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Atmospheric Research 53 2000 231–250
www.elsevier.comrlocateratmos
Correction of precipitation based on off-site
weather information
Peter Allerup
), Henning Madsen, Flemming Vejen
Danish Meteorological Institute, LyngbyÕej 100, DK-2100 Copenhagen, Denmark
Received 19 April 1999; received in revised form 7 October 1999; accepted 7 December 1999
Abstract
Owing to aerodynamic errors, correction of precipitation measurements, liquid, mixed or solid, is often carried out by means of an explicit mathematical statistical model. The magnitude of correction, e.g., calculated as a correction factor to the observed amount of precipitation, is the dependent variable, and wind speed, temperature and a measure of precipitation intensity are independent variables entering the correction model. Usually these independent variables are accessible on site where the precipitation measurements are recorded. However, in many standard precipitation networks the gauge measurements are not accompanied by on site information of these variables, and information from remote stations must be used. The aim of the paper is to describe how the statistical models can be used in evaluating if the set of remote information of the independent variables can be used for the estimation of a reliable correction factor on site. The technique is then applied to an example of precipitation network in Denmark where the typical situation for the standard gauges is that only the amount of precipitation is available. It is expected that not only can the methods be adapted to other countries, but for landscapes similar to Denmark even the conclusions can be applied directly: extrapolation of all independent variables from remote sites should be conducted with caution, wind speed information can be extrapolated from remote sites not farther away than approximately 50 km, while information on rain intensity and temperature can be safely extrapolated across longer distances. Published by Elsevier Science B.V.
Keywords: Precipitation; Spatial variability; Statistical correction model
)Corresponding author.
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E-mail addresses: [email protected] P. Allerup , [email protected] H. Madsen , [email protected] F. Vejen .
0169-8095r00r$ - see front matter Published by Elsevier Science B.V.
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1. Introduction
Numerous national and regional studies have been conducted to assess measurement
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errors in solid, mixed and liquid precipitation e.g., Hamon, 1973; Tammelin, 1975; Goodison, 1978; Allerup and Madsen, 1979,1980,1986; Allerup et al., 1997; Aune and Førland, 1985; Golubev, 1986; Sevruk, 1986; Goodison and Yang, 1995; Yang et al.,
.
1995 . In 1971, an intercomparison study dealing with errors of liquid precipitation was
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initiated by the World Meteorological Organization WMO , and the analyses resulted in construction of a statistical correction model generally applicable for correction of liquid
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precipitation WMO, 1982 . A similar WMO intercomparison study was initiated in 1986 leading to various suggestions for statistical correction models generally applicable
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for correction of solid and mixed precipitation WMO, 1998 . Within the framework of
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the above mentioned WMO studies, a statistical model for correction of liquid Allerup
. Ž .
and Madsen, 1979 as well as solid and mixed precipitation Allerup et al., 1997 was developed, in the following referred to as ‘‘the Comprehensive Model for Correction of Precipitation’’.
In the present study, and as required by the comprehensive model, the meteorological
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parameters needed for the correction to be carried out are: i rain intensity, ii
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proportion of snow, and averages of continuous measurements during precipitation, iii
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wind speed and iv air temperature, all measured at the gauge station. If continuous measurements during precipitation are not available at the meteorological stations, some
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studies use daily averages of maximum and minimum temperature instead Yang et al.,
. Ž .
1999 . WMO 1998 recommends that the meteorological parameters should be mea-sured at the gauge station, and wind speed be recorded at gauge level. However, in many countries existing networks of rain gauge stations do not offer such on-site measure-ments, and measurements must be extrapolated from other nearby weather stations. An important question when using this extrapolation procedure is how can the error on the estimated correction factor be evaluated. The aim of this paper is to answer, on the one
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hand whether missing on-site observations of wind speed V , temperature T , rain
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intensity I and snow fraction a , i.e. the fraction of precipitation amount fallen as snow, can be substituted by remote measurements, and, if so, how far away these data can be collected if the accuracy must be within certain confidence limits.
The motivation behind the study is that there is a practical need for correcting daily precipitation amounts in Denmark, operationally as well as historical data. For this reason the comprehensive correction model was implemented, using a system of gauges for operational correction of point precipitation. For the application of this system, Denmark was subdivided into 12 sub-regions each of them being as homogeneous as possible with respect to wind speed, temperature, and precipitation patterns. This assumption seems reasonable, considering the limited geographical extent of Denmark
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and the structure of a typical atmospheric pressure system Petersen et al., 1981 .
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An automatic weather station basic station is placed in the center of each of the
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P. Allerup et al.rAtmospheric Research 53 2000 231–250 233
Fig. 1. Map of automatic weather stations in Denmark.
Data from the 12 automatic weather stations were used. Wind speed recorded at 10 m level, air temperature, and amount, as well as duration of precipitation were recorded hourly. The shelter conditions were well described at all stations and taken into account when estimating the true wind speed V at gauge level. The precipitation type for estimation of a was observed every 3 h at nearby weather stations.
The criteria for ‘‘accepting’’ remote a, V, T and I information will be defined in detail below. Both temporal and spatial aspects enter into such analyses, and it could be anticipated that non-isotropic properties in the spatial distribution of wind speed will influence ‘‘where’’ missing wind information can be collected adequately.
It is, nevertheless, the intention to make these substitution procedures operational in ordinary databases in meteorological offices, and therefore the ambition is to create general rules for substitution of a, V, T and I information, depending only on distance.
2. Methods
The comprehensive model for correcting precipitation values was recently tested
ŽAllerup et al., 1997 against data collected under a study mainly dealing with solid.
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i.e., the ratio of true precipitation R over measured precipitation H , at a point of timeq q
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q a specific day , as follows:
K
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a saeb0qb1Vqb2Tqb3V TqŽ
1ya.
eg0qg1Vqg2log Iqg3V log IK
Ž .
a saPS V ,TŽ
.
qŽ
1ya.
PL V , IŽ
.
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1Ž . Ž .
Vswind speed mrs at gauge level during precipitation; Tstemperature 8C during
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precipitation; Israin intensity mmrh ; asfraction of precipitation falling as snow; PSssnow part; PLsrain part;b,gsconstants; values for various rain gauge types are given Appendix A.
It can be read immediately from the mathematical model that the correction proce-dure assumes simultaneous recordings of precipitation and independents variables. Other correction procedures are based on past cumulated knowledge concerning the distribu-tion of the independent variables, but, evidently, higher precision on the estimadistribu-tion of
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the correction factor K a can be achieved by calculations based on simultaneous recordings.
Another point of clarification is that the model in its written form is not a statistical model. In fact, the combined expression for solid and liquid precipitation arises from
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two separate statistical models PS V,T Pe and PL V, I1 Pe where the residual errors,2
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e , e , enter the models in a multiplicative structure. see Allerup et al., 1997, for further1 2
.
details . It is proposed later how to deal with the residual variances on e and e in the1 2
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evaluation of the combined correction factor K a .
The accuracy of the calculated correction factors depends on the general fit of the model. However, systematic components outside the variables included in the model influence the accuracy. The fact, for instance, that some gauges are exposed directly to wind, while other gauges are situated on protected sites is reflected in the wind speed V. If, e.g., reduced wind speed is a consequence of wind protection, this variable should be
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corrected accordingly before it enters the model 1 Førland et al., 1996 . The reduction
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of wind speed V has been found equal to V sV 1yka , where a is the average
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vertical angle 8 of obstacles around the gauge, and k is a constant equal to 0.024
ŽSevruk, 1988 . Also, changes in the roughness of terrain near the gauge can be dealt.
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with by correcting the wind speed accordingly Petersen et al., 1981 .
The model was originally designed for daily or semi-daily precipitation values, and it is therefore recommended to calculate, e.g., monthly corrected values by summing up corrected daily values. The 1st term of the model represents the corrections of solid
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precipitation, while the 2nd term takes care of liquid precipitation Allerup and Madsen,
.
1980 . An impression of the attained level of correction can be gained from Table 1
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where correction factors K a are displayed for selected values of the four controlling variables a, V, T and I. It is seen, e.g., that for wind speed Vs3 mrs, temperature mmrh. Commonly, the averages of V, T and I during precipitation are within these limits. In fact, a recent study confirmed that 99.5% of daily average values of V, T and
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P. Allerup et al.rAtmospheric Research 53 2000 231–250 235 Table 1
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Correction factors K a applicable for the National Danish Standard Rain Gauge Hellmann for selected
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values of Vswind speed mrs at gauge level during precipitation, Tstemperature 8C during precipitation,
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Israin intensity mmrh , and asfraction of solid precipitation
I T as0.00 as0.20 as0.50 as0.80 as1.00
The correction values are based on estimated b- and g-parameters for the National
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Danish Standard Rain Gauge Hellmann gauge, see Appendix A . Data from the WMO study were used, and the comprehensive model was tested with all combinations of a,
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V, T and I available in the data, although more powerful data for the fluid part as0
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were earlier obtained Allerup and Madsen, 1980 . The parameters are estimated from the Hellmann gauge, but it is expected that the general structure of the model will fit data from other rain gauges.1 In fact, for liquid precipitation several gauge types were
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earlier analyzed Allerup and Madsen, 1986 , and differentg-sets were established for the different gauge types.
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For each of the 12 basic stations the series of correction values K a form a
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12-column table. These values are calculated from Eq. 1 across the days with measured precipitation )0.0 mm at least at one of the basic stations. Table 2 displays an example in order to illustrate the steps of statistical analysis. The basic station
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selected as ‘‘on-site’’ is S20209 see the map in Fig. 1 and the ‘‘off-site’’ delivering remotea, V, T and I-information: S20501. This site is situated some 50 km away from S20209. The displayed data refer to observations from the beginning of the study period, i.e. February and March 1989. It is, e.g. seen from the table that the actual correction factor S20209 on February 25 was K2s1.32 based on local observations: as14%
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snow, Vs3.88 mrs , Ts2.4 8C and Is0.96 mmrh . Using a, V, T, I input from S20501 whereas19%, Vs2.87, Ts1.5 and Is1.03 gives rise to a correction factor
K1s1.28.
The principal problem is to compare K1 and K2 values considering the distance between the on-site and off-site as the major independent variable of interest for the analysis of the difference between K and K .1 2
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Table 2
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Correction factors K1 off-site , K2 on-site calculated from Eq. 1 based on on-sitea, V, T, I input S20209 and on off-sitea, V, T, I input S20501 . A ‘‘.’’ indicates missing values
Year Month Day Off-site Correction-factors On-site
a V T I K1 K2 a V T I
1989 2 20 0.00 3.40 8.6 1.23 1.12 1.19 0.00 5.66 7.1 1.45
1989 2 21 0.00 3.86 3.7 0.81 1.16 1.17 0.00 5.05 2.9 1.49
1989 2 22 0.08 2.40 3.2 1.11 1.14 1.12 0.02 3.22 2.8 1.94
1989 2 23 0.00 2.77 3.5 1.28 1.10 1.14 0.00 3.39 3.2 0.79
1989 2 24 0.02 2.35 2.6 0.91 1.11 1.21 0.08 3.51 3.2 1.28
1989 2 25 0.19 2.87 1.5 1.03 1.28 1.32 0.14 3.88 2.4 0.96
1989 2 26 0.01 3.47 3.9 0.94 1.15 1.17 0.00 5.02 4.1 1.41
1989 2 27 0.16 2.52 3.0 0.76 1.22 1.31 0.40 2.14 2.7 1.86
1989 2 28 . . . 1.10 0.00 2.15 4.2 0.57
1989 3 1 0.00 2.36 2.6 1.27 1.09 1.13 0.05 2.49 2.7 1.19
1989 3 2 0.00 2.07 2.4 0.66 1.09 1.09 0.00 2.28 3.0 0.94
1989 3 3 . . . 1.07 0.00 1.53 3.1 0.79
1989 3 4 . . . .
1989 3 5 . . . .
1989 3 6 0.00 2.87 6.2 0.98 1.11 1.14 0.00 4.06 6.3 1.36
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1989 3 8 0.00 1.47 7.3 1.18 1.06 . . . . .
1989 3 9 0.00 1.51 4.7 0.84 1.06 1.10 0.00 2.59 4.3 0.98
1989 3 10 0.00 3.14 6.1 0.80 1.13 1.12 0.00 2.94 5.0 0.99
1989 3 11 0.00 2.86 7.0 0.78 1.12 1.14 0.00 3.12 7.1 0.69
1989 3 12 0.00 2.65 7.1 0.79 1.11 1.14 0.00 3.45 7.0 0.98
1989 3 13 . . . .
1989 3 14 0.00 4.47 5.7 1.14 1.17 1.20 0.00 5.59 5.8 1.28
1989 3 15 . . . 1.18 0.00 4.27 3.7 0.87
1989 3 16 0.00 3.17 3.4 1.04 1.12 1.14 0.01 3.66 3.4 1.24
1989 3 17 0.00 3.40 3.5 1.05 1.13 1.10 0.00 2.78 3.6 1.14
1989 3 18 . . . .
1989 3 19 0.00 5.21 3.7 0.88 1.22 1.28 0.00 6.70 3.3 0.86
1989 3 20 0.00 3.29 5.5 0.89 1.13 1.20 0.00 5.63 4.6 1.29
1989 3 21 0.00 2.62 6.0 0.82 1.11 1.10 0.00 2.91 5.7 1.30
1989 3 22 0.30 1.97 2.3 1.12 1.24 1.32 0.31 2.59 2.9 1.32
1989 3 23 0.00 3.83 4.8 1.49 1.13 1.21 0.06 4.28 4.3 1.54
1989 3 24 0.19 4.08 2.2 1.33 1.40 1.85 0.35 5.26 2.5 1.45
1989 3 25 0.00 3.79 4.4 1.82 1.12 1.15 0.00 4.88 4.5 1.73
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Fig. 2. Correction factors K1 x-axis and K2 y-axis from Table 2.
A straight empirical comparison between these two columns of correction factors is displayed in Fig. 2. Here the complete set of approx. 2500 values, i.e., 2500 days with precipitation )0 at either S20209 or S20501 are used in the graph.
In order to evaluate the relationship between K and K and, in turn the difference2 1 between K1 and K , various techniques could be considered. In fact, measures of2 correlations or fit-statistics from linear regression techniques could be applied to summarize how close the K2 values are to the K1 values — the closer, the more acceptable it seems to substitute the on-site information on a, V, T, I by the off-site values.
It is clear from Fig. 2 that variance on the raw correction values K1 and K2 is increasing with increasing level of K , K . This feature is well known from earlier1 2 analyses of liquid and solid precipitation considering that K and K represent ratios of1 2
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precipitation Allerup and Madsen, 1980; Allerup et al., 1997 and, consequently, leads to a statistical analysis of log transformed values of K1 and K . In fact, for varying2 values of V, T and I the residual errors ss and sr of each part of the comprehensive
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correction model are then consistent homoscedastic : logPS V ,T
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.
;NormalŽ
b qb Vqb Tqb VT ,s2.
0 1 2 3 s
logPL V , I ;Normal g qg Vqg log Iqg V log I,s2
2
Ž
.
Ž
0 1 2 3 r.
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s2
f0.08 s
s2
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P. Allerup et al.rAtmospheric Research 53 2000 231–250 239
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Fig. 3. Distribution and a Box Plot of daily differences Dslog K2rK1 between on-site correction values
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K2 and off-site corrections K , 50 km away. All data N1 s1268 have been used.
Statistical analysis of the difference between the off-site K1 and on-site K2 should
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therefore be based on analysis of log K , log K1 2 rather than the raw correction values. Two separate issues are of interest when evaluating the general difference between
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K1 and K : 1 will off-site2 a, V, T, I information result in systematically biased
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correction values compared with what would be estimated locally? and 2 can K2 values in general be substituted by K values with a satisfactory degree of precision?2
1
For the example given in Table 2, the complete distribution of differences given as
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Dslog K2 ylog K1 is displayed in Fig. 3.
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Fig. 3 summarizes the complete distribution of D as an usual Box Plot 10, 25, 50,
.
75, 90-percentiles , and from the calculations it is found that medians0.02046, means0.02454 and standard deviations0.087422. This example shows a fairly sym-metrical distribution of D, a feature that might allow a brief characterization using the mean and standard deviation. It is, however, clear that the displayed distribution in Fig. 3 has a kurtosis exceeding that of a normal distribution.
When choosing proper criteria for the evaluation of the D-differences, one has furthermore to include measures of original statistical precision inherent in the basic
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correction model 1 . From the development of the model final estimates of the residual errors s2s0.06 and s2s0.08 exist and must enter into the evaluation of a
correspon-r s
2 Ž .
In theory a statistically efficient estimator i.e., small variance could be rejected because its values are
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dence between the K and K1 2 values. For values of snow fraction a close to 0.00 or
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1.00 confidence limits for K a relate simply to properties of fit for each of the
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sub-models PS V,T snow and PL V, I rain .
A suitable measure for evaluating the discrepancy between K and K can therefore1 2
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be based on the 95% level prediction intervals Pred95% log PS V,T and Pred95% log
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PL V, I arising from the regression analysis with V, T and I as independent variables. This step towards a general method of evaluating the D-differences suggests that the
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actual knowledge although complex concerning simultaneous deviations between on-site and off-site measurements of wind speed V, temperature T and rain intensity I be ignored, letting the resulting deviation between the two correction values K and K1 2 be considered stochastic. Evaluation of this stochastic difference will therefore follow the rules for evaluation of residual error embedded in s and s . Hence, under thes r
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hypothesis that the on-site missing values can be substituted by simultaneous off-site values the distributions of D must match the usual prediction limits calculated from the regressions analysis.
For the example given in Table 2 and Fig. 3, a t-test evaluation and a non-parametric
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test Wilcoxon of the mean value Ds0.024536 both show significant positive devia-tion in favor of the on-site values K3 using s2
s0.07 as an average value for the 1
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residual error for mixed precipitation. Transformed to a normal distribution approxi-mate 10–90% width for the stochastic errors, it results in 2=1.28ss0.67. This interval is on the same width if the empirical variance s2
s0.087422 is used for D. A conclusion for this example would therefore be that the off-site values are biased
Žapproximately 2% and followed by a precision of magnitude that matches the basic. Ž .
residual error of the model 1 . It should, however, be noticed that the distribution of D
fails to be normal, and the raw 10–90% Box Plot percentiles result in an interval:
wy0.02791,q0.09061 , i.e., of lengthx s0.12, or of length only 20% of the s-based length.
The judgment of general deviation between K and K , viz. the judgment of spatial1 2 distance between on-site and off-site will therefore be based on evaluations of
per-Ž . Ž .
centiles in D-distributions Box Plots and non-parametric tests Wilcoxon rather than
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on mean values and standard deviations t-tests in the D-distribution.
Analysis will focus on extrapolating the values of all controlling variables from a distant off-site station to the on-site station. It is, however, of interest for the general evaluation procedure to study what happens if only one of the controlling variables
assnow fraction, Vswind speed, Tstemperature and Israin intensity is to be substituted with values collected off-site. From a strictly practical point of view, it seems that on up-to-date equipment either none or all four variables are missing on-site.
This is one reason why focus will be on the attempt to substitute all four variables at a time with remote observations. Evaluation of the influence of each of the four variables considered separately will be conducted as marginal analyses, and recommen-dations as to how far off-site measurements can be taken to substitute one variable are, therefore, derived marginally.
3
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P. Allerup et al.rAtmospheric Research 53 2000 231–250 241 3. Results
The 12 stations are situated in Denmark so that typical regions are represented. The
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data are daily observations see details above from the period January 1989 to December 1996. Each of the 12 stations acts in turn as on-site while the other stations
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act as off-site stations. Twelve charts fixed on-site station are derived, each one
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containing eleven groups of Dslog K2rK1 slog on-siteroff-site ; each D-group consists of simultaneous observations from the eight-year series. The D-groups can be studied across the whole 8-year period or be split into sub-groups defined by the season
Žmonth ..
It is characteristic for the distributions of D-differences that they all have distinctly higher kurtosis than allowed by the normal distribution. In fact, small deviations between on-site and off-site corrections emerge with very high frequency.
Fig. 4 summarizes the basic D-chart for all possible combinations of on-site off-site stations since each of the 12 weather stations acts in turn as on-site station against the other eleven stations. This results in 132 combinations of stations or distances. For a given distance, i.e. a given combination of stations a vertical Box Plot displays 25–75
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percentiles extended by whiskers tick marks . Scattered points below and above the Box Plots indicate the range of D-observations outside the one covered by the Box Plots. The vertical limits"0.30 are introduced in order to magnify the central parts of the D-distributions; only few points are found outside these limits.
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Investigations of Fig. 4 supplemented by numerical evaluations Wilcoxon,
non-para-.
metric of the 132 D-distributions testing the hypothesis that these distributions are centered around zero everywhere result in rejections.
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The 25–75 percentile limits and even the approximate 10–90 percentile limits set by the whiskers are all within one standard deviation; "0.25 derived froms2
s0.07,
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the residual variance in the mixed model 1 . These limits are included as vertical reference lines in Fig. 4. In conclusion, D-values arising from extrapolating all four controlling variables: wind speed, temperature, rain intensity and snow fraction from another off-site seem to lead to systematically biased K2 correction values. However, the level of bias is within the one-standard deviation limits given by the original model
Ž .1 . As an average across all 132 D-groups, 4.8% of the D-observations are above 0.25, 4.2% of the observations fall below y0.25, leaving 91% of the D-observations to be
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covered by the"0.25 limits derived from the model 1 .
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The D-group marked at distances9.6 km between stations 29451 and 29439 attracts special attention because of the short interdistance. The median value is 0.00, means0.02 and 25–75 percentiles are y0.02, 0.03. The majority of extrapolation events therefore leads to discrepancies between on-site and off-site corrections of the
Ž Ž . .
order ;2–3% exp 0.02 ;1.02;2% . For the general level of corrections in the left side of Table 1 these 2–3% have only little numerical influence, and a practical position defending that extrapolation across these 9 km could anyhow be accepted. Unfortu-nately, further details for short interdistance are not available. The twelve stations are
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P. Allerup et al.rAtmospheric Research 53 2000 231–250 243 spread evenly across the country, which is confirmed by the dense distribution of points from approximately 50 to 250 km resulting in multiple Box Plots for a given distance. Only one pair of stations has an interdistance below 20 km.
Figs. 5–8 summarize the D-charts for the marginal analyses, i.e. analyses where the remote information concerns only one of the four controlling variablesa, V, T, I. Fig. 5 is largely a repetition of Fig. 4, indicating that wind speedsV is the variable with the
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most marginal influence in Fig. 4. In fact, the values of b1 and g1, regression
.
coefficients to wind speed result in high relative marginal changes of the correction
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level Allerup et al., 1997 .
The conclusion from analyses of Fig. 5 is, therefore, the same as for Fig. 4: for all distances significant deviations between on-site and off-site levels of the corrections are
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found. The test statistics non-parametric Wilcoxon clearly show significance probabili-ties close to zero, but again practical considerations about the actual level of discrepancy between on-site and off-site corrections for 9 km distances could lead to acceptance of the discrepancy. In fact, the actual values are: medians0.01, means0.02 and 25–75 percentiless y0.01,0.03.
In Fig. 6 the marginal analyses of extrapolating rain intensity I is displayed. An immediate comparison with Fig. 4 and Fig. 5 shows that the general variability of correction values due to off-site use of rain intensity information is much smaller compared to off-site use of wind speed information. This is in accordance with the
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smaller impact on the correction value through theg2-parameter of Eq. 1 and with the
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Fig. 7. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of temperature sampled off-site.
fact that the spatial variability of rain intensity is anticipated to be small. Another marked difference between Fig. 4 and Fig. 5 is that all Box Plot 25–75 limits cover the
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zero-line. Still, very significant test statistics non-parametric Wilcoxon for distances above 75 km indicate systematic differences from zero in these D-distributions. Up to 50–60 km the 25–75 percentile limits are generally y0.01, 0.01, and for distances above 50–60 km these limits arey0.02, 0.02.
The effect of measuring temperaturesT off-site is displayed in Fig 7. It is seen that
the central 25–75 percentile limits of the Box Plots are not distinguishable. In fact, all
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calculations of 25 and 75 percentiles and thereby the median are equal to 0.00 on
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second decimal place. The relative influence on the corrections calculated in Eq. 1 through b1,b3 and g1,g3 is not small, but here Fig. 7 reflects generally consistent temperature conditions within a given day in Denmark. The use of the off-site temperature information seems, irrespective of the interdistance, not to pose any problem.
Regarding the off-site use of snow fractionsa the statistical analysis must be restricted to days where the possibility of snow is positive, otherwise the Box Plots will include false D-zero values and will be artificially too close. Fig. 8 displays the
D-distributions for the winter season December through March and only for days where temperatures t-0 are considered. The impression from Fig. 8 and the test statistics
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Fig. 8. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of snow fraction sampled off-site. Temperatures T-08C are considered.
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Fig. 10. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of snow fraction sampled off-site. Temperaures T-y28C are considered.
between. A possible sign of anisotropy, which is confirmed in Fig. 8, is broken down into twelve sub-graphs, each having a fixed on-site station. The 25–75 percentile varies greatly across the distance groups.
If, however, analysis is further restricted to temperatures t-y18C and t-y28C, a more consistent distance relation will emerge. In fact, Fig. 9 and Fig. 10 display attempts at extrapolating off-site snow fraction information under these conditions. For
Ž .
distances above 100 km the numerical analyses non-parametric Wilcoxon tests demon-strate D-groups systematically biased away from zero, although 25–75 percentiles of the Box Plots generally are within"0.02 levels, i.e., less than"2%.
A possible conclusion would consequently be that information concerning snow can safely be extrapolated from off-site measurements situated less than 100 km away if the temperature is t-y18C.
4. Discussion
In this study, the defined on-site off-site D-differences were created to reflect spatial
Ž .
variability of the four controlling variables Vswind speed during precipitation ,
Ž . Ž .
Israin intensity liquid precipitation , Tstemperature during precipitation , and
Ž .
assnow fraction through the correction factors K , and K1 2 from Eq. 1 . Other
Ž .
( )
P. Allerup et al.rAtmospheric Research 53 2000 231–250 247 measurements at a number of stations constitute the data. While such analyses create stochastic models to study temporal and spatial variability based solely on wind measurements, it has been an important practical aspect of the present analyses to conduct and define analyses of differences in terms of observed consequences for the calculations of the correction factors. cf. the aim of analyses stated earlier: how far away from a gauge can values of the controlling variables be sampled with only minor
Ž .
consequences for the level of correction factors calculated through the model 1 ? It is assumed that the on-site measurements of precipitation and controlling variables are collected under conditions specified for a standard synoptic weather station.
The analysis strategy can be viewed as a simple transformation of the
four-dimen-Ž . 2
sional independent variable a, V, T, I utilizing knowledge of residual errors ss and
s2 from the original fitting of the model. r
During the analyses, which aimed at revealing relations dependent on distance only, the data were sub-grouped according to two important background factors: time and year
Žmonth and region West. Ž rEast . The structure revealed in Figs. 4–10 should be,.
ideally, invariant towards such subdivisions of data.
The effect of regionalization vanished. However, in the general residual errors when test statistics were calculated across the 8-year study period 1989–1996, and only in combination with a further restricting of the observations to the winter months, this regionalization becomes weakly visible.
The effect of subdividing data according to month of year can be visualized in Fig. 11 and Fig. 12. These figures display a subdivision of Fig. 4, in which all four variables
a, V, T, I are extrapolated from off-site into two seasons: summer and winter. It is
Ž . Ž .
( ) P. Allerup et al.rAtmospheric Research 53 2000 231–250
248
Ž . Ž .
Fig. 12. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of all four independent variables sampled off-site. Summer season.
characteristic for these subdivisions that the distance related structure remains the same,
Ž .
but ‘‘summer’’ shows less variability shorter Box Plots compared to ‘‘winter’’.
5. Conclusions
In order to overcome aerodynamic measurement errors, correction of precipitation measurements, liquid, mixed or solid, can be undertaken using statistical models where the correction factors as dependent variables are derived from known values of four
Ž
independent variables: wind speed, rain intensity, temperature, and snow fraction mix
.
fraction liquidrsolid . These values usually enter the calculations from observations at a gauge site.
The present analyses have demonstrated how the situation with missing observations on one or all of the four independent variables at a gauge site can be handled by referring to simultaneous information being collected at another remote site. A general procedure using the correction model has been presented, so that evaluations as to how far away from the gauge site one can adopt values of the four variables can be conducted in other precipitation networks.
( )
P. Allerup et al.rAtmospheric Research 53 2000 231–250 249 It appeared from the analyses of the Danish data that:
Ø in case of all four variables missing, information should not be sampled beyond 50 km from the gauge site,
Ø in case of missing wind speed information, the distance should not exceed 50 km, Ø in case of missing rain intensity information, the distance should not exceed 75 km, Ø in case of missing temperature information, the information can be adequately
substituted by any simultaneous temperature measurement,
Ø in case of missing information concerning the mixture snowrrain of precipitation, the distance should not exceed 100 km.
Acknowledgements
The authors are grateful to Lotte Abel for review of this paper.
Appendix A
Coefficients in the comprehensive model for correcting precipitation on condition that
Ž .i wind speed is measured at gauge level mŽ rs. Ž .ii rain intensity unit is mmrhour and Ž .iii temperature units is 8C. The values are related to the use of the Hellmann gauge without wind shield.
Solid precipitation Liquid precipitation Intercept b0s0.04587 g0s0.007697 Wind b1s0.23677 g1s0.034331 TemprIntensity b2s0.017979 g2s y0.00101 Product b3s y0.015407 g3s y0.012177
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