Adjustment of Discharge Measurements Made at a Distance from the Gaging Station During Periods of Changing Stage and Discharge
In Reply Refer To: June 17, 1992
WGS-Mail Stop 415
OFFICE OF SURFACE WATER TECHNICAL MEMORANDUM 92.09
SUBJECT: Adjustment of Discharge Measurements Made at a Distance
from the Gaging Station During Periods of Changing Stage
and Discharge
Stage-discharge relations at streamflow gaging stations are
defined by correlating recorded gage heights (stages) with
measured discharges. Practical considerations frequently require
that the discharge be measured at some place other than the gage.
When the flow is steady (unchanging in time) and there are no
lateral or tributary inflows, the measurement may be made at any
convenient location, because the flow also is constant with
respect to location along the stream course. When the flow or
stage is changing with time, however, the flow also is varying
from place to place along the stream. In this case, if the
measurement cannot be made at the gage, an adjustment must be
applied to make the measured flow and the recorded gage height
hydraulically comparable.
Procedures for determining and applying such adjustments are
described in Techniques of Water-Resources Investigations (TWRI)
Book 3, Chapter A8 (p.54-57), in Water Supply Paper (WSP) 2175
(p.177-179), and in a former stream-gaging manual, WSP 888 (p.79).
The purpose of this memo is to explain the hydraulic basis for
these procedures, to provide guidance for their practical
application, and to correct an error in the description of one of
the procedures.
The fundamental hydrologic principle that governs this adjustment
is the equation of continuity or hydrologic storage equation for a
reach of stream channel between the measurement section and the
gage:
storage change = inflow - outflow.
As explained in the TWRI and in WSP 2175, the change of storage in
this reach during a measurement can be computed directly from
knowledge of reach length, top width, and stage changes at the
upstream and downstream ends of the reach. If there are no
lateral or tributary inflows, the inflow and outflow are the
measured discharge and the discharge at the gage; which one is
which depends on whether the measurement is made above or below
the gage. In either case, the equation can be solved for the flow
at the gage as follows:
Qg = Qm + WLdh/dt
where Qg is the flow at the gage, Qm the measured flow, dh/dt the
average rate of change of stage at the gage and measurement
sections during the measurement, W the average channel top width,
and L the distance from the gage to the measurement section,
measured positive in the direction of flow. Both dh/dt and L have
algebraic signs: dh/dt is negative if the stage is falling,
positive if rising; L is negative if the measurement is made
upstream from the gage, positive if downstream. This equation,
computed with the rules for arithmetic with signed quantities, is
equivalent to the sign rules given in the TWRI (p. 55-56) and in
WSP 2175 (p.177) for the storage-change adjustment.
The storage-change adjustment is based on the assumption that the
storage change volume is adequately approximated by a rectangular
slab of length L, average width W, and average height dh.
Suitable average values of W and dh sometimes are not known with
certainty. In this case, alternative methods of adjustment may
be sought.
An alternative adjustment method is based on the general
observation that, in upland (non-tidal) stream channels of
generally uniform geometry and not excessively affected by
backwater or lateral or tributary inflows, streamflow peaks
normally move downstream by pure translation, without appreciable
change in shape, over moderate distances. That is, the discharge
profile (plot of discharge versus distance along the channel) at a
time t+dt can be found by translating the profile at time t a
distance L downstream, where L = cdt, and c is the speed of
propagation (celerity) of the flood wave. If the wave celerity is
known, a discharge measured at a particular time and location can
be related to the discharge (and gage height) that occurred at the
gage at an earlier or later time.
The speed of the flood wave, c, can be determined by direct
measurement of travel times, but this is not usually feasible
unless discharge or stage hydrographs are available at upstream
and downstream sites. A theoretical estimate may be obtained by
applying the continuity equation to a short segment of the rising
limb of the flood wave. Because the segment is moving along with
the wave, and the wave shape is not changing, the volume of water
in the segment also is not changing. Thus, the continuity
equation for the moving segment is:
0 = (V + dV - c)(A + dA) - (V - c)(A)
where c is the velocity of the wave (and the moving segment), V
and A are the water velocity and area at the downstream face of
the segment, and V+dV and A+dA are the area and velocity at the
upstream face. This equation can be solved for c:
c = dQ/dA = (dQ/dh)/W = f'/W
where dQ = (V+dV)(A+dA) - VA, h = gage height, and W = mean top
width. The quantity f' = dQ/dh is the slope, in cubic feet per
second per foot of stage rise, of the stage-discharge rating curve
drawn on arithmetic (rectangular) plotting paper. This derivation
of the wave speed can be found in many standard open-channel
hydraulics texts under the heading of monoclinal or uniformly
progressive wave. The result is often called Seddon's (Trans.
ASCE, 1900) principle. It also can be derived by similar analysis
of the receding limb of the hydrograph.
Although the quantity f' = dQ/dh is defined by usual gaging-
station data, a suitable value for top width W, as previously
noted, is not always readily determinable. However, under
channel-control conditions, Manning's or Chezy's formula can be
used with a suitable cross-sectional shape to approximate the
stage-discharge rating curve. The quantity (dQ/dh)/W can be
evaluated for such a theoretical rating, and the wave speed c can
be expressed as a factor times the mean water velocity. As shown
in WSP 2175, p.415, the ratio of wave speed to water speed ranges
from about 1.3 to 1.7 depending on the cross section shape and
choice of Manning or Chezy formula. The value of 1.3 seems to be
the most probable (WSP 2175, p. 415, TWRI, p.57).
If the wave velocity c is known, a discharge measurement made at a
distance L downstream from the gage during a period of changing
stage can be adjusted as follows. The segment of the flood wave
that was measured at the measuring section at time t passed by the
gage at time t-dt, where dt = L/c, and thus is hydraulically
associated with the stage recorded at that time. It is immaterial
whether the stage is rising or falling (i.e., whether the
measurement is made on the rising or the falling limb of the flood
wave). A similar analysis applies to measurements made upstream
from the gage. All cases can be summarized by the formula:
Qm(t) = f( h(t - L/c) )
in which Qm(t) is the discharge measured at the measurement
section at time t, h(t-L/c) is the recorded gage height at time t-
L/c, c is the wave speed, L is the distance from the gage to the
measurement section, measured positive in the direction of flow,
and f is the stage-discharge relation, which expresses the
discharge at the gage at any time as a function of the
corresponding stage h.
The above equation implies that Qm(t), the discharge measured at
time t, is to be plotted on the stage-discharge rating curve in
association with h(t-L/c), the stage observed at time t-L/c.
Under conditions of changing stage, the stage plotted on the
rating is computed as a discharge-weighted average of stages
observed at selected times ti' during the course of the
measurement. For measurements made at a distance from the gage,
the weighted stage can be computed as a discharge-weighted average
of the time-adjusted recorded stages, h(ti'-L/c). Alternatively,
a weighted gage height hw can be computed in the usual way without
any time adjustment, such that hw = weighted average of the
h(ti'); the time of occurrence, t", of hw then can be looked up in
the stage record, adjusted by L/c, and used to read off the time-
adjusted weighted gage height from the stage record as h(t"-L/c).
This computation can be summarized by the formula
ha(t) = h(t" - L/c) = hw - (L/c)h'
in which ha(t) is the time-adjusted gage height associated with
the measured discharge Qm(t), hw is the weighted mean gage height
omputed without time adjustment, h' = dh/dt is the average time-
rate of change of stage between times t" and t"-L/c, and L and c
are as defined previously.
The travel-time and storage-change adjustments are alternative
methods for accounting for the single fact that the flow at the
measurement site is not the same as the flow passing the gage at
the time of measurement. A measurement may be adjusted by one
method or the other, but not both. The two methods can be
illustrated and compared by an example. A measurement was made on
a falling stage at a site 3.3 miles upstream from the gage.
During the approximately 2-hour measurement period, the stage
dropped at a rate of 1.0 ft/hr at the measurement site and at 0.3
ft/hr at the gage. The measured discharge was 43,610 cfs, the
weighted-mean gage height (without travel-time adjustment) was
8.76 ft, and the mean velocity (Q/A) was 8.8 ft/s. The average
top width of the channel, scaled at 10 points on a topographic
map, was 340 ft. The slope of the stage-discharge relation, which
plots as a straight line between 8 and 10 ft on rectangular
coordinates, is 8500 cfs/ft. The adjustment by the storage-change
method is as follows:
Qg = Qm + WLdh/dt
= 43,610 + (340)(-3.3x5280)(-(1+.3)/2)/(1x60x60)
= 43,610 + 1070 = 44,680 cfs (GH = 8.76 ft)
For adjustment by the travel-time method, two estimates of the
flood wave speed are available: c = 1.3V = 1.3x8.8 = 11.4 ft/s =
7.8 mph, and c = f'/W = 8500/340 = 25.0 ft/s = 17.0 mph. The
corresponding calculated gage-height adjustments are:
ha = hw - h'L/c = (8.76) - (-0.3)(-3.3)/(7.8) = 8.76 - 0.13 = 8.63 ft
ha = hw - h'L/c = (8.76) - (-0.3)(-3.3)/(17.0) = 8.76 - 0.06 = 8.70 ft
For wave speed c = 1.3V (travel time 25 min), the discharge-
weighted mean of time-adjusted recorded stages is 8.67 ft; the
recorded gage height 25 min after the time of occurrence of the
unadjusted weighted gage height was 8.68 ft. For wave speed c =
f'/W (travel time 12 min), the corresponding gage heights were
8.72 and 8.70 ft. These values differ from the above ha-values
because the rate of change of stage was not perfectly uniform.
The adjusted gage heights are plotted against the measured
discharge (Qm = 43,610 cfs) on the rating curve. The results may
be compared with the rating curve as follows:
Method GH (ft) Q (cfs) Qr (cfs) dQ/Qr (pct)
Storage 8.76 44,680 42,540 5.0
Travel (1.3V) 8.63 43,610 41,440 5.2
Travel (f'/W) 8.70 43,610 42,030 3.8
Travel (wtd 25) 8.67 43,610 41,780 4.4
Travel (t"+25) 8.68 43,610 41,860 4.2
Travel (wtd 12) 8.72 43,610 42,200 3.3
Travel (t"+12) 8.70 43,610 42,030 3.8
These results illustrate that the travel-time and storage-change
adjustments give generally comparable results. For a perfect
translatory wave in a uniform channel the travel-time adjustment
(with c = f'/W) and the storage-change adjustment yield the same
difference between the plotted measurement and the rating curve.
The results also illustrate, however, that the travel-time
adjustment is sensitive to judgements and assumptions that may be
hard to verify and subject to dispute. The difference in rate of
change of stage at the gage and measurement sections in this case
introduces additional uncertainty. The storage-change adjustment,
on the other hand, involves only straightforward calculations on
measurable data. Therefore, as stated in the TWRI, p.57, the
storage-change method generally is preferred; WSP 2175 does not
even mention the travel-time adjustment for discharge
measurements.
The formulas given in this memo for adjustment of measured
discharge, time, and stage are computed using the normal rules of
algebraic signs, where L is a signed quantity, positive if the
measurement is made downstream from the gage, negative if
upstream. The computation of corrected times does not depend on
whether the stage is rising or falling. The computation of
corrected gage height, ha(t), involves in addition the signed
quantity h', which is negative if the stage is falling and
positive if the stage is rising. The standard rules of arithmetic
with signed quantities yield the proper signs for the adjustments.
The discussion of the travel-time adjustment in the TWRI and WSP
888 includes a verbal rule for determining the algebraic sign of
the correction to times observed at the stage recorder. This
rule, which includes consideration of whether the stage is rising
or falling, is incorrect. (The rule would have given the correct
sign for adjustment to gage height, but does not give the correct
sign for time.)
To prevent any future confusion or error, the rule for application
of the travel time adjustment should be corrected on page 57 of
the TWRI to read as follows:
In applying the time adjustment, the time of travel is
subtracted from the observed time at the measurement section
if the measurement is made below the gage and is added if
the measurement is made above the gage. The resulting
adjusted time is used to look up the corresponding gage
height in the gage-height record. (Corrected per OSW
Technical Memorandum 92.09, June 17, 1992.)
A similar correction should be made to the corresponding statement
on page 79 of any copies of WSP 888 that may still be in use. In
addition, necessary steps should be taken to ensure that all
personnel who make, use, or review discharge measurements are
familiar with the priciples and computations summarized in this
memo.
Charles W. Boning
Chief, Office of Surface Water
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