Environmental effects on propagation
5.2 Atmospheric refraction
5.2.1 The problem
The dielectric constant, £, of a vacuum, such as free space, is defined as 1.0. (This is the relative dielectric constant, relative to the absolute dielectric constant, prop- erly called the electric constant, £o, ^8.854 x 10~12 F/m. (Farads per metre)) The air molecules and water vapour in the atmosphere very slightly increase the dielectric constant, increasing the refractive index, n (n — *J~e) to a value a few hundredths of 1 per cent above 1.0. This very slightly reduces the velocity of propagation from the free-space velocity of light, c, to c/n, reducing the wavelength. If the refractive index were uniform, radar rays would continue to travel in straight lines, just as they do in free space, and the phenomenon would be of trivial importance. However, the rays between marine scanner and target traverse the bottom few tens of metres of the atmosphere. Here temperature and moisture content vary with height, making refractive index also change with height, curving the rays in the vertical plane by an amount dependent on meteorological factors and on range, as indicated in Fig- ure 5. l(a). Refraction therefore affects the ranges at which events such as the horizon occur. At very short range refraction and the associated curvature can usually be ignored, just as Earth curvature is ignored when using plane trigonometry to solve short-range navigational problems. At long range, the relative phasing of direct and indirect rays is very significantly affected, and detection range of distant targets can vary by a factor of two or more as the weather changes from hour to hour. Propagation through the atmosphere is important to radio reception, particularly at the lower fre- quency broadcast bands and has been closely studied since the 1920s, well pre-dating radar. Air masses are sometimes identified by the thermal and moisture properties of the source region: Antarctic (AA), arctic (A), polar (P) and tropical (T); continental (c) relatively dry, maritime (m) relatively moist.
We assume the refractive index, n, of a lamina sheet of atmosphere of constant height /ia above sea level remains constant through the whole scanner/target range bracket. (This usually reasonably represents reality although local inhomogeneities can arise, causing reflection in the form of distributed angel echoes, a form of clutter.) Laminae of differing heights, ha, have differing refractive index.
Figure 5.1 Refraction and ray paths. Dense, moist air near the surface slows the lower portions of the rays, curving them downwards. Path calculations assume straight direct and indirect rays above an Earth whose radius is assumed to be k times actual (c)
As mentioned in Chapter 2, Section 2.4.1, Eq. (2.3), wavelength, A, varies with frequency, / , velocity of propagation, c, and refractive index, n:
X= — m.
fn
(c) Rays straightened Earth effective radius, E = he Curvatures unwound Earth true radius, e
True surface
Effective .surface
Curved rays Target
Rays straightened Flat-Earth approximation (dotted)
Scanner Surface
Geometric horizon Grazing point,
specular reflection Earth true radius, e
(b) Curved ray paths
Grazing angle
Indirect ray
Destination (e.g. target) Low k factor, more curvature Rays curved by refraction
Earth's surface Direct ray
Source (e.g. scanner)
(a) Changing refractive index curves rays Air dense, n high
Direction of propagation normal to fronts Slowed by dense air, bottom of wavefronts are retarded, curving ray downwards Wavefronts All curvatures exaggerated Air less dense, refractive index, n, low
For a given frequency, raising n from 1 by inserting the atmosphere reduces A., the distance travelled per cycle, hence reducing the velocity of propagation below the free space value. A trivial range error ~300 mm/km arises, much less than other sources of range error such as detection cell quantising, and readily corrected by a small adjustment to the display scaling factor.
Much more important than the absolute value of n is its rate of change with height, the refractive gradient, caused by changes of density and moisture content with height. The refractive gradient curves the rays by the same order of magnitude as the curvature of the Earth. At high altitude and over land masses, the refractive gradient is reasonably uniform and unvarying. Unfortunately, within the first few tens of metres above the sea surface, meteorological conditions strongly affect the gradient, in turn varying ray curvature. Neither refractive index nor its gradient is easy to measure.
Refraction of course also affects the indirect ray between source and destination and therefore significantly influences the interference between direct and indirect rays.
5.2.2 Equivalent geometries
We now quantify these statements. To accord with meteorological usage, some expres- sions use km rather than metres. In some textbooks, definitions of certain terms differ between km and metres, introducing powers of 103. We let:
/za = height of lamina of atmosphere in question, km n = refractive index of this lamina
dn/dha = refractive gradient (rate of change of refractive index with respect to height, km)
pr = ray curvature, rad/km
e = true Earth radius, 6.371 x 106 m
£km = e expressed in kilometres, 6371 km E = effective Earth radius
k = effective Earth radius factor N = radio refractivity
p=barometric pressure, hPa (hectopascal, numerically identical to and superseding the millibar (mbar). 1 hPa = 0.7473 mm of mercury) T = absolute temperature, K
w = partial pressure of water vapour, hPa (usually written e, but we prefer w to avoid confusion with dielectric constant and Earth radius).
The algebra describing the curved ray paths of Figure 5. l(b) is difficult. It is usual to shorten the odds by an approximation which assumes the rays travel in straight lines above an Earth whose hypothetical radius, E, differs from the true radius, e, by an effective Earth radius factor, Earth radius correction factor or (atmospheric)
refraction factor, k:
* = - . (5.1) e
This approximation is shown in Figure 5.1(c) (for k ~ 2). The geometry underlying Eq. (5.1) is valid because the ranges and heights used in marine radar are very much less than the Earth's radius. The process is akin to unwinding the curvatures until the ray path is linear. Factor k is a pure number and is independent of radar frequency.
However k differs at optical frequencies, which are not affected by moisture.
A simpler approximation, discussed more fully in Section 5.6, assumes the Earth is flat and the rays travel in straight lines. This flat-Earth approximation (indicated in Figure 5.1(c)) is tantamount to putting E = oo, with k = oo also. It is valid under some circumstances and we shall use it from time to time, but is inaccurate at long range.
5.2.3 Calculation of refraction factor from meteorological parameters The following treatment is based on Hall etal [1] and on Skolnik [2]. It enables factor k to be calculated in the rather unlikely event that the meteorological parameters in the atmospheric volume traversed by the rays are accurately known. Meteorological measurements taken at a shore station may not reliably apply above the water surface off a coast. Especially near a coast, k may also change along the scanner/target path, vitiating our calculations.
As already noted refractive index depends on the dielectric constant:
n = y/e. (5.2a) For the lower troposphere (the atmosphere up to 11 km), n ~ 1.0003. To magnify the small changes of n which are of interest, we use a radio refractivity, N:
N = (n-l)x 106. (5.2b)
When, say, n = 1.000315, N = 315, a typical mid-latitude value. Refractivity at the sea surface is sometimes written Ns. The rate of change is also magnified so
dha dha
The value of N depends on the partial pressure of the atmospheric water vapour, w; absolute temperature, T9 and barometric pressure, /?, which are meteorologically inter-related (Reed and Russell [3]):
W = 0.373 x 106 ( ^ ) + 77.6 ( ^ ) . (5.3)
The contribution of water molecules gives the first, 'wet', term, which is usually dominant. Nitrogen and oxygen molecules give the second, 'dry', term. The constants were empirically derived from observations. Except in very calm weather, either the wind or waves make the air turbulent, moisture plucked from the sea surface being
stirred into the bottom few metres of the air column, moistening it. Usually, the following conditions apply.
• The air gets drier at height.
• The rate of change ofp with Aa is negative, p falling exponentially to 35 per cent of the surface value at a scale height, H, around 8 km and not varying much with the weather.
• The rate of change of T with Aa is negative, except in fog, typically falling at l°/100 m, the dry adiabatic lapse rate. Lapse rate is the magnitude of the decrease of an atmospheric parameter with height.
• The rate of change with Aa of the wet term is usually, but not always, negative.
This governs the gradient dAf/dAa, which in turn governs ray curvature, pr. Up to Aa > 0.1 km, N usually decreases exponentially; n — nSurface exp(—h/H) and near the surface dn/dha ~ —39.2 x 10~6 km"1. The lapse rate —dN/dha often differs from its standard value of 40 'N units'/km (12 'N units'/lOOO ft).
The lower parts of wavefronts encounter higher n and propagate more slowly, curving the rays downward as shown in Figure 5.1 (a). Alternatively, we could say that a ray transmitted obliquely upwards encounters decreasing refractive index, making it curve downwards in conformance with Snell's law in optics. The curvature is approximated by
*~ \--wY- < 54 °>
L ndha] As n ~ 1,
- 1 06
& ~ AAT ,Ai rad/km. (5.4b)
From the geometry,
Jk = - ^ - . (5.5a)
PT - ^km
Substituting Eqs (5.4) into Eq. (5.5a) and putting e^m = 6371 km
k = (l+6371drc/d/za) = (1+6.371 x 10~3dA^/dha)' ( 5'5 )
When dN/dha = —39.2 (sometimes taken as —40), which is approximately the standard lapse rate, k = 4/3. This k value has been adopted for general radio/radar use, giving E = 8495 km. By assuming this fictitious radius, E9 for the Earth, the rays may be regarded as travelling in straight lines in a standard atmosphere, simplifying the ray geometry.
Some textbooks use a variant approach. If the Earth is assumed flat, k = oo, a fictitious ray curvature can be adopted to preserve the actual curvature relative to the surface, using fictitious air having a modified refractivity, M as follows:
M = N + - x\03 = N + 157Aa4N units'. (5.6a)
e
Figure 5.2 Radio refractivity profiles. Idealised, differing weather conditions.
Refractive gradient is the reciprocal of the slope of the curves. Not to scale
The 103 term arises from the definition of ha in kilometres. The rate of change with height is the modified refractivity gradient. For the standard atmosphere dM/dha = 117.
a r - ^ + 157 - <"•»
dha dha
5.2.4 Standard atmosphere; four-thirds Earth approximation
The radar standard atmosphere when the lapse rate gives 39.2 'N units'/km and k = 4/3 is also called the four-thirds Earth condition. The vulgar fraction seems better than writing 1.333, which confers a quite spurious sense of highly refined scientific precision. At optical frequencies, which are unaffected by the wet term of Eq. (5.3), the equivalent k value is 1.163 (E = 7407.4 km). Horizon range is propor- tional to \/&, Section 5.5.7, so when k = 4/3, the radar horizon lies about 7 per cent beyond the visual horizon. The refraction profile of Figure 5.2(a) depicts the radar standard atmosphere with k = 4/3 and dN/dha = —39.2 'N units'/km. Figure 5.2 is plotted to a base of N9 so high dN/dha represents a shallow slope with low —dN/dha. The 'four-thirds Earth' or US Central Radio Propagation Laboratory convention is appropriate to airborne radar. It represents the mean condition at 700 m over the continental United States and often approximates the atmosphere at high altitude over land. It does not adequately represent conditions just above a sea surface and is too often automatically and unthinkingly chosen for marine radar purposes. Putting k ~ 2 might be a better all-round choice.
5.2.5 Anaprop
Anomalous propagation, anaprop, occurs when the local weather substantially changes the refraction gradient from the standard value, lapse rate and k significantly differing from 39.2 and 4/3, respectively. Sub-refraction occurs when k <& 4/3 (say k < 1.0) and dMldha > 117; super-refraction is when k ^> 4/3 (say k > 2.0) and 0 < dM/dha < 117. Sub-refraction occurs when the coastal surface loses heat by
(a) Standard atmosphere
Super-refraction (b) High A: or flat Earth Radio refractivity, N
Sub-refraction (c) Low k
Ducts shaded
(d) Surface duct (e) Elevated duct Two slope reversals 6N/6ha positive
Flat Earth Light line Radar standard
Height, hA
radiation on a clear, still night and inversion (temperature and/or moisture increasing with height) causes w to rise with height. In the extreme, radiative fog may occur.
On the other hand, if the air is dry, as in North Africa and the Middle East, the inver- sion causes super-refraction and ducting. The inversion is rapidly destroyed by solar heating after sunrise. Effective Earth radius E and k rise as lapse rate rises. Anaprop chiefly occurs in relatively calm conditions, when there is little turbulence to stir the atmosphere, and is commonly caused by a considerable difference between the wind and sea temperatures near land masses, although some anaprop is almost always present at low altitude under marine conditions. As lapse rate and k are difficult to determine in practice, anaprop has been the all-too-convenient scapegoat for many an unexpected performance shortfall over the years. For example, when a Canadian Coast Guard buoy tender observed a racon on the St Lawrence River to have half its expected maximum range during very calm autumn weather, the author had to call anaprop in aid. There was nothing obviously at fault in the radar, and the racon oper- ated to full specification in subsequent laboratory tests. Maritime weather forecasts currently do not warn of anaprop.
In a typical atmosphere where T = 293K(20°C),P = lOOOhPaandw = 15hPa, the variation in N, 8N9 is
8N = 0.265/7 + 4.38w - IA8T.
Within a quite small height increment, 8 T may change by several degrees and 8 why several Pascals, causing dN/dha to change by several 'N units', forming a waveguide called a duct, the waveguide walls being the points of inflexion or the sea surface.
Evaporation ducts are commonest and result from 8w, change of partial pressure of water vapour. A surface duct is bounded below by the surface. An elevated duct is bounded above and below by changes of dN/dha. Within ducts, dM/d/ia is negative.
Evaporative surface duct heights, according to Hall et al. [1, Section 6.3.2], who go into considerable detail, are typically: 5-6 m in the North Sea; 13-14 m in the Mediterranean and >20 m in the Arabian Gulf.
Advection ducts occur in coastal regions of seas such as the North Sea, and when the sea is enclosed by hot, dry land. Examples are again the Mediterranean and the Gulf. A warm, dry, air mass flows off the land onto the cooler sea. When a summer cyclone exists over northern Europe, warm dry air may be carried by advection to the North Sea, where it overlies cooler and moister air, giving high humidity lapse rate and temperature inversion, leading to thick (to 200 m) surface ducts which may persist near coasts for several days and are most pronounced in early evening.
The following Sections 5.2.6-5.2.9 are all examples of anaprop. Rays having obliquity < 1° (nearly all marine radar rays) can rarely cross duct boundaries.
5.2.6 Super-refraction; high k; super-standard surface layer
When, at low altitude, 10"6^k1n x dN/dha = 1, lapse rate = 157 N units/km (48 N units/1000 ft), k = oo and the Earth appears flat; E = oo (Figure 5.2(b)).
Here the rays curve down just enough to match the curvature of the Earth and the horizon recedes to infinity. Absence of horizon effects can cause targets to be detected
at unusually long range, often beyond the nominal horizon. Performance may equal or even exceed the free space value. For example, in the Gulf, radar operating on an 89 km (48 nmi) range scale at 1000 pps (maximum unambiguous range 150 km) often disconcertingly displayed spurious islands in the sea at 75 km. These were 'second time around' echoes of inland mountains at 225 km, the echoes not arriving until the next display timebase sweep. PRF stagger would have decorrelated and broken up these false echoes. Wylie points out that in anticyclonic weather there is usually a subsidence of warm dry air which, settling over the surface of the sea, may produce super-refraction conditions. In areas like the Red Sea, super-refraction conditions may be present more often than not. A temperature difference of 3°C-6°C, often well exceeded, may suffice.
5.2.7 Negative k
If lapse rate exceeds 157 or M is negative, the rays would curve downwards, making horizon range fall again. The equivalent Earth would become concave with negative k.
The author is not aware of any reported cases of this phenomenon.
5.2.8 Sub-refraction; low k; sub-standard surface layer
Particularly in fog, at night or early forenoon and in winter, a body of cold surface air may move under a body of moisture-laden air. According to Wylie [4], sub-refraction also occurs when cool air blows over a relatively warm sea, as in the Polar regions and near very cold land masses. The refraction gradient at the surface is depressed and k is less than 4/3, shown in Figure 5.2(c); Af is negative and k can fall to 0.8 or less. The rays curve upwards. This condition is potentially hazardous; the effective Earth radius is reduced, the radar horizon is nearer and maximum detection ranges may be much less than expected under just those conditions when radar is needed most.
5.2.9 Ducts
The following description of ducting mainly follows Kerr [5] and Barton and Leonov [6, p. 151]. When warm dry air passes over a cool ocean surface, the water cools and moistens the bottom air layer. Temperature increases unusually rapidly with height. The resulting stability prevents mixing and the water vapour content decreases rapidly with height. As shown in Figure 5.2(d), lapse rate may be negative in the first tens of metres above the surface before turning positive. This change forms a surface or evaporation duct, in which rays can be trapped and propagate long distances, alternately reflected from the sea surface and the layer rather like waveguide transmission, giving anomalously long detection ranges.
In calm weather or thunderstorms, a meteorological inversion layer often forms some metres above the sea surface, modifying the Af profile to the form shown in Figure 5.2(e). Figure 5.3(a) depicts ray paths when the scanner lies within the duct.
Elevated ducts are often found seasonally in the tradewind region between the mid- ocean high pressure cells and the equatorial doldrums. The most famous tradewind
Figure 5.3 Rays in elevated duct. Refractivity profile as Figure 5.2(e). Target illumination is strongly dependent on scanner, duct and target heights.
Not to scale
areas lie between Brazil and the Ascension Islands, and between southern California and Hawaii.
Some ducting is likely to be present over all water surfaces. When ducts are strong and the scanner is above the layer, rays may be deflected upwards; low targets are shut out and may be detected weakly if at all. Rays may also become trapped in elevated ducts to the detriment of targets beneath. Maximum wavelength which can be trapped varies with duct height d metres and is:
W = O.OOllVd (5.7)
• if d is less than 17.3 m, neither 3 nor 9 GHz band can be trapped;
• when d is between 17.3 and 165 m, the 9 but not 3 GHz band can be trapped;
• when d is higher still, both bands can be trapped.
Rays within a duct, Figure 5.3(a), or below an elevated duct, Figure 5.3(b), are constrained and partially focussed in elevation and the usual R~4 echo strength law changes to R~2 because the beam spreads out only in the horizontal plane. In principle and disregarding any 'waveguide loss', the echo strength would reduce 6dB when range doubles, rather than the free space 12 dB, explaining the exceptionally long detection ranges sometimes observed. If both the scanner and target are above the duct, the illumination intensity is more or less its normal value, but targets across a boundary from the scanner are at best weakly illuminated.
Wylie states that surface ducts are more significant to marine radar than elevated ducts, and ducting is usually associated with a sharp decrease of moisture content which is often accompanied by a temperature inversion, that is, a sudden increase in temperature.
5.2.10 Conditions causing anaprop
In calm tropical waters in daytime k can rise to 10 or more, falling at night. In temperate latitudes k usually lies between 1 and 2, although extremes are wider, perhaps 0.5-10.
When trialling shipborne defence radars in UK waters in daylight, fair weather,
(a) Scanner within duct
Range
(b) Scanner below duct
Scanner Low target illuminated
Low target may be weakly illuminated High target shut out Steep rays escape