WMOD00547

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

DECEMBER 1981

SILVERMAN, ROGERS AND DAHL

1473

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limitations in estimating raincell area by summing
grid subareas with nonzero gage precipitation and
2) the inability to determine and account for the
precipitation gradient in estimating the subarea pre-
cipitation amounts. The effects of both shortcomings
tend to be amplified as the gage density decreases.
When the precipitation gradient is approximately
linear, the fit of the distribution of total precipitation
estimates in' a sample set to a normal distribution is
quite good. When the precipitation gradient is non-
linear, the distribution is bimodal with about equal
probability of over- and underestimating the true
precipitation amount, depending on whether the par-
ticular gage configuration locates a gage(s) in the
high-precipitation region of the pattern or not, re-
spectively.
The sampling variance numerical experiment gen-
erated a total of 530 data points-five gage densities
for each of 100 isohyetal patterns and two gage den-
sities for each of 15 isohyetal patterns. Before re-
lating the sampling variance to the isohyetal pattern
characteristics, the input parameters were general-
ized by removing the scale factors. Thus, the pattern
shape E, which implicitly defines the raincell area
in the simulations, was combined with the appro-
priately scaled gage network densities to form the
more general variable called GPR, the number of
gages per raincell. Thus, for example, a raincell that
has an area of 100 km2 (E = 2) and is sampled by
a gage network with a density of 10 km2 per gage
(approximately 4 mi2 per gage) will have an average
GPR = 10. The range of variation of GPR corre-
sponding to the range of variation of E and gage
density used in this study is 0.157 to 714.0. In ad-
dition, the maximum precipitation point offset from
the geometrical center, originally expressed as X off-
set and Y offset, was vectorially combined into a
single variable which was called F.
Using the coefficient of variation six as the de-
pendent variable, a linear regression model was fitted
by a stepwise routine to the independent variables.
Since six -+ 0 as GPR -+ co, regression was forced
through the origin and the independent variables
used in the regression were GPR, powers and roots
of GPR, and combinations of the products of GPR
and its power functions with G and F and their power
functions. The following regression equation was ob-
tained as the best fit over the range of variation of
, the input data:

six = 0.71051GPR + 0.5079GIGPR

- 0.1381G/GPR2 + 0.012IG/GPR3

- 0.0531/GPR2. (2)

The terms in Eq. (2) are presented in the order with
which they contribute to the explanation of the vari-
ance of six, The multiple correlation coefficient for
this equation is 0.992 such that 98.4% of the variance

of six is accounted for by the regression equation.
The first term in the equation alone accounts for 80%
of the variance. Thus, it can be seen that the data
are very well fitted by Eq. (2). The coefficient of
variation is primarily a function of the number of
gages per raincell and secondarily, but importantly,
a function of the spatial precipitation gradient. It
should be noted that the coefficient of variation is
independent of the location of the precipitation max-
imum in the isohyetal pattern.
According to random sampling theory, the stan-
dard deviation (and coefficient of variation) should
be inversely proportional to the square root of GPR.
Independent variables involving this term were,
therefore, included in the search for an appropriate
regression equation. The term GPR-O.5 was not se-
lected early in the stepwise regression although it
alone (six = L429GPR-o.5) had a correlation coef-
ficient of 0.916. However, the first term selected,
(GIGPR)O,5, included this functional relationship [sf
i = 1.119(GIGPR)O,5] and had a correlation coef-
ficent of 0.978. ,In fact, the regression equation six
= 0.8659(G/GPR)O,5 + 0.2549/GPR fit the data
quite well and had a multiple correlation coefficient
of 0,990. This equation was not selected as the best
fit of the data because it had a higher standard error
of estimate than Eq. (2) and it systematically over-
estimated the values of s / x at very high values of
GPR. Attempts to improve the fit by the inclusion
of additional terms resulted in an equation which
predicted negative values of s / x for some of the high
values of GPR. Gabriel (1981) evaluated the relative
variability of simple random sampling from single
cell storms defined by Eq. (I) and indicated that it
provided an upper bound to the corresponding rel-
ative variabilities from systematic uniform grid sam-
pling given by Eq. (2). His comparison of the relative
variabilities from the two network sampling methods
revealed that Eq. (2) did, indeed, give smaller values
of the network sampling standard deviation for all
values of GPR ~ 1 but gave higher values for some
values of GPR < L The reason for the relative be-
havior of the results from the two sampling methods
is probably due to the way the area of the raincell
is treate:d. In Gabriel's evaluation of simple random
sampling, the area of the raincell is explicitly taken
into account in his analytical solution. His equation
for the network sampling standard deviation is a
function of his variable b, the proportion of the region
gaged that is covered by the area of the rain cell. In
this study, on the other hand, all information on the
characteristics of the raincell are deduced from only
the gage catches in accordance with usual observa-
tional procedure. As was mentioned earlier in this
section, the network sampling standard deviation is
caused, in part, by the limitations in estimating the
raincell area from the gage network data. The esti-
mates of raincell area become increasingly less ace