By Glen James,2014-10-29 18:59
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    CP 38

    6 March 2001

    Malcolm Hamer


1. Summary

    At the last meeting of WP3J in Geneva in 2000, document 3J/74 (1998-2000 series) from China presented a new method for calculating the specific attenuation due to rain by expressing the k and (

    coefficients as mathematical formulae. Unfortunately the proposal was not adopted because the differences between the specific attenuation calculated from the existing table of values and the new formulae exceeded 12% in places. This paper examines the reasons for these large differences. An estimate of the accuracy with which the k and ( coefficients have to be determined to achieve a

    desired accuracy for the specific attenuation is made. It is shown that residual noise existing in the tabulated values of the k and ( coefficients, together with the partial correlation between them

    prevents a formula based approach from achieving a greater accuracy than approximately 4%. Given that the goal is to achieve accurate values of the specific attenuation, a new approach to modelling the k and ( coefficients is then presented. Finally, it is shown that the current interpolation procedures for untabulated values of the k and ( coefficients can lead to large errors.

2. Introduction

    The current method for calculating the specific attenuation due to rain is documented in ITU-R P.838. In essence a set of tabulated k and ( coefficients, as a function of frequency, is provided for both

    horizontal and vertical polarisations. The specific attenuation for a given rain rate, R, is then

    calculated by the use of the coefficients in a simple formula. Guidance on an interpolation procedure for the specific attenuation at untabulated frequency values is given.

    It is inconvenient to calculate the specific attenuation from tables since some form of interpolation is always required and this is often subject to an interpretation error. A formula based approach would lend itself to an easier software implementation, but for this to be acceptable, the discrepancy between the two methods must be very small.

Document 3J/74 (1998-2000 series) proposes two part equations for the k and ( coefficients for both

    the horizontal and vertical polarisation. A number of graphs are presented that show the absolute and relative errors of the individual k and ( coefficients as a function of frequency compared to the

    tabulated values. In summary, the relative errors of the ( coefficients are less than 3% whereas those

    for the k coefficients are as high as 12.5%. The overall error of the specific attenuation is minimised because there is some degree of inverse correlation between the corresponding k and ( coefficients.

    However, even allowing for this, the best overall errors obtained are in the range from 8 to 11%. This was not deemed acceptable at the last WP3J meeting and the work was submitted into the chairman’s report, 3J/1, until such time that improvements were forthcoming.

    This document examines the equation for specific attenuation and applies the theory behind the uncertainty in a function of a number of variables, where estimates for the errors of those variables are 29/10/10 Page 1 of 17 C:\convert1\temp\92528697.doc

    ( coefficients are then made. Finally a known. Predictions for the necessary accuracy of the k and

    method of optimising the mathematical fit to modified k and ( coefficients is provided in order to achieve increased accuracy in the final resulting specific attenuation at the expense of a decreased

    accuracy in either of the individual k and ( values. A linear sum of Gaussian distribution curves is shown to be an excellent equation to model both the k and ( coefficients. The optimised model coefficients for all four k and ( coefficients is provided.

    3. The Uncertainty in the Specific Attenuation Equation

    Standard statistical texts show that, provided the variables, u, are independent or uncorrelated, the uncertainty in some function:

     vf(u,u,;,u) (1) 12r

    is given by:

    2r?f~2( (2) ??vuj(u~1jj)?


     standard deviation of v. v

     standard deviation of u. ujj

Whilst this formula can be applied directly to the equation that relates specific attenuation,;;, to rain

    rate, R:

    (k?R (3)

    it is best to work with the logarithm of the k coefficient. Since k varies by over 6 decades in value across the range of frequency, the only sensible way to achieve an accurate fit over the whole range of

    k is to work in log space. This leads to an uncertainty on the value of log(k) and not k itself. By

    rewriting equation (3) as follows:

    (10?R (4)


     k10 (5)

the uncertainty in log(k) can be entered directly into the formula as .

The uncertainty in the value of specific attenuation, , can now be written as:

    22(;?;?;?10?R?ln10?ln(R)? (6) (

    Furthermore, the relative error with respect to the value itself in percentage terms is given by:

    22;?;?;?100?ln10?ln(R)? % (7) acc(

It is helpful to plot this equation as a function of and , for 4 different values of rain rate, R. The (

    values of R chosen span the extremes of values that are likely to be used.

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    R = 1 mm/hrR = 5 mm/hr2020




    Relative Error in Gamma %Relative Error in Gamma %


    Std Devn of AlphaStd Devn of Alpha

    R = 50 mm/hrR = 200 mm/hr2020




    Relative Error in Gamma %Relative Error in Gamma %


    Std Devn of AlphaStd Devn of Alpha

     Figure 1 Relative error in the value of specific attenuation for different absolute uncertainties in the values of ( and . The six lines on each plot correspond to uncertainties in of 0, 0.01, 0.02, 0.03, 0.04 and 0.05 reading upwards respectively.

    The subjective behaviour of these graphs is easily understood. For an R=1 mm/hr there is no variation

    ( since one raised to the power of any value is still one. It is thus only the uncertainty in with

     = log(k) that matters. However, at high rain rates, say R = 200 mm/hr, the uncertainty in the (

    value soon dominates the overall uncertainty in ;!;

It is instructive to enter the uncertainties quoted in document 3J/74 into equation (7). Figure 2 shows

    the absolute uncertainties of the k and ( coefficients as functions of log frequency. It is seen that worst case figures for and , are 0.05 and 0.015 respectively. Comparing these values to the (

    graphs in Figure 1 gives an approximate uncertainty in of 12% at R=1mm/hr rising to 14% at 200 mm/hr. These figures are somewhat higher than seen in practice since figure 3 shows that the residual

    errors in figure 2 are partially inversely correlated with a slope of 2.49 and this will lead to some lowering in the relative errors but, unfortunately, not enough.

It is now very apparent that in order to keep the relative errors of the specific attenuation within 5%,

    the absolute errors in the model fits to the k and ( terms must be extremely small. Ideally, the worst case figures for and , are required to be lower than 0.015 and 0.005 respectively. This constraint (

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    will in all probability mean that more model coefficients will be required than are currently being

    proposed within document 3J/74.







    0.01Absolute Error in EtaAbsolute error in Alpha0.04



    Figure 2 Absolute differences between the model and tabulated values of and ( as a function of

    log(frequency) taken from document 3J/74. These are for the horizontal polarisation.

    Everything that has been discussed up to

    now has been aimed at ensuring that the Correlation between Alpha & Eta0.06data points are fitted well by the model

    equations. This could lead to an

    additional subtle constraint that now 0.04

    needs to be examined. Since the

    ( tabulated values for the k and 0.02coefficients are relatively sparse in some .regions, the behaviour of any model 0equation away from the data points is

    liable to become oscillatory as the 0.02number of degrees of freedom within the Absolute Error in Etamodel increases. Signs for this 0.04behaviour need to be looked for. The

    next section shows that oscillatory

    behaviour is a real problem at the high end of the tabulated values. It Absolute Error in Alphais for this reason that the novel approach

    of modifying the high frequency (

    coefficients before the model fit is made Figure 3 Correlation between the residual model is proposed. This has the effect of differences from the tabulated data of the damping out oscillations in the specific and ( values in document 3J/74. The attenuation values at the expense of ( correlation line has a slope of 2.49. The coefficients that diverge from the scatter of the points indicates that there tabulated values. will not be much reduction in the relative error of as calculated from equation (7) assuming no correlation.

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    ( coefficients 4. Problems in modelling the k and

    Figures 4a) and b) show an example of the initial fits made to the k coefficient. These show that, although plotted with a log y scale the fit at high frequencies appears good, in reality there is

    considerable noise relative to the precision required.

     10Figure 5 shows that the (

    values at the same

    frequency region have 1been adjusted in the

    1.2tabulations to smooth out

    the fluctuations that would 0.1ensue in the resulting

    values. This is not

    unreasonable since the 10.01coefficients derive from k coefficientfitting to experimental data alpha coefficient3and would have not have 101been determined 0.8independently of one 4another. The most 101important consideration

    was that the final value of 0.653specific attenuation was 1011101001103110100110the intended value and Frequency (GHz)Frequency (GHz)these only had to be correct

    at the points chosen for the

    tabulation. (Figure 6). The

    eye assumes that straight-

    line segments then join the

    points. As figure 6 shows

    the model equations do not

    1exhibit such good

    behaviour with the

    oscillatory nature of final

    curve clearly evident

    particularly at high values k coefficientof rain rate values, R.


    It is thus clear that fitting a

    model equation to the

    individual k and (

    coefficients to an

    ever-higher precision in

    0isolation is not the whole 3110100110answer. A consideration Frequency (GHz)must be given to the

    behaviour in regions (log(k)) Figure 4a) & b) Initial model fits made to the between the data points coefficient. Although not easily seen on a also. The key to doing this log y- axis, the non-systematic behaviour of the is not to treat the k and ( top few data points is clearly evident on a linear coefficients in isolation but scale. These graphs are for the horizontal to deliberately correlate polarisation but apply equally to the vertical their behaviour such that a case.

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fitting error introduced into one can be compensated for in the other. Fortunately, the desired value of

    specific attenuation at each of the data points can be calculated and used to enforce such a correlation

    between them at least in the high frequency region of the tabulations. The next section illustrates one

    way that this can be achieved.







    k coefficientalpha coefficient3101



    0.6531011101001103110100110Frequency (GHz)Frequency (GHz)

    ( coefficient. The compensatory behaviour of the data Figure 5 Initial model fits made to the

    points to those of k above 100 GHz is evident. The graph illustrated is for the horizontal 60polarisation.





    Specific Attenuation (dB/km)



    Frequency (GHz)

    Figure 6 Model fit (solid line) calculated from the k & ( coefficient fits in figures 4 & 5 compared

    to data points calculated from tabulated k & ( coefficients. The graph illustrated is for

    the horizontal polarisation and rain rates of 1, 2, 5, 10, 20, 50, 100 and 200mm/hr.

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    5. Proposed modification to the Alpha coefficients prior to model fitting

     coefficient (log(k)) illustrated in figure 4 is within the Since the absolute error in the model fit to the

    tolerance required, as determined in section 3, a decision to use this curve as the starting point for the

    fit to the specific attenuation was made. The equality shown in parameterised form in equation 8 was

    then used to recalculate the values of the ( coefficients for the top 11 data points where the oscillatory behaviour is most pronounced. Although this calculation can be done for any value of rain rate, R,

    200mm/hr was chosen, since the higher the R, the worse the oscillations. In practice it was found that the new set of modified ( coefficients produced acceptable final results of specific attenuation for all other rain rate values.

    ;?;?R,k,(?R,k,( (8) datadatafitnew


     k & ( are the actual tabulated data points for the coefficients datadata

     k is the value of the model fit to the k coefficient evaluated at a data point fit

     ( is the modified ( coefficient that corresponds to the fitted k coefficient. new




    0.75Alpha coefficient




    Figure 7. The modified ( coefficient data points (circles) compared to the original tabulated (

    coefficient points (squares) for the top part of the frequency range. The data points

    remain unchanged over the remaining region of the frequency range. The graph

    illustrated is for the horizontal polarisation.

    Figure 7 shows the way that this process alters the ( coefficients at the data points. It is these modified ( coefficients that are now fitted by the ( model equations. This model fit for (, together

    with the original k fit are now used as the pair of equations to calculate the specific attenuation, .

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    ( coefficients 6. The model equations for the k and modified

    Although superficially very different, both the k and modified ( coefficients are best modelled by a linear sum of Gaussian distributions on a straight-line background. Experimentation has shown that

    three Gaussian distributions are optimal for the k coefficient whilst four are required for the modified ( coefficient. Equations 9 and 10 represent the final forms chosen. The various coefficients for both

    horizontal and vertical polarisations are given in Tables 1 and 2.

    2???3?;?logfb(j(?;?ka?expm?logfc (9) ?jkk((c(?j1j)???)?


    2?4???;?logfb(i(;?(a?expm?logfc (10) ??i((mod((ci1?i)???)?


     k model for the original k and k coefficients HV

     ( model for the modified ;( and ( coefficients modHV

TABLE 1 Model Coefficients for the HORIZONTAL Polarisation

     a b c m c m c kk((

    0.3364 1.1274 0.2916 1.9925 -4.4123 - - j = 1

    0.7520 1.6644 0.5175 - - - - 2

    -0.9466 2.8496 0.4315 - - - - 3

    0.5564 0.7741 0.4011 - - -0.08016 0.8993 i = 1

    0.2237 1.4023 0.3475 - - - - 2

    -0.1961 0.5769 0.2372 - - - - 3

    -0.02219 2.2959 0.2801 - - - - 4

TABLE 2 Model Coefficients for the VERTICAL Polarisation

     a b c m c m c kk((

    0.3023 1.1402 0.2826 1.9710 -4.4535 - - j = 1

    0.7790 1.6723 0.5694 - - - - 2

    -1.0022 2.9400 0.4823 - - - - 3

    0.5463 0.8017 0.3657 - - -0.07059 0.8756 i = 1

    0.2158 1.4080 0.3636 - - - - 2

    -0.1693 0.6353 0.2155 - - - - 3

    -0.01895 2.3105 0.2938 - - - - 4

Figure 8 shows the absolute differences between the (log(k)) and modified ( data points and the

    model equations evaluated at these values as a function of log(frequency). The maximum extent of

    these uncertainties is 0.015 for and 0.0015 for the modified (. Figure 9 shows very little correlation between the ,; and modified (; uncertainties remains. The expected relative error in the specific attenuation will be largely dominated by the value. Plotting these uncertainties on figure 1, shows that the relative error in specific attenuation will be almost completely independent of the rain


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    Absolute Error in Eta0.020.002

    Absolute Error in Modified Alpha0.030.003012012


    Figure 8. Absolute differences between the model and tabulated values of and modified ( as a

    function of log(frequency) using equations 9 and 10. These are for the horizontal

    polarisation. Similar results are obtained for the vertical case.




    Absolute Error in Eta



    Absolute Error in Modified Alpha

    Figure 9 Correlation between the and modified ( values and the model equations 9 and 10. Very

    little correlation exists and the uncertainties can be combined using equation 7 to predict the

    relative error on the specific attenuation

    ( coefficients. The original ( Figures 10a) and b) show the model fits to the k and modified

    coefficients are plotted also for comparison purposes. The differences between the original and modified ( coefficients are only visible at the high frequency end and the small differences are indicative of the precision that needs to be exercised to achieve the desired results. The individual component Gaussian distributions for each coefficient are shown as dotted lines. The figures are for the horizontal polarisation but similar results are obtained for the vertical polarisation.

7. Comparison of specific attenuation taken from the tables and model

    Figures 11 and 12 illustrate the overall fits between the model and the data points calculated from tabulated values. Both logarithmic and linear scales are shown for a range of rain rates from 1 to 200mm/hr since the results span six decades. The agreement is excellent at both a low level and high level of specific attenuation with all signs of the earlier oscillation being damped out. 29/10/10 Page 9 of 17 C:\convert1\temp\92528697.doc





    k model fit




    Frequency (GHz)





    0.9Modified alpha model fit




    Frequency (GHz)

    Figures 10a) & b). Final model fits for the k and modified ( coefficients as a function of frequency

    using equations 9 and 10. The squares are the k and modified ( points whilst the

    circles in 10b) are the original ( points. The individual Gaussian distributions are

     shown as dotted lines. These results are for the horizontal polarisation.

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