WMOD00547

• Title: On the Sampling Variance of Raingage Networks
• Date: 12/12/1981
• Weather Modification - Doc Type: Report

1470

JOURNAL OF APPLIED METEOROLOGY

VOLUME 20

curacy of point rainfall measurements by comparing
the measurements of collocated gages and found that
the percentage uncertainty (maximum rain differ-
ence divided by maximum point rainfall) is - 5% for
maximum rainfalls near 25.4 mm, increasing to 12%
for maximum rainfalls of 2.54 mm. He felt that this
uncertainty was random variability probably due to
subtle differences in gage exposure. Huff (1955)
compared the measurements of gages 6 ft ( -1.8 m)
apart and found a relative variability (average dif-
ference divided by areal mean rainfall) of 4% for
mean rainfalls of 0.25-5 mm, decreasing to 1% for
rainfalls> 12.70 mm. Since great care was devoted
to gage exposure, gage maintenance, and observation
techniques, Huff concluded that these differences
were the minimum to be expected in shower-type
rainfall. Both Woodley et al. (1975) and Huff ( 1955)
found that the variability tends to decrease with in-
creasing rainfall.
In view of the potential magnitude of measure-
ment-related errors, it is difficult to conceive of an
approach to estimate the magnitude of local, storm-
related random variability and to estimate its con-
tribution to the sampling variance of raingage net-
works. The larger the local random variability is, the
dense,r the gage network needs to be to achieve a
desired level of accuracy in estimating total rain vol-
ume. According to the framework of this paper, we
can consider that the local variability is caused by
perturbations of a scale much smaller than that of
a raincell and that they are randomly superimposed
on the storm and raincell isohyets. For the purposes
of this paper, we assume that these small-scale per-
turbations do not represent meaningful components
of the total rainfall of raincells, and basing the anal-
ysis on raincells is valid. Since there is only limited,
indirect evidence to support this assumption, the im-
portance of local random variability is still an open
question which must be left to future additional work
to resolve.

3. Procedure

Area-depth and isohyet-area formulas relate the
average precipitation depth inside an isohyet, or the
value of the isohyet itself, to the area within (or dis-
tance to) the isohyet. They are usually obtained
graphically from isohyetal maps. A number of area-
depth formulas have been proposed (e.g., Court,
1961; Huff, 1968a; Fogel and Duckstein, 1969)
which are based on various powers of storm area, its
logarithm or its exponential, with each implicitly
having a unique precipitation gradient. The isohyet-
distance formula used in this work to model raincells
is different from previously proposed formulas in that
it is not based on any particular set of data. The
approach in this paper is more general in that more
degrees of freedom are permitted to independently

express the various characteristics of a surface rain-
cell, namely shape, area, location of precipitation
maximum and, in particular, precipitation gradient.
A model isohyetal pattern for a convective raincell
is shown in Fig. 2. The spatial distribution of pre-
cipitation for such patterns is given by ,

P(x) = PM[0.5 + 0.5 cos(1l'rx/ro)]G, (1)

where P(x) is the precipitation amount at point x;
PM is the maximum precipitation amount at point
M (not necessarily at the geometrical center of the
pattern); 'x is the distance from point M to point x;
ro is the distance from M to point 0 (on the zero
isopleth) through point x; and G is the spatial gra-
dient index. The location of M is externally defined
by the parameters X Offset and Y Offset, the co-
ordinate offset distances as proportions of the
semiminor and semimajor axes, respectively. The
shape of the pattern is determined by the parameter
E (labeled ellipse in Fig. 2), the ratio of major to
minor axes of the elliptical pattern. To facilitate gen-
erality of the results, the major axis of the isohyetal
pattern is arbitrarily scaled to always be 16 km (10
mi) long.
Fig. 3 shows the relationship between G and Aso,
the percentage of raincell area containing 50% of the
total precipitation volume (labeled half area in Fig.
2). It can be seen that Aso is a nonlinear function of
G. A value of G equal to 0.6 (Aso = 25) corresponds
to an approximately linear spatial gradient of pre-
cipitation. Both parameters are carried throughout
the paper because Aso is an easily derived value from
isohyetal maps and, by use of Fig. 3 can be converted
to G which, as we shall show later, is related to the
sampling variance of raingage networks.
A cursory check was made on the suitability of
simulating convective raincells by Eq. (1), which for
the purposes of the following comparison is referred
to as modell, by fitting an exemplary Montana rain-
cell by a nonlinear least-squares-regression method
and comparing the model-derived rainfall volume
with that calculated from the point raingage data
using the polygon method of Thiessen (1911). The
same procedure was followed for other proposed iso-
hyet models based on logarithmic and exponential
functions of area, i.e., model 2: P(x) = Pm exp( -kAn)
and model 3: P(x) = Pm(1 - kAn). The results of
this comparison are shown in Table I. It can be seen
from Table 1 that, for the case shown, both models
I and 2 are reasonable approximations of the ob-
served rainfall pattern, with model 2 being slightly
better. In other rainfall patterns examined, model 1
fit the data best. Therefore, we shall proceed with
the analysis of sampling variance using model 1 but
it should be recognized that apy model that reason-
ably fits the data could be used for this analysis, and
the results (discussed in Section 7) with respect to

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