Full-wave solvers become too slow when the object is many wavelengths long. Asymptotic methods replace the solve with rays and currents. Planning models replace the geometry with statistics. Drag the size to see which method is valid, and why.
Electrical size L/λ and the methods that are valid
good choicepossible, with care or large compute
Physical size at
Trace rays in a street. See reflection, multipath and the GO shadow.
Drag the transmitter, the receiver or a building. Each ray is one path. The receiver adds the paths.
ReflectionsSum
Path gain at Rx (isotropic antennas)
Free space, same distance
Paths
RMS delay spread
Power-delay profile
Polarization
Fresnel reflection |Γ| against the angle of incidence
GO assumes a local plane wave. Each ray carries amplitude, phase and polarization. A wall multiplies the ray by ΓTE or ΓTM.
Image method: mirror the source in each wall to get exact paths. The cost grows as (walls)reflections. Ray launching (SBR): shoot many rays and catch them in a sphere at Rx. The cost grows with the number of rays.
GO gives zero field in the shadow. Set Reflections to 0 and move Rx behind a building. Real fields are not zero there: that is diffraction.
Top view: the walls are tall and the rays stay in the horizontal plane (2.5-D), so the field spreads as 1/r. TE: the scalar field is vertical E. TM: the scalar field is vertical H (ΓTM is the H ratio). Walls are opaque here; real tools also add transmission through walls. Materials: ITU-R P.2040-3, Table 3 (now superseded by P.2040-4). Its frequency ranges are indicative: most materials are fitted from 1 GHz, brick to 10 GHz, ground 1–10 GHz only, so 900 MHz and 28 GHz values are extrapolated.
Diffraction fills the shadow
At an edge, one incident ray makes a cone of diffracted rays. UTD gives each diffracted ray its amplitude.
UTDA building corner
Polarization
Field on the dashed arc around the corner
GO onlyGO + diffraction
Field at ISB, GO+UTD vs incident
Tx to edge s′
ISB: incident shadow boundary. RSB: reflection shadow boundary. GO jumps at both. UTD adds a diffracted field that cancels each jump. At the ISB the total is about 6 dB below the incident field: the ISB term gives exactly half the field; the other wedge terms add a little, most when the ISB is near the RSB. Keller's GTD is infinite at the boundaries; the UTD transition function F removes the singularity.
Ed = Ei(Q) · D(φ, φ′, L, n) · √(s′ / (s (s + s′))) · e−jks · D: Kouyoumjian–Pathak (1974), lossy faces after Luebbers (1984)
P.526Knife-edge over a hill
Side view (vertical scale exaggerated). Drag the top of the obstacle.
Diffraction loss J(v), exact
Fresnel parameter v
P.526 eq. approx.
First Fresnel radius r₁
h / r₁
exact (Fresnel integrals)P.526 approximation
v = h √(2/λ · (1/d₁ + 1/d₂)) · J(v) = 6.9 + 20 log10(√((v−0.1)²+1) + v − 0.1) for v > −0.78 · r₁ = √(λ d₁ d₂ / d)
At grazing (h = 0) the loss is 6 dB. Keep 0.6 r₁ clear (v ≈ −0.85) and the loss is about 0 dB.
The knife-edge is the planning-tool form of diffraction. ITU-R P.526 adds rounded obstacles and multiple edges (Deygout, Bullington).
Physical optics: currents from the tangent plane
PO does not trace the field to the observer. It puts the current of an infinite flat mirror on each lit point, then radiates that current.
Js = 2 n̂ × Hinc on lit faces · Js = 0 in the shadow
CurrentPO current on a cylinder vs the exact series
exact (Bessel series)PO: 2|cosφ| lit, 0 shadow
At the shadow boundary, exact
Deep shadow, exact
Lit point, exact / PO
TMz plane wave (E along the axis), |Js| / |Hinc|. PO is good on the lit side when ka is large. It misses the creeping wave in the shadow and the smooth change at the shadow boundary.
RCSFlat strip: PO vs MoM vs PO + PTD
MoM (reference)POPO + PTD
Echo width at θ, MoM
PO error
PO + PTD error
MoM unknowns
2-D strip, TMz (E parallel to the edges), PEC. MoM: EFIE, pulse basis, 20 cells/λ, solved in your browser. PTD adds the fringe field of each edge: Keller's half-plane coefficient minus the PO edge term.
σ2DPO = k w² cos²θ · sinc²(k w sinθ)
3-D plate a × b, rotating in the plane of side a: σPO = (4πA²/λ²) cos²θ · sinc²(k a sinθ)
Near specular, PO is correct. The peak k w² matches MoM within 0.1 dB.
Away from specular, edges dominate. PO gives deep false nulls and the wrong level. Near edge-on it is far too low.
PTD corrects the edges. PO + PTD follows MoM within a fraction of a dB here. Real tools add PTD or MEC on every wedge.
SpherePO vs the exact Mie series
Mie (exact)PORayleigh 9(ka)⁴ · optical limit 1
Mie σ/πa²
PO σ/πa²
PO error
σPO/πa² = 1 − sin(2ka)/ka + sin²(ka)/(ka)²
Rayleigh region (ka < 0.5): PO scales as (ka)², not 9(ka)⁴; the error is 10 dB at ka = 0.1. Resonance region: the creeping wave interferes with the specular return. Optical region: PO is within about 1 dB for ka > 10 and about 0.4 dB for ka > 20; both tend to πa². Mie ripples because of the creeping wave, which PO does not have. PO ripples a little too, from its abrupt shadow boundary.
Shooting and bouncing rays
SBR shoots a dense grid of GO rays at the target. Each ray bounces. At each hit, a PO integral over the ray tube gives that ray's share of the scattered field.
One bounce is PO with shadowing. It gives the two plate flashes of the dihedral, but not its wide return.
Two bounces add the double reflection. The 90° dihedral retro-reflects over ±45°: σ2D = 2k w² on the bisector, falling as the aperture 2w sin(45° − |φ|) shrinks. SBR follows MoM within about 1 dB over ±30°. Near ±40°, edge diffraction (not in SBR) gives errors of 2–3 dB.
SBR has no edges. Production codes add PTD or MEC for wedges and UTD for creeping waves (Ling, Chou & Lee 1989; Xpatch).
2-D, TMz, PEC, plates 5λ long, 10 rays per λ. Tools on SBR: Ansys HFSS SBR+ (SBR with PTD and UTD edge corrections and creeping waves), CST Studio Suite Asymptotic Solver (SBR). Remcom Wireless InSite X3D shoots and bounces rays for propagation, then moves each path to the exact receiver point. Remcom XGtd uses GO, UTD, PO and MEC. Sionna RT uses SBR to find candidate paths, then the image method.
Planning models: statistics instead of geometry
Empirical models fit curves to measured path loss. They need no building map. They give a median and a spread, not a field.
Environment
free spacetwo-ray (ground εr 15, σ 0.005 S/m, horizontal pol.)Okumura–HataCOST-231 Hataoutside the valid range
Path loss at d =
Free space
Two-ray
Breakpoint 4hbhm/λ
Median ± 1σ
Hata (urban): L = 69.55 + 26.16 log f − 13.82 log hb − a(hm) + (44.9 − 6.55 log hb) log d
f in MHz, d in km, heights in m. Hata: 150–1500 MHz, hb 30–200 m, hm 1–10 m, d 1–20 km. COST-231 Hata: 1500–2000 MHz, constant 46.3 + 33.9 log f, plus 3 dB in large-city centres. Large-city a(hm) is defined for f ≤ 200 and f ≥ 400 MHz; between them this page interpolates. Drag on the plot to move d.
Two-ray: past the breakpoint the direct and ground rays cancel, and the loss grows 40 dB per decade instead of 20.
Hata grows about 30–35 dB per decade over its valid hb of 30–200 m (slope 44.9 − 6.55 log hb). Real signals scatter about the median: lognormal shadowing, σ ≈ 6–10 dB.
Empirical model (Hata, COST-231, P.1546)
Inputs
f, d, antenna heights, an environment class. For terrain models: a terrain profile.
Outputs
Median path loss. A location spread σ. No delay, no angle.
Speed
Microseconds per point. A country in minutes.
Limit
Valid only inside its fitted range. Calibrate with drive tests.
Ray tracing (GO + UTD, SBR)
Inputs
A 3-D map of buildings and terrain, materials, antenna patterns.
Outputs
Each path: power, phase, delay, angle of departure and arrival.
Speed
Milliseconds to seconds per point. A city in hours, or faster on a GPU.
Limit
Only as good as the map and the materials. Misses foliage, people and small clutter.
Terrain and point-to-area models
ITU-R P.154630–4000 MHz · 1–1000 km
Point-to-area curves of field strength from measurements. For broadcast and mobile coverage.
ITU-R P.181230 MHz–6 GHz · 0.25–3000 km
Path-specific: uses the terrain profile, clutter and diffraction (P.526 Bullington).
ITU-R P.452about 0.1–50 GHz
Interference between stations: diffraction, ducting, troposcatter, small time percentages.
ITU-R P.526diffraction
Knife-edge, rounded obstacles, multiple edges, spherical earth. Used inside the others.
Longley–Rice (ITM)20 MHz–20 GHz · 1–2000 km
Irregular terrain model from NTIA/ITS. Used by SPLAT! and many planning tools.
Which method for my problem?
Select a task. Methods and tools are typical choices, not recommendations.
Task
References
C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed., Wiley, 2012. Fresnel coefficients, PO, cylinder series, GTD/UTD.
E. F. Knott, J. F. Shaeffer, M. T. Tuley, Radar Cross Section, 2nd ed., Artech House, 1993. Plate and sphere PO, edge diffraction.
J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52(2), 116–130, 1962.
R. G. Kouyoumjian, P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62(11), 1448–1461, 1974.
R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag. 32(1), 70–76, 1984.
P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction, Wiley, 2007.
H. Ling, R.-C. Chou, S.-W. Lee, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propag. 37(2), 194–205, 1989.
G. T. Ruck et al., Radar Cross Section Handbook, Plenum, 1970. Mie series for the PEC sphere.
W. C. Gibson, The Method of Moments in Electromagnetics, 2nd ed., CRC, 2014. 2-D TM EFIE and its self term.
M. Hata, “Empirical formula for propagation loss in land mobile radio services,” IEEE Trans. Veh. Technol. VT-29(3), 317–325, 1980.
COST Action 231, Digital mobile radio towards future generation systems, final report, EUR 18957, 1999.