PIERS ONLINE, VOL. 3, NO. 4, 2007 526 Building Optimal Statistical Models with the Parabolic Equation Method M. Le Palud CREC St-Cyr Telecommunications Department (LESTP), Guer, France Abstract In this paper, we present a new application of electromagnetic propagation modeling methods like Parabolic Equation (PE) algorithms: these tools can be used to obtain optimal statistical propagation models i. e., best-fitted statistical model for a given propagation environment. Statistical propagation models are widely used because of their convenience to make predictions that are valid for different types of propagation environments and operation conditions. However their accuracy is often questionable because they are obtained either empirically with a limited amount of experimental data collected in few propagation environments or deduced using rough and general theory. As a result the model doesn t square satisfactorily with real operation. Our goal is to improve the preceding process by designing a method that would give an optimized statistical model (OSM) for a given environment. The process that leads to the OSM for a particular application involves several steps that are described in detail in the paper. Sample results are also given and comparisons between OSM and standard statistical models are discussed. DOI: 10.2529/PIERS061007103643 1. INTRODUCTION For several years strong interest has been shown for the study of ground propagation and transmission in urban as well as in rural and forested environments. An accurate characterization of the propagation channel is needed in digital communication systems as well as in RADAR systems in order to minimize error rates and to optimize system costs. In previous works, we have described a method, based on a Parabolic Equation (PE) algorithm, which allows complete computation of characteristics of stationary determinist channels. We have applied this method in various types of situations, including canonical problems and real radio links over irregular terrain with or without vegetal coverage. However, we have observed that a fully deterministic approach gives often deceiving results since it is most of the time almost impossible in practice to fully implement the features of real propagation environments, considering its high complexity (i. e., for instance: inaccurate localization and characterization of vegetation and man-made obstacles). A more basic and widely used approach is based on the use of standard statistical/empirical propagation models which gives the advantages of more simplicity and convenience but also lowers significantly prediction accuracy targets because standard statistical models will not correctly reflect the actual propagation environments. In this paper we propose an intermediate approach that is based on the following idea: the goal is to build statistical models that fit at best the actual propagation environments by using deterministic modeling tools. The first part of the paper gives a general presentation of standard statistical/empirical models. The second part of the paper gives a description of the approach used to build optimized statistical models (OSM). This is illustrated with sample results and also with comparisons and discussions in the third part before some conclusions. 2. STANDARD STATISTICAL MODELS Statistical models are mainly used for path loss prediction. They are usually elaborated using significant amount of experimental data, collected during long and costly campaigns, together with some theoretical considerations. Some other models that give field distribution predictions or correlation laws for random and/or time-dependent channels may also be ranked among statistical models (i. e., for example, Rayleigh and Rice distributions for random channel). Statistical models are dedicated to be used with restricted types of propagation environments and operational conditions but this context isn t precisely defined which may lead to important
PIERS ONLINE, VOL. 3, NO. 4, 2007 527 Figure 1: Plots of path loss versus range for 3 standard statistical models (EGLI, ITS and Modified Plane Earth) used for terrestrial propagation over weakly undulating terrain with sparse vegetation (f = 50 MHz). deviations. This last point is illustrated with Fig. 1 where three models (EGLI, ITS and Modified Plane Earth) used for terrestrial propagation over weakly undulating terrain with sparse vegetation are compared. It can be noticed that important discrepancies exist between predicted values of path loss and also between rates of increase with range. Another important point, which is illustrated on Fig. 2, is that statistical models may be incorporated in various ways inside a modeling strategy. In level-2, the model will replace the real environment with an effective flat ground takes globally into account both vegetation coverage and terrain irregularities while statistical models can be used at each step of multiple point-to-point propagation modeling (i. e., for example knife-edge diffraction algorithm) in level-1. This level-1 approach can give very efficient propagation prediction tools; it is notably used by the well-known TIREM software [1]. Considering Fig. 1 plots, it can be concluded that standard statistical models generally misestimate the lack of knowledge and the uncertainties of propagation environments. This means consequently that the propagation channel is often underutilized and that things will be improved if the model is optimized for the actual propagation environment. 3. BUILDING AN OPTIMIZED STATISTICAL MODEL (OSM) In order to correct the shortcomings of standard statistical models mentioned above, we propose the following approach to obtain optimized statistical models (OSM) i. e., models that fit at best the actual propagation environments by handling electromagnetic propagation with deterministic tools. 3.1. Parabolic Equation Algorithm Our choice has been to use a Parabolic Equation (PE) algorithm as the heart of the method that s to say as the piece of software that handles electromagnetic propagation. The PE method [2] is a full-wave method used for solving some continuous-wave propagation problems in acoustics and electromagnetism and, in particular, ground propagation of radiowaves over irregular terrain. One of the main limitations of the method is the fact that back scattering is entirely neglected, only one-way propagation off the transmitter is taken into account. The PE equation itself applies to a field function u(x, z), where x is range and z is height, which is related to the amplitude of the horizontal component of the magnetic field (TM polarization case) or of the electric field (TE polarization case). This equation allows approximate computation of the field in the conditions mentioned above when the amplitude is a slow-varying function of range. In its simplest form, it can be written as: u x = 1 2jk [ 2 u z 2 + k2 ( n 2 1 ) Our PE algorithm has been tested in canonical CW situations and its results have been compared with exact solutions with good agreement. We have also achieved computations for links over ] (1)
PIERS ONLINE, VOL. 3, NO. 4, 2007 528 Level 1 Level 2 Figure 2: This diagram illustrates different levels of complexity in terrestrial propagation modeling using statistical models. real irregular rural and forested terrain and we have made comparisons with experimental data, obtaining satisfactory agreement. 3.2. OSM Processing Three steps are necessary to create an optimized statistical model. The global flowchart of the process is summarized on Fig. 3. The first step, which is often the trickiest and the most difficult part of the job, consists in replacing the actual propagation environment, which is generally randomly defined, by an optimal set of deterministic ones that is its statistical equivalent (i. e., a set of possible realizations of terrain and troposphere that gives the same statistical averages, standard deviations, etc. of observable electromagnetic properties). Of course this equivalent set is a theoretical limit and is out of reach in practice but the idea is to obtain a reasonably sized set, sufficiently close to that limit. In order to build an optimized statistical model, one must carefully see that all specific features of the actual terrain are faithfully reflected in this optimal set. The main useful tools at this stage are Monte Carlo techniques and statistical processing. During the second step of the process, deterministic propagation model is iterated for each element of the previous optimal set and for all required values of the parameters (i. e., frequency, transmitter height, etc.). Last, statistical processing is performed to obtain the desired results that may be fitted to simple algebraic functions of the parameters: this gives the final product of the OSM procedure.
PIERS ONLINE, VOL. 3, NO. 4, 2007 529 Generation of statistically equivalent set Iteration of PE algorithm for each element of statistically equivalent set Obtaining of final results by statistical processing Figure 3: Global flowchart for OSM processing. 3.3. Sample Results We show sample results in order to illustrate the interests of our approach. First of them concern path loss predictions for radio links with geometry shown in Fig. 4: transmitting and receiving antennas are located above the ground surface which presents sinusoidal irregularities of period L and peak-to-peak amplitude H. In our present computations, L was the only random parameter and was supposed to be uniformly distributed on an interval of width L = 300 m around the central value L c. Figure 4: Geometry of radio links for sample results. A 70 MHz frequency and corresponding standard values of electromagnetic constants for dry ground were taken for the computation. It can be observed in Fig. 5 that global tendencies of OSM plots and also of Modified Plane Earth model are rather close to each other but that noticeable deviations exist between the 4 plots. These results also demonstrate the sensitivity of OSM results to parameters of propagation environments, the most influent parameters being here the heights of antennas and terrain undulation amplitude. 4. SAMPLE COMPARISONS Experimental path loss data for CW radio link at 61 MHz have been collected between fixed 6mhigh transmitter and mobile 4.7 m-high receiver. The terrain of the experiments was weakly undulating and covered with sparse vegetation. A statistically equivalent set for the terrain was generated following the process described in Fig. 3; for reasons of simplicity, we presumed sinusoidal modulation of ground height versus range (see Fig. 4) for each element of the set and uniform distribution of random parameters. Rough estimations of terrain parameters give: L c = 500 m (mean value of L) L = 300 m and similarly for H, which was also treated as a random parameter H c = 75 m (mean value of H) H = 25 m. The 2 continuous curves on Fig. 6 correspond to max and min path loss values in the statistically equivalent set at a given range; they are separated by a gap of the order of 20dB. The global tendency of the experiments and of the 2 curves is found to fit rather well. It can also be observed that most of the experimental dots are located inside the region delimited by these 2 curves, which means that the obtained statistically equivalent set represents rather accurately the actual propagation environment.
PIERS ONLINE, VOL. 3, NO. 4, 2007 530 Figure 5: Plots of path loss versus range: plot 1: OSM h R = h T = 0 m H = 8 m L c = 500 m plot 2: OSM h R = h T = 0 m H = 8 m L c = 700 m plot 3: OSM h R = h T = H = 8 m L c = 500 m blue line: modified plane earth model. Figure 6: Path loss versus range; red dots are experimental data; upper blue curve and lower green curve give respectively max and min path loss values in the statistically equivalent set at a given range (f = 61 MHz). 5. CONCLUSION In this paper, we have presented a novel application of electromagnetic propagation modeling methods algorithms like the PE algorithm: these tools can be used to obtain optimal statistical propagation models i.e. bestfitted statistical model for a given propagation environment. The process that leads to the OSM for a particular application involves several steps that have been described and illustrated with sample results. Comparisons between OSM and standard statistical models have been discussed. We have also displayed some results and comparisons focussed on CW radio propagation in rural environment that showed good agreement with experimental data. We will shortly apply this method to obtain better modeling of radio channels in this type of environment with application to OFDM modulated transmissions. We will also soon be able to obtain results with a hybrid model that mixes OSM together with deterministic processing, following the same approach as the well-known TIREM software [1]. REFERENCES 1. Eppink, D. and W. Kuebler, TIREM/SEM Handbook, Department of Defense, Electromagnetic Compatibility Analysis Center ECAC-HDBK-93-076, U. S., Mar. 1994. 2. Barrios, A. E., A terrain parabolic equation model for propagation in the troposphere, IEEE Trans. on Ant. and Prop., Vol. 42, No. 1, 90 98, 1994.