Comparison of the use of dual-frequency and single-frequency attenuation for the measurement of path-averaged rainfall along a microwave link A.R. Holt, G.G. Kuznetsov and A.R. Rahimi Abstract: It is shown that, for certain combinations of frequency and polarisation, the difference in microwave attenuation between two frequencies along a terrestrial link can give a much better estimate of path-averaged rain rate than can be obtained from a single frequency. An example is given which shows that attenuation difference can successfully provide rainfall estimates in practical applications. 1 Introduction The use of microwave attenuation to estimate rainfall, and vice versa, has been the subject of many studies [I-31. In the course of a study on data from the Olympus satellite, it was found [4] from both theoretical studies and measured data that the difference in attenuation between 12.5GHz and 19.77GHz, both vertically polarised, seemed to provide a good estimate of path-averaged rainfall. The measured data related to a slant path of approximately 30" elevation, with a path through the rain region of no more than 3km, together with a path through the melting layer. Actual rainfall on path was obtained from a raingauge at the receiver and from Chilbolton radar data. Whereas the specific attenuation-rainrate relationship is well represented by the power law form A = aRb, it was seen that the attenuation difference between the two frequencies seemed to be, to a very good approximation, a linear function of rainrate. This means that, although the actual rainrate may vary along the path, the principle of superposition shows that a true path-averaged rainrate can be obtained. This does not happen with a power law relationship. The study [4] was necessarily limited in both its theoretical investigations, and in its path length. However, it provided the stimulus for a more detailed study of the use of dual-wavelength attenuation on terrestrial paths [5]. The purpose was to obtain a method of measuring rainfall, which could be more easily implemented in an urban setting than either raingauges or radar. This paper reports theoretical calculations on attenuation difference as a function of rainrate, and it is shown that in principle it can be applied at a number of frequency combinations, though the real measurements will also be subject to other effects. We report experimental results from one event for a particular pair of frequencies, and note some of the problems to be solved in a practical implementation. 0 IEE, 2W3 IEE Pmceedhys online no. 20030616 doi:lO. IM9/ip-map:2W30616 Paper first received 2nd August 2W2 and in revised form 27th March 2003. Ohe publishing date: 5 Auyst ZW3 The authors are with the Department of Mathematics, University oi Essen, Wivenhoe Park, Colchester CC4 3SQ, UK IEE Proe-Microw Anrm Propog. &?I. 150, No. S, October 203 2 Theory If a linearly polarised electromagnetic wave with wavelength i. (cm) and wavenumber k,(= 2n/A) an-', is scattered by a raindrop of equivolume diameter D (an), then the copolar scattering amplitude is fw(D) or fHH(D), dependent on whether the incident wave was vertically (V) or horizontally (H) polarised. If the wave passes through a precipitation region where the dropsize distribution is N(D)dD( crK4) for a distance of 1 km, then the specific attenuation A(dB/km) experienced by the wave is given by (61 AY,H = (8.686)10sImKv,~ (1) where One difficulty encountered in the remote sensing of rain is that we do not have precise information on the temperature or shape of raindrops, and very little information on the dropsize distribution. Indeed we should expect these quantities to vaty from one region of space to another. Consequently, it is important to ascertain whether a proposed technique is sensitive to these unknown para- meters. In this study, we consider various values of the temperature, three models of the dropshape, and a number of different dropsize distributions. For temperature, we initially considered 0, IO, 20"C, and subsequently, for some cases, we also considered 5, IO, 15°C. For drop shape, assumed to be oblate spheroidal, with three semi-diameters of lengths a, a and h (an) and equivolume diameter D = 2(0'b)''~, we have used the following: (i) axial ratios derived by Goddard and Cherry [7] for D<O.3an, and shapes given by Pmppacher and Pitter [XI for Dz0.3cm (we denote these as RAL shapes); (ii) axial ratios given by Beard and Chuang [9]; (iii) axial ratios as assumed by Momson and Cross [IO]. In all calculations we have assumed 28 different drop diameters between 0.025 and 0.7cm. The axial ratios (i) and (ii) are believed to be good approximations for drop shapes when the drops are relatively close to the ground, whereas (i) is believed to be somewhat extreme, and is included to 315
Transcript
Comparison of the use of dual-frequency and single-frequency attenuation for the measurement of path-averaged rainfall along a microwave link
A.R. Holt, G.G. Kuznetsov and A.R. Rahimi
Abstract: It is shown that, for certain combinations of frequency and polarisation, the difference in microwave attenuation between two frequencies along a terrestrial link can give a much better estimate of path-averaged rain rate than can be obtained from a single frequency. An example is given which shows that attenuation difference can successfully provide rainfall estimates in practical applications.
1 Introduction
The use of microwave attenuation to estimate rainfall, and vice versa, has been the subject of many studies [I-31. In the course of a study on data from the Olympus satellite, it was found [4] from both theoretical studies and measured data that the difference in attenuation between 12.5GHz and 19.77GHz, both vertically polarised, seemed to provide a good estimate of path-averaged rainfall. The measured data related to a slant path of approximately 30" elevation, with a path through the rain region of no more than 3km, together with a path through the melting layer. Actual rainfall on path was obtained from a raingauge at the receiver and from Chilbolton radar data. Whereas the specific attenuation-rainrate relationship is well represented by the power law form A = aRb, it was seen that the attenuation difference between the two frequencies seemed to be, to a very good approximation, a linear function of rainrate. This means that, although the actual rainrate may vary along the path, the principle of superposition shows that a true path-averaged rainrate can be obtained. This does not happen with a power law relationship.
The study [4] was necessarily limited in both its theoretical investigations, and in its path length. However, it provided the stimulus for a more detailed study of the use of dual-wavelength attenuation on terrestrial paths [5]. The purpose was to obtain a method of measuring rainfall, which could be more easily implemented in an urban setting than either raingauges or radar. This paper reports theoretical calculations on attenuation difference as a function of rainrate, and it is shown that in principle it can be applied at a number of frequency combinations, though the real measurements will also be subject to other effects. We report experimental results from one event for a particular pair of frequencies, and note some of the problems to be solved in a practical implementation.
0 IEE, 2W3 IEE Pmceedhys online no. 20030616 doi:lO. IM9/ip-map:2W30616 Paper first received 2nd August 2W2 and in revised form 27th March 2003. O h e publishing date: 5 Auyst ZW3 The authors are with the Department of Mathematics, University oi Essen, Wivenhoe Park, Colchester CC4 3SQ, UK
IEE Proe-Microw A n r m Propog. &?I. 150, No. S, October 2 0 3
2 Theory
If a linearly polarised electromagnetic wave with wavelength i. (cm) and wavenumber k,(= 2n/A) an-', is scattered by a raindrop of equivolume diameter D (an), then the copolar scattering amplitude is f w ( D ) or fHH(D), dependent on whether the incident wave was vertically ( V ) or horizontally (H) polarised. If the wave passes through a precipitation region where the dropsize distribution is N(D)dD( crK4) for a distance of 1 km, then the specific attenuation A(dB/km) experienced by the wave is given by (61
AY,H = (8.686)10sImKv,~ ( 1 )
where
One difficulty encountered in the remote sensing of rain is that we do not have precise information on the temperature or shape of raindrops, and very little information on the dropsize distribution. Indeed we should expect these quantities to vaty from one region of space to another. Consequently, it is important to ascertain whether a proposed technique is sensitive to these unknown para- meters. In this study, we consider various values of the temperature, three models of the dropshape, and a number of different dropsize distributions.
For temperature, we initially considered 0, IO, 20"C, and subsequently, for some cases, we also considered 5, IO, 15°C. For drop shape, assumed to be oblate spheroidal, with three semi-diameters of lengths a, a and h (an) and equivolume diameter D = 2(0'b)''~, we have used the following:
(i) axial ratios derived by Goddard and Cherry [7] for D<O.3an, and shapes given by Pmppacher and Pitter [XI for Dz0.3cm (we denote these as RAL shapes); (ii) axial ratios given by Beard and Chuang [9]; (iii) axial ratios as assumed by Momson and Cross [IO].
In all calculations we have assumed 28 different drop diameters between 0.025 and 0.7cm. The axial ratios (i) and (ii) are believed to be good approximations for drop shapes when the drops are relatively close to the ground, whereas ( i ) is believed to be somewhat extreme, and is included to
315
Table 1: Semidiameters and axial ratios for the drop shapes considered
Equivolume Drop shape model diameter (cml
RA L a b
0.05
0.1
0.2
0.3 0.4
0.5
0.6
0.7
0.025
0.050
0.102
0.158 0.219
0.281
0.345 0.410
0.025
0.'050
0.096
0.135
0.167
0.197
0.226
0.255
bla
1.000
1.000 - 0.943
0.851
0.762
0.701
0.655
0.621
Beard and Chuang a b
0.025 0.025
0.050 0.050
0.102 0.096
0.158 0.134
0.218 0.168
0.280 0,199
0.348 0.223
0.419 0.244
bla
1.000
1.000
0.940
0.849 0.787
0.708
0.642
0.581
Morrison and Cross a b
0.0252 0.0246
0.051 0.048 0.104 0.093
0.158 0.135
0.215 0.172
0.275 0.206
0.338 0.236
0.404 0.263
bla
0.975
0.950
0.900
0.850
0.800
0.750
0.700
0.650
help with an investigation of the sensitivity of the results to drop shape. These axial ratios are summarised in Table 1, For dropsize distributions we have assumed a Gamma distribution model of the fonn
N ( D ) = N,D"exp[-(3.67 + p)D/D,] ( 3 ) where seven different values of p have heen taken: p = - 1, 0, 1, 2, 3, 4, 5. We have allowed for two different Gamma distribution models-the original description by Ulbrich [ I I ] and the recent normalisation given by Testud ef al. [12]. Again, in considering a wide range of dropsize distributions, we are seeking to obtain extreme values. This enables us to see whether values of attenuation difference are sensitive to the various unknown parameters of the rain being sensed.
We have generally restricted our attention to frequency bands available for terrestrial propagation within the United Kingdom. The frequencies included in the study are 7.7, 10.5, 12.8, 13.9, 17.6, 19.0, 22.9, 24.1, 28.0, and 38.0GH.z.
In order to give information over a wide range of frequencies, while restricting the information presented to that which can be readily absorbed, we first present full details for one pair of frequencies, and then summarise the results for other pairs of frequencies.
3.1 Frequency pair 12.8 GHz and 17.6 GHz Initial calculations were carried out at On, lo", 20°C to give a wide spread of temperatures, covering the whole range of drop temperatures Likely to be encountered in practice in the UK. Later, it was realised that below about YC, it was likely that sleet could be encountered, and that the attenuation in sleet could be very different to that in rain. Hence we performed the calculations again at 5". lo", 15°C and thus have information on the sensitivity to temperature of the derived attenuation difference-rain-rate relationship.
An.Diff.(dBikm) 17.6V 12.8H (GHz) Fits: 0.010 + 0.0266R or 0.0270R
OL' 0 i n 20 30 40 0 10 20 30 40
R mmlh R. mm/h
A~.Oiff.(dBikm) 17.6H 12.8V (GHr) Fits -0.046 + 0.0445 or 0.0427R
An.oiff.(dBikm) 17.6H 12.8H (GHr) Fits: 4.027 + 0.0385R or 0.0375R
0 10 20 30 40 0 10 20 30 40 R, m m R, mmlh
Fig. 1 Spec@ attenuation dflermce ( d B l h ) hetween I7.6GHz and 12.8GHz. as a function of rain-rate and polarisution combination Each figure consists of 63 curves as described in the text, the gamma distributions being those due to Ulbrich. The fits are least squares linear fits either unconstrained, or constrained to pass through the origin
316 IEE Proc.-M;crow A n t e m Propay. VoL ISO, No. 5, 0crobpr2003
Figure 1 plots the attenuation difference as a function of rain-rate for the Ulbrich form of the Gamma dropsize distribution for the four possible combinations of vertical and horizontal polarisation, and o", IO", 20°C. Each diagram consists of 63 curves (three temperatures, three shapes and seven dropsize distributions). We have obtained the least squares lincar fit to these 63 curves for two cases: (i) where there is no constraint on the linear fit, and (ii) where the linear fit is constrained to pass through the origin.
It will be seen that the results are very dependent on which polarisation pair is used. The top right-hand graph, for 17.6GHz vertical and 12.8 GHz horizontal polarisation_ is clearly the worst and the attenuation difference for this combination cannot be a good predictor of rain-rate. The best combinations are clearly either both polarisations horizontal (bottom right graph), or both vertical (top left). At both frequencies, horizontal polarisation shows the least sensitivity to the various parameters, but the unconstrained linear fit has an intercept on the attenuationdifference axis, which is comparable to the rain-rate. The unconstrained V- V linear fit almost passes through the origin, but there is a greater sensitivity among the individual curves to the various parameters. Since the majority of rainfall events in the UK are likely to be dominated by low rainfall rates, for both po1aris;rtions vertical polarisation appears to be preferable.
Figure 2 shows the same data assuming the Testud form of the Gamma distribution. There is less sensitivity to the unknown parameters hut for H-H polarisation the axis intercept is gtedter. In general, the gradients of the least squares linear fit vary by ahout 2% between Figs. 1 and 2. For the Ulbrich distribution, an attenuation difference of 0.75 dB/km corresponds to a rain-rate range of 2&27 mm/h with the least squares fit estimating 23.5mm/h The maximum error across the 63 cases considered is therefore 15%. For the Testud distribution, the range is IP-26mm/h with a least squares estimate of 23.2mm/h. For 5", lo", IYC, the corresponding curves for the Ulbrich distribution give a rain-rate range of 21-27mm/h with a least squares
An.Ditf.(dBikm) 17.6V 12.8V (GHr) Fits: -0.009 + 0.0331R or 0.0328R
estimate of 23.5mm/h, and for the Testud distribution a range of 2@26mm/h with a fit of 23.0mm/h. The least squares estimates are insensitive to which fit (constrained or unconstrained) is used.
Overall, given that we have deliberately chosen extreme cases of both drop shape and dropsize distribution, it is reasonable to conclude tkat using 17.6GHz and 12.8GHz, both vertically polarised, we can measure path-averaged rainfall rate to about 10%. This is in spite of the fact that we have no information about the dropshapes, dropsize distribution or temperature on path, save for the assump- tion that the precipitation is all water (no melting particles).
For the single-frequency specific attenuations, we con- sider a fit to the individual curves of the form A = aR '. For temperatures of S", IO", 15"C, and a rain-rate of 20mm/h, the Ulbrich distribution fit predicts an attenuation of 0.66dB/km at 12.8GHz and I.3OdB/km at 17.6GHz. The Testud distribution predicts 0.62 dB/km and I .25 dB/km respectively.
It can be seen that attenuation is more sensitive than attenuation difference to the form of the dropsize distribu- tion. A comparison of the different coefficients a, h for vertical attenuation at these frequencies is given in Table 2. The power-law fits have each been obtained from the
Table2 Power law fit (A=aR? to vertical specific attenuation as a function of rain-rate R (mmlhl
Type of gamma Frequency, Temperature drop-size GHz distribution
0". 10". 20" 5". lo", 15"
a b a b Ulbrich 12.8 0.020 1.14 0.020 1.14 Testud 12.8 0.018 1.19 0.019 1.19 Ulbrich 17.6 0.046 1.10 0.046 1.10 Testud 17.6 0.043 1.14 0.043 1.14
An.DiW.(dBikm) 17.6V 12.8H (GHz) Fils: 0,015 + 0.0265R or 0.0271 R
1.2
1
0.8
0.6
0.4
0.2
0
An.Diff.(dBikm) 17.6H 12.8v (GHz) Fils: -0.060 + 0.0467R or 0.0444R
An.oin.(dBhm) 17.6H 12.8H (GHz) Fils: 4.035 + 0.0401R or 0.0387R
R,mmih
Fig. 2 As Fig. I , bet the gumma distributions ore those of Trrtud er U(.
IEE Proc-Microw A n l e m s Propg . . Vol. ISO, No 5. Ocroher 2003
R,mmih
317
attenuation curves corresponding to the ensemble of 63 cases considered in Figs. 1 and 2.
It will be seen that b differs significantly from 1.0 at both frequencies, meaning that path-averaged attenuation does not necessarily correspond to path-averaged rain-rate, since the principle of superposition does not hold.
lower end of the rain-rate range is approximately 20mm/h for the polarisation pair with the narrowest range of rain- rate.
We give these results in Table3, where we have also included summarised results for 17.6 GHz/12.8 GHz. We have rounded rain-rates to the nearest integer. It will be seen that the least uncertainty in rain-rate due to the unknown
3.2 Other pairs of frequencies The best pair of frequencies to use for a particular link will depend on the fade margin available at the two frequencies and on the hnk length. The attenuation to be measured at each frequency must be above the noise level of the system for the rainfall rates of interest. For this study, we have concentrated on a range of rain-rates of W m m / h , since this was suggested as being the rain-rate range of most interest in urban hydrology. We present here summary findings for each of the frequency pairs 13.9/7.7, 17.6/10.5, 22.9/13.9,24.1/13.5, 28.0/19.0 and 38.0/24.1 GHz. To try to gwe some comparison between the pairs of frequencies, we have selected a specific attenuation difference where the
varbdbleS of d r o p s i distribution, drop shape, and tem- perature is at the frequency pairs 22.9/13.9GHz and 24.1/ 13.5 GHz. This is evident from the fact that for these pairs, the range of estimated rain-rates is smallest for both forms of the gamma distribution. In the worst case, at the low frequencies of l3.9/7.7 GHz, the possible error about the linear fit is around 20%. For the combination of 38.0 and 24.1 GHz the situation is also clearly not satisfactory since there is a large difference between the results for the two gamma drop-size distribution models, while the 4.5dB/km specific attenuation for the fit to the Ulhrich model (23 mm/h) seriously underestimates the maximum rain-rate (35mm/h) obtained from the various curves.
Table 3: Comparison of the use of specific attenuation difference to estimate rain-rates at approximately 20mmIh
Rain-rate range, mmih Fit (mmlhrl Specific attenuation Frequency, GHz Polarisation
1 2 1 2 Ulbrich Testud U T dBikm
13.9 7.7 V V 2W30 1527 24 22 0.75 17.6 10.5 V V 21-28 2W28 25 24 1.1
22.9 13.9 H H 20-23 19-22 21 20 1.1
24.1 13.5 H V 20-24 2&22 22 21 2.2 28.0 19.0 H V 20-24 1 M 4 22 21 2.4 38.0 24.1 H V 1S35 19-29 23 24 4.5
38GHz Horiz. Anenualion(dB1km) Fit: 0.288 RA 1.027
Fig. 3
318
Specgc attrnuution for horizontal polarisation for 7.7, 13.9, 24.1, 38.0 GHz. 7 7 ~ c2(1ue.~ are as in Fig. I
IEE Proc.~M;crow. A i r l e m Pwpog.. Yo/. 150, No. 5, Ocrober 2W3
The uncerlainty in the difference is largely due to the differing sensitivity in the individual attenuation curves to the various unknowns. For low frequencies, the attenuation curves are significantly non-linear, and attenuation is known to be particularly sensitive to temperature in the 10-14GHz frequency range. It should be noted, however, that this is not reflected in the power law fits given in Table 2. At 38 GHz, the attenuation appears to be very sensitive to the choice of drop size distribution. However, at around 2S24GHz the attenuation seems to he leas1 dependent on the various parameters. This is illustrated in Fig. 3, where we show the attenuation curves for 7.7GHz, 13.9 GHz, 24.1 GHz and 38 GHz at horizontal polarisation. As before, each graph consists of 63 curves.
3.3 Atmospheric and other effects Whereas we have hitherto discussed the attenuation in rain, an important factor also is attenuation caused by atmo- spheric gases-in particular, water vapour. To estimate this, we have used the formula given by the ITU with the assumption of a temperature of I o"C, barometric pressure of 1010mb, and a humidity of 90%. The values at the various frequencies are given in Table 4. The attenuation is Seen to be greatest at 22.9GHz. Examining the differences between the pairs of frequencies given in Table 3, it is clear that at low rain-rates the attenuation due to atmospheric gases is an important factor in any use of attenuation- difference to measure path-averaged rain-rate, particularly at frequencies close to the water vapour absorption line at 22GHz. Indeed, examination of 18 months data on a 17.6/ 12.8GHz 23km link has shown a seasonal variation particularly evident when rainfall is absent. However, statistical processing of the attenuation data has resulted in excellent agreement between attenuationdifference derived rain-rates, and an average of four rain gauges placed underneath the link. This suggests that the variations in drop-shdpe, drop size distribution and temperature assumed in this study are more extreme than those normally occurring. An example is discussed in the next section.
Table 4 Specific attenuation IdBIkm) in atmospheric gases according to ITU formula for a temperature of 10°C. pressure of 1010mb. and 90% relative humidity
Frequency. GHz Gaseous absorption dBkm
1.1 10.5
12.8
17.6
19.0
22.9
24.1
28.0
38.0
0.012
0.017
0.023
0.060
0.089
0.212
0.190
0.119
0.132
4 Experimental results
Figure 4a shows raw attenuation data at frequencies 12.8GHz and 17.6GHz from the 23km link near Bolton, UK. This link runs approximately North-South with the receiver some 266m lower than the transmitter. The data illustrated cover a 16 hour period, from 14.00 on 7 October 2001 until 06.00 on the following day. We can see that the
IEE Proc.-Mkms~. Alrfeniwr Propug. Vol l5ll No. 5, October ZW3
Fig. 4 Affenution and rainfall esrimate cram for etienf from 14:W on 07 October 2001 ro OdW on 08 October 2001 a raw attenuations at 12.8GHz and I7.hGHz b attenuation difference (17.6GHz-12.8GH.z) c estimated rain rates derived from the attenuation difference method and the weighted average from four rain gauges d corresponding cumulative estimates, including those derived from the two single frequencies
higher frequency experiences nearly 20 dB of attenuation at the peak of the event (around 01.00 on 8 October). Figure 46 shows that the corresponding maximum difference in the attenuation approaches 9dB.
Figure 4c compares the rain-rates estimated by a weighted combination of the results recorded by the rain gauges nearest to the link, with the estimate obtained from
319
For practical applications, we believe that rainfall can be deduced to within approximately 15%. This figure is based both on the theory and on experimental results. It must be remembered that the gauge estimates relate to a weighted average of four point estimates over a 23 km link, and that gauges themselves have a possible error of around 10%. The accuracy of the link method may therefore be rather better than the figure 7 ven.
6 Acknowledgments
This work was supported by the European Union, under project EVKI-CT-200060. The data collection was funded by the United Kingdom Natural Environment Research Council under grant GR3/C0035, and we thank the Rutherford Appleton Laboratory and all our partners in that project. We also gratefully acknowledge help from Graham Upton with a figure, and for comment on the manuscript.
the attenuation differences. It is evident that the agreement is generally excellent. Perhaps the most evident differences are those at the beginning of the event (from 15.15 to 15.45, and from 16.15 to 17.00 on 7 October). The link provides a path-averaged rain rate of between 0 and 3mm/h and is generally higher than the gauge estimates. The gauge measurement of low rain rates is inherently difficult since the inter-tip times increase, and the assignment of rainfall to a particular minute is less certain. However, the impact of the differences is slight, as can be seen in Fig. 4d, which shows the cumulative rainfall estimates.
Figure 4dalso shows the estimates obtained from the two single ,frequencies. The accumulations deduced from the single frequencies exceed those from the other methods throughout the event. The final estimated totals are 42" for the low frequency estimate and 38" for the high frequency estimate. The rain gauge estimate, based on four gauges, is 26" (with the individual gauges recording between 20" and 31 mm). It should be noted that the gauges represent 4.0km, 6.5km, 4.4km and 8.4km respectively, hence the gauges can only provide an estimate and not a reference figure. The dual-frequency estimate of 30" lies within the range of the results from the individual gauges.
The example shown in Fig. 4d is not a 'best case', but is typical of results obtained in rain. Of 112 events analysed on this link, less than a third of the link and gauge cumulative estimates differ by more than 15% [13]. Generally, it is observed that the difference in the estimates decreases with increasing rain-rate.
5 Conclusions
It has been shown that, for a range of frequencies, attenuation-difference can be assumed to be a linear function of rain rate, even when there may be considerable variation in raindrop temperature and shape, and dropsize distribution. The restriction is that no melting hydrometeors must be present along the microwave propagation path.
The frequency combination to be used in any particular situation must depend on the length of the link, and on the equipment specification, since it is necessary to have sufficient attenuation along the path so that noise is not a major problem, and yet the attenuation must not be so great that the receivers are saturated. For a given frequency pair, the choice of polarisations may be crucial. It must be remembered that the attenuation due to atmospheric gases has a peak around 22GHz. The study has shown that frequencies above around 30GHz are not suitable for this work.
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References
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