Radio Science

Prediction of the degradation of the carrier-to-noise plus interference ratio concerning a site diversity system suffering from differential rain attenuation

Authors


Abstract

[1] Rain-induced attenuation is a factor of utmost importance in designing a satellite link since it can strongly deteriorate the availability and performance of an Earth-space path. Availability and performance are key criteria for the reliable design process. In the present paper the statistics of carrier-to-noise plus interference ratio under rain fade conditions are examined, given that the interference is caused by an adjacent satellite operating at the same frequency, and taking also under consideration double site diversity protection as a countermeasure technique. The method presented herein is based on a model of convective rain cells and the lognormal assumption for the point rainfall statistics. Numerical results have been obtained for heavy rain climatic regions and very large availability times, to examine how effective site diversity can be, not only for lowering fade margins but for compensating severe interference effects. Moreover, the elaboration of the numerical results has revealed how the optimum design of the satellite link can be achieved by selecting proper values for some critical parameters of the system.

1. Introduction

[2] In the band of frequencies above 10 GHz rain attenuation is considered to be a principal factor affecting the availability and performance of a satellite communication system. Interference effects must be carefully taken into account, also for the reliable design of an Earth-space path. In the present work, we assume that the dominant interference source comes from the differential rain attenuation from an adjacent Earth-space system operating at the same frequency.

[3] So far, the outage performance analysis has been based on the assumption that the thermal noise is dominant [Ha, 1986], and consequently, the outage time is mainly dictated by the fact that the rain attenuation Ac of the wanted carrier signal should exceed a certain rain fade level, called margin M (in dB). The contribution of the interference to the total outage time effects can be further taken into account by calculating the degradation of the carrier-to-interference ratio (C/I), under the condition that the system is working properly (Ac < M). For this reason, some models have been so far proposed [Kanellopoulos and Houdzoumis, 1990; Kanellopoulos et al., 1993; Kanellopoulos and Margetis, 1997] dealing with the prediction of the differential rain attenuation ΔA = Ac − AI, under the above condition of proper operation (Ac < M). Validation of the above models has been made by comparing the theoretical results with a set of simulated data in Montreal area [Rogers et al., 1982]. Some recent experimental results derived from rain attenuation measurements in Italy for two frequency bands (11.6 and 20 GHz) [Matricciani and Mauri, 1996; Matricciani, 1997] have been also used for this purpose, and the agreement is found to be quite satisfactory [Kanellopoulos et al., 2000].

[4] In recent years, taking into account the expected aggravation in the frequency and orbital congestion, the complicated problem of predicting the degradation of the total carrier-to-noise plus interference ratio (CNIR), under the presence of rain fades, has been carefully examined [Kanellopoulos and Livieratos, 1997; Livieratos and Kanellopoulos, 2000]. This is an important subject contributing toward the much desirable accurate estimation of the total outage time of an interfered satellite link without any condition regarding the thermal noise. This is even more imperative for systems working under the interference domination consideration, a quite probable situation for modern satellite paths as previously mentioned.

[5] The subject of the present paper is an extension of the previous analysis in order to include interfered Earth-space systems located in regions representing heavy rain climatic conditions. In this case, the demand of using the possible low fade margins can lead to the employment of the site diversity technique. The extended method assumes again the model of convective rain cells for the spatial rainfall structure and the lognormality for the point rainfall statistics. The more complicated case, where another interference source such as the cross-polarization is also present, will be examined in a future work. The numerical results presented in the last part of the paper are concerned with the proper use of the CNIR statistics toward the reliable design of an interfered site diversity system suffering from differential rain attenuation.

2. Analysis

[6] The configuration is shown in Figure 1a. Two Earth-stations E1 and E2 are in communication with a satellite S1, forming a double-site diversity system. A second satellite S2, operating at the same frequency, is in close orbit to S1, the two subtending the same angle θd to E1 and E2. The elevation angles of the various Earth-space paths pointing toward satellites S1 and S2 (EiSj, i,j = 1,2) are considered to be different, denoted by φ1 for the E1S1, E2S1 paths and φ2 for the E1S2, E2S2, respectively. For reasons of simplicity, we assume that both satellites S1 and S2 operate in the single polarization mode. As a result, the main source of interference relates to the differential rain attenuation induced by the adjacent path. Our objective is the evaluation of the time during which the carrier-to-noise plus interference ratio (CNIR), under rain fade conditions, does not exceed a specified level r (in dB), for both paths E1S1 and E2S1. Following the definitions and using a straightforward algebra, the CNIR (in dB) for the ith path (i = 1,2) can be expressed as

equation image
equation image

where equation image is the attenuation of the received signal for the EiS1 path and equation image the attenuation of the potential interfering signal for the EiS2 path. Moreover, (CNR)i,nom, (CIR)i,nom represent the nominal values (clear-sky conditions) for (C/N) and (C/I) respectively. Details of the analytical derivations of the expressions (1)(2) can be found elsewhere [Kanellopoulos and Livieratos, 1997]. Further, the (CNR)i,nom and (CIR)i,nom are expressed in terms of the basic link parameters, as follows [Ha, 1986]:

equation image

and

equation image

Here EIRPs,i is the effective isotropic radiated power of the satellite S1 in the direction of the Earth-station Ei In addition, EIRP′s,i is the effective isotropic radiated power of the interfering satellite S2 in the direction of the Earth-station Ei and Gr,i(dB) is the on-axis received antenna gain of station E1 The last term of the expression (4) specifies a sidelobe envelope relative to the normalized peak gain (0 dB) [Ha, 1986]. The other parameters encountered in (3) have the following definitions:

equation image

: down link slant range (in m) for the EiS1 path;

fd

: down link carrier frequency (Hz);

L′I

: antenna tracking loss and atmospheric attenuation (in dB) for the EiS1 path;

c

: speed of light (2.997925 × 108 m/sec);

Ti

: system noise temperature of the Earth-station Ei (oK);

B

: noise bandwidth of satellite channel (Hz);

K

: Boltzmann's constant: 1.38 ×10−23 J/oK.

Figure 1a.

Configuration of the problem under consideration.

[7] For the following analysis we assume a balanced diversity system and consequently we have

equation image

and

equation image

(i = 1,2).

[8] The main point of the analysis is the calculation of the following joint nonexceedance probability,

equation image

[9] The analysis of (7) is then based on some fundamental assumptions, which are as follows:

  1. The lognormal form is adopted for both the unconditional (including nonraining time) point rainfall rate (R) and attenuation (A) distributions.
  2. The convective rain cell model, suggested by Lin [1975], will be employed for the description of the horizontal variation of the rainfall spatial structure. The other assumptions concerning the vertical variation of the rainfall structure [Crane, 1980] and the formulation of the specific rain attenuation Ao (dB/km) in terms of rain rate R, are the same as those presented elsewhere [Kanellopoulos et al., 1993].

[10] As a result of the above considerations and following its definition, the joint nonexceedance probability can be written as

equation image

where

equation image
equation image
equation image
equation image

and

equation image
equation image
equation image

(i = 1, 2)

equation image
equation image

In the above expressions equation image and equation image are the joint probability densities of the random variables, equation image and equation image. For the definition of equation image and equation image, see Appendix A. Details for the derivation of the expressions (8)(17) can also be found in Appendix A.

[11] The next step is the calculation of the probabilities (9)(12). This can be achieved by using the above considerations (particularly assumption (1) for the lognormality), and following a straightforward mathematical analysis (see Appendix A). It should be emphasized here that the reliable and accurate evaluation of the multidimensional integrals involved is the keypoint of the whole analysis. The final results are presented here. (The Pascal code concerning the numerical calculation of the multidimensional integrals can be obtained by sending a request via e-mail to Spiros.Livieratos@oteglobe.gr.)

equation image
equation image
equation image
equation image

where

equation image
equation image

(i = 1,2) The analytical forms of the one-dimensional normal density function equation image, the two-dimensional function equation image as well as the three-dimensional one equation image can be found in Appendix B. The parameters μ3/1,2, μ4/1,2, μ4/1,2,3 and σ3/1,2, σ4/1,2, σ4/1,2,3 correspond to the proper conditional mean values and standard deviations in each case and are also presented in Appendix B.

[12] In the above expressions equation image are the lognormal statistical parameters of the attenuation equation image or equation image given by [Kanellopoulos and Koukoulas, 1987]

equation image

In the same way, equation image and equation image are the lognormal statistical parameters of the attenuation equation image or equation image, given by

equation image

in terms of the point rainfall parameters Rm,Sr the constants a and b of the specific rain attenuation (Ao = aRb) and the factors H1C,H1I which have the form,

equation image

In the previous expressions, G is a characteristic parameter depending on the rainfall spatial structure suggested by Lin [1975], and the path lengths LCD,LID are defined in Appendix A along with the attenuations equation image and equation image.

[13] Summarizing the procedure, the joint nonexceedance probability under consideration (expression (7)) can be evaluated by means of the expressions (8)(23). The lognormal statistical parameters equation image and equation image involved in the previous relationships can be calculated by using expressions (24)(26). Finally, the statistical parameters μ3/1,2, μ4/1,2, μ4/1,2,3 and σ3/1,2, σ4/1,2, σ4/1,2,3 are presented in Appendix B.

3. Numerical Results and Discussion

[14] As a next step to the previous analysis we proceed on the prediction of the carrier-to-noise plus total interference ratio (CNIR) degradation for Earth-space systems suffering from adjacent satellite interference and using double site diversity protection. As noted previously, the proposed methodology is quite general including also the case of interference-dominant operating systems, a scenario quite probable in the near future. On the other hand, we point out that experimental data for this kind of problems exists for the single-site interfered case, concerning a noise dominant operating system, and that these are the only available published results so far [Matricciani and Mauri, 1996; Matricciani, 1997]. In a very recent publication [Kanellopoulos et al., 2000] an extensive comparison between theoretical results and the above experimental data has been presented. As shown there, a clear tendency of coincidence between the experimental data and the numerical results is quite obvious, and this is more emphatic with respect to the data concerned with the more statistically significant sample size. These remarks provide a sufficient degree of credibility for the proposed methodology, because the single site interfered case constitutes the basis of the present analysis.

[15] Our attention is further concentrated on the analytical examination of the outage performance due to degradation of the CNIR for some simulated interfered site diversity systems, located in various indicative climatic zones. Comparison is also made with theoretical results referring to the same interfered systems but operating under no diversity protection [Livieratos and Kanellopoulos, 2000]. The deduced conclusions are expected to be very useful toward the optimum design of modern satellite systems where frequency and orbital congestion are becoming a serious problem. The climatic zones examined in this section are the K, Q ones, representing the Mediterranean and tropical regions, respectively.

[16] In Figure 2, the probability that the CNIR will never exceed a specified level r (in dB) is drawn for an interfered 20 GHz single site system or using diversity protection and located in the K zone. The numerical values for CNRnom and INRnom are given in the caption. As denoted, two extreme values for INRnom are used corresponding to a noise dominant ((INR)nom = −3dB) or an interference dominant system ((INR)nom = 6dB) respectively, operating under clear-sky conditions. The implementation of the procedure requires the knowledge of the parameters LC, LI, Rm, Sr, G, a, b, φ1 and φ2 with respect to the slant path under consideration. A list of appropriate numerical values for these parameters is presented in Table 1. The rainfall parameters Rm and Sr have been taken from Crane [1980], where the values of the specific attenuation for a and b have been estimated by using the International Radio Consultative Committee (CCIR) [1999] recommendation. As far as the characteristic distance G is concerned, a value of 1.5km is employed for the K zone, in accordance with that suggested by Lin [1975]. On the other hand, a 1km characteristic distance has been considered to be appropriate for tropical regions belonging to Q zone, where the spatial rainfall structure is more limited. As a first remark, we may observe the significant improvement due to the site diversity protection, particularly for the low probability levels, which are of utmost importance for engineering purposes in practice. This is obvious for both noise and interference dominance operation. For example, taking an outage probability level of the order of 10−4 (99.99% of the year system availability) an improvement of about 10 dB for (INR)nom = 6dB and 12 dB for (INR)nom = −3dB is obtained. Another interesting point is the abrupt variation of the joint nonexceedance probability for the site diversity system. As may be seen, slight changes of the threshold r (in dB) create drastic variations of the nonexceedance probability for both values of (INR)nom. Further, the parameter INRnom seems to play, for the site diversity system, a quite significant role on the achievable CNIR threshold, even for low probability levels where the rain attenuation probably dominates. On the contrary, no such conclusion can be obtained for the single site system, where the carrier-to-noise plus interference ratio is mainly degraded due to the strong rain attenuation occurring in the wanted signal. It is also worthy to notice that all the curves asymptotically tend to the corresponding (CINR)nom as the nonexceedance probability goes toward one.

Figure 2.

CNIR nonexceedance probability versus level (r) in dB for an interfered single and double site diversity system located in K-climatic zone. The other parameters are: f : 20 GHz, θd = 3°, (CNR)nom = 20 dB.

Table 1. Parameters of the Hypothetical Reference Systems
 K ZoneQ Zone
LC(km)7.637.63
LI(km)7.637.63
Rm0.016740.13415
Sr2.02001.85680
G (km)1.51.0
φ129°29°
φ229°29°
 12GHz20GHz
a0.017920.07204
b1.21921.0993

[17] Another set of diagrams very useful for the system designer, exhibit the variation of the (CNIR) versus the angular separation θd for various levels of nonavailability times. In Figures 34, (CNIR) curves (in dB) versus θd are presented for the same Earth-space system considered above, generalized by considering both Ku (12GHz) and Ka (20GHz) bands and using various separation distances S for the diversity reception. The determination of the interference level under clear-sky conditions can be taken into account by means of the following expression:

equation image

where (INR)*nom represents the difference (CNR)nom−(CIR)nom of the interfered system under consideration for θd = 1°. For all the cases, that were examined, the (INR)*nom = 20dB level was selected. As a consequence, the interfered system behaves as interference-dominant system under clear-sky conditions for θd less than about 4°. On the other hand, for higher values of θd the system tends to behave as a thermal-noise dominant one. In this case, a relative independence of the allowable (CNIR) threshold from θd is obvious, and this clearly characterizes a noise dominant operation. Therefore, the appropriate rain attenuation fade margins can be estimated by taking the differences of the allowable (CNIR) thresholds from the system clear-sky curve for sufficiently large θd. For example, for the Ku-band (12 GHz) the resulting margins are 7 dB for the single site and 3 dB for the corresponding diversity protected system. In the Ka-band case, the respective margins are 17 and 5 dB, and the option of using site diversity protection in the Ka-band to achieve low fade margin operation clearly arises. Examining now the system performance for small values of θd, an intense noise dominant behavior is also obvious for the single site system in the Ka-band. A similar conclusion cannot be drawn for the Ku-band system operating under no-diversity protection, where the variation of (CNIR) threshold as a function of θd depends significantly upon the interference level (INR)nom. Subsequently, considering the site diversity system, the dependence of (CNIR) upon θd is quite pronounced, even in the case of Ka-band, where higher levels of rain attenuation are encountered. As a matter of fact, the (INR)nom level or, equivalently, the value of θd is a critical parameter for the determination of the achievable (CNIR) threshold for each level of nonexceedance time. Another useful conclusion can be derived by examining the dependence of the minimum value of θd, in the attempt to achieve a particular (CNIR) threshold, on the distance S. As may be seen (Figure 4), increasing S leads to a subsequent decrease of the minimum allowable value of θd. This is an important conclusion that can be used toward improving the exploitation of the available orbital spectrum. For example, taking in Figure 4 an allowable (CNIR) level of about 12 dB the resulting minimum θd for S = 30km decreases by 1° as computed with that achieved for S = 10km. Similar results may be obtained by considering an interfered Earth-space system located in the tropical region (zone Q). However, in this case (Figure 5), due to heavier rainfall phenomena encountered and in order to achieve allowable (CNIR) thresholds, similar to those for climatic zone K, we have only used Ku-band operational features combined with quite larger nonexceedance times (of the order of 240 min/year). On the other hand, the combination of Ka-band operational features with nonexceedance times of several tens of minutes per year for the Q-zone yields enormous rain fade margins, leading this way to the consideration of the triple site diversity scheme as a potential countermeasure. The analysis of such problem is planned be carried out in a future work. However, the cost elements and the practical necessity of such a solution should be very carefully considered in a real system planning. Complicated error-correction coding algorithms and uplink power control mechanisms could also be combined in order to meet strict practical requirements. That is why triple site diversity protection should be examined on a case-by-case basis, and after having verified that the above alternative countermeasure schemes have been practically proved insufficient.

Figure 3.

Carrier-to-noise plus interference ratio (CNIR) (in dB) versus angular separation between adjacent satellites (θd) for an interfered single/double site diversity system located in K-climatic zone for various frequencies (12 and 20 GHz). The other parameters are: (CNR)nom = 20 dB,(INR)*nom = 20 dB outage time: 60 min/year.

Figure 4.

Carrier-to-noise plus interference ratio (CNIR) (in dB) versus angular separation between adjacent satellites (θd) for an interfered double site diversity located in K-climatic zone for various separation distances S. The other parameters are: (CNR)nom = 20 dB, (INR)*nom = 20 dB, outage time: 60 min/year.

Figure 5.

The same as in Figure 4 but for Q-zone, f : 12 GHz, outage time: 240 min/year, (CNR)nom = 20 dB, (INR)*nom = 20 dB.

[18] In the remaining Figures 6 and 7, the total gain due to diversity reception, denoted as GDR (Gain Differential Reception), is drawn versus the distance S for various (INR)nom levels and nonexceedance times. The interfered Earth-space system examined here is the same as before (see Figures 2, 3 and 4). By the term GDR, we define the difference (in dB) of the ratio (C/N + I) for a single site system from the corresponding (C/N + I) for the same system but operating under the site diversity scheme. The physical meaning of GDR implies how closer to (C/N + I)nom than in the single site reception the achievable (C/N + I) is, adopting the diversity technique. As may be seen, the GDR increases along with the separation distance S. The dependence from S is quite pronounced for systems operating under the following specifications: Ka-band of frequencies, noise dominant operation under clear-sky conditions (INRnom < 3dB) and very large availability times (see corresponding diagram of Figures 6 and 7). In this case, it is worthy to notice the tendency of GDR saturation for large values of S, in the range of the mean diameter of rain cells. Obviously, the rain cell size depends on its definition, also pointed out by Lin [1975]. In the present work a “rain cell” based upon a definition of ρo (spatial correlation coefficient) ≥0.05 within the cell is adopted. This value along with a characteristic distance G ranging from 1 to 1.5 km, as previously mentioned, yields a mean rain cell size of the order of 20km up to 30km, complied with what suggested by Drufuka and Torlaschi [1977] and Sasaki and Nagamune [1983]. In any case, modeling effectively the spatial structure of the rainfall medium is of high importance since it has a significant influence on the projected results. Considering, at this point, systems operating in the Ka-band but with interference-dominant conditions (INR > 3dB) under clear-sky the GDR tends to saturate for small values of S (see corresponding curve of Figure 6). These conclusions are useful for choosing an appropriate value of S that optimizes GDR.

Figure 6.

The gain due to differential reception (GDR) (in dB) as a function of the separation distance S (in km) for various interference levels. The other parameters are: climatic zone: K, f : 20 GHz, θd = 3°, outage time: 90 min/year, (CNR)nom = 20 dB.

Figure 7.

The gain due to differential reception (GDR) (in dB) as a function of the separation distance S (in km) for various outage time levels. The other parameters are: climatic zone: K, f : 20 GHz, θd, = 3°, (CNR)nom = 20 dB, (INR)nom = 3 dB.

[19] Finally, the GDR increases with decreasing INRnom and in the limit INRnom→−∞, tends to the well known diversity gain G = Ms − Md, for a noise-dominant operation. In this case, Ms and Md are the rain attenuation fade margins for single and double site diversity operation, respectively.

4. Conclusions

[20] The prediction of the degradation of the carrier-to-noise plus interference ratio (CNIR) has been examined in this paper, for systems located in heavy rainy climatic zones and operating under very large availability times. In such cases, the site diversity is suggested as a countermeasure technique, and our analysis is concerned with the evaluation of the total outage time of such Earth-space systems suffering from differential rain attenuation. Numerical results taken from the proposed methodology examine the availability performance due to the (CNIR) degradation for a simulated site diversity system located in K (Mediterranean) and Q (tropical) zones. The conclusions that may be drawn are summarized as follows:

  1. The interference level under clear-sky conditions (INRnom) for an interfered site diversity system is a significant parameter and seems to play a decisive role toward the achievable CNIR threshold, even for low probability levels, for both Ku and Ka bands (see Figure 2).
  2. Further, the angular separation θd is another key parameter for the determination of the achievable (CNIR) threshold for each level of nonexceedance time (see Figure 3).
  3. The increase of the separation distance S between the Earth stations of a site diversity system results in a subsequent decrease of the minimum permittable value of angular separation θd. The quantitative relation between those two, as exposed herein, can be used regarding the available orbital spectrum efficient exploitation (see Figure 4).
  4. Finally, the total gain due to the use of the diversity reception, GDR, increases as long as the separation distance S also increases. For systems operating in the Ka-band of frequencies, a remarkable tendency of GDR saturation is observed for large values of S, under the assumptions that the thermal noise power is dominant over the interference in the reception process and very large availability times should be applicable. On the other hand, for a diversity system operating in the same band but under interference-dominant conditions the GDR tends to saturate for small values of S (see Figures 67). All the above remarks provide potentially useful guidelines for the system designer.
Notation
φi

Elevation angle of the slant paths pointing toward satellite Si (i = 1, 2).

θd

Differential angle between two satellites.

S

Site separation between stations E1 and E2.

D

Distance between the projections of the slant radio paths E1S1 and E2S1.

D′

Distance between the projections on the slant radio paths E1S2 and E2S2.

equation image

Rain attenuations on the wanted signal referring to Earth-space slant paths EiS1 (i = 1, 2).

equation image

Rain attenuations on the interfering signal referring to Earth-space slant paths EiS2 (i = 1, 2).

(CNR)i,nom

Carrier-to-Noise Ratio at the Earth station Ei receiver (i = 1, 2) under clear-sky conditions.

(CIR)i,nom

Carrier-to-Interference Ratio at the Earth station Ei receiver (i = 1, 2) under clear-sky conditions.

(INR)i,nom

= (CNR)i,nom − (CIR)i,nom.

(CNIR)i,nom

= (CNR)i,nom − 10logequation image

(CNIR)i

Carrier-to-Noise plus Interference Ratio at the Earth-station Ei receiver (i = 1, 2).

LC

Effective average length of the Earth-satellite paths E1S1, E2S2 corresponding to the wanted signal.

LI

Effective average length of the Earth-satellite paths E1S2, E2S2 corresponding to the interfering signal.

LCD, LID

The projected LC and LI, respectively.

equation image(

Rain attenuations calculated for the projections of the slant paths EiS1 (i = 1, 2).

equation image

Rain attenuations calculated for the projections of the slant paths EiS2 (i = 1, 2).

a, b

Constants of the specific rain attenuation Ao (in decibels/kilometer).

G

Characteristic distance used to model the spatial inhomogeneity of the rainfall medium.

Rm, Sr

Statistical parameters of the point rainfall distribution.

r

Nonexceedance level of the (CNIR) ratio (in decibels).

equation image

Statistical parameters of the lognormal distribution concerning equation image and equation image random variables.

equation image

Statistical parameters of the lognormal distribution concerning equation image( and $\rm A_{I_{2}}^{\prime}$ random variables.

equation image

One-dimensional normal p.d.f.

equation image

Two-dimensional normal p.d.f.

equation image

Three-dimensional normal p.d.f.

equation imageu3

Correlation coefficient between equation image and equation image random variables.

equation image

Correlation coefficient between equation image and equation image random variables.

equation image

Correlation coefficient between equation image and equation image random variables.

equation image

Correlation coefficient between equation image and equation image random variables.

equation image

Correlation coefficient between equation image and equation image random variables.

equation image

Correlation coefficient between equation image and equation image random variables.

Appendix A:: Derivation of the Expressions (8)(26)

[21] As pointed out in the main text, the probability under consideration can be calculated as

equation image

Following Crane's simplified considerations for the vertical variation of the rainfall structure [Crane, 1980], we may write

equation image

where equation image and equation image are the attenuations calculated for hypothetical terrestrial links with path lengths LCD = LC cos φ1 for equation image and LID = LI cos φ2 for equation image respectively. Moreover, LC and LI are the effective average lengths of the Earth-satellite paths with respect to the wanted and interfering signals expressed in terms of the rain height H [Crane, 1980]. The latter parameter is given by means of the latitude of the location Λ [CCIR, 2001]. Substituting (A2) into (A1), we have

equation image

Considering now the planes equation imageA'I1 and A'c2equation image (see Figure 1b), we draw the curves

equation image
equation image

By setting 0, ∞ for x1, x2 into (A4) and (A5) we can determine the respective roots equation image and equation image given by

equation image

while the (CNIR)nom is given by expression (17) of the main text.

Figure 1b.

Regions of integrations in the A′c − A′I planes.

[22] Following the definition of the double integral and after confirming that the functions y1(x1) and y2(x2) are positive increasing functions, the following nonexceedance probabilities can be expressed as

equation image
equation image

The levels yk, yp are given by means of the expressions (13)(14) of the main text. The levels equation image and equation image are the roots of the equations (A4) and (A5) for equation image and equation image respectively (see expression (15) of the main text).

[23] It is now obvious that the joint nonexceedance probability given by (A1) can be expressed as

equation image

where the probabilities P1, P2, P3, P4 are given by means of (9)(12) of the main text.

[24] Adopting now assumption (1) (section 2) and after using the following definition for the normal variables

equation image

we have

equation image
equation image
equation image
equation image

The definition of equation image and equation image is given in the main text (see expressions (22)(23)). Using now the Bayes' theorem [Papoulis, 1991], the joint density functions equation image and equation image, can be expressed as

equation image

Substituting (A15) into (A11) ÷ (A14) and using the result

equation image

where f(x) is a normal density function with parameters (m,σ), one gets expressions (18)(21) of the main text.

Appendix B:: 1,2,3 - Dimensional Normal Density Functions and Parameters μ3/1,2, μ4/1,2, μ4/1,2,3 and σ3/1,2, σ4/1,2 and σ4/1,2,3

[25] The normal density functions equation image and equation image can be defined as [Papoulis, 1991]

equation image
equation image
equation image
equation image
equation image
equation image

[26] In addition, the conditional mean values and standard deviations encountered in expressions (18)(21) can be evaluated as

equation image
equation image
equation image
equation image
equation image

In the above expressions, equation image and equation image are the logarithmic correlation coefficients given by [Kanellopoulos et al., 1993]

equation image
equation image
equation image
equation image
equation image
equation image

in terms of the path correlation coefficients equation image and equation image respectively. The latter coefficients can be expressed analytically in terms of the geometrical parameters of the configuration along with the spatial correlation coefficient ρ(z, z′) between two points of the rainfall medium, as given by Lin [1975]. Analytical details for their derivation can be found elsewhere [Kanellopoulos and Ventouras, 1996]. We present here the final results

equation image
equation image
equation image
equation image
equation image
equation image

where the factors Hij (i = 1,2,3 and K = 2,3,4,i < j) can be defined as follows (see Figure 1c):

equation image
equation image
equation image
equation image

and

equation image
equation image
equation image
equation image
equation image
equation image

(see Figure 1c).

Figure 1c.

The projections of the slant paths corresponding to the desired and interfering signals.

[27] In the above formulas ρ(z, z′) denotes the spatial correlation function of the rainfall structure [Lin, 1975] and is a function of the distance equation image between two points belonging to the projections of the slant paths under consideration, whereas ΔΨ is the projected differential angle (see Figure 1c). Finally, the factors HIC, H1I are given by means of the expressions (26) of the main text in terms of the path lengths LCD, LID.

Ancillary