the coefficient predictions. SMITH and MILLER (1980) showed that non-Gaussian profiles measured by rockets implied non-linear wind shears in the generation mechanism. In the most extreme example shown in their paper, one &-layer exhibited an almost square profile. Very large gradient at the edges of the layer would be expected in this case and it is instructive to see how large the partial reflections produced by these edges can become. The use of non-linear shears, involves using an odd value ofp, greater than 1, in equation (14). The peak
concentration No is kept equal to the value it has with p = 1. This implies that different values of p have different metallic-ion contents, A. A layer with f& = 5 MHz like the one used to produce Fig. 5, but with p = 25, gives the layer a squarish looking profile (Fig. IO). For north to south propagation through this profile, in the southern hemisphere, the reflection coefficient analysis produced the results shown in Fig. 1I. Partial reflections were enhanced substantially to the point where an ionosonde would see a significant band of frequencies exhibiting partial reflections. Very sharp concentration profiles like the one in Fig. 10 are believed to exist in the D-region and to produce most of the partial reflection observed from that region. The sharp gradient used in this simulation produced quasi-periodic variations in the coefficient profiles which may be due to interference of partial reflections from different heights, but they could be due to
387
Reflection coefficients for model &-layers .-.__..L_.IR; Rex-Rxo
letz =40d 4
5
6
hetz
1
I
5
6
7
‘P I
=60degr
8
8
-‘.
Roe-RXX
10
8
10 12 14 16
I -
ROX-RXO ‘... ~~~-...._ Roo-Rx?---
.xtz =82degs 1
15
20
25
15
20
25
30
I
wtz =84degs I
20 25 30 35
TRANSMITTED
FREQUENCY
I
I
30
40
50 30
40
50
60
(MHZ)
Fig. 8. Oblique incidence reflection coefficients for east to west propagation through an &-layer 1 km wide and at a dip angle of 45” in the southern hemisphere. The zenith angle of the wave is varied from 0 to 85’. Mode coupling reaches near mode reversal conditions at zenith angles greater than 60”.
numerical instability, although the Runge-KuttaFehlberg adaptive method is a precaution against this. Interference of o mode reflections from different heights has been described by CHBELL (197la) and has been reported to be observed by JALONEN (198 1). In this case, some o mode is converted to x mode at the coupling point near where X = I, and then travels up to where X = 1+ Y and reflects. Then, on the way down a portion is recoupled to o mode as it passes the coupling point and finally at the receiver, it interferes with the o mode reflected from where X = 1. As a test of the stability of the integration process in this particular case, the error tolerance in the Rung+ Kutta-Fehlberg algorithm was varied and the convergence to the result shown in Fig. 11 was examined. Figure 12 shows that when the error tolerance is set to 10. ’ or less, changes in the reflection coefficients are negligible for our purposes. For safety an error
tolerance examples
of lo-’ was used in the calculation shown in this paper.
6.
of all the
CONCLUSIONS
Predicted reflection and transmission coefficients have been numerically calculated for radio waves obliquely incident on &-layers. The profiles of the layers were derived using wind shear theory and care was taken in placing the layers in a background Elayer. The highest frequency at which these simulated ESlayers strongly reflect increases as the elevation angle of the wave decreases, closely obeying the secant rule as does the region of frequencies that produce partial reflections. The departure from the secant rule increases as the width of the layer is decreased, but this only becomes significant for layers of 400 m width
0
-1
2
w-2 0 LL LL E o-3
0 Rex-Rxo
I
F
RowRxo
Row-Rxx
b/2
20
30
40
width
=200
20
, 30
40
TRANSMITTED
20
30
FREQUENCY
40
20
30
(MHZ)
Fig. 9. Oblique incidence reflection coefficients for east to west propagation with a zenith angle of 80” and at a dip angle of 45’ in the southern hemisphere. The half-width of the layer is varied from 50 to 5000 m. Partial reflections are quite significant for very thin layers.
I
-
I
,
I
10"
I
3x10”
profile of a synthetic
I
1
I
2x10" Electron
Fig. 10. Concentration
I
I
Concentration
&layer
4x10”
I
5x10”
I
I
6X10”
Lm-‘1
produced
by a non-linear
wind shear.
40
Too
1RANSMl
I IEU
or less. These very thin model layers produce detectable partial reflections over a significant frequency range as do sharp gradients produced by non-linear wind shears. &-layers which have been observed to have squarish profiles are likely to produce significant amounts of partial reflection over a relatively wide band of frequencies. Mode coupling, upon reflection, between o and .Y modes is predicted in several examples. The effect reaches a maximum near where the magnetic field and the wave normal are perpendicular but becomes negligible when the wave normal and the field are parallel or anti-parallel. The degree of coupling also appears to depend on the nature of the concentration gradient. Decreasing the concentration gradients in the layer reduces the amount of coupling seen. and it may be that since the E-region is a much thicker layer than the &-layer. measurement of polarization could be used to distinguish between the two on oblique ionograms.
;
rf
(Ml
Ir)
Fig. 1I. Reflection and transmission coelhcients for the E,layer shown in Fig. 10. Although the layer is relatively wide the sharp gradient produced by the non-linear shear has produced significant partial reflections.
20
25
30
20
20
25
20 30 TRANSMITiEt>
Ackno,~lecrlqc,ments-I wish to thank all members of the Ionospheric Effects group at DSTO for helpful discussion, criticism and support.
25
30
25 30 t REOUCNCY
20
25
30
70
25
30
(Mt tz)
Fig. 12. Reflection coefficients for the &layer in Fig, IO, calculated with a varying value of error tolerance in the Runge-Kutta-Fehlberg algorithm. Once the value of the tolerance is below lo- ’ the changes in reflection coefficients are negligible. For safety a value of 10eh was used for all the examples in this paper.
390
R. I. BARNEY REFERENCES
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