ESTABLISHING STAGEDISCHARGE RELATION 1 WHY A STAGEDISCHARGE RELATION

ESTABLISHING STAGE-DISCHARGE RELATION (1) • WHY A STAGE-DISCHARGE RELATION? – FLOW IS THE VARIABLE OFTEN REQUIRED FOR HYDROLOGICAL ANALYSIS – CONTINUOUS MEASUREMENT OF FLOW USUALLY IMPRACTICAL OR PROHIBITIVELY EXPENSIVE – STAGE OBSERVATIONS CONTINUOUSLY OR AT REGULAR SHORT TIME INTERVALS – STAGE OBSERVATION COMPARATIVELY EASY AND ECONOMICAL – RELATION BETWEEN STAGE AND DISCHARGE CAN BE ESTABLISHED: * THE DISCHARGE RATING CURVE OHS - 1

ESTABLISHING STAGE-DISCHARGE RELATION (2) GENERAL: – RATING CURVE ESTABLISHED BY CONCURRENT MEASUREMENTS OF STAGE h AND DISCHARGE Q COVERING EXPECTED RANGE OF RIVER STAGES AT SECTION OVER A PERIOD OF TIME – IF Q-h RATING CURVE NOT UNIQUE, THEN ADDITIONAL INFORMATION REQUIRED ON: * SLOPE OF WATER LEVEL (BACKWATER) * HYDROGRAPH h(t) (UNSTEADY FLOW) OHS - 2 – Q-h EXTRAPOLATION MAY BE REQUIRED TO COVER FULL RANGE OF STAGES – RATING EQUATION IS USED TO TRANSFORM h(t) INTO Q(t)

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Analysis of stage-discharge data Station name : CHASKMAN Data from 1997 1 1 to 1997 12 30 Single channel Gauge Zero on 1997 7 29 =. 000 m Number of data = 91 Power type of equation q=c*(h+a)**b is used Boundaries / coefficients lower bound upper bound 594. 00 595. 19 595. 95 600. 00 a b -592. 170 -593. 866 -594. 025 9. 709 2. 770 2. 531 c. 7147 E-03. 1507 E+02. 2263 E+02 Number W level M Q meas M 3/S Q comp M 3/S DIFf M 3/S Rel. d. IFf 0/0 Semr 0/0 1 2 3 4 5 6 594. 800 595. 370 596. 060 596. 510 598. 080 597. 700 9. 530 36. 480 127. 820 231. 400 738. 850 583. 340 8. 541 46. 661 136. 679 226. 659 783. 019 610. 359 . 989 -10. 181 -8. 859 4. 741 -44. 169 -27. 019 11. 58 -21. 82 -6. 48 2. 09 -5. 64 -4. 43 3. 75 2. 12 2. 90 2. 06 3. 63 3. 03 Overall standard error = 5. 904 Statistics per interval Interval Lower bound Upper bound Nr. of data Standard error 1 594. 000 595. 192 38 7. 20 2 595. 192 595. 950 27 5. 24 3 595. 950 600. 000 26 4. 84 OHS - 4

THE STATION CONTROL GENERAL: – THE SHAPE, RELIABILITY AND STABILITY OF THE Q-h RELATION ARE CONTROLLED BY A SECTION OR REACH OF CHANNEL AT AND/OR D/S OF GAUGING STATION = STATION CONTROL – ESTABLISHMENT OF Q-h RELATION REQUIRES UNDERSTANDING OF NATURE AND TYPE OF CONTROL AT A PARTICULAR STATION – ESTABLISHING A Q-h RELATION IS NOT SIMPLY CURVE FITTING OHS - 5

TYPES OF STATION CONTROLS • CHARACTER OF RATING CURVE DEPENDS ON TYPE OF CONTROL, GOVERNED BY: – GEOMETRY OF THE CROSS-SECTION – PHYSICAL FEATURES OF THE RIVER D/S • STATION CONTROLS CLASSIFIED IN MANY WAYS: – SECTION and CHANNEL CONTROLS – NATURAL and ARTIFICIAL CONTROLS – COMPLETE, COMPOUND and PARTIAL CONTROLS – PERMANENT and SHIFTING CONTROLS OHS - 6

CONTROL CONFIGURATION IN NATURAL CHANNEL OHS - 7

SECTION CONTROL OHS - 8

CHANNEL CONTROL (1) OHS - 9

FITTING RATING CURVES (3) MAIN CASES: – SIMPLE RATING CURVE * SINGLE CHANNEL * COMPOUND CHANNEL – RATING CURVE WITH BACKWATER CORRECTION: * NORMAL FALL * CONSTANT FALL – RATING CURVE WITH UNSTEADY FLOW CORRECTION – RATING CURVE WITH SHIFT ADJUSTMENT OHS - 22

FITTING SINGLE CHANNEL SIMPLE RATING CURVE (1) TO BE CONSIDERED: – – – EQUATIONS USED PHYSICAL BASIS EQUATION PARAMETERS DETERMINATION OF DATUM CORRECTION NUMBER AND RANGE OF RATING SEGMENTS DETERMINATION OF RATING CURVE COEFFICIENTS – ESTIMATION OF UNCERTAINTY IN RATING CURVE OHS - 23

FITTING SINGLE CHANNEL SIMPLE RATING CURVE (2) • EQUATIONS: – PARABOLIC TYPE: Q = c 2(h + a)2 + c 1(h + a) +c 0 – POWER TYPE: Q = c(h + a)b log Q = log c + b log(h + a), Y = A + BX OHS - 24

FITTING OF SINGLE CHANNEL SIMPLE RATING CURVE (3) RELATION BETWEEN POWER TYPE RATING CURVE AND MANNING EQUATION MANNING: Q = Km. AR 2/3 S 1/2 FOR RECTANGULAR X-SECTION: A = B. H R H MANNING: Q Km. BS 1/2. H 5/3 POWER: Q = c(h + a)b OHS - 25 SO: c = Km. BS 1/2 h+a=H and b = 5/3

FITTING OF SINGLE CHANNEL SIMPLE RATING CURVE (4) • POWER b IN POWER TYPE RATING CURVE VARIES WITH SHAPE OF CROSS-SECTION: – – – OHS - 26 RECTANGULAR: TRIANGULAR: PARABOLIC: IRREGULAR: COMPOUND: b = 1. 7 b = 2. 5 b = 2. 0 1. 2 <b<3 (TYPICALLY) b>5 ( , , )

FITTING OF SINGLE CHANNEL SIMPLE RATING CURVE (5) DATUM CORRECTION a: Q = c(h + a)b so: Q = 0 for a = - h METHODS TO DETERMINE a: – TRIAL AND ERROR – ARITHMETIC PROCEDURE – COMPUTER-BASED OPTIMISATION OHS - 27

FITTING OF RATING CURVES IN HYMOS FOLLOWING STEPS ARE REQUIRED: OHS - 44 * SELECT THE REQUIRED PERIOD AND STATION * CHECK THE MAXIMUM RANGE OF WATER LEVELS IN THE TIME PERIOD * INSPECT THE AVAILABLE STAGE DISCHARGE DATA TOGETHER WITH A REPRESENTATIVE CROSS-SECTION OF THE CONTROL * IDENTIFY THE BREAKS IN THE SCATTER PLOT * ELIMINATE OUTLIERS IF UNRELIABLE (MIND OTHER REASONS FOR SCATTER!!!) * SELECT EQUATION TYPE AND ‘a’ FORCED OR FREE * SELECT THE INTERVALS WITH OVERLAPS TO FORCE INTERSECTIONS * INSPECT THE PLOT AND THE TABULAR OUTPUT * REPEAT IF RESULT IS UNSATISFACTORY * SAVE THE CURVE PARAMETERS IF ACCEPTABLE

COMPOUND CHANNEL RATING CURVE (1) hf hr Br B Qriver = (hr. Br)(Kmrh 2/3 S 1/2 and Qfp = hf(B-Br)(Kmf hf 2/3 S 1/2 Qtotal = Qriver + Qfp OHS - 45

COMPOUND CHANNEL RATING CURVE (2) OHS - 46

COMPOUND CHANNEL RATING CURVE (3) • COMPUTATIONAL PROCEDURE (1): – FIRST THE RATING CURVE IS FITTED FOR THE MAIN CHANNEL UP TO BANKFULL LEVEL – THIS CURVE IS EXTENDED TO RIVER STAGES ABOVE BANKFULL LEVEL = Qr – ABOVE BANKFULL LEVEL: OBSERVED FLOWS Qobs ARE CORRECTED FOR MAINCHANNEL FLOW Qr TO OBTAIN FLOOD PLAIN FLOW ONLY = Qf: Qf = Qobs - Qr OHS - 47

COMPOUND CHANNEL RATING CURVE (4) COMPUTATIONAL PROCEDURE (2): – LAST WATER LEVEL RANGE IS USED TO FIT THE CURVE FOR THE FLOOD PLAIN FLOW Qf ALONE HENCE: – h < BANKFULL: Q = c 1(h + a 1)b 1 – h BANKFULL Q = c 1(h + a 1)b 1 + c 2(h + a 2)b 2 OHS - 48

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RATING CURVE WITH BACKWATER CORRECTION NO UNIQUE STAGE-DISCHARGE CURVE WHEN STATION CONTROL IS AFFECTED BY OTHER CONTROLS DOWNSTREAM CAUSES: * FLOW REGULATION D/S * LEVEL IN MAIN RIVER OR TRIBUTARY AT CONFLUENCE * WATER LEVEL IN RESERVOIR D/S * VARIABLE TIDAL EFFECT * D/S CONSTRICTION WITH VARIABLE CAPACITY DUE TO WEED GROWTH * RIVERS WITH RETURN OF OVERBANK FLOW OHS - 50

BACKWATER EFFECT hx h 0 Lx S OHS - 51

CHANNEL CONTROL EXTENT OF CHANNEL CONTROL: – FIRST ORDER APPROXIMATION OF BACKWATER EFFECT (rectangular channel): at x = 0: h 0 = he + h 0 at x = Lx: hx = he + hx hx = h 0. exp[(-3. S. Lx)/(he(1 -Fr 2)] Fr 2 = u 2/(gh) often << 1 Q = Km. Bhe 5/3 S 1/2 he = {q/(Km. S 1/2)}3/5 ln( hx/ h 0) = -3. S. Lx/he at: hx/ h 0 = 0. 05: Lx = he/S Backwater: Froude: Manning: So with q = Q/B: OHS - 52

BACKWATER VARIABLE BACKWATER: CAUSES VARIABLE ENERGY SLOPE FOR THE SAME STAGE HENCE: DISCHARGE IS A FUNCTION OF BOTH STAGE AND OF SLOPE: SLOPE-STAGE-DISCHARGE RELATION GENERALLY: OHS - 53 ENERGY SLOPE APPROXIMATED BY WATER LEVEL SLOPE

BACKWATER CORRECTION (1) • FALL BETWEEN MAIN AND AUXILIARY STATION TAKEN AS MEASURE FOR SURFACE SLOPE • • • OHS - 54 m = MEASURED r = REFERENCE S = SLOPE F = FALL VALUE OF POWER P THEORETICALLY 0. 5

BACKWATER CORRECTION (2) • TWO PROCEDURES FOR BACKWATER CORRECTION: – CONSTANT FALL METHOD * STAGE-DISCHARGE RELATION IS AFFECTED BY BACKWATER AT ALL TIMES – NORMAL (OR LIMITING) FALL METHOD * STAGE-DISCHARGE AFFECTED ONLY WHEN THE FALL REDUCES BELOW A GIVEN VALUE OHS - 55

CONSTANT FALL METHOD • MANUAL PROCEDURE – SELECT AN AVERAGE FALL, CALLED THE REFERENCE FALL Fr – CREATE A RATING CURVE h-Qr WHERE: Qr = Q/ (Fm/Fr) – CREATE A SECOND RELATION FOR Qm/Qr = f(Fm/Fr) – USE SECOND RELATION TO UPDATE Qr AND THE STAGE-DISCHARGE RELATION h-Qr, etc. • USE: Q = Qr(Fm/Fr)p OHS - 56 with Fm from observations Fr from procedure Qr from rating curve

CONSTANT FALL METHOD OHS - 57

CONSTANT FALL RATING OHS - 58

CONSTANT FALL COMPUTATIONAL PROCEDURE FITTING: – FIRST A REFERENCE FALL IS SELECTED – A RATING CURVE IS FITTED BETWEEN h AND Qr – VALUE OF p IS OPTIMISED USE: – FOR GIVEN h AND FALL Fm, Qr AND Fr FROM THE STORED INFORMATION – DISCHARGE FROM SECOND RELATION OHS - 59

CONSTANT FALL METHOD WITH HYMOS (1) OHS - 60

CONSTANT FALL METHOD WITH HYMOS OHS - 61

NORMAL FALL METHOD FOR BACKWATER CORRECTION (1) MANUAL PROCEDURE: – PLOT STAGE AGAINST DISCHARGE AND MARK THE BACKWATER FREE MEASUREMENTS – FIT A RATING CURVE FOR THE BACKWATER FREE MEASUREMENTS: Qr-h RELATION – PLOT FALL VERSUS STAGE AND DRAW A LINE FOR THE NORMAL OR LIMITING FALL Fr – COMPUTE Qm/Qr AND Fm/Fr FOR EACH OBSERVATION AND DRAW AVERAGE CURVE – ADJUST THE CURVES BY HOLDING TWO CONSTANT AND PLOTTING THE THIRD, ETC. OHS - 62

NORMAL FALL METHOD FOR BACKWATER CORRECTION (2) OHS - 63

NORMAL FALL METHOD FOR BACKWATER CORRECTION (3) OHS - 64

NORMAL FALL METHOD FOR BACKWATER CORRECTION (4) OHS - 65

NORMAL FALL METHOD FOR BACKWATER CORRECTION (5) USE OF THE PROCEDURE WITH h AND Fm GIVEN: – – – OHS - 66 READ Fr FROM Fr - h CURVE CALCULATE Fm/Fr READ Q/Qr FROM Qm/Qr - Fm/Fr RELATION READ Qr FROM Qr - h RELATIONSHIP MULTIPLY Q/Qr WITH Qr TO COMPUTE Q

NORMAL FALL METHOD FOR BACKWATER CORRECTION • COMPUTATIONAL PROCEDURE: – COMPUTE BACKWATER FREE RATING CURVE – DERIVE Fr FROM Fm, Qm AND Qr – FIT PARABOLA TO Fr - h DATA – OPTIMISE PAR. p • USE: – WITH ABOVE REATIONS FOR Qr-h AND Fr-h APPLY LAST EQUATION OHS - 67

RATING CURVE WITH UNSTEADY FLOW CORRECTION (1) • NOTE: – WATER SURFACE SLOPE ON FRONT SIDE OF FLOOD WAVE STEEPER THAN ON BACK SIDE – DISCHARGE PROPORTIONAL WITH ROOT OF SLOPE • HENCE: – FOR THE SAME STAGE, THE DISCHARGE IS LARGER FOR RISING STAGES THAN FOR FALLING STAGES – RATING CURVE HAS TO BE ADJUSTED TO ACCOMMODATE FOR THESE EFFECTS OHS - 68

RATING CURVE WITH UNSTEADY FLOW CORRECTION (2) Qm = measured discharge Qr = steady state discharge c = flood wave celerity S 0 = bed slope (energy slope for steady flow) dh/dt = change of h per unit of time Procedure: – trial Qr - h relation is established from measurements where dh/dt = 0 – compute 1/c. S 0 and fit a relation for 1/c. S 0 = f(h) OHS - 69

RATING CURVE WITH UNSTEADY FLOW CORRECTION (3) • CORRECTION REQUIRED IF FACTOR (1+1/c. S 0. h/ t)1/2 < 0. 95 OR >1. 05 • CORRECTION FACTOR HIGH WHEN: – BED SLOPE IS SMALL – CELERITY IS SMALL – h/ t IS LARGE • USE: – – OHS - 70 OBTAIN Qr VIA Qr-h FROM OBSERVED h OBTAIN 1/c. S 0 VIA 1/c. S 0 -h FROM OBSERVED h OBTAIN h/ t FROM HYDROGRPAH APPLY JONES FORMULA TO COMPUTE ACTUAL (UNSTEADY) FLOW

EXAMPLE UNSTEADY FLOW CORRECTION(1) OHS - 71

EXAMPLE UNSTEADY FLOW CORRECTION(2) OHS - 72

EXAMPLE UNSTEADY FLOW CORRECTION(3) OHS - 73

EXAMPLE UNSTEADY FLOW CORRECTION(4) OHS - 74

EXAMPLE UNSTEADY FLOW CORRECTION(5) OHS - 75

UNSTEADY FLOW WITH HYMOS (BEFORE CORRECTION) OHS - 76

UNSTEADY FLOW WITH HYMOS (WITH CORRECTION) OHS - 77

SHIFTING CONTROL (1) • CONSIDERATION: – A STABLE CONTROL IS A DESIRABLE PROPERTY OF A GAUGING STATION – ALLUVIAL STREAM-BEDS ARE NOT STABLE DUE TO SILTATION AND SCOUR (MOVING DUNES AND BARS) – AS A CONSEQUENCE THE STAGE-DISCHARGE RELATION WILL VARY – EXTENT AND FREQUENCY OF VARIATION DEPENDS ON: * TYPICAL BED MATERIAL SIZE * FLOW VELOCITIES OHS - 78

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SHIFTING CONTROL (3) INDETERMINATE Q-h OHS - 80

SHIFTING CONTROL (4) ALTERNATIVE: u-R PLOT OHS - 81

SHIFTING CONTROL (5) APPROACHES • FOUR POSSIBLE APPROACHES: – FITTING A SIMPLE RATING CURVE BETWEEN SCOUR EVENTS – VARYING THE SHIFT PARAMETER – APPLICATION OF STOUT’S SHIFT METHOD – FLOW DETERMINED FROM DAILY GAUGING OHS - 82

SHIFTING CONTROL (6) SIMPLE RATING BETWEEN EVENTS • USE: – WHERE RATING SHOWS LONG PERIOD OF STABILITY – WHERE SUFFICIENT GAUGINGS PERIOD ARE AVAILABLE – WHERE SHIFTS IN RATING ARE EASILY IDENTIFIABLE: * PLOT DATA WITH DATE * FLOOD EVENTS CAUSE CHANGE * NOTES IN THE FIELD RECORD BOOK ON REASONS FOR SHIFT OHS - 83

SHIFTING CONTROL(7) VARYING SHIFT PARAMETER Q=c 1(h+a 1)b 1 Q = c 1(h+a 1+ a)b 1 OHS - 84 • USE: – WHERE RATING SHOWS PERIODS OF STABILITY BUT INSUFFICIENT DATA ARE AVAILABLE FOR NEW RATING – THEN PARAMETER “a” IS ADJUSTED AS SHOWN LEFT: hr = rated h for Qm hm = observed stage for Qm CHECK APPLICABILITY OF a FOR FULL OR PARTIAL RANGE OF h

SHIFTING CONTROL (8) STOUT’s METHOD (1) PROCEDURE: hr =(Qm/c)1/b - a h = hr - hm ht = f( hi, hj) Qt = c 1(ht+ ht+a 1)b 1 OHS - 85 – FIT A MEAN RELATION FOR ALL POINTS IN PERIOD – DETERMINE hr FROM Qm – DETERMINE h FOR INDIVIDUAL MEAS. – DETERMINE ht BY LINEAR INTERPOLATION BETWEEN h’s – ht ARE USED TO CORRECT RATING

SHIFTING CONTROL(9) STOUT’s METHOD (2) OHS - 86

SHIFTING CONTROL (10) STOUT’s METHOD (3) • WHEN: – GAUGING IS FREQUENT – MEAN RATING IS REVISED PERIODICALLY – IF PREVIOUS METHODS DO NOT APPLY • ASSUMPTION: – SHIFTS GRADUAL CHANGES IN RATING • DRAWBACK: – ERRORS IN MEASUREMENT ARE MIXED DEVIATIONS DUE TO SHIFTS IN CONTROL – INDIVIDUAL MEASUREMENT ERRORS HAVE SEVERE CONSEQUENCES DIFFERENT FROM ORDINARY RATING CURVE OHS - 87

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SHIFTING CONTROL (11) DAILY GAUGING • WHEN: – IF BROAD SCATTER IS AVAILABLE NEITHER FROM BACKWATER NOR FROM SCOUR – CALCULATED SHIFT IS ERRATIC – HENCE WHEN NON OF OTHER PROCEDURES APPLY • NOTE: – IMPORTANT PARTS OF THE HYDROGRAPH MAY BE MISSED – BETTER TO RELOCATE THE STATION UNLESS URGENT NEED OHS - 91