# Ground Reflection Model

For uniform flat ground, the two-ray approach (see Chapter 10) can be used. The contribution of the ground reflection, relative to free field sound propagation, is given by

where *R _{d}* is the direct path length between source and receiver, R

_{s}is the ground-reflected path length,

*k*is the wave number and Q is the spherical ground reflection coefficient (see Chapter 2) and

*C*is a coherence factor. For fully coherent summation,

_{coh}*C*1, while, for completely incoherent interactions between both paths,

_{coh}=*C*0 (see Section 12.7.3.7 for more information on quantification of this coherence factor).

_{coh}=In the presence of ground impedance discontinuities, the H2P2 model uses (linear) Fresnel weighting:

where *w*, is the Fresnel weight of ground in zone *i, N,„* is the number of different ground types encountered in between source and receiver and *AL _{G} *

*fi*the partial ground reflection contribution as if ground type

_{a},^_{uni}f_{orm}^*i*would appear everywhere between source and receiver.

The Fresnel zone concept is used to account for the fact that the reflection at the ground is not limited to the exact specular reflection point but involves a wider zone, and thus potentially interacts with different ground types in case of impedance discontinuities. The latter becomes more pronounced with increasing wavelength. An ellipsoid is constructed with foci at the image source and receiver, where each point on the Fresnel ellipsoid is defined by the following distances relationship:

where *S _{jnig}* is the location of the image source, R is the position of the receiver and

*n*is the Fresnel parameter, defining the fraction of the wavelength involved, which is a parameter to be tuned (a value of 8 might be appropriate [45]). The extent of the intersection line between the Fresnel ellipsoid and (each) ground segment is calculated to find the appropriate weights

_{f}*w*If this intersection line fully falls within a particular segment,

_{r }*w,*becomes 1 (and consequently zero for all other segments). Mathematical expressions for the geometrical parameters related to Fresnel ellipses and their intersections with the ground can be found in Chapter 8, section 8.3.4.

Improvements have been proposed (the so-called modified Fresnel weighting) to increase accuracy in the higher frequency range, including a frequency-dependent transition function. A low-pass filter approach was considered appropriate, which needs the definition of a cut-off frequency [43,44]. Basically, this modified approach makes the Fresnel parameter, *ti _{F}, *frequency dependent.

If relevant ground reflections in a concave sequence of segments (such as would be the case for ‘valley-shaped’ terrain) can be identified, the procedure ro be used is based on Fresnel weighting. A transition function *F _{g }*is applied for the final ground effect, interpolating between the one for a flat ground and valley-sloped terrain:

If the direct path between source/receiver or intermediate diffracting edges is not blocked by a sequence of convex-shaped ground segments, a transition model procedure is proposed which balances ground reflections and diffractions (see Section 12.7.3.4).