WSP11764

• File Number: 8273.100
• Description: Colorado River Basin Salinity Control - Federal Agencies - Bureau of Reclamation
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 1/1/1984
• Author: BOR
• Title: Water Quality of the Colorado River System - Historical Trends in Concentration/ Load/ and Mass Fraction of Inorganic Solutes
• Water Supply Pro - Doc Type: Report/Study

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w

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0.10
0.10

1,0
, Q
-=--..-.
r kA

Figure 5. - Residence time equation on 10g~log scale. (The
power equation would yield a straight line with the slope
equal to the power term b2 in equation 6.)

stations on the Colorado River (see tables 4 and 5
in appendixes B through 0). Nonlinear regressions
of equations 6 and 11 used the SPSS (Statistical
Package for the Social Sciences) subprogram
NONLINEAR [11, 12]with the Marquardt algorithm
[13]. Values only from periods of minimum flow
regulation upstream of the gaging stations are
included in the regression analyses. By dividing
the year into three seasons, assumptions 2 and 4
forthe residence time model imply that b, and b2 in
equation 11 are constant only for one season.
Although the solute solubility parameter (b, in
equation 11) may be independent of the season,
seasonal values were estimated to maintain the
same degrees of freedom used in the power
regression.

Because precision in the chemical analyses is
generally proportional to the magnitude of the
solute concentrations, a weighting factor of Cm -, is
used in the nonlinear regressions. With this
weighting factor, the RMS (root-mean-square)
error, or standard deviation of the residuals, is
dimensionless and becomes the fractional standard
deviation. For the weighted regression, the RMS
error is defined as:

(f ( Cj.Ob~~ Ci.pre,,\2)O'5
J=1 _ j,ObS-1.
RMS error = n-k (12)

where:

Ci.obs = Observed monthly value of
concentration

Cj.pred = Monthly value of
concentration estimated from
streamflow by the power or
residence time equation
n = N umber of observations
k = Number of estimated
parameters (k = 2 for
equations 6 and 11)
n-k = degrees of freedom

An RMS error of 0.1 0 indicates(assuming random
normally distributed residuals and greater than 30
degrees of freedom) an approximately 68-percent
probabilitythatthe estimated and observed values
of concentrations differ by less than 10 percent.

10.0

Although complete model ev.aluation requires
examination of the residuals for any systematic
bias in estimating concentration, residual exami-
nation is omitted from this report because of the
large number of regressions. A preliminary com-
parison ofthe accuracy ofthe models can be made
from the RMS errors summarized in table 2. The
missing values in table 2 result from difficulties in
computing that regression. Based on the RMS
errors, the power equation is more accurate than
the residence time equation for the peak flow
months of May and June. For the other two
seasons the RMS errors differ little. Neither equa-
tion is accurate for the Dolores River, near Cisco,
Utah, where the RMS errors are generally greater
than 0.20. Both equations are used in the multiple
regression analyses in the next section to estimate
long-term changes in concentration.

The regression results of the power equation (see
table 4 in appendixes B through 0) indicate that
for several solutes at several locations, the stream-
flow has only a minor effect on the concentration.
Based on the estimated values of the power term
(b2 in equation 6) that are greater than -0.20, the
bicarbonate concentration is relatively indepen-
dent of the streamflow for all stations (see table 3).
By the same criteria, the calcium concentration
varies little with the streamflow at 7 of the 12
stations.

LONG.TERM CHANGES FOR
PERIODS OF MINOR FLOW
REGULATION

The object of this sectioh is to determine whether
the monthly values of concentration, load, or mass
fraction of any of the six major ions (calcium,
magnesium, and sodium cations; and chloride,

10