Two-Ray Ground Reflection Model

A propagation model accounting for both the direct path and a ground-reflected path between transmitter and receiver.

What is the Two-Ray Ground Reflection Model?

The two-ray ground reflection model extends free-space path loss by adding a second signal path: the signal that reflects off the ground between the transmitter and receiver. At the receiver, these two paths (direct and reflected) can add constructively or destructively depending on their relative phases.

When the reflected path arrives nearly in phase with the direct path, the received signal is stronger than FSPL alone would predict (up to +6 dB). When they arrive out of phase, deep nulls can cause received power to drop far below FSPL predictions (potentially 20–40 dB below FSPL). This interference pattern is highly sensitive to antenna heights, link distance, and frequency.

The model is particularly relevant for maritime links (over sea water), terrestrial microwave paths, drone-to-ground links, and any scenario where one or both antennas are close to a reflective surface.

Why Does It Matter?

For links over flat, reflective ground or water, FSPL can severely underestimate or overestimate actual received power
The two-ray model predicts deep fading nulls that can occur at specific distances — critical for fixed link planning
Maritime links over sea water experience very strong reflections and must use the two-ray or more advanced models
Antenna height selection can be optimized to avoid destructive interference zones
The stability factor determines when the model is valid vs. when FSPL is a better approximation

Quick Two-Ray Calculator

Frequency (MHz)

Distance (km)

Tx height (m)

Rx height (m)

Formulas Used by LinkBudgetPro

\[ \text{Stability Factor} = \frac{h_{\text{tx}} \cdot h_{\text{rx}}}{\lambda \cdot d} \]

If stability factor \(> 3.0\): ground reflection negligible, fall back to FSPL

\[ \phi = \frac{2\pi \cdot h_{\text{tx}} \cdot h_{\text{rx}}}{\lambda \cdot d}, \qquad E_{\text{field}} = \frac{\lambda}{4\pi d} \cdot 2\sin(\phi) \]

\[ L_{\text{2-ray}} = -10\log_{10}(E_{\text{field}}^{\,2}) \quad \text{bounded to } [\text{FSPL} - 6,\ \text{FSPL} + 40] \text{ dB} \]

Constructive interference (gain) capped at +6 dB over FSPL; deep nulls capped at +40 dB over FSPL

Parameter Explanation

Parameter	Symbol	Unit	Description
Tx Antenna Height	h_tx	m	Height of transmit antenna above the reflecting surface
Rx Antenna Height	h_rx	m	Height of receive antenna above the reflecting surface
Link Distance	d	m	Total one-way path length in metres
Wavelength	λ	m	c / f — determines how phase changes with geometry
Stability Factor	—	—	Ratio (h_tx × h_rx)/(λ·d). >3.0 means FSPL is adequate
Phase	φ	rad	Phase difference between direct and reflected path at the receiver
Two-Ray Loss	L_2ray	dB	Propagation loss accounting for direct + reflected path interference

Worked Example

900 MHz maritime link, 5 km, antenna heights 10 m (Tx) and 5 m (Rx). Check stability and find loss:

λ = 300/900 = 0.333 m; d = 5000 m
Stability Factor = (10 × 5) / (0.333 × 5000) = 50 / 1667 = 0.030
Stability < 3.0 → Two-Ray model applies (ground reflection significant)
Phase = (2π × 10 × 5) / (0.333 × 5000) = 314.16 / 1667 = 0.1885 rad
sin(0.1885) ≈ 0.1874
Field Factor = (0.333 / (4π × 5000)) × 2 × 0.1874 = 5.31×10⁻⁶ × 0.3748 = 1.99×10⁻⁶
FSPL (900 MHz, 5 km) = 32.44 + 13.98 + 59.08 = 105.5 dB
Raw Two-Ray Loss = −10·log₁₀((1.99×10⁻⁶)²) = −10·log₁₀(3.96×10⁻¹²) = 114.0 dB
Bounded Two-Ray Loss = min(max(114.0, 99.5), 145.5) = 114.0 dB (+8.5 dB over FSPL)

When Should You Use It?

Maritime / over-water links — sea water is a near-perfect reflector; two-ray is essential
Low-altitude terrestrial links — when both antennas are within a few metres of the ground
Drone-to-ground control links — drone altitude changes continuously, varying the two-ray interference pattern
Microwave backhaul over flat terrain — large flat areas produce significant ground reflections
Any time stability factor < 3.0 — use the two-ray model; otherwise FSPL is sufficient

Do not use two-ray for indoor links, heavily obstructed paths, or when the reflecting surface is irregular terrain — more complex models are needed in those cases.

Related Calculations

Free Space Path Loss (FSPL) — baseline model used when stability factor > 3.0
Fresnel Zone & Clearance — complementary analysis for obstacle clearance
Fade Margin Calculator — use two-ray loss as the path loss input
EIRP Calculator — effective transmitted power overcoming the two-ray loss
RF Documentation Index — all RF engineering reference pages

Select the Two-Ray Ground Reflection model in the full calculator for maritime and low-altitude link analysis.

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