1536-1225 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2016.2557960, IEEE Antennas and Wireless Propagation Letters Abstract— A novel prediction model of joint exceedance probability of rain attenuation in Earth-satellite site diversity systems based on Gaussian copula is presented. The proposed model is based on a number of experimental data of two-site diversity systems with different configurations. The proposed model outperforms the existing model for two-site diversity given in Recommendation ITU-R P.618-12 and the recently presented model based on Clayton copula. The method is generalized to multiple-site diversity prediction. Results for a few existing three- site diversity experiments are evaluated. Index Terms— Gaussian Copula, modeling, rain attenuation, satellite, site diversity. I. INTRODUCTION N SATELLITE backhaul networks, site diversity technique is proved to be the most efficient fade mitigation technique to reduce attenuation due to rain [1]. In order to design and evaluate a site diversity system, channel characterization must be performed in advance. Various empirical and physical- mathematical models for prediction of site diversity performance have been proposed. An empirical model for prediction of diversity gains is given in [2]. Physical models in [3] and [4] predict the joint exceedance probability of rain attenuation assuming that the single and joint rain attenuation follow lognormal distributions. The physical-mathematical model proposed in [5] assumes that rain attenuation on slant paths follows Inverse Gaussian distribution. In [6], a new model is proposed for prediction of joint exceedance probability of rain attenuation for two-site diversity systems based on Archimedean copula functions. In this paper, a new model for prediction of joint exceedance probability of rain attenuation for dual and multiple site diversity systems based on Gaussian copula function is presented. Gaussian copula has been recently used in [7] for modeling of joint exceedance probability in time diversity and for new rain attenuation time series synthesizer. Manuscript received December 24, 2015; revised February 11, 2016; accepted April 12, 2016. A. Kelmendi is with Jozef Stefan International Postgraduate School, Ljubljana, Slovenia (e-mail: arsim.kelmendi@ijs.si). A. Hrovat, G. Kandus and A. Vilhar are with Jozef Stefan Institute, Ljubljana, Slovenia, (e-mail: andrej.hrovat@ijs.si; gorazd.kandus@ijs.si; andrej.vilhar@ijs.si). C. Kourogiorgas and A.D. Panagopoulos are with the school of Electrical and Computer engineering at National Technical University of Athens, Greece, (e-mail: harkour@mail.ntua.gr; thpanag@ece.ntua.gr). Modeling through copulas is a simple method that does not require specific distribution of rain attenuation. Moreover, lower number of parameters is required compared to other models. The advantage of Gaussian copula model is improved accuracy and easy generalization to multiple-site diversity. The remainder of this paper is the following: in Section II, the probability theory of calculating multivariate exceedance probability using Gaussian copulas is briefly presented and in Section III, the accuracy of the proposed modeling of joint rain attenuation exceedance probability for a dual site diversity is investigated. In Section IV, a model is proposed for predicting the joint exceedance probability of rain attenuation for dual and multi-site diversity systems and in Section V, the proposed model is compared to ITU-R prediction model and to Clayton copula based model using experimental data. In the last section the paper is concluded. II. PROBABILITY THEORY AND GAUSSIAN COPULA In probability theory, Copulas are functions that are used to describe the dependence between random variables. According to Sklar's theorem any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables [8]. Considering n random variables with continual marginal distributions, i.e., CDFs, P[X i ≤x i ]=F i (x i )=u i , i=1,…,n defined in the interval [0,1], there exists a unique copula function C(u 1 ,u 2 ,...,u n ) which is equal to the joint CDF of the n random variables: [ ] 1 1 2 2 1 2 , , ( , ,..., ). n n n PX xX x X x Cuu u … ≤ = ≤ ≤ (1) The joint Complementary Cumulative Distribution Function (CCDF) for n variables can be expressed from CDFs as [9]: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 1 1 1 1 2 2 1 1 2 2 3 3 1 1 1 1 2 2 3 3 1 1 2 2 2 2 1 1 1 1 1 2 2 , , 1 , , , , , , , , , , ( 1) , , , n n n n n n n n n n n n n n n n n n n n PX x X x PX x PX x PX x X x PX x X x PX x X x PX x X x PX x X x X x PX x X x X x PX x X x X x PX x X x X − − − − − − − ≥ … ≥ = − ≤ −…− ≤ + ≤ ≤ +…+ ≤ ≤ +…+ ≤ ≤ +…+ ≤ ≤ − ≤ ≤ ≤ −…− ≤ ≤ ≤ −…− ≤ ≤ ≤ −…− − ≤ ≤ … ≤ [ ]. n x (2) Equation (2) can be expressed through copulas as: Prediction of Joint Rain Attenuation Statistics Induced on Earth-Satellite Multiple Site Diversity Systems Using Gaussian Copula Arsim Kelmendi, Charilaos Kourogiorgas, Member, IEEE, Andrej Hrovat, Member, IEEE, Athanasios D. Panagopoulos, Senior Member, IEEE, Gorazd Kandus, Senior Member, IEEE, and Andrej Vilhar, Member, IEEE I