All of these computations are nested, with

ness as the primary random variables. The mean

of *H*ia 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 *Q*ice. With *Q*st and *Q*ice

known from eq 7 and 4, and *Q*in and *Q*out known

presented by Ashton (1989). Ice density and ther-

from gage records, we can obtain *Q*sub with eq 8.

mal conductivity are correlated random variables,

Finally, tributary inflows are used with *Q*sub 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 *Q*gw

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

333,400 J/kg (Ashton 1986). Data were unavail-

Mean values can now be determined for *Q*ice,

able to quantify the snow depth on the ice. How-

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, *h*s was assumed to be negligible.

channel width. In addition, many of the measured

In computing *Q*ice 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 *Q*st 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 *Q*st is generally negli-

ice heat transfer coefficient, and the initial ice thick-

7

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