Two-ray ground reflection model — path loss with LOS and ground bounce

The Two-Ray Ground Reflection Model (also known as the Plane-Earth Loss Model) is a foundational tool in wireless propagation.

When a cell tower transmits a signal in deep space, power drops off with distance squared ($1/d^2$) following Friis’ free-space law. But put that same tower on Earth, and suddenly the signal decays at $1/d^4$ ($40\text{ dB/decade}$) over long distances.

Why the sudden penalty? The ground reflection.

In terrestrial wireless environments (such as cellular networks, Wi-Fi, or vehicular communications), antennas are elevated above the ground. The receiving antenna picks up two distinct waves:

1. The Line-of-Sight (LOS) Ray: Travels directly through the air over distance $d_1 = d_{los}$.
2. The Ground-Reflected Ray: Bounces off the Earth’s surface over distance $d_2 = d_{ref}$.

When the receiver is far away, both rays travel nearly identical distances and arrive almost parallel. Because bouncing off the ground flips the reflected wave’s phase by roughly $180^\circ$, the two waves arrive out of phase and nearly cancel each other out.

The farther away you go, the closer they get to total cancellation—driving the signal down at $1/d^4$.

Geometric Path Difference ($\Delta d$)

Let $h_t$ be the transmit antenna height, $h_r$ be the receive antenna height, and $d$ be the horizontal separation distance between the two towers.

Using the Pythagorean theorem and a basic Taylor series expansion ($d \gg h_t, h_r$):

$$d_1 \approx d + \frac{(h_t – h_r)^2}{2d}, \quad d_2 \approx d + \frac{(h_t + h_r)^2}{2d}$$

$$\Delta d = d_2 – d_1 \approx \frac{2 h_t h_r}{d}$$

Electric Field Superposition

The physical distance gap creates a phase delay $\Delta \phi = \frac{2\pi \Delta d}{\lambda}$. Combined with the ground’s $180^\circ$ ($\pi$) phase flip:

$$\vert{}E_{\text{tot}}\vert{} \approx \frac{2 E_0}{d} \sin\left( \frac{2\pi h_t h_r}{\lambda d} \right)$$

This $\sin(\cdot)$ term is the mathematical heart of the Two-Ray model:

• Because received power is proportional to field squared ($P_r \propto \vert{}E_{\text{tot}}\vert{}^2$), power varies as $\sin^2\left( \frac{2\pi h_t h_r}{\lambda d} \right)$.
• When $d$ is small to moderate, the multiplier $\frac{2\pi h_t h_r}{\lambda}$ is very large. As you walk away from the tower, the angle inside $\sin(\cdot)$ winds through multiple multiples of $\pi$, causing power to swing rapidly between constructive peaks ($+6\text{ dB}$) and destructive nulls ($-\infty\text{ dB}$).

Far-Field Approximation ($d \gg d_c$)

For long distances, the argument inside the sine function becomes tiny. Using $\sin(\theta) \approx \theta$:

$$\vert{}E_{\text{tot}}\vert{} \approx \frac{2 E_0}{d} \left( \frac{2\pi h_t h_r}{\lambda d} \right) = \frac{4\pi E_0 h_t h_r}{\lambda d^2}$$

Since received power $P_r \propto \vert{}E_{\text{tot}}\vert{}^2$, squaring this expression yields:

$$P_r = P_t G_t G_r \frac{h_t^2 h_r^2}{d^4}$$

Notice that the $\sin(\cdot)$ term—and all interference oscillations—completely vanish, leaving a smooth $1/d^4$ curve.

This has two surprising takeaways:

• Wavelength ($\lambda$) disappears! At long distances, a $2.4\text{ GHz}$ and $5\text{ GHz}$ signal suffer the exact same two-ray loss over flat ground.
• Height is king ($h_t^2 h_r^2$). Doubling your antenna height quadruples your received power (+6 dB gain).

Real-World Reality Checks

Textbook derivations assume a flat mirror, isotropic antennas, and perfect $100\%$ reflection ($\Gamma = -1$). In the field, three physical realities change the story:

• Ground isn’t a mirror (Brewster’s Angle): For vertically polarized cellular signals, the reflection coefficient $\Gamma_V$ drops to zero at Brewster’s Angle ($\theta_B \approx 14^\circ$). At this distance, the ground absorbs the signal instead of reflecting it—preventing cancellation and creating a localized signal spike.
• Ground isn’t flat (Surface Roughness): Grass, asphalt, and ocean waves cause diffuse scattering. By the Rayleigh criterion ($h > \frac{\lambda}{8 \sin\theta_g}$), rough ground weakens the reflected ray. Less reflection means less cancellation, pulling path loss back toward $1/d^2$.
• Antennas aren’t isotropic (Beam Tilt): Base station antennas tilt downward. Because the direct and reflected rays leave the antenna at different angles, they arrive with unequal power. This prevents complete “zero-signal” blackouts in real networks.

The 3 Operational Regions

The exact Two-Ray formula, while mathematically elegant, is impractical for day-to-day radio network engineering. If you are designing a mobile network, calculating link budgets, or setting cell coverage radii, evaluating rapid trigonometric oscillations ($\sin(\cdot)$) at every meter is neither helpful nor realistic. So there exists a piecewise graph that describes how a system engineer plans coverage. It turns complex wave physics into three distinct rule-of-thumb zones used in modern 4G/5G Radio Network Planning (RNP) software. It proves to be an useful tool for

• Separating Large-Scale Path Loss from Small-Scale Fading – The piecewise line strips away the high-frequency “jitter” of the exact curve and presents the true underlying mean signal trend.
• Identifying the Cell Edge (The Breakpoint $d_c$) – Knowing $d_c$ tells network planners exactly where a cell tower’s reliable coverage ends and where handover to the next tower must occur.
• Simplifying Link Budgets to Simple Addition – the piecewise approximation converts complex exponential field math into three straight lines:
  • Region 1: $0\text{ dB/decade}$ slope (Flat line)
  • Region 2: $20\text{ dB/decade}$ slope (Free-space slope)
  • Region 3: $40\text{ dB/decade}$ slope (Two-ray far-field slope)

Python script

import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

# --- 1. System Parameters ---
f = 900e6
c = 3e8
lambda_val = c / f
h_t = 50.0
h_r = 2.0
P_t = 1.0
G_t = G_r = 1.0

d_c = (4 * np.pi * h_t * h_r) / lambda_val
distances = np.logspace(0, 5, 2500)

# --- 2. Exact Two-Ray Model Calculation ---
d_1 = np.sqrt(distances**2 + (h_t - h_r) ** 2)
d_2 = np.sqrt(distances**2 + (h_t + h_r) ** 2)
delta_phi = (2 * np.pi / lambda_val) * (d_2 - d_1)

E_term = np.abs(1 / d_1 - np.exp(-1j * delta_phi) / d_2)
P_r_exact = P_t * G_t * G_r * (lambda_val / (4 * np.pi)) ** 2 * E_term**2
P_r_exact_dBm = 10 * np.log10(P_r_exact * 1000)

# --- 3. Piecewise 3-Region Approximation ---
P_r_approx = np.zeros_like(distances)

for i, d in enumerate(distances):
    if d <= h_t:
        P_r_approx[i] = (
            P_t * G_t * G_r * (lambda_val / (4 * np.pi * h_t)) ** 2
        )
    elif h_t < d < d_c:
        P_r_approx[i] = (
            P_t * G_t * G_r * (lambda_val / (4 * np.pi * d)) ** 2
        )
    else:
        P_r_approx[i] = P_t * G_t * G_r * (h_t**2 * h_r**2) / d**4

P_r_approx_dBm = 10 * np.log10(P_r_approx * 1000)

# --- 4. Plotting Configuration ---
plt.figure(figsize=(11, 6.5), dpi=300)

plt.semilogx(
    distances,
    P_r_exact_dBm,
    color='#1f77b4',
    linewidth=1.8,
    label='Exact Two-Ray Model',
)
plt.semilogx(
    distances,
    P_r_approx_dBm,
    color='#d62728',
    linestyle='--',
    linewidth=2.2,
    label='3-Region Piecewise Approximation',
)

# Vertical boundary lines
plt.axvline(x=h_t, color='magenta', linestyle=':', linewidth=1.5)
plt.axvline(x=d_c, color='magenta', linestyle=':', linewidth=1.5)

# Boundary labels at the bottom of the plot
plt.text(
    h_t * 0.85,
    -123,
    r'$d = h_t$' + f'\n({int(h_t)}m)',
    color='magenta',
    fontsize=10,
    fontweight='bold',
    ha='right',
    va='bottom',
)
plt.text(
    d_c * 1.15,
    -123,
    r'$d = d_c$' + f'\n({int(d_c)}m)',
    color='magenta',
    fontsize=10,
    fontweight='bold',
    ha='left',
    va='bottom',
)

# Region text boxes
plt.text(
    4,
    -110,
    'Region 1 ($d \\leq h_t$)\nDirect Path Dominates\nSlope: 0 dB/dec',
    fontsize=9,
    ha='center',
    bbox=dict(boxstyle='round,pad=0.5', facecolor='white', alpha=0.85),
)
plt.text(
    400,
    -110,
    'Region 2 ($h_t < d < d_c$)\nInterference Oscillations\n'
    'Avg Slope: 20 dB/dec ($1/d^2$)',
    fontsize=9,
    ha='center',
    bbox=dict(boxstyle='round,pad=0.5', facecolor='white', alpha=0.85),
)
plt.text(
    25000,
    -110,
    'Region 3 ($d \\geq d_c$)\nGrazing Phase Cancellation\n'
    'Slope: 40 dB/dec ($1/d^4$)',
    fontsize=9,
    ha='center',
    bbox=dict(boxstyle='round,pad=0.5', facecolor='white', alpha=0.85),
)

# Chart styling
plt.title(
    'Two-Ray Ground Reflection Model & 3-Region Approximation',
    fontsize=13,
    fontweight='bold',
    pad=15,
)
plt.xlabel('Distance $d$ [meters] (log scale)', fontsize=11, fontweight='bold')
plt.ylabel('Received Power [dBm]', fontsize=11, fontweight='bold')
plt.grid(True, which='both', linestyle='--', alpha=0.4)
plt.legend(fontsize=10, loc='upper right', framealpha=0.9)
plt.xlim([1, 100000])
plt.ylim([-125, 0])

plt.tight_layout()
plt.savefig('two_ray_ground_reflection.png', dpi=300)
print(f'Breakpoint distance: {d_c:.1f} m')
print('Saved plot: two_ray_ground_reflection.png')

Topic in this chapter

• Introduction to Large scale propagation models
• Friis free space propagation model
• Log distance path loss model
• Two ray ground reflection model
• Modeling diffraction loss
  • Single knife-edge diffraction model
  • Fresnel zones
• Hata Okumura model for outdoor propagation

FAQ

When is the two-ray ground-reflection model a good approximation? It works best in large open areas with a strong ground bounce and a clear line-of-sight path — rural macros, highways, flat desert links — not in rich indoor multipath where many scatterers dominate.

What is the breakpoint distance in the two-ray model? It marks the transition from a regime that still resembles free-space-like decay (with interference ripples) toward the large-distance asymptote where the direct and reflected rays combine so that average power falls about 40 dB per decade of range ($latex 1/d^4$).

How does two-ray differ from Friis free-space loss? Friis models a single ray in free space ($latex 1/d^2$ power decay). Two-ray adds the ground-reflected path, so the received field is the coherent sum of two phasors and the distance law changes once the geometry reaches the far breakpoint region.

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