Two-ray ground-reflection model

The two-rays ground-reflection model is a multipath radio propagation model which predicts the path losses between a transmitting antenna and a receiving antenna when they are in line of sight (LOS).

From the figure the received line of sight component may be written as and the ground reflected component may be written as where

is the length of the direct line-of-sight (LOS) ray,

is the combined antenna gain along the LOS path,

is the combined antenna gain along the ground-reflected path,

depending if the signal is horizontal or vertical polarized, respectively.

is the relative permittivity of the ground (or generally speaking, the material where the signal is being reflected),

is the angle between the ground and the reflected ray as shown in the figure above.

From the geometry of the figure, yields: and Therefore, the path-length difference between them is and the phase difference between the waves is The power of the signal received is where

If the signal is narrow band relative to the inverse delay spread

is very large relative to the height of the antenna we may expand

: and taking the first two terms only, The phase difference can then be approximated as When

using Taylor series and retaining only the first two terms it follows that so that and path loss is which is accurate in the far field region, i.e. when

(angles are measured here in radians, not degrees) or, equivalently, and where the combined antenna gain is the product of the transmit and receive antenna gains,

[3] Note that the power decreases with as the inverse fourth power of the distance in the far field, which is explained by the destructive combination of the direct and reflected paths, which are roughly of the same in magnitude and are 180 degrees different in phase.

As distance increases, these waves add up constructively and destructively, giving regions of up-fade and down-fade.

An approximation to critical distance may be obtained by setting Δφ to π as the critical distance to a local maximum.

, which may be not the case in many scenarios, e.g. when antenna heights are not much smaller compared to the distance, or when the ground cannot be modelled as an ideal plane .

and more refined analysis is required, see e.g.[4][5] The above large antenna height extension can be used for modeling a ground-to-the-air propagation channel as in the case of an airborne communication node, e.g. an UAV, drone, high-altitude platform.

When the airborne node altitude is medium to high, the relationship

does not hold anymore, the clearance angle is not small and, consequently,

This has a profound impact on the propagation path loss and typical fading depth and the fading margin required for the reliable communication (low outage probability).

[4][5] The standard expression of Log distance path loss model in [dB] is where

is selected for convenience of measurements and to have clear line-of-sight.

This model is also a leading candidate for 5G and 6G systems[6][7] and is also used for indoor communications, see e.g.[8] and references therein.

The path loss [dB] of the 2-ray model is formally a special case with

The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance.

Using the free-space and two-ray model above, the propagation path loss can be expressed as

is a minimum path loss (at smallest distance), usually in practice;

This should be kept in mind when using these approximations at small distances (ignoring this limitation sometimes produces absurd results).