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


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 

                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 

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 

                                 Charles W. Boning
                                 Chief, Office of Surface Water