Groundwave Propagation, Part One

Groundwave Propagation, Part One 1 Planar Earth groundwave 2 Planar Earth groundwave example 3 Planar Earth elevated antenna effects Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 1 / 39

Groundwave propagation For antennas located above a PEC plane, we can use image theory to solve the problem Non-PEC plane, Chapter 7 approach was to still use image theory but multiply image by reflection coefficient This is an approximation because real antennas do not generate only direct and reflected plane waves Real antennas generate a plane wave spectrum consisting of plane and evanescent waves propagating in all directions Consideration of this shows we need another term to satisfy boundary conditions on conductor This additional term is the groundwave Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 2 / 39

direct wave groundwave reflected wave Figure: Schematic depiction of direct wave, reflected wave, and groundwave excitated by a source near ground. The groundwave is guided at the air/ground interface decays exponentially away from it. At large distances and relatively low frequencies, the direct and reflected waves tend to cancel each other and the groundwave can become the dominant contribution. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 3 / 39

Groundwave usually smaller than direct + reflected, so it only is observable when direct + reflected cancel, i.e. near the ground when Γ 1. Near is defined by the wavelength! Groundwave has a complicated form both for variations along the boundary and away from the boundary. More poorly conducting grounds cause groundwave to attenuate faster. Groundwave also attenuates more rapidly for horizontal polarization. Requirements for groundwave usually mean HF or lower frequencies. Need to consider ionosphere for these frequencies too; usually groundwave is mainly important for local (still over-the-horizon) reception, ionosphere for longer distance. A skip zone may exist. At higher frequencies, groundwaves can be neglected and the direct + reflected formalism is usually applicable. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 4 / 39

Snells law incident wave reflected wave air ground groundwave transmitted wave Figure: An incident plane wave at a planar ground interface produces reflected and transmitted waves as shown in blue. Since the ground has larger permittivity than air and is also conductive, a basic application of Snell s law shows that the transmitted wave bends closer to the normal direction. Hence, the groundwave apparently cannot be excited by any incident angle regardless of the angle of incidence. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 5 / 39

The mathematical theory behind the groundwave is complex, so we will touch it only briefly. There are two solutions that are applied: Sommerfeld Planar Earth problem: first solved by Sommerfeld, applies so long as Earth curvature is negligible. Has a long history of disucssion in EM community due to confusion in calculation. Applies for d < 80/f 1/3 MHz km. Van der Pol and Bremmer spherical Earth theory: same theory we used in Chapter 7, again pretty complicated. Used for getting groundwave predictions beyond limits of planar theory. Basic configuration usually considered has both antennas on the ground, often a short vertical monopole transmitting 1 kw. We will also consider height-gain effects for elevated antennas both in and beyond line of sight. Next, we will cover a bit of the underlying math and physics behind the existence of groundwaves. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 6 / 39

Groundwave mechanism Take a scalar point source, which produces a spherical wave in free-space. A spherical wave admits the following integral representation e jkr r = j jkx x jky y jkz e z dk x dk y (1) 2π k z where k z = (k 2 k 2 x k 2 y ) 1/2. This is called Weyl identity and shows that a spherical wave admits a spectral representation as a superposition (integral) of plane wave components. Note that the domain where k 2 x + k 2 y < k 2 correspond to propagating waves and the domain k 2 x + k 2 y k 2 to evanescent waves. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 7 / 39

Groundwave mechanism Groundwaves are excited by the evanescent spectrum such that the incident angle is complex-valued. Take, for simplicity k y = 0, then e jkx x j(k2 k 2 x ) 1/2 z = e jkx sin θ i +jkz cos θ i (2) in the Weyl integral, when k x > k, we can have sin θ i > 1 corresponding to a complex-value θ i. Since sin θ i > 1, we can have a groundwave excited with θ t = 90 0 (sin θ t = 1) and compatible with Snell s law. The evanescent field contribution is stronger for lower frequencies and when the antenna is close to the ground. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 8 / 39

Groundwave mechanism In the presence of the planar ground interface, the field on the air produced by a point source located at a vertical distance d from the ground writes as the following modification of the Weyl integral j jkx x jky e [ ] dk x dk y e jk 1z z + R 12 e jk 1z z 2jk 1z d (3) 2π k z where the second term is caused by the presence of the ground. The above integral can be solved numerically on a case-by-case basis but does not have exact analytical solution. For a real antenna (not a point source), the expression should be generalized to a vector-equivalent and incorporate the antenna patterns. Good approximate solutions can be obtained at low frequencies (i.e., few-mhz range and below) and for antennas close to the ground. We will use those next. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 9 / 39

Planar Earth groundwave A good approximation for the planar Earth region is E tot z = E [ 0 cos 3 ψ 1 e jk 0R 1 + Γ cos 3 ψ 2 e jk 0R 2 d ] +(1 Γ) cos 2 ψ 2 e jk 0R 2 F 1 (R 2, ɛ) (4) Note that we discussed E 0 (the unattenuated field intensity at unit distance) in Chapter 7 cos 3 ψ dependencies result due to (1) antenna pattern (2) z component of field and (3) changing d into path distance Note we are adding direct + image times reflection coefficient exactly as in Chapter 7 Groundwave term is (1 Γ) cos 2 ψ 2 e jk 0R 2 F 1 (R 2, ɛ) This is not just an image contribution due to additional F 1 (R 2, ɛ) factor Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 10 / 39

Tx ψ 1 ψ 2 R 1 h 1 air R 2 R 2 Rx h2 ground d h 1 R 2 =R 2 +R 2 Figure: Geometrical parameters for a planar Earth groundwave link. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 11 / 39

For antennas near ground, Γ 1, and direct and reflected waves cancel so that the field expression in (4) simplifies to E tot z 2E 0 R 2 F 1 (R 2, ɛ)e jk 0R 2 (5) Field amplitude thus varies with distance as F 1 (R 2, ɛ)/r 2. The factor F 1 (R 2, ɛ) is less than one and produces an excess loss. If antennas are very near ground, the factor F 1 (R 2, ɛ) does not depend on R 2 and ɛ independently; it can be simplified to F (p) in terms of a numerical distance p defined through p = j k 0R 2 2ɛ ( ɛ cos 2 ) ψ 2 ɛ Vertical polarization (6) p = j k 0R 2 2 (ɛ cos2 ψ 2 ) Horizontal polarization (7) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 12 / 39

The function F (p) decreases for increasing p For antennas very near ground, where cos 2 ψ 2 1 and R 2 d, we can simplify the expressions for p as p j k 0d 2ɛ ( ) ɛ 1 ɛ Vertical polarization (8) p j k 0d (ɛ 1) Horizontal polarization (9) 2 Furthermore, for a typical ground σ 10 3 to 10 2 S/m, and σ/ωɛ 0 18/f MHz to 180/f MHz. Hence, for frequencies below a few MHz, we have ɛ 1 and p j k 0d 2ɛ Vertical polarization (10) p j k 0d ɛ Horizontal polarization (11) 2 Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 13 / 39

p is thus distance scaled by wavelength and a function of ɛ; becomes larger at higher frequencies and for less conducting grounds. p is much larger in horizontal polarization than vertical. As a result, vertical polarization antennas are preferred. p also written as p e jb. b depends only on ɛ. 0 < b < 90 for vertical, 90 < b < 180 for horizontal F (p) does not have a simple form, but series approximations are available. A good approximation for p < 10 is given by the series F (p) = 1 j (πp)e p 2p + (2p)2 1 3 (2p)3 1 3 5 +... (12) For p > 20, the approximation F (p) = 1/(2p) is useful. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 14 / 39

For p > 20, if more accuracy is required, the asymptotic series below can be used: F (p) = 1 2p 1 3 (2p) 2 1 3 5 (2p) 3... (13) A plot of F (p) with b as a parameter is on the next page. Notice near constant nature followed by 1 2p roll off Groundwaves thus transition from a 1/d dependence to 1/d 2 dependence with distance in the planar Earth region. Change from one dependence to the other varies with the value of b (conductivity of ground.) Larger σ/(ωɛ 0 ) implies larger b and a more gradual transition between these two dependencies. Remember this is in terms of the numerical distance, so behavior in actual distance will be a scaled version of the plot. Also applies only to 80/f 1/3 MHz km. Beyond that the spherical Earth approach should be used, as discussed later. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 15 / 39

Excess Propagation Loss F(p) (db) 0 20 40 b=0 b=45 b=90 b=180 1/(2p) 60 0.01 0.1 1 10 100 Numerical Distance p Figure: Excess groundwave propagation loss versus p. Excess here means w.r.t 1/R 2. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 16 / 39

Planar Earth groundwave example 1 kw, 1 MHz vertical monopole on ground, ɛ R = 30, σ = 0.01 S/m ɛ I = σ ωɛ 0 = 18000σ f MHz = 180 Planar Earth methods should apply for d < 80/f 1/3 MHz = 80 km For both antennas on the ground, the field is E tot z 2E 0 d F (p)e jk 0d (14) For a short vertical monopole transmitting 1 kw of power, 2E 0 is 300 mv for distance d in km. Numerical distance here is p = j k 0R 2 2ɛ ( ) ɛ 1 = ( 5.7411 10 2) d exp ( j0.1708) (15) ɛ with d in km. Reaches value of p = 4.6 at planar Earth region boundary. b is 9.8 (highly conducting.) Next page plots groundwave field amplitude in decibels above 1 µv per meter. Spherical Earth curve also included. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 17 / 39

Field amplitude (db above 1 µ V/m rms) 120 100 80 60 40 20 0 20 Planar Earth Spherical Earth 40 60 1 10 100 1000 Distance (km) Figure: Groundwave field amplitude in 1 MHz example Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 18 / 39

Planar Earth elevated antenna effects When transmit or receive antennas are not on the ground, elevated antenna effects can potentially become important. If either antenna height is greater than 610/f 2/3 MHz meters, planar Earth theory does not apply. See spherical methods. When applicable, numerical antenna heights q 1 and q 2 are used instead of physical heights h 1 and h 2 in the planar Earth region to simplify calculations: (vertical antennas) q 1,2 = k 0 h 1,2 ɛ cos 2 ψ 2 ɛ (16) Again these are the antenna heights scaled by the wavelength and a function of the ground conductivity. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 19 / 39

Planar Earth elevated antenna effects For q 1 + q 2 < 0.01, antenna elevation can be neglected. For q 1 + q 2 > 0.01, the calculation proceeds by adding the direct, reflected, and groundwave terms keeping track of the phases, but with the groundwave term modified to uses F (P), where P = 4p/ (1 Γ) 2. Computation can be quite tedious, so it is typically done through a numerical code However, a useful approximation is available for sufficiently apart antennas such that p > 20, p > 10q 1 q 2, and p > 100(q 1 + q 2 ). In this case, the resulting total field amplitude is simply that of the groundwave along (assuming both antennas on the ground) multiplied by height functions f (q 1 ) and f (q 2 ), where f (q) = [ 1 + q 2 2q cos( π 4 + b 2 ) ] 1/2 (17) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 20 / 39

A plot of the height-gain function f (q) is shown below. Height gain function f(q) (db) 40 20 0 b=0 b=45 b=90 b=180 20 0.01 0.1 1 10 100 Numerical antenna height q Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 21 / 39

Planar Earth groundwave: another example 1 kw, 10 MHz vertical monopole on ground, ɛ R = 15, σ = 0.003 S/m ɛ = 15 j5.4 Planar Earth methods should apply for d < 80/f 1/3 MHz = 37 km Numerical distance is p j k ( ) 0d ɛ 1 = 6.18 d e j1.24 (18) 2ɛ ɛ At the distance where planar earth theory becomes invalid, p = 229, much larger than in the 1 MHz example before Furthermore, b = 71.5 0, indicating a relatively low conductivity Next page plots groundwave field amplitude in decibels above 1 µv per meter. Both antennas are assumed located on the ground. Spherical Earth curve also included for comparison. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 22 / 39

Field amplitude (db above 1 µ V/m rms) 100 80 60 40 20 0 20 Planar Earth Spherical Earth 40 60 1 10 100 Distance (km) Figure: Groundwave field amplitude in 10 MHz example. Planar Earth results become inaccurate beyond about 37 km. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 23 / 39

Groundwave Propagation, Part Two 1 Spherical Earth groundwave 2 Spherical Earth groundwave example 3 Computer program 4 Summary of groundwave methods Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 24 / 39

Spherical Earth groundwave When distances exceed 80/ 3 f MHz km or when either antenna height is greater than 610/f 2/3 MHz m, Earth curvature must be taken into account. This theory is only applicable for these larger distances. Does not converge well in the planar Earth region or for antennas high enough to produce line of sight. For high antennas in the line of site, a direct-plus-reflected approach for a sphere can be used. Available in references, not considered in detail here. A new scaled distance parameter is used: ( ) d 3 x = π (λ/a eff ) 2 (19) λ where a eff is the effective Earth radius. Note does not depend on ground dielectric properties. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 25 / 39

A new ground dielectric descriptor τ is used: τ = j 3 π a eff ɛ 1 λ ɛ (20) τ = 0 is PEC; larger values indicate less conducting grounds. Elevated antennas are described by new numerical antenna heights y: y 1,2 = 2 h 1,2 3 π λ 2 λ (21) a eff Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 26 / 39

Field in the Spherical Earth region is then expressed as the summation below E tot z = 2E 0 d e jk 0d πx j s=1,2,3,... G s (y 1 )G s (y 2 ) exp ( jxt s ) (t s τ 2 ) Here s is an index to a set of complex roots t s. Roots have negative imaginary parts so total field is exponentially decaying. Roots are a function of τ; series expressions provided in the Appendix of Chapter 9. G s is a spherical height-gain function that, as indicated, also depends on s. They have complicated expressions, with approximate forms given in the Appendix of Chapter 9. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 27 / 39

Results transition from planar Earth to an exponential falloff with distance. A sample plot of the excess spherical Earth propagation loss is: Excess Propagation Loss (db) 0 20 40 60 80 100 120 140 160 180 τ=0 τ=2 j2 τ=100 j100 200 0.1 1 10 Spherical Earth distance parameter X Figure: Spherical Earth excess loss, determined by the field amplitude divided by 2E 0 /d Number of series terms needed decreases with distance. For distances d > 36 3 λ km, only one term is good enough. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 28 / 39

G s (y 1 ) and G s (y 2 ) are new height gain functions; both have value unity when antennas are on ground: 100 Height gain function (db) 90 80 70 60 50 40 30 20 τ=0 τ=1 j1 τ=2 j2 10 0 10 0.01 0.1 1 10 Scaled antenna height y Figure: Spherical Earth height gain functions Again show initial decrease, but later exponential increase with height. Low conducting grounds show larger increases due to smaller initial value Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 29 / 39

Spherical Earth approximation When both antennas are on the ground and d > 36 3 λ km, a single term is enough and the series becomes: Ez tot = 2E 0 π x exp [xim{t1 d t 1 τ 2 }] (22) = 2E 0 d Q x exp [xim{t 1 }] (23) Plots of Im{t 1 } and field scaling factor Q are provided on the next page versus τ. Exponential attenuation is obtained for all larger distances. Field amplitudes fall of very rapidly so groundwave less usable in this region. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 30 / 39

Imaginary part of t 1 0.5 1 1.5 2 45 degrees 60 degrees 75 degrees 90 degrees Field Scaling Factor Q (db) 10 0 10 20 30 40 50 60 45 degrees 60 degrees 75 degrees 90 degrees 70 2.5 0.1 1 10 100 Magnitude of τ 80 0.1 1 10 100 Magnitude of τ Exponential attenuation is obtained for all larger distances. Field amplitudes fall of very rapidly so groundwave less usable in this region. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 31 / 39

Spherical-Earth groundwave example 1 MHz vertical monopole on ground, ɛ R = 30, σ = 0.01 S/m ɛ I = 180, and the planar Earth theory applies for d < 80 km. Assume 1 kw power transmitted, if more or less we can just scale field values according to P T, with P T in kw. Find the spherical Earth distance parameter (with d in km): ( ) d 3 x = π (λ/a eff ) 2 = ( 5.256 10 3) d (24) λ This obtains a value of 0.42 at distance 80 km where we transition between theories. In this case, τ = j 3 π a eff ɛ 1 = 2.13 j2.53 = 3.3e j0.87 (25) λ ɛ Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 32 / 39

Spherical-Earth groundwave example (cont d) Reading previous plots, Im{t 1 } = 1.77 and Q = 14.6 db. Single term solution should be good for d > 3 λ = 241 km. Below are results using full series. Field amplitude (db above 1 µ V/m rms) 120 100 80 60 40 20 0 20 Planar Earth Spherical Earth 40 60 1 10 100 1000 Distance (km) Figure: Groundwave field amplitude in 1 MHz example Single term results is within 0.5 db for d > 241 km. Note: between 80 and 241 km, curve is very smooth; a simple smooth curve drawn between two regions would yield reasonable accuracy. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 33 / 39

ITU information and other resources ITU-R recommendation P.368-9 provides several curves for predicting groundwave intensities. The reference is a 1 kw vertical monopole transmitter at ground level Curves are presented for various ground properties and frequencies. For elevated antennas, the height-gain functions should be used on top of those predictions. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 34 / 39

ITU information and other resources The figure in the next page is an example of such a plot. It should be noted that the models described here apply to groundwave propagation predictions on average. For a given link, site-specific terrain effects may cause deviations form these estimates. These deviations are more significant at higher frequencies and urban or mountain regions. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 35 / 39

Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 36 / 39

ITU information and other resources Alternative methods for computing groundwave propagation over mixed paths also exist. Paths that are part ground and part sea. Paths over layered surfaces (such as ice-covered seawater). Paths over some urban areas Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 37 / 39

Summary of groundwave methods Groundwave most important for near-ground or low-height (in terms of wavelength) antennas. This is by far the primary case when we will use these techniques Usually antennas will be low-height because frequencies are low. Vertical polarization is typically used. Programs provided and ITU curves are therefore sufficient for most practical predictions. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 38 / 39

Summary of groundwave methods (cont d) Important especially at the MF band, from 300 KHz and 3 MHz Dominant propagation mechanism for daytime AM-band broadcasts. Propagation mechanism explored by some over-the-horizon (OTH) radars, including over seawater (a high conductive ground ). Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 39 / 39