3. Results
The 12 stations are situated in Denmark so that typical regions are represented. The Ž
. 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 Ž
. act as off-site stations. Twelve charts fixed on-site station are derived, each one
Ž .
Ž .
containing eleven groups of D s log K rK s log on-siteroff-site ; each D-group
2 1
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
Ž .
month . 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
Ž .
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.
Ž .
Ž .
Fig. 4. 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.
Ž 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. Ž
. 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 from s
2
s 0.07, Ž .
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 K
correction values. However,
2
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
Ž . covered by the 0.25 limits derived from the model 1 .
Ž .
The D-group marked at distance s 9.6 km between stations 29451 and 29439 attracts special attention because of the short interdistance. The median value is 0.00,
mean s 0.02 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
Ž .
Ž .
Fig. 5. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of wind speed sampled off-site.
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 variables a , V, T, I. Fig. 5 is largely a repetition of Fig. 4, indicating that wind speed s V is the variable with the
Ž most marginal influence in Fig. 4. In fact, the values of b
and g , regression
1 1
. coefficients to wind speed result in high relative marginal changes of the correction
Ž .
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 Ž
. 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: median s 0.01, mean s 0.02 and 25–75 percentiles s 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
Ž . smaller impact on the correction value through the g -parameter of Eq. 1 and with the
2
Ž .
Ž .
Fig. 6. General distance relations. Difference Dif, y-axis vs. distance x-axis between on-site and off-site correction factors in case of rain intensity sampled off-site.
Ž .
Ž .
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
Ž .
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 are y0.02, 0.02.
The effect of measuring temperature s T 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
Ž .
calculations of 25 and 75 percentiles and thereby the median are equal to 0.00 on Ž .
second decimal place. The relative influence on the corrections calculated in Eq. 1 through b , b
and g ,g is not small, but here Fig. 7 reflects generally consistent
1 3
1 3
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 fraction s a 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
Ž
. non-parametric Wilcoxon is that of an inconsistent distance relation. In fact, D-groups
can be accepted to be centered around zero at various distances with clear rejections in
Ž .
Ž .
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.
Ž .
Ž .
Fig. 9. 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 -y18C are considered.
Ž .
Ž .
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
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