WMOD00547

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

DECEMBER 1981

SILVERMAN, ROGERS AND DAHL

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dient [P(x) oc kA where A is the area within the
isohyet of interest and k is a constant] is most fre-
quently found with steady~type rains in the colder
part of the year, an increasing gradient toward the
storm center [P(x) oc kAI/2] most commonly occurs
in midwestern storms, and a very steep gradient
[P(x) oc exp( -kA1/4)] occurs in single-celled air
mass storms of strong intensity but small areal ex-
tent. The tendencies suggested by Court and Huff
are not obvious in the data derived from the area-
depth formulas. The lack of consistency may, in part,
result from the varying densities of the gage networks
used in the various studies and from the subjective
construction of the isohyetal maps. Also, the fit of
the area-depth formulas to the data may only be
approximate. The data derived graphically from the
isohyetal maps, however, do tend to support the gen-
eralizations of Court and Huff.

6. Comparison with previous studies

We now compare the results of this study with
those of other investigators who have considered this
problem. As stated earlier, previous investigators
(Light, 19472; Linsley and Kohler, 1951; Mc-
Guinness, 1963; Huff, 1970; Woodley et aI., 1975)
defined the average error in determining areal mean
rainfall as the difference in rainfall calculated from
their highest gage density, assumed to be the "true"
rainfall, and that calculated from subnetworks of
lesser density, They related these errors to some or
all of the following parameters: mean precipitation,
gage density, area, .and storm duration. There are
two points to be considered in light of the results of
this study, namely the accuracy of their assumed
"true" rainfall and the meaning and utility of the
empirically determined sampling error relationships.
Light derived his empirical relationship for areas
in the range 500-8000 mi2 (1295-20720 km2) with
a primary gage density of 18 mi2 (47 km2) per gage.
Linsley and Kohler derived their error relationship
for a fixed area of 220 mi2 (570 km2) and a basic
gage density of 4 mi2 (10 km2) per gage. McGuinness
used data from a 7.16 mi2 (19 km2) watershed having
a gage density of 0.11 mi2 (0.28 km2) per gage to
derive his mean rainfall accuracy expression. Huff
derived his comprehensive error function with data
from gage networks covering 400 and 550 mi2 (1036
and 1424 km2) having gage densities of 8 and 11 mi2
(21 and 28 km2) per gage, respectively. Woodley et
al. (1975) derived their error curves for an area of
220 mi2 (570 km2) having a gage density of 1.18 mi2
(3.1 km2) per gage. The maximum gage densities
used by these investigators were examined in the
context of Eq, (2). The coefficient of variation for
the largest number of gages per storm (maximum
gage density for the area gaged) and the precipitation
gradient of the storms (Table 2) considered in each
study was calculated. These results indicate that,

"I

although the assumption that the "true" rainfall is
represented by the highest gage density becomes
weaker as the precipitation gradient becomes in-
creasingly steeper and the number of gages per storm
decreases, the maximum gage densities in all of these
, studies was sufficiently high to render this assump-
tion valid for practical purposes. The worst case re-
sulted from the smallest area investigated by Light
(GPR = 27.8 and G = 1.71) for which the coefficient
of variation was <6%.
The second issue involves the interpretation of the
average absolute errors derived from the empirical
relationships. In Section 4 it was pointed out that
the sample mean precipitation is always unbiased but
the sample standard deviation increases with de-
creasing gage density or increasing precipitation gra-
dient. This, of course, is the case because we have
been able to simulate all possible samplings of the
storm by the gage networks. This was not the case
in the data sets used to construct the empirical re-
lationships. These relationships were based on a fi-
nite, insufficient number of samples which randomly
fell on the spectrum of possibilities. Huff (1970) rec-
ognized this limitation when he stated, "Except for
mean precipitation, duration, and average intensity,
it is difjicult, if not impossible, to express these sam-
pling error factors in quantitative terms. Thus, unless
huge samples are available to permit grouping of the
data according to all of these various factors, the
sampling error with a given gage density in a par-
ticular storm cannot be defined with a high degree
of accuracy." He supported this point by showing
the great amount of variability about the average
sampling error from his empirical relationship. It is
likely that the values of average sampling error are,
in reality, part of this sampling variance and not
statistical bias. In any given situation, the error could
be in any direction and of any magnitude. The em-
pirical relationships do not cover the full spectrum
of possibilities. In this paper we have proposed a sta-
tistical solution through model simulation studies.

7. Discussion

The quantitative results of this study of the sam-
pling variance of precipitation gage networks has
many potential hydrometeorological applications.
Consider, for example, its application in estimating
the sample size requirements in evaluating precipi-
tation augmentation experiments. The sample size
requirements for a 50/50 randomized precipitation
augmentation experiment may be calculated from
the following expression

N = 4(ZI-<> + ZI_p)2S2 (3)
D2

where D = dx; normal distribution, loge 1 + d); log-
normal distribution and N is the total sample size,