Radio Science

Rain fade slope predicted from rain rate data

Authors


Abstract

[1] A key parameter of the statistics of the rate of change of rain attenuation (“fade slope”) on satellite links is the variance of relative fade slope. This paper shows how this parameter can be derived from rain rate measurements without the use of satellite beacon measurements. Relating rain rate directly to attenuation would give unrealistic results. A theoretical model to estimate the variance of relative fade slope is derived, using the integrating effect of rain rate variations along the propagation path and using as inputs several meteorological parameters only. The theoretical values are compared to measured results from a link in the United Kingdom at 50 GHz. The agreement is good on average and also in their correlation with the type of rain and with the rain height. With the wind speed, the theoretical values increase more strongly than the measured results, which may be due to the limited spatial resolution of the meteorological data.

1. Introduction

[2] In frequency ranges at V band and above, rain attenuation on satellite links becomes so severe that static fade margins are not feasible anymore, and dynamic adaptive fade countermeasures are necessary, usually referred to as fade mitigation techniques (FMT) [Sweeney and Bostian, 1999; European Cooperation in the Field of Science and Technology, 2002]. For the design of FMTs, information on the dynamic behavior of rain attenuation is essential. Knowledge of the rate of variations of attenuation is useful to design a control loop of an FMT system that can follow signal variations, and also to allow a better short-term prediction of the propagation conditions. Information on the dynamic behavior of rain attenuation is also useful for the development of models of this dynamic behavior [e.g., Maseng and Bakken, 1981; Sweeney and Bostian, 1992].

[3] Commonly, the dynamic behavior of rain attenuation is studied in terms of the rate of change of attenuation, or “fade slope.” In order to provide information on rain fade slope, various experiments have been carried out to analyze its statistical properties [e.g., Matricciani, 1981; Stutzman et al., 1995; Feil et al., 1997; Schnell and Fiebig, 1997; van de Kamp, 2003a]. An empirical prediction model of fade slope, based on measurements from various sites [van de Kamp, 2003a] is currently recommended by International Telecommunication Union (ITU) [2005]. An overview of published measurements and models is given by van de Kamp and Castanet [2002].

[4] To calculate the rain fade slope from measured time series, the rapid component due to tropospheric scintillation is filtered out; this component is unpredictable and not useful in the design of FMTs [Poiares Baptista and Davies, 1994]. Commonly used filter bandwidths are in the order of 0.01–0.02 Hz [Poiares Baptista and Davies, 1994]. The resulting fade slope of the filtered signal is dependent on the filter bandwidth used [van de Kamp and Clérivet, 2004; ITU, 2005].

[5] The fade slope ζ can be calculated from two attenuation sample values A (dB) and the time interval Δt between them:

equation image

This definition is used in most (but not all) studies cited above. Values of Δt used in practice vary between 2 s [van de Kamp, 2003a] and 10 s [Feil et al., 1997]. The resulting fade slope depends also on Δt [van de Kamp and Clérivet, 2004; ITU, 2005].

[6] Since the rain fade slope ζ is stochastic, measured results are studied statistically. The distribution of ζ is generally found to be symmetrical around a zero mean. In many studies the distribution of ζ is evaluated conditional to the coinciding attenuation value A. This conditional distribution is also symmetrical, and has, for all attenuation values above a certain minimum threshold Ath, the following properties [van de Kamp, 2003a; ITU, 2005]: a zero mean, a constant shape (independent of A), and a standard deviation approximately proportional to A.

[7] The minimum value Ath must be regarded because of the following: since the standard deviation of ζ is proportional to A, when A becomes small ζ also becomes small. So even after low-pass filtering, the fluctuations may be dominated by residual scintillation and noise, which show different dynamical characteristics.

[8] The “relative fade slope” is defined as

equation image

From the abovementioned properties of the distribution of ζ conditional to A, it follows that the distribution of ζr conditional to A also has a zero mean and a constant shape, but also an approximately constant standard deviation. This means that the unconditional distribution of ζr, for all values of A > Ath, also has the same standard deviation. The square of this standard deviation, the “variance of the relative fade slope” σζr2, will be used as the characterizing parameter of fade slope in this paper. It is defined as

equation image

where E denotes an average. From the above follows that the standard deviation of the distribution of fade slope ζ conditional to attenuation A is (approximately) given by equation imageA. The relation between σζr2 and the current ITU-R model of fade slope is given as

equation image

where S is a link-dependent parameter and F(fB, Δt) is a factor expressing the dependence on the low-pass filter bandwidth fB and the time interval Δt [van de Kamp, 2003a; ITU, 2005].

[9] The variance of relative fade slope σζr2 is useful for more purposes than the prediction of fade slope statistics: with Δt = 10 s it is also one of the parameters from which the input parameters of the “two-sample model” can be derived, as is shown by van de Kamp [2003b, 2005a]. The two-sample model [van de Kamp, 2002; van de Kamp, 2003c] predicts the probability distribution of rain attenuation a short time after a measured value, dependent on the values of two previous samples of rain attenuation. This model can be applied in the design of FMT systems, for the short-term prediction of rain attenuation. It can also be used to generate simulated rain attenuation time series, which can be used for testing FMT systems. Sweeney and Bostian [1999] showed that 10 s is within the range of useful time constants for the design of FMTs. It is therefore clear that σζr2 is an invaluable parameter in the design of FMT systems.

[10] The variance of relative fade slope σζr2 is a site- and climate-dependent parameter. There are indications that the rain fade slope depends on the elevation angle [Feil et al., 1997], and on meteorological parameters such as the wind speed, the path length through rain, and the type of rain (widespread/convective) [Buné et al., 1988]. Therefore the parameter σζr2 should be determined for every separate link and meteorological condition for which FMT systems are to be designed.

[11] In this paper relations are studied between the dynamic properties of rain attenuation and those of the rain itself. The main purpose of this study is to predict σζr2 for specific sites from rain gauge data, without having to use rain attenuation data.

2. Measured Attenuation Dynamics and Rain Rate Dynamics

[12] As a first step, the variance of rain fade slope σζr2 derived from measured data will be compared to an equivalent parameter determined from measured rain rate data [van de Kamp, 2005b].

[13] σζr2 has been determined from beacon data from the satellite Italsat, which had been measured by Rutherford Appleton Laboratory in Sparsholt, United Kingdom. The signals, at 18.69, 39.59 and 49.49 GHz were recorded during 43 months from April 1997 to January 2001. The elevation angle of the satellite was 29.9°. In this paper, only the measurements at 49.49 GHz are considered (hereafter referred to as 50 GHz).

[14] For this analysis, the measured rain attenuation data were divided in the periods: 0300–0900, 0900–1500, 1500–2100 and 2100–0300 of each day. For each of these 6-hour periods, the data were low-pass filtered to reduce the effect of scintillation, using a bandwidth fB of 0.02 Hz, as recommended by Poiares Baptista and Davies [1994, p. 91]. The relative fade slope ζr was calculated as in equation (2) with Δt = 10 s, and from this, the variance σζr2 of relative fade slope was calculated as

equation image

where the integral is evaluated over the measured data in the 6-hour period for which A > Ath as mentioned in section 1, and T is the accumulated time of these data. The optimum value for Ath depends on frequency, since both scintillation and attenuation increase with frequency. It was found that σζr2 shows stable results (i.e., becoming constant with respect to Ath) for Ath = 5 dB at 50 GHz. The resulting total amount of data used was 2,018,068 data points (seconds), 2.09% of the total measured time.

[15] It was checked that the results were free from residual scintillation, by calculating σζr2 from selections of data with different values of scintillation measured in a higher frequency band (0.1–0.5 Hz) [van de Kamp, 2005b]. No scintillation dependence was found from this analysis; the values of σζr2 calculated as described above can therefore be trusted to be due to rain attenuation only. The effect of the low-pass filter with fB = 0.02 Hz was compensated by scaling the results to an infinite filter bandwidth by the factor F(fB, Δt) (see equation (4)) [van de Kamp and Clérivet, 2004; ITU, 2005].

[16] From the values of σζr2 and T for each 6-hour period, the long-term values of σζr2 were calculated as

equation image

where σ2ζri and Ti indicate the values of σζr2 and T for 6-hour period i.

[17] These results were analyzed concurrent with meteorological data, which was obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) (http://www.ecmwf.int/) and the British Atmospheric Data Centre (http://badc.nerc.ac.uk/, from their Operational Data database, which is a continuation of their ERA-40 project). Various meteorological parameters are available recorded at reference times 0000, 0600, 1200 and 1800 (UTC) of each day, and for grid points around the globe 2.5° apart in longitude and latitude.

[18] The main meteorological parameter of interest for the characteristics of fade slope is the wind speed [van de Kamp, 2003a]. From ECMWF, the wind speed in two orthogonal directions is available on various height levels, expressed as pressure levels. The relation of σζr2 with wind speed was studied using the velocity of wind in eastward direction vu (m/s) and in northward direction vv (m/s) at the pressure level 850 mbar, which is approximately 1.5 km high. This being the approximate height at which rain originates, it is expected that the wind speed at this height will show some correlation with the dynamics of rain attenuation. These data were downloaded for the period 1997 to 2001, and two-dimensionally linearly interpolated between the grid points to obtain the values in Sparsholt (latitude 51.07°; longitude –1.43°). This grid square is about 175 km (E-W) × 250 km (N-S), and contains much of southern England including its south coast, but no mountains to form orographic barriers. Thus, although the interpolated values cannot give exactly the momentary values on one spot, on average they are expected to be representative for the meteorological conditions at Sparsholt.

[19] From vu and vv, the absolute value of wind speed vw was calculated as

equation image

[20] Next, the concurrent analyses was performed as follows: the values of vw were classified in bins with bin size 2 m/s, and for every bin, the long-term values of σζr2 were calculated over all values in the 6-hour periods coincident with vw being in the respective bin. The results represent σζr2 as a function of wind speed. The result is shown in Figure 1 (solid line).

Figure 1.

Variance of relative fade slope as a function of wind speed and under different conditions of the vertical velocity.

[21] Another possible meteorological parameter which is likely to influence the statistics of fade slope, is the type of rain. To study this, the data were analyzed concurrent with the vertical velocity of the air vver, also obtained from ECMWF. Because convective rain is caused by vertical updraft of air, the vertical velocity can be an indicator for the degree of convectivity and thus for the type of rain. The unit of vver is Pa/s, so it is expressed as the change in pressure that an air package encounters in an updraft (negative values indicating upward movement). The vertical velocity vver was also obtained for the 850 mbar pressure level.

[22] The concurrent analyses with vw and vver together was performed by calculating the long-term values of σζr2 over all values in the 6-hour periods coincident with vver < −0.15 Pa/s and for every value of vw; the same was done using the condition of vver > −0.05 Pa/s. These threshold values were arbitrarily chosen, to cut off approximately the upper and lower one-third part of the distribution. The results of this are included in Figure 1 (dotted lines).

[23] Figure 1 shows that σζr2 increases with wind speed, for all data, as well as for both categories of vver. Furthermore, the values for vver > −0.05 Pa/s (supposedly corresponding to stratiform rain) are higher than those for vver < −0.15 Pa/s (convective rain). All curves have a relatively small offset.

[24] The results for σζr2 increasing with wind speed, with a small offset, suggests that the major part of variations in rain attenuation observed are influenced by wind, and are most likely caused by horizontal variations in the structure of the rain storm, which is moved across the propagation path. The gradient of the variance as a function of the wind speed is likely to be an indication of the variance of spatial horizontal variations of the rain intensity in the structure of the rainstorm. The observation that this gradient is larger during stratiform rain suggests that although convective rainstorms are smaller in horizontal size, stratiform rain contains more spatial variations.

[25] A relation of this parameter with rain rate dynamics may be searched as follows. According to the ITU-R recommendation P.618-8 [ITU, 2003a], rain attenuation is related to the rainfall intensity, or “rain rate” R (mm/hr) as

equation image

where LE is the effective path length through rain, and a and b are frequency-, elevation- and polarization-dependent coefficients; b takes values around 1 in the centimeter wave region. Although a and b can be expected to depend on the drop size distribution, which may vary between rainstorms and even within a rainstorm, this model with static values of a and b is considered adequate to predict attenuation statistics [ITU, 2003b]. It may therefore be expected that a similar dynamic behavior as for the attenuation would be found for the rain rate.

[26] In order to study this, 94 months of rain rate measurements in Sparsholt and Chilbolton (8 km from Sparsholt) have been analyzed. The rain measurements were performed in three different measurement campaigns by Rutherford Appleton Laboratory, using drop counter rain gauges. These rain gauges release the collected rainwater as drops of a constant size, which are then counted. More details are given in Table 1. Even though these measurements are not completely concurrent with the rain attenuation data from Sparsholt, they come from (nearly) the same site, and the relation between the dynamics of rain and climatic characteristics can be expected to be the same for both data sets.

Table 1. Details About the Measurements of Rain Rate Used
LocationPeriodLength, monthsIntegration Time, sQuantization Level, mm/hr
SparsholtApr 1997 to Dec 1999 and Jan 200131101.44
SparsholtJan 2003 to May 2004 and Jan to Dec 200529101.44
ChilboltonJan 2003 to Oct 2004 and Jan to Dec 200534101.44

[27] The rain rate data were analyzed in the same way as the attenuation data, and the variance of relative rain rate slope was calculated similarly as equations (5) and (2), where A(t) is replaced by the rain rate R(t), and Δt = 10s. The integral of equation (5) was evaluated only over samples in each 6-hour period which exceed 8 mm/hr, in order to avoid effects of the quantization level of 1.44 mm/hr (Table 1). The resulting total amount of rain data used was 34,637 data points (= 346,370 s), 0.146% of the total measured time. This relative portion of data is smaller than for the attenuation data; this is because a rain rate of 8 mm/hr causes at 50 GHz about 22 dB of attenuation, so this threshold is effectively higher than Ath, the threshold used for the attenuation.

[28] The measured variance of relative rain rate slope was compensated for the filtering effect caused by the 10 s integration time of the rain gauge, using the same procedure as the attenuation data. The data were analyzed concurrent with the wind speed vw and vertical velocity vver as obtained from ECMWF, in the way described above for the attenuation data. The results are shown in Figure 2.

Figure 2.

Variance of relative rain rate slope as a function of wind speed and under different conditions of the vertical velocity.

[29] On the basis of the almost linear relation between rain rate and attenuation as given in equation (8), it would be expected that the variance of relative rain rate slope will be similar to that of relative fade slope. Figure 2 shows however a quite different picture. The variance of relative rain rate slope also increases with wind speed, and, although slightly less pronouncedly, with the vertical velocity. However, note the different vertical axis scales: the variance of relative rain rate slope is much larger than that of fade slope. For instance, at vw = 10 m/s the variance of relative fade slope is 8 × 10–5 s–2, while that of rain rate slope is 1.4 × 10–3 s–2, over 17 times larger!

[30] Such a difference must mean that the dynamic relation between rain rate and attenuation is less straightforward than suggested by equation (8). If rain rate measurements are to be used for the prediction of the variance of relative fade slope, a more elaborate relation between these two will have to be used. This will be addressed in the next section.

3. Theoretical Expression for the Variance of Relative Fade Slope

3.1. Relation With the Rain Rate

[31] A possible cause of the large difference in dynamic behavior between rain rate and rain attenuation, is that rain rate is spatially variable, and is therefore not uniform along the propagation path through rain. While the measurements of a rain gauge represent the rain rate at one spot, the attenuation observed on a link received on the same site is caused by an integration of the rain rates along the path.

[32] Assume the configuration as depicted in Figure 3. The rain rate at measurement point xs at time of measurement tm is equal to R(xs, tm), while the attenuation A(tm) is caused by the rain all along the path up to the rain height hr. This path has a length hr/sinɛ, where ɛ is the elevation angle (this approximation is valid as long as ɛ > 5° [ITU, 2003a]).

Figure 3.

Configuration with a path length through rain.

[33] In equation (8) the path length LE can then be replaced by an integral along the path:

equation image

[34] Although the coefficients a and b can also vary within the rain event, their variations along the path will be averaged out by the integral and be represented by the long-term effective values as given by the ITU [2003b], so they can be represented as constant in this equation. An integral representation as in equation (9) was also used by Matricciani in the “synthetic storm technique,” used to predict attenuation statistics [Matricciani, 1996] and fade duration statistics [Matricciani, 1997] from rain rate time series. However, several major differences between the synthetic storm technique and the approach of this paper exist, as will become evident.

[35] Assume the rain is uniform in vertical direction, so the rain rate at any location l on the path is the same as on its horizontal projection. In Figure 4, the situation is depicted seen from above. The integral of equation (9) is performed along the line S1-P1. The orientation of this line is defined by the azimuth angle θl of the link.

Figure 4.

Configuration (seen from above) with the wind moving a rainstorm across the measurement site, located at S1.

[36] The rain attenuation at time tm + Δt is found by using the rain intensity R(x, tm + Δt) in equation (9). Now suppose the “frozen storm” hypothesis [Taylor, 1938]: any measured variation in rain rate is caused by the wind, moving a geometrically stable rainstorm across the measurement site. Say the wind velocity vector is vw, its azimuth angle is θw and the magnitude of its speed is vw. Under the frozen storm hypothesis, the rain intensity at time tm + Δt is equal to the rain intensity that was at time tm present on the line S2-P2 in Figure 4. This line is a translation of the line S1-P1 by the distance vwΔt, in the direction opposite of vw.

[37] In the synthetic storm technique approach, the wind direction was assumed to be along the propagation path, i.e., θw = θl. This assumption gives reasonable results for statistics of attenuation and fade duration, as shown by Matricciani. However, for the fade slope this assumption cannot be made because it would have a major effect on the result, as will be shown later.

[38] Using the assumptions of this section, Appendix A shows that the variance of relative fade slope can be found as [van de Kamp, 2005b]

equation image

where equation imageAt) = the autocorrelation coefficient of attenuation for a time lag Δt. Appendix B shows that this can be found as [van de Kamp, 2005b]

equation image

with equation imageequation image = the spatial autocorrelation coefficient of Rb for a distance d12:

equation image

and d12 being the distance between X1 and X2 as indicated in Figure 4. From Figure 4 can be derived that this distance is equal to

equation image

3.2. Autocorrelation of Rain Rate

[39] For the theoretical expression of the autocorrelation of attenuation in equation (11), the autocorrelation of rain rate equation imageequation image(d12) is needed as a parameter. This will be derived from theory in this section.

[40] Maseng and Bakken [1981] derived a dynamic model of rain attenuation using stochastic theory. Since in the derivation of this model, they did not take into account the path integration effect described in equation (9), this model may be expected to be also valid for rain rate. Supposing the Maseng/Bakken model is valid for rain rate, the probability distribution of instantaneous rain rate R2, dependent on the value R1 a time τ earlier, is lognormal:

equation image
equation image
equation image

where

m2

median value of R2, conditional to R1;

σ2

standard deviation of lnR2, conditional to R1;

mR

long-term median value of R;

σlnR

long-term standard deviation of lnR;

B

parameter describing the dynamics of rain rate (called β by Maseng and Bakken).

[41] This model describes the dynamic behavior of rain rate measured in time. This can be converted to a model describing the spatial irregularities of rain using the frozen storm hypothesis (see section 3.1). The dependence on τ can be converted to dependence on distance d along the wind direction by substituting τ = d/vw, where vw is the magnitude of the wind speed. Furthermore, according to the frozen storm hypothesis, the spatial variations should be independent of the wind speed, so B/vw must be constant, and can be defined as Bx. Equations (15)–(16) can then be replaced by

equation image
equation image

[42] This way equation (14) describes the probability distribution of rain rate R2 at location X2 and a certain time, given that the rain rate R1 at location X1 at the same time is known, with d being the distance between X1 and X2.

[43] If the rainstorm is isotropic, the dependence of R2 on R1 is independent of direction, and the distance d can be measured in any direction.

[44] From the above equations can be derived that the long-term distribution of rain rate (unconditional to another value) is also lognormal:

equation image

[45] Using equations (14) and (19), the autocorrelation coefficient of rain rate R can be calculated as [Maseng and Bakken, 1981]

equation image

[46] The autocorrelation coefficient of Rb can be found by converting the Maseng/Bakken model to a model for Rb. The resulting probability distribution of the instantaneous value of R2b, dependent on the value R1b a time τ earlier, is also lognormal, with the parameter values mRb substituted for mR and lnR for σlnR, and with the same parameter B. The spatial autocorrelation coefficient of Rb is then [van de Kamp, 2005b]

equation image

[47] It results that to obtain information on the spatial dynamics of rain rate, as input into the theoretical expressions (10)–(13), the parameters σlnR and Bx are needed. This is another advantage of the approach of this section compared to estimating the dynamics of rain attenuation using the synthetic storm technique. The synthetic storm technique, because it converts measured rain rate time series to spatial dynamics of rain rate using the frozen storm hypothesis, always requires a large amount of rain rate data as input. In the approach of this section, once the rain parameters σlnR and Bx are known for a specific site (possibly dependent on climatic parameters), the inputs for the theoretical expressions (10)–(13) can be obtained without the use of rain rate time series.

3.3. Assessment of Parameter Values

[48] In this subsection, the values of the parameters Bx and σlnR, needed in equation (21), will be derived from rain rate measurements.

[49] From equations (14) and (16) can be calculated that

equation image

where mR2 and σR2 are the mean and standard deviation of R2 conditional to R1 (not to be confused with m2 and σ2). This equation can help to derive σlnR and B from σR2 and mR2, obtained from rain rate measurements.

[50] In order to derive σR2 and mR2, the distributions p(R2R1) have been derived from the rain rate measurements in Sparsholt and Chilbolton, described in Table 1. Because the rain rate measurements have an integration time of 10 s, τ = 10 s. As in section 2, the filtering effect of the integration time was compensated by scaling the measured standard deviations to an infinite filter bandwidth.

[51] Figure 5 shows the standard deviations σR2 versus the mean values mR2. Here, each value was derived from a distribution for one value of R1, with bin size 1.44 mm/hr. It is evident that σR2 is approximately proportional to mR2, making their ratio σR2/mR2 approximately constant (the deviation from this for small values of mR2 is mainly due to quantization). Since this proportionality is a major basic assumption for the Maseng/Bakken model, this result makes it likely that this attenuation model can be assumed valid for rain rate.

Figure 5.

Standard deviation versus the mean of the distribution of rain rate R2, conditional to the rain rate R1 10 s earlier, with a fitted line.

[52] The results were analyzed together with concurrent data of vertical velocity vver and wind speed vw at 1.5 km height from ECMWF (see section 2 for more info about the data and the joint analysis procedure).

[53] Here σR2/mR2 has been determined by fitting a zero-offset straight line to these for every value of vw and vver, weighting the values with the amounts of data used for each. Values for mR2 < 8 mm/hr were excluded, in order to reduce quantization distortion.

[54] Furthermore, the temporal autocorrelation coefficient of rain rate has been calculated from the measured rain rate data, for different values of vw and vver. For each vw and vver, because σR2/mR2 is known, B and σlnR are one-to-one related according to equation (22). For each value of vw and vver, these have both been derived by least squares fitting of the theoretical temporal autocorrelation function, as can be derived from equation (20) by substituting d = vwτ:

equation image

to the measured autocorrelation coefficient, in the range 10 ≤ τ ≤ 200 s. Some of the results of B are shown in Figure 6. Because the results of this showed no significant dependence on vver, the graph also shows the results for all values of vver (solid line).

Figure 6.

Values of B as derived from rain rate measurements, as functions of wind speed vw, for three values of the vertical velocity vver (dotted line) and independent of vver (solid line).

[55] The parameter Bx = B/vw was found by least squares fitting of a zero-offset line to the solid line in Figure 6, weighting for the amounts of data used for each point. The fitted line is included in the graph (dashed line). The result of this is

equation image

The values of σlnR were then calculated for each value of vw and vver by fitting equation (23) to the measured autocorrelation of rain, using B = Bxvw with the value of Bx found above. Since wind is expected to influence the dynamics of rain, but not as much its long-term statistics, it is not expected that σlnR is significantly dependent on wind speed. Also Matricciani [1996] found that attenuation statistics do not significantly depend on wind speed. Indeed, the resulting values of σlnR as functions of vw did not show any significant dependence on vw. These results were averaged for each value of vver, weighting for the amounts of data used for each point. The results of this are shown in Figure 7.

Figure 7.

Values of σlnR as a function of vver.

[56] The results in Figure 7 show an increase of σlnR with vver. The relative importance of the points in the graph is indicated in Figure 8, giving the amounts of data for each value of vver. This shows that the decrease for vver > 0.1 Pa/s is hardly significant. An empirical model for the relation between σlnR and vver was derived by fitting an exponential curve (in order to be monotonically increasing and positive) to all results in Figure 7, weighting for the amounts of data. The result of this is

equation image

and is included in Figure 7 (dashed line).

Figure 8.

Amounts of rain rate data points used in the derivations of section 3.3, as a function of vertical velocity.

[57] With B and σlnR determined as functions of vw and vver, the modeled autocorrelation of rain rate is now given for every wind speed and vertical velocity. As an example, Figure 9 shows the measured autocorrelation coefficient of rain rate for a few wind speeds and vertical velocities, and the modeled functions according to equation (23), with Bx and σlnR as found above. Figure 9 shows a reasonable, but not perfect agreement between the theoretical functions and the measured ones.

Figure 9.

Measured autocorrelation coefficient of rain rate as a function of time lag (dotted lines with pluses) and the modeled relations according to equation (23) (solid lines) for vver = −0.2 Pa/s, vw = 8 m/s (lines marked “A”); vver = −0.1 Pa/s, vw = 12 m/s (lines marked “B”); and vver = 0.0 Pa/s, vw = 14 m/s (lines marked “C”).

[58] This discrepancy, and the irregularities in the measured results, show that the measurements of the autocorrelation are not accurate enough for a representative autocorrelation for different wind speeds and vertical velocities. This means that much more data are needed if the theoretical autocorrelation function of rain rate (equation (23)) is to be verified, dependent on wind speed and vertical velocity.

[59] Nevertheless, since the derived dependencies of B and σlnR on vw and vver approximately agree with the measurements in this section, these relations can be assumed for the moment. With these relations, the autocorrelation of rain rate as derived in section 3.2, can be calculated dependent on vw and vver, and from these, the theoretical expressions for the variance of relative fade slope as derived in section 3.1. In the next section, these relations will be compared to measured results, as functions of meteorological data.

4. Comparison With Measurements

[60] The theoretical expression for the variance of relative fade slope σζr2, composed of equations (10), (11) and (21) will now be tested by comparing to measurements. For this purpose, the measured results of rain attenuation from Sparsholt at 50 GHz (see section 2) are again analyzed. A more extensive comparison can be found elsewhere [van de Kamp, 2005b]. The theoretical expression depends on the following parameters (with the values of the system parameters of the link in Sparsholt given): the magnitude of wind speed vw, the azimuth angle of the wind direction θw, the azimuth angle of the link θl = 161.7°, the rain height hr, the elevation angle of the link ɛ = 29.9°, the parameter indicating spatial variability of rain Bx, the long-term standard deviation of the log rain rate σlnR, the parameter b of the ITU-R rain attenuation model ≈0.87 for 50 GHz, and the time length Δt = 10 s.

[61] To obtain all necessary input parameters for the theoretical expression, the following meteorological parameters were obtained from ECMWF, for a 6-hour time resolution and 2.5° horizontal resolution: the wind speeds vu and vv in (respectively) eastward and northward direction, at the 850 bar pressure level (around 1.5 km height), the vertical velocity vver at the 850 bar pressure level, and the temperature profiles up to 16 km height.

[62] From these, the magnitude of wind speed vw (see equation (7)) was calculated, as well as the azimuth angle of wind direction:

equation image

and the rain height [ITU, 2001]:

equation image

where h0C is the height at 0°C temperature, linearly interpolated from the temperature profile. All parameters were linearly interpolated to obtain the values in Sparsholt.

[63] In the comparison of the theoretical σζr2 with measured values dependent on all these parameters, the dependence on wind speed and direction will here be represented as a dependence on effective wind speed, in order to increase the amount of data used in each bin and therefore the statistical significance. The usefulness of this is shown in the following.

[64] Figure 10 shows the theoretical σζr2 calculated according to the procedure derived in section 3 (solid lines) as a function of wind azimuth angle θw, for three values of vw, and specified values of the other parameters. Figure 10 shows that σζr2 depends strongly on both vw and θw. This also demonstrates that the assumption θw = θl, as used in the synthetic storm technique [Matricciani, 1996], would introduce large errors in the calculation of the variance of relative fade slope: for θw = 70° (the average value for Sparsholt), the calculated values would be more than 10 times too small.

Figure 10.

Theoretical variance of relative fade slope as a function of wind azimuth angle, for three values of the wind speed and θl = 161.7°, ɛ = 29.9°, f = 50 GHz, Δt = 10 s, Bx = 1.05 × 10−3 m−1, vver = −0.1 Pa/s, and hr = 2600 m (solid lines), and the variance calculated using the effective wind speed vw,eff (dashed lines).

[65] To express the dependence on wind speed and direction in the comparison of this section, the effective wind speed, derived by trial and error, is defined as

equation image

Figure 10 also includes the values of σζr2 calculated with vw,eff substituted for vw, a constant value θl + 90° substituted for θw, and the rest of the parameters the same (dashed lines), as functions of the value of θw used in equation (28). In Sparsholt, during 70% of all rainy time θw was between 20° and 100°. Figure 10 shows that in this region, the result using vw,eff is very similar to that using the true vw; for ∣θw–θl∣ > 10° (±180°), the relative error is less than 5%. This means that for the validity check with the rain attenuation measurements in Sparsholt, the dependence of σζr2 on the effective wind speed vw,eff can be used to represent the dependence on both vw and θw. Note that equation (28) is valid for the specific parameter values of the Sparsholt link only. Note also that this parameter is only for the sake of comparison in this paper; it is not needed in the application of the model.

[66] Now, first the dependence of σζr2 on vver (an indication of rain type) will be verified with the measurements. In order to check any correlation between this parameter and other meteorological parameters, the average hr and the average vw,eff as functions of vver were evaluated during the measured attenuation time on the link in Sparsholt and for ∣θw–θl∣ > 10° (±180°). The results showed that both are fairly constant over the range of vver, with overall averages of hr = 2550 m and vw,eff = 13.6 m/s. This first result is a little surprising, considering that convective rainstorms usually are much higher than stratiform storms [Rogers and Yau, 1989], so a correlation between hr and vver would be expected. Possibly this means that during convective rain, the rain height cannot really be assumed to be 360 m above the 0°C height. However, in the absence of a better rain height model, this result is kept as it is.

[67] The measured long-term σζr2 was calculated concurrent with different values of vver and for ∣θw–θl∣ > 10°, using the procedure described in section 2 (with A > Ath). Figure 11 shows the result as a function of vver. Also included is the result of the theoretical expression, with hr = 2550 m and vw,eff = 13.6 m/s. Figure 11 shows that the model agrees well with the measurements, both on average and in their variation with vver.

Figure 11.

Variance of relative fade slope as a function of vver as measured in Sparsholt (dotted line with asterisks) and the theoretical expression, where hr = 2550 m and vw,eff = 13.6 m/s (solid line).

[68] Next, the theoretical σζr2 is verified as dependent on hr. It was checked that there is not much correlation between this parameter and vw,eff: in the most significant region (more or less between 1700 and 3300 m), the average vw,eff is fairly constant with respect to hr, around the long-term average of 13.6 m/s.

[69] The measured long-term σζr2 was calculated concurrent with different values of hr and vver and for ∣θw–θl∣ > 10°. Figure 12 shows the result as a function of hr for different values of vver, and compared to the results of the theoretical expression, with vw,eff = 13.6 m/s. Figure 12 shows that the resulting values of the model are on average close to the measured ones, although the expected decrease with hr is slightly recognized in only two of the measured curves.

Figure 12.

Variance of relative fade slope as a function of rain height for three ranges of vver, as measured in Sparsholt (dotted lines), and the theoretical expression with vw,eff = 13.6 m/s (solid lines).

[70] However, it should be noted that the fact that the theoretical values are close to the values observed is already an indication that the dependence of the theoretical model on rain height is in agreement with the measurements. As was seen in Figure 1 and 2, the variance of relative rain rate slope is much larger than that of relative fade slope. If the theoretical model would be based on attenuation calculated from point rain rate, its result for the relative fade slope would be similar to that of rain rate, and hence much larger than the measured results. This can be seen by applying the model using a zero rain height and therefore zero path length. What happens to the result of this can be seen in Figure 12 by the strongly increasing curves for decreasing rain height. It is the integrating effect along the path length which reduces the variance of relative fade slope to the values observed. The results in this paper therefore show that the incorporation of the rain height in the model is well in agreement with measurements.

[71] Figure 12 also shows that the fact that during convective rain the calculated rain height might be too small (see above), may have some effect on the result. The bottom two measured curves are on average slightly to the left of the theoretical ones, so the rain height assigned to the measured results could be slightly too small.

[72] Finally, the model is verified as dependent on effective wind speed vw,eff. The measured long-term σζr2 was calculated concurrent with different values of vw,eff and vver and for ∣θw–θl∣ > 10°. Figure 13 shows the result as a function of vw,eff for different values of vver, and compared to the results of the theoretical expression, with hr = 2550 m. Figure 13 shows again that the average values of the theoretical model are in agreement with the measured values, for the different values of vver, and also, the theoretical values increase with effective wind speed, as do the measured values. However, the theoretical increase with vw,eff is stronger than measured.

Figure 13.

Variance of relative fade slope as a function of effective wind speed for three ranges of vver, as measured in Sparsholt (dotted lines), and the theoretical expression with hr = 2550 m (solid lines).

[73] Especially for small wind speeds, it appears that the measured σζr2 is larger than predicted by the model. This is possibly due to the fact that the frozen storm hypothesis is not entirely correct: for small wind speeds, the variation of rain attenuation cannot entirely be ascribed to the wind moving a stable rainstorm across the propagation path.

[74] Furthermore, it was found that the use of a long-term average vw,eff of 13.6 m/s (as in Figures 11 and 12) gave a good agreement with measurements. This suggests that the cause of the discrepancy is that the instantaneous wind speed as obtained by interpolation from ECMWF data, with a 2.5° spatial resolution and a 6-hour time resolution, is not a good representative of the true wind speed on the propagation path during the measured rain event. Given the generally strong variability of wind, it seems likely that meteorological data with a higher resolution would improve the instantaneous correlation between the variance of relative fade slope and wind speed.

5. Conclusions

[75] A theoretical model has been developed to predict the variance of relative fade slope from rain rate measurements. This model uses the property that the attenuation is a result of the integrated effect of rain along the propagation path. This property strongly reduces the variance of relative fade slope, and is therefore important to be taken into account. The model assumes the frozen storm hypothesis, supposing that all measured dynamic behavior of rain rate is caused by the wind, moving a stable rainstorm across the measurement point. The model uses as inputs two parameters of the autocorrelation of rain rate and several meteorological parameters only.

[76] The theoretical model shows a good agreement with measurements in a comparison as a function of the vertical velocity and of the rain height. As a function of wind speed, the model predicts an increase of the variance of relative fade slope which is stronger than the one observed. This may be partly due to the frozen storm hypothesis, and partly due to the spatial and temporal resolution of the meteorological data. Using a long-term average wind speed, the model is well in agreement with the measurements.

[77] The performance of the model may be improved using meteorological input data with a higher resolution, or a different estimate of the rain height of convective rain. Possibly, the model itself can still be improved by including an offset in the relation between σR2 and mR2 (Figure 5), or in that between B and vw (Figure 6). However, these alterations are not straightforward, because they imply modifications in two basic assumptions of the model: the Maseng-Bakken model setup and the frozen storm hypothesis, respectively.

[78] The model has been verified with rain attenuation data above a threshold Ath, to reduce the effect of scintillation and noise. However, it can also be applied to predict attenuations below Ath, assuming the fade slope is proportional to attenuation similarly as it is for larger attenuation values.

[79] This model allows to predict fade slope statistics from rain gauge data, without the use of rain attenuation data. This can prove very useful for obtaining necessary information for the design of FMT systems at sites where satellite measurements have not been performed, but rain rate measurements, which are much cheaper and easier to do, have.

Appendix A:: Variance of Relative Fade Slope Related to Rain Rate

[80] Assume the configuration as depicted in Figure 3. According to equation (9), the attenuation A(tm) can be written as an integral along the path:

equation image

where

hr

rain height (km);

ɛ

elevation angle (°);

R

rain rate (mm/hr);

a, b

coefficients ((dB/km) and ()) of the ITU-R specific rain attenuation model;

l

coordinate along the path (km);

tm

time of measurement (s).

[81] Because the rain is assumed uniform in vertical direction, the rain rate at any location l on the path is the same as on its horizontal projection, given by x = xs + l cos ɛ. So change the integration variable to x:

equation image

[82] In Figure 4, the situation is depicted seen from above. The integral of equation (A2) is performed along the line S1-P1. The orientation of this line is defined by the azimuth angle θl of the link, the wind velocity vector is vw, its azimuth angle is θw and the magnitude of its speed is vw. As explained in section 3.1, under the frozen storm hypothesis, the rain intensity at time tm + Δt is equal to the rain intensity that was at time tm present on the line S2-P2 in Figure 4. Assuming that a and b do not vary significantly within Δt, the attenuations at times tm and tmt can then be written as

equation image
equation image

[83] The relative fade slope is defined in equation (2). Using the property that the average of relative fade slope is zero [van de Kamp, 2003a], its variance can be found as follows:

equation image

Because this becomes very hard to calculate, the following approximation will be made.

[84] It was concluded in section 1 that the conditional variance of the relative fade slope is, approximately, independent of A1 and therefore equal to the unconditional variance:

equation image

However, within the distribution conditional to A1, A12 is constant, so

equation image

As a seemingly pointless step, A12 is equal to the average of A12 conditional to A1:

equation image

Since, as stated before, the result of this is independent of A1, it will be equal to the unconditional expression

equation image

Summarizing equations (A6) to (A9) gives

equation image

Therefore equation (A5) can be approximated by

equation image

where

equation imageA*(Δt)

the autocorrelation of attenuation for a time lag Δt;

equation imageAt) = equation imageA*(Δt)/equation imageA*(0)

the autocorrelation coefficient of attenuation for a time lag Δt.

Appendix B:: Autocorrelation of Attenuation

[85] Using equations (A3) and (A4) in Appendix A, the autocorrelation of attenuation for a time lag Δt can be found as follows:

equation image

Assume, for the moment, that b = 1:

equation image

Here, E[R(x1, t)R(x2, t)] is the long-term time average of the product of two values of the rain intensity, at the locations X1 and X2 and at the same point in time. Using the frozen storm hypothesis, this long-term time average will be equal to the wide-range spatial average of the same function: Es[R(x, t)R(x + d12, t)] (Es denoting a spatial average), with d12 being the distance between X1 and X2, in the direction as indicated in Figure 4. By definition, this average is equal to the spatial autocorrelation of rain rate for a distance d12:

equation image

[86] Assuming the rainstorms are isotropic, this autocorrelation will be independent of direction, and the two values of R in equation (B3) can be displaced by the distance d12 in any direction. Equation (B2) can therefore be written as

equation image

[87] Both integrations in equation (B4) are performed for x1 and x2 from 0 to the horizontal projection of the path length, which is hr/tanequation image (see equation (A2))

equation image

with the distance d12 given in equation (13). The autocorrelation coefficient equation imageA of attenuation is found by normalizing equation (B5) by the autocorrelation for Δt = 0:

equation image

[88] This can be written in terms of the autocorrelation coefficient equation imageR of rain rate, defined as

equation image

With this, equation (B6) becomes (using the fact that equation imageR*(0) is independent of x1 and x2)

equation image

[89] If b ≠ 1 (but does not vary much over a timescale of Δt), in equation (B8)equation imageR is replaced by the spatial autocorrelation coefficient equation imageequation image of Rb:

equation image

with equation imageequation image defined as

equation image

Acknowledgments

[90] This study was part of an external Research Fellowship for the European Space Agency (ESA), performed at the University of Bath, United Kingdom. I would like to thank ESA for granting this fellowship and the University of Bath for hosting me. I would like to thank Rutherford Appleton Laboratory (RAL), Chilton, United Kingdom, and the Italian Space Agency for providing the Italsat data measured in Sparsholt, RAL also for the rain rate data from Sparsholt and Chilbolton, and the European Centre for Medium-Range Weather Forecasts (ECMWF) for the global meteorological data.

Ancillary