All of these computations are nested, with
ness as the primary random variables. The mean
of Hia was taken as 20 W/m2 C with a standard
changes in ice thickness obtained for a given month
deviation of 5 W/m2 C, representative of the data
and then used to find Qice. With Qst and Qice
known from eq 7 and 4, and Qin and Qout known
presented by Ashton (1989). Ice density and ther-
from gage records, we can obtain Qsub with eq 8.
mal conductivity are correlated random variables,
Finally, tributary inflows are used with Qsub to
but their variability is minor. The mean ice den-
sity was taken as 900 kg/m3 with a standard de-
obtain the groundwater discharge or recharge Qgw
from eq 5.
viation of 15 (Mulherin et al. 1992). The mean
thermal conductivity of ice was taken as 2.17 W/
m C with a standard deviation of 0.1, and the
latent heat of fusion was assumed constant at
POINT ESTIMATE METHOD
333,400 J/kg (Ashton 1986). Data were unavail-
Mean values can now be determined for Qice,
able to quantify the snow depth on the ice. How-
Qst, Qsub, and Qgw, but they would not account
ever, because the overflow of water on grounded
for the known variability of input parameters such
ice often incorporates the snow into the ice sur-
as air temperature, heat transfer coefficient, and
face, hs was assumed to be negligible.
channel width. In addition, many of the measured
In computing Qice with eq 4, the independent
or estimated parameters contain uncertainty that
random variables are river distance between the
contributes to the uncertainty of the correspond-
gages, channel width at each gage, and correlated
ing dependent variable. We will use the Rosen-
initial and final ice thicknesses. The estimated
blueth (1975) point estimate method (PEM) to ac-
mean river distances between the gages are given
count for and quantify the uncertainty in our de-
in Table 1. Based on multiple trials, the measure-
terministic winter water balance. The indepen-
ment error in obtaining these distances from maps
dent variables in each deterministic equation that
was about 2% of the distance, and in addition, the
contain uncertainty are considered random vari-
movement of the river within the floodplains could
ables. The first two or three moments of each ran-
alter the distances from those shown on the maps
dom variable and the correlation coefficients be-
by a few percent of the length. Therefore, we as-
tween variables are given as input, quantifying
sume a coefficient of variation of 0.05 for reach
the variability or uncertainty. The PEM provides
length. Ice formation causes the flow in the wide,
the mean, variance, and limits of the dependent
flat channel to shift its location. Ice then freezes to
variable, uniquely specifying a beta distribution
the bed in nearshore and bar areas, and water
(Harr 1977) that describes the uncertainty of a
flow is restricted to only a portion of the apparent
function of random variables. The estimated mean
width. The channel width of the ice/water inter-
value is equivalent to a second-order Taylor se-
face was measured by the USGS each time a dis-
ries approximation, and the variance is a first-
charge rating was done at a gage. The channel
order estimate. The method is algebraic, replac-
cross section and discharge at which these mea-
ing the distribution of each random variable by
surements were made varied. We used all avail-
point estimates and not requiring the computa-
able measurements during ice-covered flow con-
tion of derivatives. The PEM offers several ad-
ditions to obtain the mean width and its variance
vantages over a deterministic approach. A com-
near each gage. The mean width varied from 4 m
puted mean value has much greater importance
on the Little White River at Martin to 23 m on the
White River at Oacoma. The coefficient of varia-
when the variance is small, but variance is un-
tion of the river width varied between 0.11 and
known in a deterministic model. The random vari-
0.46. These data indicate that the width of the
ables contributing most of the uncertainty to the
White River at low flow can vary significantly
results can be readily identified, which can help
over short distances. Systematic analysis of aerial
to refine data collection. In addition, the interpre-
photographs of the river taken at low flow just
tation of PEM results is straightforward. For ex-
prior to ice formation or extensive ground mea-
ample, though a river reach may have positive
surements along the river would best quantify
inflow from its subbasin based on mean values,
there may be a significant probability that the flow
the distribution of width.
The Qst computation in eq 7 has reach length
direction is the opposite.
and stream width and correlated depth changes
We apply the PEM to the ice growth or melt in
at the ends of the reach as independent random
eq 2 or 3 by considering air temperature, the air
variables. We assume that Qst is generally negli-
ice heat transfer coefficient, and the initial ice thick-
7
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