Hydrological Sciences–Journal–des Sciences Hydrologiques, 48(6) December 2003
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Grubbs revisited: a statistical technique to differentiate measurement methods of the degree of exposure of raingauges in a network BERT DE SMEDT*, BERNARD MOHYMONT & GASTON R. DEMARÉE Royal Meteorological Institute of Belgium, Ringlaan 3, B-1180 Brussels, Belgium
Abstract Precipitation measurement by raingauges is biased by the systematic error resulting from deformation of the local wind field due to the presence of the gauge itself. To correct for this error, it is necessary to know the degree of site exposure. Four measurement methods are dealt with to determine site exposure. It is shown how statistical techniques developed by Grubbs can be used to select the most precise measurement method. Key words precipitation measurement; wind-field deformation; site exposure; Grubbs technique
Grubbs revisité: une technique statistique pour différencier les méthodes de mesure du degré d’exposition de pluviomètres dans un réseau Résumé La mesure des précipitations au moyen de pluviomètres est entachée de l’erreur systématique qui résulte de la déformation du champ de vent local, due à la présence de l’instrument lui-même. Pour corriger cette erreur, il est nécessaire de connaître le degré d’exposition de site. Quatre méthodes de mesure sont considérées, pour déterminer l’exposition de site. Nous montrons comment des techniques statistiques développées par Grubbs peuvent être utilisées pour sélectionner la méthode de mesure la plus précise. Mots clefs mesure de précipitations; déformation du champ de vent; exposition de site; technique de Grubbs
THE WIND FIELD DEFORMATION For centuries, precipitation has been measured by raingauges in climatological networks. However, this kind of ground-based monitoring is subject to appreciable systematic errors (see e.g. Sevruk, 1982). The most important of these errors results from the local wind-field deformation caused by the presence of the gauge itself (Fig. 1). This deformation gives rise to a deflection of the original raindrop trajectories towards the leeward side of the gauge. Due to this deflection, a small fraction of the precipitation that would normally fall within the gauge rim, is taken past the gauge rim. The magnitude of this aerodynamic error varies—on average—from 2 to 10% (Sevruk, 1982), but may amount to 60% under some conditions (Goodison et al., 1998). Many factors are involved in this aerodynamic effect. One can discern instrumental, site-related and meteorological factors (Table 1), which are more or less correlated. The most important component is the wind speed above the gauge rim. The higher the wind speed, the higher the precipitation loss. *Now at: Department of Geography, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium;
[email protected] Open for discussion until 1 June 2004
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Gauge rim diameter Diameter of rain section caught Raindrop trajectory Wind
Fig. 1 Illustration of the wind-field deformation error (after Perrin de Brichambaut, 1977).
Table 1 Factors and their main components influencing the wind-field deformation error. Instrumental Gauge shape and dimension Gauge height above ground Gauge rim dimensions Wind shield
Meteorological Wind velocity Precipitation intensity Precipitation type
Site Exposure Surface slope Ground cover
To reduce this aerodynamic effect, the WMO prescribes guidelines for sites and instruments (WMO, 1996). Often however, these guidelines are not fully respected because of logistic and physical restrictions. The last decades have seen the development of precipitation correction models, with the aerodynamic correction as their main component (see e.g. Sevruk, 1982; Sevruk & Hamon, 1984; Legates & Willmott, 1990; WMO, 1996; Goodison et al., 1998). The required input for these models is: (a) the precipitation series to be corrected, (b) a precipitation series by a local reference instrument, (c) local wind speed measurements, and (d) the degree of site exposure, SE. The latter two are used to derive the wind speed at the raingauge level, Uh, from which the correction coefficient is calculated by means of an empirical relationship between (a) and (b). Figure 2 illustrates why it is necessary to know SE to derive Uh. The top panel shows a completely exposed site, for which the wind velocity is known to increase logarithmically with elevation above the ground. The bottom panel shows a partly protected site. Here, the logarithmic wind profile is modified by the surrounding obstacles. So, to assess the perturbation of the wind-speed profile and to calculate the wind-speed measurements at gauge level, knowledge of the site exposure is needed. Conventionally, SE is derived from the angular elevation of the surrounding obstacles, here symbolically represented as α1 and α2. These elevations can be measured by a simple optical instrument (meridian) or theodolite above the gauge orifice. MEASURING THE DEGREE OF EXPOSURE Four methods to determine SE are considered here. The method developed by Sevruk (Sevruk & Zahlavova, 1994) is the most popular. Here, SE is the average of the mean
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(a)
(b)
Fig. 2 Wind speed profiles for (a) open and (b) partly protected gauge sites (after Sevruk & Zahlavova, 1994).
angular elevation for each of the eight principal sectors of wind direction. To reduce the weight of obstacles higher than 25°, their angular elevation is taken as 25° in the calculation of the sector mean value. Also, the angular elevation of obstacles with a horizontal extent less than one-tenth of their distance to the gauge, is reduced by half. The Sevruk weighted method (Sevruk & Zahlavova, 1994) is the same, but mean sector values for dominant wind directions are given a double weight. Taking theodolite set-up time into account, the average time period required for one SE measurement with the Sevruk method is about 20 min. The method introduced by Harrison (1988) considers only the angular elevations of the eight principal wind directions. Also, its reference inclination is –5° instead of 0°. The Harrison SE is calculated from: SE =
8
å n =1
α n ⋅ α n +1 8
(1)
where n is a principal wind direction (α9 = α1). One SE measurement takes about 15 min. The last method is developed by Leroy (1998) and considers only the elevation of the highest obstacle and the mean surface slope. Here, only 10 min are needed for one measurement. GRUBBS REVISITED A given set of measurements consists of two components: the true value, x, of each item and the measurement error, e. It is known that the variance of these measurements
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equals the sum of the item variance (or product variability) σx2, the error variance (or precision) σe2, and the covariance of items and measurement errors σx,e. Working in the US Army Ballistic Research Laboratories, Grubbs was concerned with the precision of measurement instruments such as electrical clocks used to measure the burning time of fuses. In 1948 he wrote “On estimating the precision of measurement instruments and product variability”. Therein, Grubbs (1948) describes techniques to select the most precise out of m instruments, by measuring the same n items once with each instrument, thus yielding a (n × m) matrix. This is done by separating and estimating σx2 and σe2 for each instrument and estimating the variance of the σe2 estimate. Here, these estimators are denoted sx2, se2 and var(se2), respectively. The latter is an indication of the uncertainty of se2. Grubbs’ techniques are based on the assumptions that, over a limited range of real item values, x, the value of σx,e is 0 for each instrument; that both real item values and measurement errors have a normal distribution; and that there is no correlation between measurement errors for different instruments. The precision of the estimators depends on σx2, σe2 for each of the instruments and on the sample size, n. A lower σx2 and a larger n will yield a higher estimator precision. Conventionally, Grubbs’s techniques are applied to measuring devices such as clocks and meters. Sneyers (1964) was the first to use Grubbs’ method to test the performance of raingauges. Here, it is proposed to use Grubbs’s techniques to calculate the precision of the four methods to determine SE, thus selecting the most precise “instrument”. PRACTICAL EXAMPLE As a practical illustration, the degree of exposure for 26 sites in an existing precipitation measurement network was determined according to the four methods. These sites are all horizontally sloping, so the surface slope criterion in the Leroy method becomes trivial. Table 2 gives an overview of the values obtained. Since the four methods have a different value scale, the measured values are not directly comparable between different methods. Therefore, they are also given a rank according to each method. Sites with a high exposure value and a low ranking are ones which are highly protected. Although protected sites theoretically yield a lower aerodynamic error, obstacles should not be too close to the raingauge either (to avoid blocking of raindrop trajectories). The WMO (1996) pictures an ideal precipitation measurement site as consisting of a circle of equally high obstacles with the raingauge in the centre. The angular elevation of the circular screen should be between 30 and 45°. Only rarely do sites conform to this guideline. This is why in a less rigid guideline, the WMO prescribes a distance between the raingauge and obstacles of at least twice and preferably four times the height of the obstacles above the raingauge. These limits correspond to SE values of, maximum 26.6 and preferably less than 14 for the Sevruk & Leroy methods, or maximum 53.1 and preferably less than 32 for the Harrison method. The values highlighted in Table 2 correspond to SE values exceeding the lower (italic) and upper limit (bold). It is clear that the values obtained by the Leroy method deviate significantly from the results for the other methods. This is confirmed
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Table 2 Measured site-exposure values according to four methods. Values in italic (bold) indicate sites exceeding the lower (upper) WMO limits for site exposure described in the text. Station number
Sevruk: Value
Rank
Sevruk weighted: Value Rank
Harrison: Value Rank
Leroy: Value
Rank
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
1.1 2.4 3.4 4.6 5.7 6.1 6.9 7.6 8.3 8.4 8.5 9.6 9.8 10.1 10.4 10.5 11.1 11.2 11.4 12.9 14.4 14.9 15.7 16.3 16.4 17.1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
1.0 2.4 2.8 5.7 5.0 6.6 7.8 7.8 7.4 10.4 7.8 8.7 10.9 9.5 9.5 10.3 10.5 11.1 11.1 12.6 12.1 15.3 15.3 15.4 15.8 15.7
10.4 12.8 14.6 16.3 15.4 21.2 20.2 22.9 21.4 25.6 32.1 26.0 25.8 25.2 30.7 26.2 26.5 28.5 24.2 30.3 36.6 35.3 32.6 47.5 37.0 47.8
4.0 5.0 12.0 16.0 13.5 17.0 15.0 12.0 19.0 32.0 56.0 19.0 22.0 14.5 50.5 20.5 22.0 20.0 28.0 29.5 58.0 40.0 51.0 45.5 50.0 41.0
1 2 3 8 5 9 7 3 10 18 25 10 14 6 23 13 14 12 16 17 26 19 24 21 22 20
1 2 3 5 4 6 9 8 7 15 10 11 17 12 13 14 16 18 19 21 20 23 22 24 26 25
1 2 3 5 4 7 6 9 8 12 20 14 13 11 19 15 16 17 10 18 23 22 21 25 24 26
Table 3 Spearman rank correlation between the four measurement methods of SE. The Sevruk, Sevruk weighted, Harrison, and Leroy methods are denoted with S, Sw, H and L, respectively. S–Sw 0.98
S–H 0.93
S–L 0.79
Sw–H 0.90
Sw–L 0.79
H–L 0.89
Table 4 Significance of the Kolmogorov-Smirnov normality test for each of the SE measurement methods. The Sevruk, Sevruk weighted, Harrison, and Leroy methods are denoted with S, Sw, H and L, respectively. S Sw H L 0.98 0.90 0.81 0.19
Table 5 Grubbs estimators for σe2 and the variance of σe2 for the z-scores of each of the SE measurement methods. The Sevruk, Sevruk weighted, Harrison, and Leroy methods are denoted with S, Sw, H and L, respectively. 2
se var(se2)
S 0.029 0.003
Sw 0.097 0.009
H 0.063 0.006
L 0.403 0.034
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by the Spearman rank correlation between the four methods (Table 3). Lower rank correlation values are obtained between the Leroy method and each of the other methods. In order to perform Grubbs’s analysis on these measurements, one first needs to resample the value scale of each method to a uniform scale. This was done by computing z-scores (subtracting for each measurement the sample mean and dividing by the standard deviation). One of Grubbs’s assumptions was that both real item values and measurement errors have a normal distribution. Though this is impossible to test, one can test the normality of the calculated z-scores. Table 4 shows that, at the 10% level, the hypothesis of normality remains valid. Finally, it was possible to estimate σe2 for each of the methods (Table 5). Again, the Leroy method deviates strongly from the other methods, with its σe2 value at least four times greater than that of any of the other methods. Also, one can see that Sevruk’s method performs slightly better than the other methods. Since the Harrison σe2 is only three times higher than the Sevruk σe2, it seems reasonable to measure only the angular elevation in the eight principal wind directions to reduce measurement time. CONCLUSION To conclude, it can be inferred that for the sample of 26 stations used in this study, out of the four methods, the Sevruk method is the most precise and the Leroy method is by far the least precise. The latter method is probably too simplistic because it takes into account only the angular elevation of the highest obstacle, thus neglecting a great deal of information. Measurement time can be reduced by replacing the Sevruk method with the less time-consuming, but also less precise, Harrison method. Through the efforts of this study, it appears that the Grubbs technique is a valuable tool to differentiate measurement methods of the degree of exposure of raingauges in a climatological network. Acknowledgements This work is a contribution to the project “16EB/01/10 – Doorlichting van het pluviografisch Meetnet van het HIC” carried out by the Department of Research and Development of the Royal Meteorological Institute of Belgium for Flanders Hydraulics, AWZ, Ministry of the Flemish Community. REFERENCES Goodison, B. E., Louie, P. Y. T. & Yang, D. (1998) WMO Solid precipitation measurement intercomparison, final report. WMO/TD no. 872, World Meteorological Organization, Geneva, Switzerland. Grubbs, F. (1948) On estimating precision of measuring instruments and product variability. J. Am. Statist. Assoc. 43, 243–264. Harrison, S. J. (1988) Numerical assessment of local shelter around weather stations. Weather 43(9), 325–330. Legates, D. R. & Willmott, C. J. (1990) Mean seasonal and spatial variability in gauge-corrected, global precipitation. Int. J. Climatol. 10, 111–127. Leroy, M. (1998) Meteorological measurement representativity, nearby obstacle influence. In: WMO Instruments and Observing Methods Report no. 70. (Papers presented at the WMO Technical Conference on Meteorological and Environmental Instruments and Methods of Observation (TECO-98), Casablanca, Morocco, May 1998) WMO/TD no. 877, 51–54. World Meteorological Organization, Geneva, Switzerland.
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Perrin de Brichambaut, C. (1977) Les instruments de mesures météorologiques. Cah. d’inf. Mét. Nationale 1, Trappes, France. Sevruk, B. (1982) Methods of correction for systematic errors in point precipitation measurements for operational use. WMO Operational Hydrology Report no. 21. WMO-no. 589, World Meteorological Organization, Geneva, Switzerland. Sevruk, B. & Hamon, W. R. (1984) International comparison of national precipitation gauges with a reference pit gauge. WMO Instruments and Observing Methods Report no. 17. WMO/TD no. 38, World Meteorological Organization, Geneva, Switzerland. Sevruk, B. & Zahlavova, L. (1994) Classification system of precipitation gauge site exposure: evaluation and application. Int. J. Climatol. 14, 681–689. Sneyers, R. (1964) La statistique des précipitations à Bruxelles-Uccle. Ciel et Terre 80(5–6), 145–164. WMO (World Meteorological Organization) (1996) Guide to Meteorological Instruments and Methods of Observation (Sixth edn). WMO no. 8, WMO, Geneva, Switzerland.
Received 13 March 2003; accepted 1 August 2003
