Parabolic Equations

Introduction

Most of the empirical models deliver predictions with very limited accuracy. So other models were investigated which take diffraction and forward-scattering processes into consideration. One very efficient method is the Parabolic Equation (PE) model. The PE method is not only used in electromagnetical wave propagation problems. It is also used for under water acoustics, water waves or seismic wave problems.

Consideration of Propagation Phenomena

As the PE uses numerical algorithms to consider propagation phenomena like reflection and diffraction, it requires a quite long computation time. But on the other hand, the results of this model are very accurate.

The PE model takes the following effects into consideration:

• Reflection

• Diffraction

• Forward-Scattering

It accounts for the properties of the ground by the following parameters:

• Conductivity of the ground

• Dielectricity of the ground

Computation

The Standard Parabolic Equation (SPE) is a partial differential equation and is derived from the Maxwell equations, neglecting backward propagation and assuming a rotation symmetrical problem.

Ψ is the field strength related to field of a linear source. In the far field the vertical component of the electrical field can be assumed as

and ko denotes the wave number in free space. k is the complex waver number in an inhomogeneous lossy atmosphere.
 

The results of the PE are only valid, if the propagation angle in respect to the horizon lies within -15° up to +15°. This is reason why the computation starts at the distance rini.

Some of the upper layers of the atmosphere have the effect like a reflector. As the reflected waves increase the computation time, an absorbing media below these layers is assumed. The height of the absorbing media is about dA = 150 wavelengths.

PE has three possibilities to consider the conductivity and the dielectricity of the ground soil; the Discrete terrain approximation, the Continuous terrain approximation, the Terrain profile approximation with coordinate transformation.

With an extension of the SPE, the disadvantage of the rather small propagation angle can be avoided. The so-called Wide Angle Parabolic Equation (WAPE) model leads to valid results for propagation angles between -40° up to +40°. As this extension is not noticeable in the computation time, the WAPE should be preferred.

Sample Predictions

Here some sample predictions with the PE are presented.

Vertical plane from tx to rx.	Vertical plane from tx to rx.

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