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 Ž
. Allerup et al., 1997 against data collected under a study mainly dealing with solid
Ž .
Ž . Ž .
precipitation WMO, 1998 . The model 1 predicts the correction factor K a s R rH ,
q q
i.e., the ratio of true precipitation R over measured precipitation H , at a point of time
q q
Ž .
q a specific day , as follows: K a s a e
b qb
1
Vq b
2
Tq b
3
V T
q 1 y a e
g qg
1
Vq g
2
log Iqg
3
V log I
Ž . Ž
.
K a s a PS V ,T q 1 y a PL V , I 1
Ž . Ž
. Ž
. Ž
. Ž .
Ž .
Ž .
V s wind speed mrs at gauge level during precipitation; T s temperature 8C during Ž
. precipitation; I s rain intensity mmrh ; a s fraction of precipitation falling as snow;
PS s snow part; PL s rain part; b,g s constants; 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 estimation of
Ž . 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
Ž .
Ž .
two separate statistical models PS V,T P e and PL V, I P e where the residual errors,
1 2
Ž e , e , enter the models in a multiplicative structure. see Allerup et al., 1997, for further
1 2
. details . It is proposed later how to deal with the residual variances on e and e in the
1 2
Ž . 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
Ž . Ž .
corrected accordingly before it enters the model 1 Førland et al., 1996 . The reduction
X
Ž .
of wind speed V has been found equal to V s V 1 y k a , where a is the average Ž .
vertical angle 8 of obstacles around the gauge, and k is a constant equal to 0.024 Ž
. Sevruk, 1988 . Also, changes in the roughness of terrain near the gauge can be dealt
Ž .
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
Ž 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
Ž . 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 V s 3 mrs, temperature T s y28C, a s 50 snow and rain intensities I s 1 mmrh, the correction factor is
Ž equal to 1.69. The model can be adequately used only for 0 F V F 7 mrs solid
. Ž
. precipitation , or for 0 F V F 15 mrs liquid precipitation , y128C F T and 0 F I F 15
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
I satisfied the requirements.
Table 1 Ž .
Ž .
Correction factors K a applicable for the National Danish Standard Rain Gauge Hellmann for selected
Ž .
Ž .
values of V s wind speed mrs at gauge level during precipitation, T s temperature 8C during precipitation, Ž
. I s rain intensity mmrh , and a s fraction of solid precipitation
I T
a s 0.00 a s 0.20
a s 0.50 a s 0.80
a s1.00 V s 3
V s6 V s 3
V s6 V s 3
V s6 V s 3
V s6 V s 3
V s6 1
1.12 1.24
1.32 1.86
1.62 2.79
1.93 3.71
2.13 4.33
3 1.07
1.14 1.28
1.78 1.60
2.74 1.92
3.70 2.13
4.33 5
1.05 1.10
1.27 1.75
1.59 2.72
1.91 3.69
2.13 4.33
1 y2
1.12 1.24
1.34 2.00
1.69 3.13
2.03 4.27
2.25 5.03
3 y2
1.07 1.14
1.31 1.92
1.66 3.09
2.02 4.25
2.25 5.03
5 y2
1.05 1.10
1.29 1.89
1.65 3.06
2.01 4.24
2.25 5.03
1 y4
1.12 1.24
1.37 2.16
1.75 3.54
2.13 4.92
2.38 5.84
3 y4
1.07 1.14
1.33 2.08
1.73 3.49
2.12 4.90
2.38 5.84
5 y4
1.05 1.10
1.32 2.05
1.72 3.47
2.12 4.89
2.38 5.84
1 y6
1.12 1.24
1.40 2.35
1.82 4.01
2.24 5.67
2.52 6.78
3 y6
1.07 1.14
1.36 2.27
1.80 3.96
2.23 5.65
2.52 6.78
5 y6
1.05 1.10
1.35 2.23
1.79 3.94
2.23 5.64
2.52 6.78
The correction values are based on estimated b- and g-parameters for the National Ž
. 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 , Ž
. V, T and I available in the data, although more powerful data for the fluid part a s 0
Ž .
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 Ž
. earlier analyzed Allerup and Madsen, 1986 , and different g-sets were established for
the different gauge types. Ž .
For each of the 12 basic stations the series of correction values K a form a
Ž . 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
Ž .
selected as ‘‘on-site’’ is S20209 see the map in Fig. 1 and the ‘‘off-site’’ delivering remote a , 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 K s 1.32 based on local observations: a s 14
2
Ž .
Ž .
Ž .
snow, V s 3.88 mrs , T s 2.4 8C and I s 0.96 mmrh . Using a , V, T, I input from S20501 where a s 19, V s 2.87, T s 1.5 and I s 1.03 gives rise to a correction factor
K s 1.28.
1
The principal problem is to compare K and K
values considering the distance
1 2
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
1
For national Nordic precipitation gauges the parameter estimates can be found in Førland et al., 1996.
P. Allerup
et al.
r Atmospheric
Research 53
2000 231
– 250
236
Table 2 Ž
. Ž
. Ž .
Ž .
Ž .
Correction factors K off-site , K
on-site calculated from Eq. 1 based on on-site a , V, T, I input S20209 and on off-site a , V, T, I input S20501 . A ‘‘.’’
1 2
indicates missing values Year
Month Day
Off-site Correction-factors
On-site a
V T
I K
K a
V T
I
1 2
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 1989
3 7
. .
. .
. 1.13
0.00 2.78
5.5 0.57
P. Allerup
et al.
r Atmospheric
Research 53
2000 231
– 250
237 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 1989
3 26
0.00 5.82
5.3 0.95
1.23 1.23
0.00 7.47
4.9 1.90
Ž .
Ž .
Fig. 2. Correction factors K x-axis and K
y-axis from Table 2.
1 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 difference
2 1
between K and K , various techniques could be considered. In fact, measures of
1 2
correlations or fit-statistics from linear regression techniques could be applied to summarize how close the K
values are to the K values — the closer, the more
2 1
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 K and K
is
1 2
increasing with increasing level of K , K . This feature is well known from earlier
1 2
analyses of liquid and solid precipitation considering that K and K represent ratios of
1 2
Ž .
precipitation Allerup and Madsen, 1980; Allerup et al., 1997 and, consequently, leads to a statistical analysis of log transformed values of K
and K . In fact, for varying
1 2
values of V, T and I the residual errors s and s of each part of the comprehensive
s r
Ž .
correction model are then consistent homoscedastic : logPS V ,T ; Normal b q b V q b T q b VT ,s
2
Ž .
Ž .
1 2
3 s
logPL V , I ; Normal g q g V q g log I q g V log I,s
2
2
Ž .
Ž .
Ž .
1 2
3 r
s
2
f 0.08
s
s
2
f 0.06
r
Ž .
Fig. 3. Distribution and a Box Plot of daily differences Ds log K r K between on-site correction values
2 1
Ž .
K and off-site corrections K , 50 km away. All data N s1268 have been used.
2 1
Statistical analysis of the difference between the off-site K and on-site K
should
1 2
Ž .
Ž .
therefore be based on analysis of log K , log K rather than the raw correction values.
1 2
Two separate issues are of interest when evaluating the general difference between Ž .
K and K : 1 will off-site a , V, T, I information result in systematically biased
1 2
Ž . correction values compared with what would be estimated locally? and 2 can K
2
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 Ž
. Ž
. D s log K y log K
is displayed in Fig. 3.
2 1
Ž 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 median s 0.02046,
mean s 0.02454 and standard deviation s 0.087422. 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
Ž . correction model 1 . From the development of the model final estimates of the residual
errors s
2
s 0.06 and s
2
s 0.08 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 Ž
. systematically at a wrong numerical level bias .
dence between the K and K values. For values of snow fraction a close to 0.00 or
1 2
Ž . 1.00 confidence limits for K a
relate simply to properties of fit for each of the Ž
. Ž .
Ž . Ž
. sub-models PS V,T
snow and PL V, I rain .
A suitable measure for evaluating the discrepancy between K and K can therefore
1 2
Ž Ž
.. Ž
be based on the 95 level prediction intervals Pred log PS V,T
and Pred log
95 95
Ž ..
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 Ž
. 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 K
1 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 the
s r
Ž .
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
Ž .
test Wilcoxon of the mean value D s 0.024536 both show significant positive devia- tion in favor of the on-site values K
3
using s
2
s 0.07 as an average value for the
1
Ž .
residual error for mixed precipitation. Transformed to a normal distribution approxi- mate 10–90 width for the stochastic errors, it results in 2 = 1.28s s 0.67. This
interval is on the same width if the empirical variance s
2
s 0.087422 is used for D. A conclusion for this example would therefore be that the off-site values are biased
Ž .
approximately 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:
w x
y0.02791,q 0.09061 , i.e., of length s 0.12, or of length only 20 of the s-based length.
The judgment of general deviation between K and K , viz. the judgment of spatial
1 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
Ž .
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
a s snow fraction, V s wind speed, T s temperature and I s rain 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
The use of t-test will be discussed later.
3. Results
