Measurement of liquid flow in open channels — Part 2: Determination of the stage-discharge relation

Mesurage de débit des liquides dans les canaux découverts — Partie 2: Détermination de la relation hauteur-débit

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Second edition
Measurement of liquid flow in open
channels —
Part 2:
Determination of the stage-discharge relation
Mesurage de débit des liquides dans les canaux découverts —
Partie 2: Détermination de la relation hauteur-débit
Reference number
ISO 1100-2:1998(E)
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ISO 1100-2:1998(E)

1 Scope .....................................................................................................................................................................1

2 Normative references ...........................................................................................................................................1

3 Definitions and symbols ......................................................................................................................................2

4 Units of measurement ..........................................................................................................................................2

5 Principle of the stage-discharge relation ...........................................................................................................2

6 Stage-discharge calibration of a gauging station .............................................................................................5

7 Methods of testing stage-discharge relations ...................................................................................................18

8 Uncertainty in the stage-discharge relation.......................................................................................................18

Annex A Uncertainty in stage-discharge relation and in continuous measurement of discharge ................22

Annex B Bibliography ............................................................................................................................................25

© ISO 1998

All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic

or mechanical, including photocopying and microfilm, without permission in writing from the publisher.

International Organization for Standardization
Case postale 56 • CH-1211 Genève 20 • Switzerland
X.400 c=ch; a=400net; p=iso; o=isocs; s=central
Printed in Switzerland
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ISO ISO 1100-2:1998(E)

ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies

(ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical

committees. Each member body interested in a subject for which a technical committee has been established has

the right to be represented on that committee. International organizations, governmental and non-governmental, in

liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical

Commission (IEC) on all matters of electrotechnical standardization.

Draft International Standards adopted by the technical committees are circulated to the member bodies for voting.

Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote.

International Standard ISO 1100-2 was prepared by Technical Committee ISO/TC 113, Hydrometric determinations,

Subcommittee SC 1,
Velocity-area methods.

This second edition cancels and replaces the first edition (ISO 1100-2:1982), which has been technically revised.

Annexes A and B of this part of ISO 1100 are for information only.
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Measurement of liquid flow in open channels —
Part 2:
Determination of the stage-discharge relation
1 Scope

This part of ISO 1100 specifies methods of determining the stage-discharge relation for a gauging station. A

sufficient number of discharge measurements, complete with corresponding stage measurements, is required to

define a stage-discharge relation to the accuracy required by this part of ISO 1100.

Stable and unstable channels are considered, including brief descriptions of the effects on the stage-discharge

relation of ice and hysteresis. Methods for determining discharge for twin-gauge stations, ultrasonic velocity

stations, electromagnetic velocity stations, and other complex ratings are not described in detail. These types of

rating are described in other International Standards and Technical Reports, namely ISO/TR 9123, ISO 6416 and

ISO 9213, as shown in clause 2.
2 Normative references

The following standards contain provisions which, through reference in this text, constitute provisions of this part of

ISO 1100. At the time of publication, the editions indicated were valid. All standards are subject to revision, and

parties to agreements based on this part of ISO 1100 are encouraged to investigate the possibility of applying the

most recent editions of the standards indicated below. Members of IEC and ISO maintain registers of currently valid

International Standards.
ISO 31:1992 (all parts), Quantities, units and symbols.
ISO 772:1996, Hydrometric determinations — Vocabulary and symbols.

ISO 1000:1992, SI units and recommendations for the use of their multiples and of certain other units.

ISO/TR 5168: – , Measurement of fluid flow — Evaluation of uncertainties.

ISO 6416:1992, Liquid flow measurement in open channels — Measurement of discharge by the ultrasonic

(acoustic) method.

ISO/TR 9123:1986, Liquid flow measurement in open channels — Stage-fall-discharge relations.

ISO 9196:1992, .

Liquid flow measurement in open channels — Flow measurements under ice conditions

ISO 9213:1992, Measurement of total discharge in open channels — Electromagnetic method using a full-channel-

width coil.
To be published. (Revision of ISO 5168:1978)
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ISO 1100-2:1998(E) ISO
3 Definitions and symbols

For the purpose of this part of ISO 1100, the definitions and symbols given in ISO 772 apply. Those that are not

covered by ISO 772 are given in the text of this part of ISO 1100. The symbols used in this part of ISO 1100 are

given below:
A cross-sectional area,
C a coefficient of discharge,
C Chezy's channel rugosity coefficient,
h gauge height of the water surface,
b slope of the rating curve,
Q total discharge,
Q steady-state discharge,

r hydraulic radius, equal to the effective cross-sectional area divided by the wetted perimeter (A/P)

S friction slope,
S water surface slope corresponding to steady discharge,
velocity of a flood wave,
B cross-section width,
e effective gauge height of zero flow,
H total head (hydraulic head),
is Manning's channel rugosity coefficient,

p is a constant that is numerically equal to the discharge when the effective depth of flow (h 2 e) is equal

to 1,
t is time.
4 Units of measurement

The International System of Units (SI Units) is used in this part of ISO 1100 in accordance with ISO 31 and ISO

5 Principle of the stage-discharge relation

The stage-discharge relation is the relation at a gauging station between stage and discharge, and is sometimes

referred to as a rating or rating curve. The principles of the establishment and operation of a gauging station are

described in ISO 1100-1.
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ISO 1100-2:1998(E)
5.1 Controls
5.1.1 General

The stage-discharge relation for open-channel flow at a gauging station is governed by channel conditions

downstream from the gauge, referred to as a control. Two types of control can exist, depending on channel and flow

conditions. Low flows are usually controlled by a section control, whereas high flows are usually controlled by a

channel control. Medium flows may be controlled by either type of control. At some stages, a combination of section

and channel control may occur. These are general rules and exceptions can and do occur. Knowledge of the

channel features that control the stage-discharge relation is important. The development of stage-discharge curves

where more than one control is effective, where control features change, and where the number of measurements

is limited, usually requires judgement in interpolating between measurements and in extrapolating beyond the

highest or lowest measurements. This is particularly true where the controls are not permanent and tend to shift

from time to time, resuIting in changes in the positioning of segments of the stage-discharge relation.

Controls and their governing equations are described in the following clauses.
5.1.2 Section control

A section control is a specific cross-section of a stream channel, located downstream from a water-level gauge, that

controls the relation between gauge height and discharge at the gauge. A section control can be a natural feature

such as a rock ledge, a sand bar, a severe constriction in the channel, or an accumulation of debris. Likewise, a

section control can be a manmade feature such as a small dam, a weir, a flume, or an overflow spillway. Section

controls can frequently be visually identified in the field by observing a riffle, or pronounced drop in the water

surface, as the flow passes over the control. Frequently, as gauge height increases because of higher flows, the

section control will become submerged to the extent that it no longer controls the relation between gauge height and

discharge. At this point, the riffle is no longer observable, and flow is then regulated either by another section control

further downstream, or by the hydraulic geometry and roughness of the channel downstream (i.e. channel control).

5.1.3 Channel control

A channel control consists of a combination of features throughout a reach downstream from a gauge. These

features include channel size, shape, curvature, slope, and rugosity. The length of channel reach that controls a

stage-discharge relation varies. The stage-discharge relation for relatively steep channels may be controlled by a

relatively short channel reach, whereas, the relation for a relatively flat channel may be controlled by a much longer

channel reach. In addition, the length of a channel control will vary depending on the magnitude of flow. Precise

definition of the length of a channel-control reach is usually neither possible nor necessary.

5.1.4 Combination controls

At some stages, the stage-discharge relation may be governed by a combination of section and channel controls.

This usually occurs for a short range in stage between section-controlled and channel-controlled segments of the

rating. This part of the rating is commonly referred to as a transition zone of the rating, and represents the change

from section control to channel control. In other instances, a combination control may consist of two section

controls, where each has partial controlling effect. More than two controls acting simultaneously is rare. In any case,

combination controls, and/or transition zones, occur for very limited parts of a stage-discharge relation and can

usually be defined by plotting procedures. Transition zones in particular represent changes in the slope or shape of

a stage-discharge relation.
5.2 Governing hydraulic equations

Stage-discharge relations are hydraulic relations that can be defined according to the type of control that exists.

Section controls, either natural or manmade, are governed by some form of the weir or flume equations. In a very

general and basic form, these equations are expressed as:
Q = C BH (1)
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ISO 1100-2:1998(E) ISO
Q is discharge, in cubic metres per second (m /s),
C is a coefficient of discharge and may include several factors,
B is cross-section width, in metres (m), and
H is hydraulic head, in metres.

Stage-discharge relations for channel controls with uniform flow are governed by the Manning or Chezy equation,

as it applies to the reach of controlling channel downstream from a gauge. The Manning equation is:

0,67 0,5
A r S
h f
Q = (2)
A is cross-section area, in square metres,
r is hydraulic radius, in metres,
S is friction slope, and
n is channel rugosity.
The Chezy equation is:
0,50 0,50
Q = C A r S (3)
h f
where C is the Chezy form of rugosity.

The above equations are generally applicable for gradually varied, uniform flow. For highly varied, nonuniform flow,

equations such as the Saint-Venant unsteady flow equations would be appropriate. However, these are seldom

used in the development of stage-discharge relations, and are not described in this part of ISO 1100.

5.3 Complexities of stage-discharge relations

Stage-discharge relations for stable controls such as a rock outcrop, and manmade structures such as weirs,

flumes, and small dams usually present few problems in their calibration and maintenance. However, complexities

can arise when controls are not stable and/or when variable backwater occurs. For unstable controls, segments of a

stage-discharge relation may change position occasionally, or even frequently. This is usually a temporary condition

which can be accounted for through the use of the shifting-control method.

Variable backwater can affect a stage-discharge relation, both for stable and unstable channels. Sources of

backwater can be downstream reservoirs, tributaries, tides, ice, dams and other obstructions that influence the flow

at the gauging station control.

Another complexity that exists for some streams is hysteresis, which results when the water surface slope changes

due to either rapidly rising or rapidly falling water levels in a channel control reach. Hysteresis is sometimes referred

to as loop ratings, and is most pronounced in relatively flat sloped streams. On rising stages the water surface slope

is significantly steeper than for steady flow conditions, resulting in greater discharge than indicated by the steady

flow rating. The reverse is true for falling stages. See 6.8.4 for details on hysteresis ratings.

The succeeding clauses of this part of ISO 1100 will describe in more detail some of the techniques available for

analyzing the various complexities that may arise.
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ISO 1100-2:1998(E)
6 Stage-discharge calibration of a gauging station
6.1 General

The primary object of a stage-discharge gauging station is to provide a record of the discharge of the open channel

or river at which the water lever gauge is sited. This is achieved by measuring the stage and converting this stage to

discharge by means of a stage-discharge relation, which correlates discharge and water level. In some instances,

other parameters such as index velocity, water surface fall between two gauges, or rate-of-change in stage may

also be used in rating calibrations. Stage-discharge relations are usually calibrated by measuring discharge and the

corresponding gauge height. Theoretical computations may also be used to aid in the shaping and positioning of the

rating curve. Stage-discharge relations from previous time periods should also be considered as an aid in the

shaping of the rating.
6.2 General preparation of a stage-discharge relation
6.2.1 General

The relation between stage and discharge is defined by plotting measurements of discharge with corresponding

observations of stage, taking into account whether the discharge is steady, increasing or decreasing, and also

noting the rate of change in stage. This may be done manually by plotting on paper, or by using computerized

plotting techniques. A choice of two types of plotting scale is available, either an arithmetic scale or a logarithmic

scale. Each has certain advantages and disadvantages, as explained in subsequent clauses. It is customary to plot

the stage as ordinate and the discharge as abscissa, although when using the stage-discharge relation to derive

discharge from a measured value of stage, the stage is treated as the independent variable.

6.2.2 List of discharge measurements

The first step before making a plot of stage versus discharge is to prepare a list of discharge measurements that will

be used for the plot. At a minimum this list should include at least 12 to 15 measurements, all made during the

period of analysis. These measurements should be well distributed over the range in gauge heights experienced. It

should also include low and high measurements from other times that might be useful in defining the correct shape

of the rating and/or for extrapolating the rating. Extreme low and high measurements should be included wherever


For each discharge measurement in the list the following items shall be included:

a) Unique identification number
b) Date of measurement
c) Gauge height of measurement
d) Total discharge
e) Accuracy of measurement

f) Rate-of-change in stage during measurement, a plus sign indicating rising stage and a minus sign indicating

falling stage.

Other information might be included in the list of measurements, but is not mandatory. Table 1 shows a typical list of

discharge measurements, including a number of items in addition to the mandatory items. The discharge

measurement list may be handwritten for use when hand-plotting is done, or the data may be a computer list where

a computerized plot is developed.
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ISO 1100-2:1998(E) ISO
Table 1 — Typical list of discharge measurements
ID Date Width Area Mean Gauge Effective Number Gauge
Made Discharge Method Rated
number velocity height depth verticals height
m m m/s m m m/h
m /s
12 04/08/38 MEF 36,27 77,94 1,272 2,682 2,082 99,12 0,2/0,8 22 –0,082 GOOD
183 02/06/55 GTC 33,53 78,41 1,405 2,786 2,186 110,2 0,6/0,2/0,8 22 –0,047 GOOD
201 02/04/57 AJB 28,96 21,92 1,511 2,002 1,402 33,13 0,6/0,2/0,8 21 –0,013 POOR
260 03/13/63 GMP 26,52 21,46 1,400 1,981 1,381 30,02 0,6 22 –0,020 GOOD
313 08/24/66 HFR 30,18 42,08 1,602 2,374 1,774 67,40 0,6/0,2/0,8 22 +0,006 GOOD
366 08/21/73 MAF 28,96 14,86 0,476 1,557 0,957 7,080 0,6 21 0 GOOD
367 10/10/73 MAF 28,96 13,66 0,361 1,490 0,890 4,928 0,6 21 0 GOOD
368 11/26/73 MAF 29,26 14,21 0,373 1,509 0,909 5,296 0,6 18 0 GOOD
369 02/19/74 MAF 29,87 16,26 1,291 1,838 1,238 20,99 0,6 21 0 GOOD
370 04/09/74 MAF 29,26 21,27 0,805 1,780 1,180 17,13 0,6/0,2/0,8 21 0 GOOD
371 05/29/74 MAF 29,57 19,69 0,688 1,710 1,110 13,54 0,6 21 0 GOOD
372 07/10/74 MAF 28,96 16,81 0,458 1,573 0,973 7,703 0,6 21 0 GOOD
373 08/22/74 MAF 29,26 15,79 0,481 1,570 0,970 7,590 0,6 21 0 GOOD
374 10/01/74 MAF 29,26 13,19 0,264 1,414 0,814 3,483 0,6 21 0 GOOD
375 11/11/74 MAJ 28,96 11,71 0,283 1,396 0,796 3,313 0,6 21 0 GOOD
382 10/01/75 MAF 30,48 43,76 1,598 2,432 1,832 69,95 0,2/0,8 21 +0,017 GOOD
6.2.3 Arithmetic plotting scales

The simplest type out measurements shown in figure 1. Scale subdivisions should be chosen to cover the complete

range of gauge height and discharge expected to occur at the gauging site. Scales should be subdivided in uniform,

even increments that are easy to read and interpolate. They should also be chosen to produce a rating curve that is

not unduly steep or flat. Usually the curve should follow a slope of between 30° and 50°. If the range in gauge height

or discharge is large, it may be necessary to plot the rating curve in two or more segments to provide scales that

are easily read with the necessary precision. This procedure may result in separate curves for low water, medium

water, and high water. Care should be taken to see that, when joined, the separate curves form a smooth,

continuous combined curve.

Graph paper with arithmetic scales is convenient to use and easy to read. Such scales are ideal for displaying a

rating curve, and have an advantage over logarithmic scales, in that zero values of gauge height and/or discharge

can be plotted. However, for analytical purposes, arithmetic scales have practically no advantage. A stage-

discharge relation on arithmetic scales is almost always a curved line, concave downward, which can be difficult to

shape correctly if only a few discharge measurements are available. Logarithmic scales, on the other hand, have a

number of analytical advantages as described in the next clause. Generally, a stage-discharge relation is first drawn

on logarithmic plotting paper for shaping and analytical purposes, and then later transferred to arithmetic plotting

paper if a display plot is needed.
6.2.4 Logarithmic plotting scales

Most stage-discharge relations, or segments thereof, are best analyzed graphically through the use of logarithmic

plotting paper. To utilize fully this procedure, gauge height should be transformed to effective depth of flow on the

control by subtracting from it the effective gauge height of zero discharge. A rating curve segment for a given

control will then tend to plot as a straight line with an equation form as described in The slope of the straight

line will conform to the type of control (section or channel), thereby providing valuable information to shape correctly

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ISO 1100-2:1998(E)

the rating curve segment. In addition, this feature allows the analyst to calibrate the stage-discharge relation with

fewer discharge measurements. The slope of a rating curve is the ratio of the horizontal distance to the vertical

distance. This non-standard way of measuring slope is necessary because the dependent variable (discharge) is

always plotted as the abscissa.

NOTE — Numbers indicated against plotted observations refer to ID numbers given in table 1.

Figure 1 — Arithmetic plot of stage-discharge relation

Rating curves for section controls such as a weir or flume conform to equation (1), and when plotted logarithmically

the slope will be 1,5 or greater depending on control shape, velocity of approach, and minor variations of the

coefficient of discharge. Logarithmic rating curves for most weir shapes will plot with a slope of 2 or greater. An

exception is the sharp-crested rectangular weir, which plots with a slope slightly greater than 1,5. Logarithmic

ratings for section controls in natural channels will almost always have a slope of 2 or greater. This characteristic

slope of 2 or greater for most section controls allows the analyst to identify easily the existence of section control

conditions simply by plotting discharge versus effective depth, (h-e), on logarithmic plotting paper.

Rating curves for channel controls, on the other hand, are governed by equation (2) or (3), and when plotted as

effective depth versus discharge the slope will usually be between 1,5 and 2. Variations in the slope of the rating

when channel control exists are the result of changes in rugosity and friction slope as depth changes.

The above discussion applies to control sections of regular shape (triangular, trapezoidal, parabolic, etc.). When a

significant change in shape occurs, such as a trapezoidal section control with a small V-notch for extreme low

water, there will be a change in the rating curve slope at the point where the control shape changes. Likewise, when

the control changes from section control to channel control, the logarithmic plot will show a change in slope. These

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ISO 1100-2:1998(E) ISO

changes are usually defined by short curved segments of the rating, referred to as transitions. This kind of

knowledge about the plotting characteristics of a rating curve is extremely valuable in the calibration and

maintenance of the rating, and in later analysis of shifting control conditions. By knowing the kind of control (section

or channel), and the shape of the control, the analyst can more precisely define the correct hydraulic shape of the

rating curve. In addition, these kinds of information allow the analyst to extrapolate accurately a rating curve, or

conversely, know when extrapolation is likely to lead to significant errors

Figure 2 gives examples of a hypothetical rating curve showing the logarithmic plotting characteristics for channel

and section controls, and for cross-section shape changes. Insert A in figure 2 shows a trapezoidal channel with no

flood plain and with channel control conditions. The corresponding logarithmic plot of the rating curve, when plot ted

with an effective gauge height of zero flow (e) that results in a straight fine rating, has a slope less than 2. In insert B

a flood plain has been added which is also channel control. This is a change to the shape of the control cross-

section, and results in a change in the shape of the rating curve above bankful stage. If the upper segment (above

the transition curve) were replotted to the correct value of effective gauge height of zero flow, it too would have à

slope less than 2. In the third plot, insert C, a section control for low flow has been added. This results in a change

in rating curve shape because of the change in control. For the low water part of the rating, the slope will usually be

greater than 2.

Figure 3 is a logarithmic plot of an actual rating curve, using the measurements shown in table 1. This rating is for a

real stream where section control exists throughout the range of flow, including the high flow measurements. The

effective gauge height of zero flow (e) for this stream is 0,6 metres, which is subtracted from the gauge height of the

measurements to define the effective depth of flow at the control. The slope of the rating below 1,4 m is about 4,3,

which is greater than 2 and conforms to a section control. Above1,5 m, the slope is 2,8, which also conforms to a

section control. The change in slope of the rating above about 1,5 m is caused by a change in the shape of the

control cross-section. Below about 1,4 m the control section is essentially a triangular shape. In the range of 1,4 to

1,5 m the control shape is changing to trapezoidal, resulting in the transition curve of the rating. And above about

1,5 m the control cross-section is basically trapezoidal.

The examples of figures 2 and 3 are intended to illustrate some of the principals of logarithmic plotting. The analyst

should try to use these principals to the best extent possible, but should always be aware that there are probably

exceptions and differences that occur at some sites.
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ISO 1100-2:1998(E)
Figure 2 — Relation of channel and control properties to rating curve shape

NOTE — Numbers indicated against plotted observations refer to ID numbers given in table 1.

Figure 3 — Logarithmic plot of stage-discharge relation Gauge height of zero flow

The actual gauge height of zero flow is the gauge height of the lowest point in the control cross-section for a section

control [sometimes referred to as the cease-to-flow (CTF) value]. For natural channels, this value can sometimes be

measured in the field by measuring the depth of flow at the deepest place in the control section, and subtracting this

depth and the velocity head from the gauge height at the time of measurement.

The effective gauge height of zero flow is a value that, when subtracted from the mean gauge heights of the

discharge measurements, will cause the logarithmic rating curve to plot as a straight fine. For regular shaped

section controls, the effective gauge height of zero flow will be nearly the same as the actual gauge height of zero

flow. For irregular shaped controls, the effective gauge height of zero flow is greater than the actual gauge height of

zero flow. At points where the control shape changes significantly, or where the control changes from section

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ISO 1100-2:1998(E) ISO

control to channel control, the effective gauge height of zero flow will usually change. This results in a need to

analyze rating curves in segments (separate logarithmic plots for each control condition) to properly define the

correct hydraulic shape for each control condition. The gauge height minus the effective gauge height of zero flow is

the effective depth of flow on the control.

The effective gauge height of zero flow should be determined for each rating curve segment. For regular shaped

controls, this value will be close to the actual gauge height of zero flow. For most controls, however, a more exact

determination can be made by a trial-and-error method of plotting. A value is assumed, and measurements are

potted based on this assumed value. If the resulting curve shape is concave upward, then a somewhat larger value

for the effective gauge height of zero flow should be used. A somewhat smaller value should be used if the curve

plots concave downward. Usually only a few trials are needed to find a value that results in a straight fine for the

rating curve segment. The effective gauge height of zero flow is sometimes referred to as the logarithmic scale

offset. Logarithmic equation
The equation for a straight line rating curve on logarithmic plotting paper is:
Q = p(h 2 e) (4)
(h 2 e) is the effective depth of water on the control,
h is the gauge height of the water surface,
e is the effective gauge height of zero flow,
b is the slope of the rating curve, and

p is a constant that is numerically equal to the discharge when the effective depth of flow (h 2 e) is

equal to 1.
6.3 Curve fitting
6.3.1 General

The curve fitting process for stage-discharge relations includes the actual drawing, positioning, and shaping of the

rating. Hydraulic analysis and mathematical fitting can be used to aid in the curve fitting process, but in the final

analysis, the stage-discharge relation m

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