Scattering and two-ray ground reflection model

Scattering

Scattering occurs when a radio wave meets an object whose size is comparable to or smaller than the wavelength, or a rough surface (foliage, street signs, lamp posts, rough ground), and the energy is re-radiated in many directions. It explains why received signal is often stronger than reflection and diffraction models predict. A surface is treated as rough when its height variation exceeds the critical height hc = λ/(8 sin θi) (Rayleigh criterion).

Two-ray ground reflection model

  Tx
   |\  LOS path d'
ht | \_______________________ Rx
   |  \                    _/|
   |   \  reflected d''  _/  | hr
 --+----\____________/-------+-- ground
   |<----------- d --------->|

1. Field expressions. With E0 the field at reference d0, the LOS and ground-reflected fields are:

E_LOS = (E0d0/d')·cos[ωc(t − d'/c)]
E_g   = Γ·(E0d0/d'')·cos[ωc(t − d''/c)]

For grazing incidence, Γ ≈ −1.

2. Path difference (method of images):

d'  = √[(ht − hr)² + d²] ≈ d + (ht − hr)²/(2d)
d'' = √[(ht + hr)² + d²] ≈ d + (ht + hr)²/(2d)
Δ = d'' − d' ≈ 2ht·hr/d

3. Phase difference and delay:

θΔ = 2πΔ/λ = Δ·ωc/c
τd = Δ/c

4. Total field. Taking d' ≈ d'' ≈ d in the amplitudes and Γ = −1:

|ETOT| = (E0d0/d)·|1 − e^(−jθΔ)|
       = (E0d0/d)·√(2 − 2cosθΔ)
       = 2(E0d0/d)·sin(θΔ/2)

5. Large-distance approximation. When θΔ/2 < 0.3 rad (i.e. d > 20πht·hr/(3λ) ≈ 20ht·hr/λ), sin(θΔ/2) ≈ θΔ/2:

|ETOT| ≈ (2E0d0/d)·(2π·ht·hr/(λd)) ≈ k/d²   V/m

6. Received power and path loss. Since power ∝ |E|²:

Pr = Pt·Gt·Gr·ht²·hr² / d⁴

PL(dB) = 40 log d − (10 log Gt + 10 log Gr
                     + 20 log ht + 20 log hr)

So at large distances received power falls at 40 dB/decade, compared with 20 dB/decade in free space, and the path loss is independent of frequency.